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pro Fit 6.2 User Manual

1. Editing drawing objects You can change many attributes of drawing objects such as color line thickness or Fill __ N background pattern To do this first select the desired object s Then change the at a m en tributes using the Fill Pen Dash and Arrow popup menus from the drawing tools E b b A fill pattern and a fill color can be specified for all drawing objects except simple lines Dash _ See Chapter 10 Printing for a list of limitations on patterns when printing with Post Script A line color can be specified for all drawing objects except imported pictures Arrow b The two Pen popup menus are used to select a thickness and a line color The dash pattern of a line is selected using the Dash popup menu Choose Other from this menu to design your own dash pattern and add it to the Dash menu The line thickness and dash pattern can be specified for all objects containing lines If the se lected object is a graph the line styles of the axes ticks grid and frame will be changed The color also applies to the labels More complex options are available for the graph See section Plotting in this chapter If the selected object is a legend you can change the appearance of the curves and data sets dis played in the legend and the corresponding graph See the section Plotting later in this chapter for details Arrows of various size and shape can be added to poly
2. 39 You can also use the preview window to view functions graphically edit function parameters select a range of data points compare a function to a data set etc Functions that have multiple outputs can define groups of outputs that are displayed at the same time in the preview window if the de fault output is part of the group The other outputs in the same group are displayed in a lighter color Examples of this behavior are seen when using one of the built in Peaks functions Choose Preview from the Windows menu to see pro Fit s Preview Window a00 Preview Cc E3 Q Float Show Settings Fit Data Undo Refresh Help 1 0 LS amp M Auto t Q C Log yb 1 0 1 0 M Auto O Log 1 0 Selected marker x 0 30982 y 0 248796 The preview window is primarily operated by the buttons in its toolbar In addition on the right of the window there is a tool palette with tools for moving and zooming the coordinate region dis played by the view port for graphically editing the function and the data set and for determining precise x and y coordinates The buttons in the toolbar are the following Depres the Float button to make the preview window a floating window which always stays in front of all document windows Uncheck this option to transform it into a normal window which you can be hidden by other windows The three buttons titled Show determine from left to right if the current dataset and or the curre
3. Here when you click the To Menu button pro Fit adds some stuff at the beginning and the end of the script This is because Pascal requires every variable such as i to be defined and every piece of code defining a program in Pascal needs to be incapsulated in a proper way program PowersOf2 var i integer begin NewDataWindow for i 1 to nrRows do data i l 2 i end Here the program definition starts with the keyword program followed by its name After that we first have a variable declaration for the variable i which is of type integer The body of our program i e the actual statements are placed between begin and end In contrast to the Python script the calls to pro Fit s predefined functions such as NewDataWin dow does not need to be preceded by pf Note You can dispense with the need to precede pro Fit calls with a pf in Python by import ing all symbols of the module pf by wildcard i e by adding from pf import to the beginning of a Python script but this is not recommended in order to avoid sym bol collisions A simple function As a second example we will write a simple function that you can use for fitting or functional analysis using pro Fit s various tools Since this is a function we need to distinguish it from simple script like the above and to give it a name To define the function in Python open a new function window and enter the following 1
4. Draw at Line Styles v major ticks E v minor ticks o E Use ticks of Xi Apply Cancel D In the Grid editing panel you can define where you want to have horizontal and or vertical grid lines and if you want to see them at minor ticks major ticks or both You can also choose which axes must be used as a reference to draw the grid lines The grid lines are drawn at the tick marks of their reference axis By default the ticks of the main axes X1 and Y1 are used Panel Bar charts To select the options for displaying bar charts double click the graph and select the Bar charts panel 85 Graph Settings a General X axes Y axes Z axis Plots Frame Grid Bar charts Multiple charts Side by side Bar width Group width percent B 90 percent B Base line Position of horizontal line 0 Position of vertical line 0 M Draw line E _ Frame bars with rectangles Apply Cancel Cok The pop up at the top defines how to draw multiple bar charts in a single graph They can either appear side by side or on behind each other bars side by side bars behind each other 4 2 0 0 2 2 01234567 01234567 The settings under Bar width define the width of individual bars The width can either be a per centage of the available space or an absolute value in pixels centimeters or inches The settings under Group width are used when draw
5. Hit the delete key backspace while a marker is active to delete it Click and drag a marker to move it to a new position Move a function marker to the right or left border of the viewport to delete it To move a marker click and drag it or use the left and right arrow keys A data marker jumps to the next point to its left or its right a function marker will move along the function curve pro Fit makes sure that you don t move a marker outside the ranges of the view port You can override this by holding down the option key while moving the marker with the arrow keys When you have markers on the function and you uncheck the show function checkbox all of them are deleted The same applies to markers on data points when you uncheck the show data check box Uncheck and check the show function and or show data checkboxes if you have many mark ers around and want to get rid of all of them in one rapid move Data markers store their position as the number of the data point they mark If you have data markers around and you delete or add points to the data set the data markers might move to a new data point If no new point corresponding to the old index is found for a given marker that data marker is destroyed If you have function markers and you change the ranges of the display in such a way that their x coordinates are not visible anymore those markers are destroyed Tips and tricks Using the preview window during a fit
6. 3 nititialization 131 def DoMyStuff ask for an input column default is the one that was set in initialize global inputColumn inputColumn pf Input which column inputColumn transform the data ade Alternatively when defining a function you can also use another technique Namely after having compiled a function that contains a function directive pro Fit checks if the script also defines a function called xxx_init with xxx being the name given in the function directive If sucha function exists it is called after compilation The current default input values are passed as argu ments to the function Example the name of the function it appears in the Func menu function SampleFunc def SampleFunc_init al a2 pf SetMenuCommandKey menu Func SampleFunc key F6 shiftKey True def SampleFunc x al a2 return al x a2 1 In the above example SampleFunc_init is called after the function SampleFunc is added to the Func menu It adds the key equivalent shift F6 to its menu item In Pascal programs or functions it is the procedure named Initialize that is executed after compilation Example program DoMyStuff var inputColumn integer where our data comes from procedure Initialize prints a description of the program and sets the default value of inputColumn begin Writeln This program converts a data column Writeln into normalized units inputColu
7. The second less obvious but more versatile manner to write Python programs that appear in the Prog menu is achieved by inserting one or more program directives at the beginning of the script A program directive is a single text line starting with followed by the keyword program and the name of a function defined further below Such a script looks as follows program demo def demo a 5 print a When you click the To menu button or choose Compile amp Add to Menu from the Customize menu the script is compiled and executed Execution of such a simple script does nothing but complile the function within in for more advanced examples see below Then the function listed in the program directive i e the function named demo is added to the Prog menu When you then choose the program demo from the Prog menu it is executed This second way to define programs is more versatile because it allows to define several programs or functions see below in a single function window and to execute initialization statements at compile time It also allows the programs to share global variables easily Example program won program lost initialization of a global variable myAccount 0 defining two programs def won global myAccount myAccount myAccount pf Input how much have you won 0 print You now have myAccount cents def lost global myAccount myAccount myAccount pf Input ho
8. Y axes First 2000 Last 5500 lin 5000 Draw Ticks M axis line custom oan M ticks 1st major 2000 M labels Distance 1000 _ minor 3000 minor 1 2000 Apply A Cancel Co You can change a variety of things here Often you will use the edit fields First and Last to set the range of the axis Another important field is the Distance field that defines the distance between major tick marks 4 Enter 0 for First and 6000 for Last then click OK The vertical axis of the graph now starts at 0 and ends at 6000 Double click other parts of the graph or its legend to change other attributes Try double clicking the horizontal axis the center of the plot or the dot representing the data points in the legend You can also double click any text in the drawing to change it Or you can choose any of the drawing tools to add lines polygons text etc A function to fit our data The growth of a population can often be described by an exponential function of the type 17 ptt p x0 ex 8 1 where p t is the population at time t p x0 the population at an arbitrary start time x0 and tO its growth constant Let us try to investigate the validity of this formula for the world s population We want to find the set of parameters for which equation 3 1 fits our data best 1 Choose Exp from the Func menu This selects the built in exponential function as the current function to be
9. m v m c v c 1 m m1 m2 vl v2 abs v sqr v sqr m conj v conj m compl matrix multiplication result is a matrix matrix times vector both must have the same dimension result is a vector multiplication by a scalar Every matrix or vector element is multiplied by c this is the inverse of the matrix m Produces a run time error if the matrix cannot be inverted 1 m adjoint m determinant m matrix division m1 is multiplied with the inverse of m2 scalar product between two vectors The result is a number the absolut value of a vector The result is a real number translates to v v and m m respectively the complex conjugate is obtained by taking the complex conjugate of each individual element used to define a complex number compl x y x i y The following is an example of a program doing some matrix and vector calculations 193 program SomeMatrixAndVectorCalculations var ml m2 matrix 2 two 2x2 matrices mm mm2 matrix 3 two 3x3 matrices vl v2 vector 2 two vectors of length 2 c complex a complex number begin mm1 matr3 1 ii 2 2 ii 3 4 5 6 7 8 9 3 define the elements of the 3x3 matrix mm1 mm2 1 mm1 mm2 is now the inverse of mm1 ml matr1 1 2 3 4 define the 2x2 matrix m1 m2 sqr m1 4 2 3 3 m2 is calculated from m1 vl vect2 1 2 ii define the vector v1 v2 m2 v1 matrix multiplication of v1 gives v2 c vl v2 c is the scalar dot product of v1 and v2 en
10. pro Fit Pascal is a derivative of the original Pascal programming language with some extensions and some restrictions This appendix summarized the most important syntax elements and pro gramming techniques Program definition syntax The structure of a program definition is basically identical to that of a program in standard Pascal It starts with the keyword program followed by the name of the program and a semicolon Then you can optionally define some variables constants procedures or functions for your own use The main part of the program where the execution starts is placed between begin and end at the end of the program program myProg const c 3e8 var u v real done boolean procedure MyProc begin statements end procedure Initialize begin statements end begin statements end the name of the program myProg will appear in the Prog menu optional definition of constants and variables optional definition of a local procedure or function used by the program Note that you can call local procedures recursively more definitions of functions or procedures can follow here optional the procedure Initialize that is called once when the program is compiled and added to the function menu Any initialization of global variables can be done here the main body of the program where execution starts Note the after the end After the title of the program you can defi
11. 1 constant al 1 10 a 2 l active a2 var sS function sum_of_column c var sum r begin sum 0 for r 1 to NrRows do if DataOK r c then sum sum data r c sum_of column sum end procedure Initialize begin s sum_of column 1 end procedure First begin s sum_of_column a 1 end begin y x s a 2 end Calculating derivatives Derivatives xxx_derivatives If you define a procedure called Derivatives Pascal or a function called xxx derivatives Python where xxx is the name of the function in the function directive it is called whenever the partial derivatives of you function in respect to its input values are required Defining Derivatives xxx_derivatives is optional If defined it is used during fitting with the Levenberg Marquardt algorithm This algorithm uses the partial derivatives of the function with respect to its parameters If you do not define the Derivatives procedure the derivatives are calculated numerically but this slows down the fitting process considerably If you notice that fitting 136 is particularly slow you can define this function and at least calculate some derivatives pro Fit will still calculate numerically any derivative you don t define Derivatives xxx_derivatives is called after pro Fit has asked your function to calculate output values Hence if the calculation of the output values requires the evaluation of some ex pre
12. Loading Plug ins Choose Load Plug in from the Customize menu to load a plug in You are asked to locate the plug in The command Load Plug in can also be used to load compiled Apple Scripts See Chapter 11 Apple Script for details Removing functions and programs from the menus To remove a function or a program or an Apple Script from pro Fit s menus choose Remove from Menu from the Customize menu A submenu lists all the functions and programs that can be removed from the menus Select the name of the function or of the program you want to remove Note you cannot remove any of pro Fit s built in functions Spline Polynomial etc Loading plug ins automatically on startup Imagine you have one or more plug ins or Apple Scripts that you use often You can make them available automatically whenever you start pro Fit 147 Put the plug ins you want to add permanently to pro Fit into a folder named pro Fit plug ins This folder must be located in the same folder as pro Fit s or in the Preferences folder of your System Folder When you create the folder pro Fit plug ins type the name exactly as given here other wise pro Fit will not find it Whenever pro Fit starts up it checks if a folder named pro Fit plug ins is located in the same folder as the application itself and tries to load all plug ins it finds there Then pro Fit looks for a folder pro Fit plug ins in the
13. The function s actual definition is given between the keywords begin and end The function s value is calculated and then assigned by the operator to the name of the function As an al ternative it can also be assigned to the predefined output named y The variable x contains the function s default input value and a 1 a 2 etc are two additional input values that normally play the roles of function parameters In pro Fit Pascal a is a predefined array that represent the function parameters or any additional input value past the standard x value The parameters can be accessed by their index i e a 1 a 2 etc Instead of using a i for the parameters you can also use parameter names of your own by declaring them as in standard Pascal in the header of the function Note When defining a simple function in Pascal you can just enter its expression Such as a 1 sin x ln x a 2J in an empty function menu and then hit the To Menu button pro Fit will automatically add the function declaration and the begin and 119 end statements Similarly for defining a simple program you can simply write its statements such as writeln hello world and then hit the To Menu button and pro Fit will add the program declaration variable declarations and the the begin and end statements The Python and Pascal languages can be both used to define much more complex algorithms and calculations than the simple example
14. downhill in parameter space Starting from some initial value the parameters are varied and the resulting value of xr is calculated From this the algorithm finds the direction in which xr decreases and moves that way Then it samples again the surroundings by varying the parameters It stops when a minimum is reached When the x errors are not zero the xi necessary for calculating the minimal distance between a data point and the function curve are calculated for each data point by an explicit minimization of the term IR dx R dy l in Eq 2 Minimization is performed with limited precision in order to save processing time The x will be de termined to an accuracy which is a fraction of the x error specified for each point pro Fit will also count the number of function calls it is using to determine one x and will stop after a maximum of 50 function calls normally much less function calls lt 10 are needed to find the minimum This procedure introduces a small uncertainty in the determination of xr However the statistical signifi cance of such an uncertainty will be limited because the precision with which the xi are deter mined is in any case much better then the errors of the data points 99 Note A robust fit with x errors larger than zero will be considerably slower than the same fit performed with zero x errors When for zero x errors evaluation of 9 requires a num ber of function calls equal to the number o
15. for example use sqr x instead of x 2 For Pascal programmers is used for the power operator pro Fit does not know any thing about pointers and is not used for dereferencing The order of precedence for the operators is the same as in standard Pascal But since the pro Fit definition language does not distinguish between boolean and real expres sions refer to the next chapter this order of precedence provides a dangerous pitfall a gt x and b gt y will be compiled as a gt x and b gt y Use brackets to clarify what you want a gt x and b gt y In contrast to some other programming languages all the expressions in a composite logical expression of the form condition 1 and condition2 and condition 3 will be evaluated even if condition 1 returns false Functions and procedures provided by pro Fit pro Fit provides a reach set of functions and procedures that can be called by your scripts These are documented in pro Fit s help file and not repeated here Comparison to standard Pascal The programming language used to define functions and programs in pro Fit is closely related to the Pascal programming language However to keep it simple and to allow the generation of fast code some restrictions are present There are also some extensions with respect to standard Pas cal The most important differences to standard Pascal are 200 You cannot define your own data types All numeric types except co
16. most notably the Levenberg Marquardt fitting algorithm do pro vide estimates for the error of the parameters these estimates are often not sufficient or too im precise pro Fit provides a general way for estimating the confidence intervals within which the true value of a fitted parameter can be assumed to lie with a certain probability level The influence of variations in the data points on the fitted parameters can be analyzed with the help of a Monte Carlo simulation For this purpose synthetic data sets are generated starting from the points x f x that were obtained in the fit see above For each of the original data points a simulated data point is generated by random variation around x f x within the specified errors and using the specified error distributions This produces a synthetic data set that effectively simu lates a measurement The simulation of the measurement is based on the function that was de termined in the last fit which is assumed to correspond to the underlying reality and on the measurement errors that were specified A short description of this error analysis technique is found in W H Press B P Flannery S A Teukolsky W T Vetterling Numerical Recipes the Art of Scientific Computing For each of the synthetic data sets pro Fit performs a fit Once that all synthetic data sets have been fitted the confidence intervals are calculated by analyzing the values obtained
17. stp while t lt radius do begin 142 DrawEllipse x0 t y0 t x0 t y0 t draw a circle t tt stp end MoveTo x0 radius 1 1 y0 draw the crosshairs LineTo x0t radius 1 1 y0 MoveTo x0 y0 radius 1 1 LineTo x0 y0 radius 1 1 GroupEnd end the group end Note MoveTo and LineTo can also be used to draw into a Graph if they are combined with calls to OpenCurve and CloseCurve In this way it is possible to calculate and add any curve to a graph as a new plot directly from a program pro Fit Objects Some of pro Fit s functions such as GetWindowObject return object specifiers Object specifiers are very similar to and based on apple event object specifiers They can specify objects that exist or objects that do not yet exist For example you can create a shape object by calling Get ShapeObject window Drawing 17 shape myshape even if the given shape or win dow do not exist at the time of creation of an object Look up object specifiers in pro Fit s help for more details In Python scripts object specifiers are Python classes whose properties can be accessed through the classes attributes In Pascal object specifiers are opaque objects and pro Fit provides acces sor routines Such as GetPlotProperty and SetPlotProperties for reading and modifying them This is best illustrated by the following example It obtains the name of the first column in the frontmost window writes it to th
18. the insertion mark moves horizontally within one cell The tab key moves the selection one column to the right The carriage return or enter key moves the selection to the cell below The little in out x y symbols in the header of the first two columns tell that they will be pre selected as defaults to be associated to the inputs of functions the in columns or to be associ ated with the outputs of functions the out columns The x and y columns are special they are per default associated with the current default input value of a function the x value and the hy 9d current default output value of a function the y value When using the simplest functions of plot hy 3 ting and fitting you will most likely only work with x and y columns Selecting data You can select a single cell by clicking it To select a rectangular region of data cells drag the mouse from the top left to the bottom right cell or click the top left cell and then click the bottom right cell while holding down the shift key To select all cells in a row column click the row column number field To select several rows or columns click and drag over the row or column numbers you wish to select To select all cells in a column starting from a certain row hold down the option key while clicking the topmost cell of the desired selection or click the column number field an
19. B Resolution dpi PDF Many Mac OS X applications support the exchange of PDF via clipboard This is the pre ferred format to export pictures from pro Fit because PDF data is resolution independent and al lows excellent rendering on nearly all devices Quickdraw PICT Some legacy or monopolistic applications may not support PDF via clipboard For these applications pro Fit can place Quickdraw pictures on the clipboard These are accepted by nearly all applications But they are not resolution independent For best portability pro Fit ex ports bitmap based pictures They should render fine on all devices as long as the resolution of the bitmap is equal to or much higher than the one of the device Therefore when using Quickdraw pictures you must set the resolution correctly The field Resolution gives the resolution of the bit map in dots per inch The larger the resolution the finer the bitmap will print and the more memory it will use Set this field to your printer resolution or to an integer divider of your printer resolution The pop up Color depth sets the number of bits for encoding the color of each pixel Note In most situations you can check PDF as well as Quickdraw PICT The target applica tion will usually select whatever information it handles best Uncheck one of these op tions if you want to force the target application to use a given format if it can 170 12 Preferences pro Fit offers many possibilities to cu
20. Fit uses the parameter names defined in the function header for display ing the parameters in the parameters window After adding the function to pro Fit you can change its parameters in the parameters window You can plot the function use it for fitting calculations etc To plot it you should first set its parameters to reasonable values e g 1 and 0 5 Enter these val ues in the Parameter window and choose Plot Function from the Draw menu In the dialog box that comes up select the plotting range e g the x axis from O to 5 If you already have an open drawing window you should check the option Open New Window otherwise your curve will be drawn into the existing graph Our sample function is not defined for x lt 0 If you were to calculate it for a negative x value an error would occur However the function converges to y a 2 for x 0 You may want to expand the definition range of the function by defining y x a 2 for all x lt 0 This can be done easily with the following modification Click the To Menu button or choose Compile amp Add to Menu from the Customize menu when you are finished function logSine ampl offset real begin if x lt 0 23 then y 0 else y ampl sin x ln x y y offset end Your function could even become much more complicated than this You can define functions that contain more than one statement as well as variables and procedures You can use m
21. Func menu 2 Set its parameters in the parameters window 3 Choose Function f x from the Plot menu The Plot Function dialog box appears Plot Function Use fitted parameters Coordinate X axis x1 hea Scaling lin hea from 1 to 1 X step auto X range desi Magnitude Y axis v W Scaling lin Wa __ Auto range from 0 to 8 Options _ New Window Line style _ m Plot into current graph Fil E Graph Styles Cancel Cok If Use fitted parameters is checked the function is plotted using the parameter values calculated in the last fit If Use fitted parameters is not checked the parameter values in the parameters window are used The controls under the heading Coordinates define the settings for the x axes 65 The popups titled X axis and Scaling are used to define the x axis of the graph that will be used for this plot as well as its scaling type The edit fields titled from and to define the range of the x axis If you are using linear x axis scaling the entry in the field X Step determines the distance step width between consecutive calculated x values If you are using any other x axis scaling the field has the name X and determines the number of x values that will be calculated to plot the function The default value for step is auto This invokes a specially designed plotting algorithm that automatically selects the x values at which the function is calculated If t
22. It provides com patibility with a large number of legacy applications but does not support all modern imaging features In particular it does not support transparency and poorly supports rotated text or Uni code text pro Fit allows you to set various options for exporting pictures The default options can be set in the PICT Options panel of the Preferences command pro Fit menu the options for exporting into a file can be accessed through the button Options in the Save As dialog box PDF Portable Document Format This is a very powerful imaging format recognized by a large number of applications pro Fit exports PDF files for saving drawings but it also supports the exchange of PDF data through the clipboard EPS encapsulated postscript This file format is also very common and it can e g be used for exporting drawings to LaTeX in addition to PDF Note pro Fit does not support unicode text in EPS files but only the SOLatin1 character set pro Fit generates Postscript level 2 code JPEG Joint Photographic Experts Group This is a very common file format for exchanging images It is however lossy and not well suited for line drawings GIF Graphics Interchange Format This is a no loss bitmap based file format widely used in web browsers and other applications CompuServe Inc defined it in 1987 and 1989 The image can have up to 256 colors and 16 000 x 16 000 pixels For compression the LZW Lempel Ziv Welch algorithm is u
23. You can also do this by selecting the graph in the drawing window and editing its size using the Coords window Check Autorange after updates to force the ranges of the graph s axes to be updated after an update command to match the updated plots If you do not want auto ranging to take place un check this option The button Styles allows you to save and load the current settings of a graph A more detailed de scription of graph styles is given at the end of this chapter In the following sections we discuss the various parts of a graph and how to edit them Panel Axes When you want to edit an axis double click it Alternatively you can choose Axes from the Graph submenu in the Menu Draw or you can reach the axis editing panel using the list of icons in the Graph Settings dialog box 75 The axis editing panel for x axes looks like this Graph Settings M FJ E E La Y axes Z axis Plots Frame Grid Bar charts General Range amp Ticks Labels Prefix Lines X axes First 1 Last 1 lin Draw Ticks M axis line _ custom M ticks Ist major 1 mi labels Distance 0 5 __ minor a minor 1 a ee a ee E S 10 05 00 0 5 1 0 Apply Cancel Co Use the popup menu in the top left corner to navigate between the various axes to create a new axis or to delete the current axis The X1 and the Y1 axes are the main axes and cannot be de leted The edit field in the top right co
24. a legend to change the name of a plot Double click the plot symbol to choose the color plot symbol and line styles for a plot Find more information on this topic later in this chapter To change the space allocated for the plot symbols or the distance between lines in the legend simply resize the legend by dragging its selection points A graph is logically linked to its legend and vice versa If you change the appearance of a plot the change is reflected in both the graph and its legend Notes To maintain this relationship only one legend belongs to a graph and vice vera If you duplicate a legend e g by option dragging or copy and paste operations the duplicate will be an inert picture shape not linked to any graph You also can ungroup a legend This transforms the legend into a set of simple drawing shapes which then can be copied Again this essentially deletes the legend and its functionality the result doesn t have any connection to a graph anymore If you have deleted a legend don t worry You can always create a new legend for a graph by double clicking the graph and checking Draw legend in the dialog box that appears Note that you can change the text style font and font size of a legend by selecting it and choosing an appropriate setting from the Style Font and Size submenu in the Edit menu You also can change the line styles and color of curves and lines in a legend by choosing the ap propriate se
25. about the function Advanced programmers may also use it for initialization of global static variables memory allocation etc InitializeFunc should return the following data in its parameters e hasDerivates must be set to true if you want to calculate some derivatives of your function with respect to its parameters yourself in the function Derivatives described below Any derivative you don t calculate will have to be calculated numerically by pro Fit If you set hasDerivates to 159 false all derivatives will be calculated numerically and the function Derivatives will be ignored The derivatives are used for nonlinear fitting e descristLine descr2ndLine These two strings are displayed in the parameters window and should give a short description of your function C programmers should use the function Set PascalStr described under SetUp above for setting these strings numberOfParams Here you should return the number of parameters of your function up to 128 e a0 This is a pointer to a structure that defines the default values modes names and limits of your parameters You can leave this record unchanged if you want to use the default values The following table lists the values that can be set in a0 for each parameter i Pascal notation 1 C notation 2 contains a0 value i a0 gt value i Default value a0 mode i a0 gt mode i Default mode set to active varied during fitting inactive not varied
26. algorithm can only converge very slowly towards the best parameter set Its main use is to scan parameter space in order to find good parameter starting values for one of the deterministic fit algorithms or to try to jump out of a local minimum where a deterministic fit ting algorithm is stuck Since the algorithm is normally used for a first estimate of fitted parameters it is not recommended to run it with non zero x errors this merely slows down the algorithm without substantially in creasing the accuracy of the estimates The Levenberg Marquardt algorithm The Levenberg Marquardt algorithm is derived directly from the mean square deviation expres sions 8 or 10 and cannot be used with deviation functions R other then the square deviation R d 96 The Levenberg Marquardat algorithm is in principle the fastest fitting algorithm available in pro Fit lts performance however depends strongly on the behavior of the function to be fitted as well as on the selected starting parameters The classical version of the Levenberg Marquardt algorithm does not allow for x errors and mini mizes the mean square deviation 10 The algorithm can be described in words as follows Starting from a given set of parameters the mean square deviation x is calculated Then the pa rameters are varied slightly to observe their influence on x2 From this the direction in which x2 decreases most rapidly can be evaluated and a new
27. and Minor ticks fields you can set the line style length and color of major and minor tick marks The line style used to draw the axis can be edited using the Axis line style popup menus All the options outlined above for editing axes let you create many different kinds of graphs Note that you can 79 create new axes and change their scaling tick marks etc also if you don t use them to plot any curve For example you can uncheck the Energy eV Same as X1 check box in the X2 axis hee e if us a panel and edit it to reflect a completely 9 different scaling labels style and range 5 sa than the X1 axis So R 0 4 A typical application for this is a graph g 0 2 that displays its x values on its horizontal 0 bottom axis and the reciprocal x values 400 500 600 700 800 900 1000 on its top axis Wavelength nm As an example imagine that you have a set of data that was measured for different light wave lengths between 400 and 1000 nm You would like to plot your data as a function of wavelength but you would also like to have a reading for the light energy in eV on the top axis The energy of the light is inversely proportional to the wavelength so you have to use 1 x scaling for the top axis In the example above note that the top axis which has 1 x scaling has the smallest value to its right and the largest value to its left Click Prefix in the axis dialog box to set pre and postfix for the labels to mu
28. and end try is executed Note that we are setting a property called error alert to false before opening the files This tells pro Fit that it should not show any error alerts of its own when it cannot open a file The above example simply opens each file in the folder and closes it again In practice you may e g want to run a program on each opened file For this purpose simply insert run program MyProg analyze it after the command open file where MyProg is the name of the program you want to run The following is a more complete version of the above script It not only runs a program on each opened file it also defines the program adds it to pro Fit s Prog menu and then exchanges data with it the following defines the pro Fit program run for each data file set scriptProgram to program ScriptProgram var sum i begin sum 0 for i 1 to nrRows do if DataOK i 1 then sum sum dataLi 1 globalData 1 sum store result end bring up a dialog for selecting the folder of the files to analyze set myFolder to choose folder with prompt Choose a folder with data files create a list with all files in the folder set myFiles to list folder myFolder a list of files in myFolder set myFileCount to count myFiles the number of files in myFolder now Start working with pro Fit tell application pro Fit ppc set oldErrorAlerts to error alerts save error alert status set error aler
29. and viewing variables Python is a full featured high level object oriented programming language Just as for pro Fit Pascal you can edit Python scripts in a pro Fit function window and then add the new functions and programs to pro Fit s menus Python scripts can be debugged just like pro Fit Pascal using pro Fit s built in debugger 114 Apple Script is the high level scripting language of the MacOS To script pro Fit using Apple Script you need an external application for writing and running the scripts such as the Ap pleScript Editor found in Applications Utilities Apple Script can be used for defining programs macros but not functions If you are planning to write scripts that control pro Fit together with other applications such as the Finder you are likely to use Apple Script Otherwise Python or Pascal will probably be better suited for your purposes Since Pascal is pro Fit s original scripting language there is a wider se lection of Pascal function and programs available Python however is a more powerful language than pro Fit s Pascal and there are numerous libraries examples and tutorials available for it so it may often be a better choice for new projects The following documentation will focus on Pascal and Python Apple Script programmers will be able to obtain the necessary documentation from pro Fit s dictionary and since pro Fit is record able they can automatically generate scripts by r
30. arbitrary we do not want to fit it 3 Uncheck Use for fitting The check box Use for fitting can be found in the lower right corner of the parameter window when a parameter is selected As a shortcut you can also simply click a parameter name to toggle its status The parameter name changes from bold face to plain text This indicates that this parameter is constant and will not be fitted 4 Click the parameter name const We don t want to fit this parameter either The parameter name is not bold anymore and the option Use for fitting is unchecked now Fitting 1 Choose Curve Fit from the Calc menu This is the command you use to set up the various options used for fitting such as the correspon dence between data columns and default input and output values of a function data errors etc a 00 Fitting Algorithm Levenberg Marquardt B Auto search Function Exp m Guess initial parameters Data l t dat Output data Data Window population d B Heii daki Aoa lets cody Data Column year B Data Column population i B Error type Zero B Error type Unknown 9 Plot C Show last fit 2 5e 4 I Show function O log M Show data auto 0 0 1930 0 Mauto Clog 1990 0 Error analysis _ Close window when done __ Print full description E m Print active parameters only _ Done Fit gt The data column settings are already ok 19 2 Click the button Fi
31. as a background shape and a foreground shape With this dia log box you can design both of them pro Fit offers some predefined simple shapes and lets you edit any closed polygon to define a new data point symbol In the above example both foreground and background shapes are defined using a closed polygon You can use this dialog box simply to change the size of an existing point symbol or to design more complicated point symbols Draw the foreground and background shapes in the preview area at the right of the dialog box Use the popup menu above it to set the magnification of the preview The center of the preview area defines the hot spot of the data point symbol When plotting the hot spots of data point symbols are positioned on the correct mathematical coordinate Draw a closed polygon by dragging the polygon handles the little circles or squares at the edges of the polygon To make your work easier pro Fit lets you define a rotational symmetry and mirror symmetries Choosing 5 from the Symmetry popup menu like in the above example tells pro Fit to draw the definition points at 5 positions 360 5 degree apart before connecting them with lines Use the Pts popup menu to set the number of definition points Checking the H or V check boxes tells pro Fit to draw the definition points at the 2 positions obtained by mirroring them at a horizon tal or vertical axis respectively You can achieve quite astonishing effects by combini
32. commands General drawing commands apply to all types of drawing objects You probably already know these commands if you ever used any drawing application Here we quickly review them one by one To select an object in the drawing window 1 Choose the arrow tool in the tools palette by clicking the box containing the arrow symbol 2 Click the object you want to select To select multiple objects you can either click on the desired objects while holding down the shift key or you click into an empty part and drag the mouse to generate a dotted selection rectangle every object enclosed by the rectangle will be selected Click on an object while holding down the shift key to deselect it To move an object 1 Click the object and hold down the mouse button 2 Drag If you hold down the shift key while dragging movement is constrained to horizontal or vertical directions If you hold down the command key while dragging movement is constrained to diago nal 45 directions e If you hold down the option key while dragging the object is duplicated i e a copy of the original is created at the destination instead of simply moving the original You can drag one object from one drawing window to another If you do this a copy of the ob ject is created in the destination window e You can drag objects to any other application Supporting drag and drop or to the Finders desktop In the latter case the Finder will produce a sma
33. define your own function for fitting and you find that the fitting process is too slow then you should define these derivatives explicitly in the procedure called De rivatives If you do not define the derivatives yourself pro Fit calculates them numeri cally This can make fitting considerably slower More information on how to define functions and their derivatives is given in Chapter 9 Defining functions and programs 97 Estimation of parameter errors The Levenberg Marquardt algorithm allows the determination of the standard deviations of the pa rameters These are the values that are printed in the results window after a successful fit under the heading standard deviations The standard deviation defines the region that contains 68 3 of the total integral of a Gaussian distribution Whenever fitting errors are specified the standard deviation og of the parameter value a obtained after a successful fit is found from Oa Cy 12 where Cj is the diagonal element of the covariance matrix C The full covariance matrix of the pa rameters used in the fit is the inverse of a matrix A C A When no fitting errors are specified it is in principle not possible to calculate a covariance matrix However what pro Fit does in case of unknown errors is to calculate the covariance matrix using arbitrary error values of 1 It is then possible to make sense of this seemingly arbitrary covariance matrix by consid
34. drawing mode To do this hold down the control key and choose Display As Drawing For changing the text of a shape or its name double click it Alternatively select it and choose Shape Settings from the Draw menu As a shortcut you can change some properties of the items when the window is still in dialog mode To do so hold down the command key and double click the item you want to modify 155 Working with plug ins This chapter explains how to add plug ins to pro Fit Plug ins are files containing the computer code for a pro Fit function to appear in the Func menu or a pro Fit program to appear in the Prog menu pro Fit comes with a number of ready to run plug ins containing useful functions or programs The next section tells you how you add them to pro Fit See the section Creating a plug in and Writing a plug in for a detailed explanation of how to create your own plug in Loading a plug in To add a plug in to pro Fit 1 Select Load Plug in from the Customize menu You are asked to locate your plug in 2 Choose the plug in you want to load and click Open pro Fit checks if a plug in can be found in the file you have selected If yes it is loaded If the plug in is a function it is added to the Func menu If it is a program it is added to the Prog menu Instead of loading a plug in by choosing Load Plug in you can double click its file For this the file type and creat
35. during fitting or constant cannot be fitted a0 name i a0 gt name i Parameter name a Pascal string of length maxParamLength 3 a0 lowest i a0 gt lowest i The lower limit for a parameter By default this value is INF a0 highest i a0 gt highest i The upper limit for a parameter By default this value is INF 1 In Pascal indices for these arrays run from 1 to 128 2 In C indices for these arrays run from 0 to 127 3 c programmers should set the name by calling the function SetPascalstr with a maximum string length of maxParamLength Example SetPascalStr a0 gt name 0 pname maxParamNameLength void Func double x ParamArray a double const y ExtModulesParamBlock pb This procedure is called to calculate the return value of your function It has the following parame ters x The function s independent variable a The function s parameters ali Note that the index i ranges from 1 in Pascal but from 0 in C e y The function s return value to be calculated from x and a void Derivatives double x ParamArray a ParamArray dyda ExtModulesParamBlock pb This routine calculates the partial derivatives of your function with respect to its parameters You can leave this routine empty if you don t need it or you can calculate only some derivatives You 160 don t need to calculate all of them pro Fit will check if you did not calculate a derivative and will calculate it nume
36. entry in the legend Un check this check box to hide the corresponding entry in the legend When an entry is visible in the legend you will usually change its style by double clicking it When an entry is not visible in the legend you must choose Curves from the Graph submenu to access and change the style of the corresponding curve or data points Click Tabulate to recover the original data points that were used to draw the plot In this way you can retrieve data points from a drawing when you have lost the original data set or you can obtain a list of the data points that pro Fit calculated to draw a particular function Click Delete to delete the curve or data points from the graph You can use the delete backspace key as a keyboard equivalent for this button Click Update to delete a function plot and replace it with the currently selected function using the parameters displayed in the parameters window Note that the graph does not know about the original function that you used for that particular plot The update button simply replaces that func tion with a new plot based on the current function and its parameters For data plots the name of the button changes to Data Errors and clicking it allows you to select the data set linked to the plot and to add or change the error values Panel Frame A frame is a rectangular box around your graph 83 0 8 0 8 0 4 0 4 0 0 0
37. follow ing graphic formats PDF PostScript EPS PICT high resolution bitmaps PNG TIFF GIF JPEG Scriptability and Recordability You can record your actions automatically as a pro Fit program or Apple Script Pascal or Python script for replaying them later Externally compiled code It is possible to define functions algorithms and other programs us ing an external programming environment and compiler and to import them as plug ins pro Fit can also be controlled via Apple Script On line evaluation of mathematical expressions Enter any mathematical expression wherever pro Fit expects numerical input Such as in spreadsheets or dialog boxes Drawing from a script pro Fit scripts can directly draw into pro Fit s drawing windows to create drawings with high precision coordinates These drawings are available for copying and pasting into other applications and for high resolution printing Scripting Write complete macros to perform common tasks such as opening and closing docu ment windows fitting importing and exporting files etc using either Apple Script Python or a Pascal like scripting language Debugging environment A powerful debugger provides tools for developing and debugging complex scripts Extensive on line help An on line help system provides answers hints and explanations Plug ins Various plug ins further increase pro Fit s power e g for contour plotting and 3D plot ting of functions and data s
38. for the new graph A more detailed explanation of graph styles is given below 67 To plot an output value y value of a function versus two of its input values choose the command Function f x y from the Plot menu Plot Function XYZ Use fitted parameters Coordinates X axis X1 4 Scaling lin B Input x B from 1 to 1 steps auto Y axis Y1 4 Scaling lin B Input al B from 1 to 1 steps auto Magnitude Z axis Z1 4 Scaling lin rq Mi Auto range Options m New Window vi Draw contours Plot into current graph M Color plot of magnitude color scheme y Cr eae Graph Styles A Cancel Most of the options are the same as in the previous dialog box Note the following differences though e For the x and the y axis you have to specify an Input value of the function corresponding to the axis The input value can either be the function s x value or any other input that makes sense Check Draw color plot to draw a color plot and Draw contours to draw the contour lines The two graphs below show a color plot left and a contour plot right The pop up Color scheme allows you to select a color encoding to use when drawing a color plot of the function The z axis defines the default color scheme and the location of the contour lines to be used as well as the scaling to be applied to the function s output value 68 I 1 1 i 1 Plotting a two dimensional data
39. for the predefined func tion ChiSquared which can be used in pro Fit programs to retrieve the value of 2 ob tained in the last fit Zero X errors In most experiments it is possible to determine the x coordinate much more precisely than the y coordinate In such a case the x errors can be assumed to be very small The only way to minimize the mean deviation 2 is then to have x xi The mean deviation function becomes much sim pler X V ee 9 i Oi The function is evaluated at the x coordinates of the measured points The function value and measured y coordinate directly give the normalized distance when weighted with the measure ment error The usual case Chi squared and zero x errors In many experiments it is not only possible to make the x errors so small that they can be consid ered zero It is also common to have or hope for Gaussian distributed measurement errors In this case we have to minimize a particularly simple expression for chi squared yee 10 Oi Since this case is easy to handle from an algebraic and numerical point of view many common fitting algorithms and applications work under the assumption that the mean deviation is the mean square deviation given by Eq 10 A classical fitting algorithm that works on this basis is the Levenberg Marquardt algorithm in its unmodified original form see below 94 Error analysis and confidence intervals Although some fitting algorithms
40. from another program using the call AttachProgram Working with controls in drawing windows As explained in Chapter 7 drawing windows can contain control shapes such as buttons or checkboxes The following is a list of all control shapes and of the most important properties they have These properties can be read by calling GetShapeProperty and modified through Set ShapeProperties ih Buttons These are simple objects that highlight when i Button J clicked Properties active Set to true if the button can be clicked Set to false if it is grayed and cannot be clicked value Usually 0 Set to 1 for hi lighting the button text The text that appears in the button Checkbox Checkboxes They automatically change their state when they are clicked Properties active Set to true if the checkbox can be clicked Set to false if it is grayed and cannot be clicked value 0 if not checked 1 if checked text The text that appears beside the checkbox 151 Radio button text field Static text field Popup item 1 Radio buttons They are checked when they are clicked They usually come in groups The program that manages the radio buttons is responsible for un checking all other radio buttons when one radio button is clicked Proper ties active Set to true if the checkbox can be clicked Set to false if it is grayed and cannot be clicked value 0 if not checked 1 if checked text The text that a
41. function It specifies the default input and the default output In this case the default input is the standard one x but other inputs can be made to be the default one by checking the Default Input check box using the contextual menu that appears when you control click in the window or using the hierarchical menu near the top of the Func menu when the output values are not displayed in the window there is also an additional shortcut to do this simply click and hold on the name of the default output and select a different one in the menu that pops up A list of all function outputs appears when the disclosure triangle to the left of the word Outputs is clicked For the polynomial function there is only one output but for other func tions like the Bessel functions there can be many outputs Any of the outputs can be made to be the default one using the Default Output check box the contextual menu that appears when you control click in the window or the hierarchical menu near the top of the Func menu The default input and the default output are used for all operations where a function must be used to transform one value into another like when a function is plotted in a two dimensional graph or when it is used to fit a two dimensional data set Some other operations that require more than one input like contour plotting or provide more than one output like tabulating a function define their own sets of inputs and outputs a
42. have a large number of points in one area e g between x 1 and 2 and just one or two points far out e g at x 50 it is necessary to decrease the error for these lonely points if you want to force the function to come close to them e When plotting a curve in a graph with a logarithmic y axis a deviation of the curve from a small y value appears much larger than the same deviation from a larger y value If this astonishes you it is probably because your measurement errors are proportional to the measured value 111 When plotting a fit on a graph with a logarithmic y axis the errors of the yi are often given in percent This results in smaller deviations from points with small y values Here is an example of logarithmically plotted data with fits using percent errors and constant errors 108 e data constant errors m percent errors gt 104 102 112 9 Defining functions and programs Introduction One of the most important features of pro Fit is the ease with witch it allows to define new functions and programs A function is added to the menu Func It behaves like any of pro Fit s built in functions and you can use it for fitting plotting etc see Chapter 5 Working with functions A program is added to the menu Prog A program performs a sequence of tasks Programs can be used for data analysis and calculations but they can be used as powerful scripts for auto mating
43. i gons and lines using the Arrows pop up menu Dn gt gt Pages xl Choose Other from the Arrows menu to design your own arrows and add your personal arrow styles to the A ai Arrow menu 59 Fill colors and line colors are set using the corre sponding popup menus Hold down the mouse in one of those menus then select the color and opacity you want Use the color sucking symbol in the color menu to copy colors from one object to another When you se lect this tool the cursor becomes a pipette which you can use to click an object with the desired color The shape of the cursor changes and becomes a paint bucket and with that you can click the object that the color is to be applied to To pick up a fill color for the target shape instead of a line color hold down the shift key while click ing with the color measuring tool pro Fit stores the color that was measured with the color measuring tool inside the standard color popup menu so you can also use the menu to apply the color to other shapes at a later time Exporting pictures There are a number of ways to export pro Fit drawings e using the Copy or Cut commands in the edit menu dragging them and dropping them to their destination saving the whole drawing as a PICT EPS GIF TIFF PDF or PNG file When using the clipboard or drag and drop for exporting pictures pro Fit generates PICT and PDF data PICT data is a legacy format compatib
44. index type its type Since types are ignored by pro Fit you can omit of type in the declaration To access an array use the syntax name index Example vararrl array 1 10 of real arr2 array 0 100 i for i 1 to 10 do arrl i 0 arr2 33 22 1 Note the maximum size of all variables in a variable list is limited to 32 kBytes This limits the size of an array to about about 4000 entries Multi dimensional arrays are not supported Note that arrays are a general purpose object and should not be confused with the built in vector types that only support vectors of length 2 3 and 4 and that are mainly used in conjunction with the matrix types to perform matrix and vector operations Loop statements The for loop A for loop takes the form for variable startValue to endValue do statement 195 A for loop executes its statement for all integer values of its variable between startValue and end Value If startValue equals endValue the for loop is executed only once If startValue is larger than endValue the for loop is never executed Example program PowersOf2 var i integer begin NewDataWindow for i 1 to nrRows do data i 1l 2 i end The end value in our for loop is given by the return value of the function NrRows which is always equal to the number of rows in the current data window The statement in our for loop is an assignment to the array element data i 1 that
45. many days fit in a year or a month 29 pro Fit stores relative times as double precision floating point numbers interpreted as a number of seconds This leads to a range of 2147483647 to 2147483647 centuries which corresponds to floating point numbers between 6 77680 8 and 6 77680 8 Permanent transformations In pro Fit data windows you can attach a permanent transformation to each individual column This means that you can have one column represent a calculation based on the data in other columns The contents of the column will be automatically updated whenever any of the data on which it is based changes If possible the recalculation is done only for the rows that have been changed If you are defining permanent calculations using formulas that contain calls like data pro Fit al ways must recalculate all data in the corresponding column because the input rows are not clearly defined anymore Permanent calculations are created either from the Data Transform dialog discussed below or via the Column Properties dialog discussed above The modification of existing permanent calcula tions is done via the Column Properties dialog Note that recursive usage of column data is not allowed E g if column 2 is automatically perma nently calculated from the contents of column1 then it is forbidden to define a permanent calcula tion for column 1 that depends on column 2 If you are defining permanent calculations using for mul
46. matr2 1 2 ii ii writeln sqr m end The compiler recognizes that sqr is called with a matrix argument Therefore a matrix version of the square function is used The output of the above program is 1 00 i 2 00 2 00 i 2 00 1 00 i 1 00 1 00 i 2 00 Type conversion from real or complex to matrix or vector To define matrices use the prede fined functions matr2 matr3 matr 4 All mathematical calculations will automatically recognize matrix and vector types and interpret them correctly when it makes sense The following are some special functions to be used on vectors and matrices determinant returns the determinant of a matrix transp returns the transposed matrix adjoint returns the adjoint matrix outer returns the matrix defined as the outer product of two vectors 192 matr2 matr3 matr4 vect2 vect3 vect4 used to define a matrix these routines take 4 9 or 16 complex parameters respectively used to define a vector these routines take 2 3 or 4 complex parameters respectively Expressions of the matrix or vector type can be used with normal mathematical operators and functions when it makes sense Mathematical operations between matrices and matrices matrices and vectors vectors and vectors and matrices vectors with numbers do the expected thing In the table below m stands for any matrix n type v for any vector n type and c stands for any complex or real number m m
47. of the axes with a color and pattern of your choice To do this check Fill and select the axis towards which the plot must be filled and the fill color and pattern using the popup menus to its right The Contour plot option is available for three dimensional plots Check it to draw contour lines for the z values of a three dimensional plot To define the location of the contour lines change the set tings of the z axis attributed to the plot Check Color plot and select a color scheme to draw a color encoded image of the z values of a three dimensional plot The Coordinate Axes popup menus define the coordinate axes used by the selected curve or data set With these pop ups menu you can change the reference axis of any given curve Note Doing this for function curves that were drawn with auto step is not recommended If the scaling of the original axis and the one of the destination axis differ considerably as when moving a function from a linearly scaled to a logarithmically scaled axis the results can be disappointing Remember that a function curve is only defined by a set of points pro Fit calculated these points in an optimized way when it plotted the func tion for the axis scaling and range on which the function was plotted If you then change scaling or range your curve may loose its smoothness In such a case it is bet ter to redraw the function curve on the new axis Check Appears in legend to make the curve or data set appear as an
48. oy 1 Choose Constant to set the standard deviation of all points to a given absolute value Choose Percent if you want to enter the error as a fraction of the data value in If you have the errors stored in a column of your data window then select In dividual and choose the appropriate column in the pop up menu that appears Note Make sure that the columns you select contain the correct error values in the correct positions For each row in the table there must be a one to one relationship between the values in the x y columns and the values in the error columns The Distribution popup menu which appears when you define errors gives the error probability distribution that will be assumed for the fit This popup menu is dimmed if the Levenberg Marquarat algorithm is used because this algorithm only works with Gaussian error distributions The checkbox Guess Initial Parameters tells pro Fit to use a function specific algorithm for choos ing appropriate starting values for the parameters Such algorithms are available for some of pro Fit s built in functions such as Exp Log Power as well as the Peaks functions The resulting fitted function is previewed directly in the dialog box section Plot Once fitting is done the numerical results of the fit are written to the results window Check Print active parame ters only if you only want to see the values of the parameters that were fitted Use this optio
49. pf SetData i 2 numpy sqrt pf GetData i 1 else pf SetData i 2 0 doAlert True if doAlert pf Alert Some data was negative This program shows some additional features of the definition syntax e Before accessing the data in a cell we test if there is really a number in this cell This is done with the function DataOK r c which returns true if the cell in row r and column c contains a valid number If the cell is empty or if it contains text it DataOK returns false e The innermost if statement if data i 2 gt 0 has two statements in its else branch In the Pascal version they are delimited by the keywords begin and end to make it clear that they both belong to the else statement In the Python version the same effect is achieved by inden tation At the end of the program an if condition checks whether any data was negative If there were negative numbers in the input column the function Alert is called Alert takes one argument a string i e a text between quotes It displays an alert box that shows this string Another more Pythonesque way to do the same takes advantage of the fact that pro Fit allows to pass whole arrays from to data windows import numpy doAlert False def process x global doAlert if numpy isnan x return numpy nan if x lt 0 doAlert True return 0 return numpy sqrt x x pf GetData pf RowRange 1 pf SetData 1 2 map process x if doAlert pf Alert Some data
50. reshape a polygon double click it 55 If the selection marks of a polygon appear at the corners a E of its enclosing rectangle the polygon is not in reshape mode If the selection marks appear at its corners it is in reshape mode z m To move one of the corner points of a polygon click and drag it To remove one of the corner points click it while holding down the option key To add a corner point click the connecting line between two corner points while holding down the option key Temporarily unsmooth a smoothed polygon to make this kind of editing easier Data Points When plotting data the data points are represented by special symbols You can create such plot symbols as individual objects anywhere in a drawing window This is useful for creating your own legends or for exporting single point symbols to other applications e g for figure captions Since the point symbols can assume a quite large size they can also be used as parts of standard drawings Data point symbols can be edited using a particular set of tools that let you achieve ef fects not easily achieved with other drawing objects below you will find more details about editing data point symbols To create a point object choose the point tool from the drawing EK OD palette Keep the mouse button down for a little while to select 4 2000 the symbol that you want to use A pop up menu with a choice of al x XX so i data points appears Its top part contai
51. the right of the cursor and in the bottom left corner of the drawing window If you now press the shift key you can select a part of the graph The ranges of the graph will be changed to display only this part This is useful for Zooming in on some part of the plotted data set 89 8 Fitting This chapter describes what pro Fit does when using the curve fitting commands It contains infor mation on the fitting algorithms it uses and on the various options that are available when perform ing a fit Fitting the parameters of a function to a data set roughly means finding those parameters that make the function s curve follow the data points as closely as possible There are various possible definitions of the term as closely as possible The correct definition is often determined by the origin and characteristics of the data set to be fitted For example a data set might be subject to large errors in the x coordinate and to smaller errors in the y coordinates The probability of incurring in a given measurement error can decrease in some known way when the magnitude of the error increases There are also various possible methods of looking for the best parameter set pro Fit provides a choice of different ways for measuring the distance of the data points from the function as well as a choice of different methods to reach the best parameter sets The first part of this chapter deals with the definition and mathemat
52. the small est distance of two data points or is a fixed value In skyline plots the steps can have varying width depending on the distance of the data points bar chart skyline AA 0O A 0O O N 012345678 012345678 When plotting multiple bar charts the bars can either be behind each other or on top of each other In skyline plots the plots are always on top of each other bars side by side bars behind each other 71 Double clicking the graph and selecting the Bar charts panel can set most of the options for bar charts This panel is described below Note You can use text columns for plotting When you e g create a plot using a text column as x column the labels of the x axis of the graph will be set to the texts in the text col umn while each new x value corresponds to the row index of the entry in the text col umn This type of category axes is especially useful for histograms Plotting a three dimensional data set There are two command for plotting data sets that have z values in addition to explicit or implicit x and y values e If your dataset is a sequence of x y and z values in three separate columns use the com mand Data z x y from the Plot menu If your dataset is a matrix of z values in a data window and the x and y values correspond to the row and column indices of the individual z values use the command Data z row col from the Plot menu Both commands bring up d
53. used for analysis fitting and plotting It also brings the parameters window of the Exponential function to the front This window gives a description of the exponential function and its parameters OO Exp Cae Inputs default x x 0 0 A 1 0 l x0 0 0 l to 1 0 const oo E Outputs default y Selected Value No input or output selected C Default input Use for fitting The window is divided into three regions The top region provides a short description of the se lected function The central part displays the input and output values of the function and lets you edit input values The bottom region displays additional information on the selected input or output value The function Exp looks like this x x0 y A expl const 3 2 which is essentially identical to equation 3 1 The parameter window also displays the default values for the parameters input values A t0 and const Starting from these parameters pro Fit can find a better set of parameters for describing our data But first you must define which parame ters you want to fit i e which parameters you want to vary in order to approximate the data with the Exponential function As mentioned above the starting time x0 is arbitrary Let us set it to 1940 2 Click the number beside x0 in the parameters window and enter 1940 This defines the parameter s value and the starting value that will be used for curve fitting 18 Since x0 is
54. user interface Applications of Check are described in Chapter 9 Defining functions and Programs Parameters e paramNo This is the index of the parameter that the user has changed 1 64 in Pascal 0 63 in C e a0 This is a record in C a pointer to a struct that defines the default values modes names and limits of your parameters as they appear in the parameters window The values that you can access or change in this data structure are listed under the routine InitializeFunc above Check should return one of the following values good if the new parameter is to be accepted 161 update if the new parameter is to be accepted but the parameters window must be redrawn because Check changed some values in a0 bad if the new parameter cannot be accepted void Last ExtModulesParamBlock pb This routine is called whenever an operation that has used your function such as a command for fitting is done In most cases you can leave this procedure empty Applications of Last are given in Chapter 9 Defining functions and Programs Predefined constants and types When writing a plug in you can and must use several predefined constants types and proce dures or functions They are defined in profit_interface h and proFit_paramBlk h This section de scribes some of the most important things defined in these files Note The definitions in these files should not be changed Doing so might cause inc
55. values true or false 190 complex complex is the type for complex numbers It consists of two real values the real and the imaginary part vector n vector n is the type for a vector with n complex elements 2 lt n lt 4 The i th element of a vector v can be accessed using v i matrix n matrix n is the type for matrices with nxn complex elements 2 lt n lt 4 The element in the i th row and the j th column of a matrix m can be ac cessed using nii j Note pro Fit 6 does not distinguish between real integer and Boolean types All these types are implemented as 8 byte floating point numbers Simple numeric types The boolean value true is represented by the real value 1 0 and false by 0 0 All non zero val ues are interpreted as true in a boolean expression Most Pascal compilers on the MacOS distinguish between the floating point types extended dou ble and real which have different accuracy All simple number types of the pro Fit definition lan guage have extended i e double accuracy The accuracy and range of numerical values in pro Fit is given in Appendix C Complex type The Complex data type is used to represent complex floating point values having a real and an imaginary part Example program ComplexTest var c Complex begin c l writeln sqrt c end The compiler recognizes that sqrt is called with a complex argument Therefore a complex version of the square root function is used which c
56. view and modify its variables set breakpoints etc Initially the program stops at the first line of code that is executed Note Some parts of your pro gram may already be called right after compilation such as the procedure Initialize In this case the debugging window will come up right after compilation to let you debug these parts of your code 145 The debug window has four sections At the very top there s a tool bar The function of each button is explained below At the top left the Calling sequence is shown It shows the chain of procedures and functions pro Fit followed before reaching this particular point in your code Note that you may step through more than one of your programs and or functions if they call each other At the top right the variables that are valid at this point are displayed You can watch and mod ify their values Just double click a value to change it Clicking onto the small triangles lets you view the elements of arrays and matrices At the bottom the source of the program or function is displayed with an arrow showing the current location In addition if you click the button Show Evaluate an edit field and result fields are inserted at the very right of the window They can be used for entering expressions or for Python code whole statements and viewing their results The buttons at the very top let you control execution of your code Click Step Into or Step Over for
57. want to tabulate a function between 0 and two times pi at intervals of pi 5 type option command T and enter the following 173 Table of Polynomial x Tabulate by varying x H4 from 0 to 2 pi step 0 5 auto points Use fitted parameters f Cancel 3 ex When typing a mathematical expression you use the same syntax elements that are available when writing a function definition in Pascal In on line mathematical expressions x is equal to the last result that was evaluated and ali is equal to the input values shown in the current parameters window You can use all the predefined functions available when writing the definition of a function As an example after a successful fit you can type ChiSquared in a data window cell This tells pro Fit to set the value of that cell to the mean deviation obtained in the last fit Let s look at a simple example that illustrates how you can use pro Fit s understanding of mathe matical expressions when you are pasting into a data window Write the following text and copy it to the clipboard a 2 gt fittedParams 2 gt a 2 fittedParams 2 gt paramSD 2 a 3 gt fittedParams 3 gt a 3 fittedParams 3 gt paramSD 3 a 4 gt fittedParams 4 gt a 4 fittedParams 4 gt paramSD 4 Where the gt stands for a tabulator character and each line is terminated with a carriage return f If you paste the above text into a data window after a succes
58. was negative 141 This script obtains the data in column 1 as an array x and then applies a mapping function in order to finally put the resulting array back into column 2 of the data window This technique yields in many cases better performance For more information about passing arrays from and to data windows see pro Fit s help file entries for the functions GetData and SetData Drawing pro Fit provides a set of drawing functions that allow to create and modify objects in a drawing window In the following we discuss a script that creates a bull s eye at the point where you last clicked in the drawing window In uses the commands MoveTo and LineTo for drawing lines DrawEllipse for drawing a circle and Get ClickedCoord for obtaining the last clicked coordinates In Python script BullsEye the name to appear in the Prog menu radius 40 step 8 pt pf GetClickedCoord coordinates of last click x0 pt x yO pt y pf GroupBegin start a group for t in range step radius l step pf DrawEllipse x0 t yO t x0 t y0 t draw circle pf MoveTo x0 radius 1 1 y0 draw the crosshairs pf LineTo x0 radius 1 1 y0 pf MoveTo x0 yO radius 1 1 pf LineTo x0 y0 radius 1 1 pf GroupEnd end the group In Pascal program BullsEye const radius 40 stp 8 var x0 y0 t real begin GetClickedCoord x0 y0 coordinate of last click GroupBegin start a group t
59. written in Python Since the window is still empty and pro Fit is unable to make a reasonable guess the de fault entered in the Preferences window as described above is used A simple program In a first step we will write a simple script that fills the first column of a data window with the pow ers of two 2 4 8 16 etc In Python such a script looks as follows pf NewDataWindow for i in pf RowRange pf SetData i 1 i 2 Enter this script into the function window and hit the button To Menu or choose Compile amp Add to Menu from the Customize menu This will add the script to the Prog menu By default it will have the name of the window e g Untitled Function 2 in the example above Choose this name from the Prog menu to run the script From the above example you can see that the script calls some functions preceded by pf this indicates that they have been defined in the Pyhton module pf which is a module that pro Fit automatically adds to all Python scripts and which contains the functions and symbols used for controlling pro Fit Thus for example the function NewDataWindow opens a new data window the function RowRange returns a range object of all rows in the window and SetData row col val sets the data cell of the given row and column to the given value 117 In Pascal the same script looks similar NewDataWindow for i 1 to nrRows do data i 1l 2 i
60. 0 1 0 0 5 0 0 0 5 1 0 1 0 0 5 0 0 0 5 1 0 without frame with frame To change the appearance of a frame either double click a graph and click the Frame icon or choose Frame from the submenu Graph in the Draw menu Graph Settings J L General X axes Y axes Z axis Plots me Grid Bar iri Frame lines and ticks M Draw frame Lines __ Draw ticks inside frame Draw ticks outside frame Apply Cancel Coa In the dialog box that appears you can edit the Line style of the frame and determine if tick marks must be drawn on it The tick positions of the main coordinate axes X1 and Y1 are used If you draw a frame with ticks you usually do not wish to draw the axes ticks as well Uncheck the corre sponding check boxes in the axis dialog box Panel Grid Grid lines are horizontal and vertical lines at the positions of the ticks 100 100 1 0 0 5 0 0 0 5 1 0 0 5 0 0 without grid 0 5 with grid 84 To add grid lines to your graph double click a graph and check Draw grid This will add horizontal and vertical grid lines To customize the grid lines click the Grid icon in the same dialog box or choose Grid from the Graph submenu Graph Settings a LJ l Z axis Plots Frame Ea Bar charts m Draw grid Horizontal grid lines Draw at Line Styles M major ticks E minor ticks _ E Use ticks of Vertical grid lines
61. 18 function myFunc import numpy def myFunc x al a2 return al numpy sin x numpy log x a2 The first statement function myFunc is ignored by the Python interpreter and tells pro Fit that this is the definition of a function named myFunc to be added to pro Fit s Func menu The next statement import numpy imports the symbols of the numpy library which defines a rich set of mathematical functions such as the sine and logarithm functions that we will use below def my Func x al a2 starts the definition of the function This function must have the same name as the one declared in the function statement The arguments of the function are used as the function s input values in pro Fit and will appear in the parameters window with the first one being the default input x value The function s value is then returned by the return statement at the end lt is calculated as al numpy sin x numpy log x a2 Again as for the script above you hit the To Menu button or choose Compile amp Add to Menu in order to add this custom function to pro Fit s Func menu In Pascal the same function looks as follows function MyFunction begin MyFunction a 1 sin x ln x a 2 end The first word of our example is function It tells pro Fit that the definition of a function follows The next word MyFunction gives the name under which the function will appear in the Func menu
62. Fit An example is the possibility of defining command key equivalents for calling a program Or 129 you can also put a program in a submenu To find out about all these possibilities and how to use them the on line help provided by pro Fit is the place to go you can also read on in this chapter but the on line help is often a more handy tool The help menus When defining functions and programs you can use a series of predefined names functions and procedures To help you use them pro Fit provides a popup menu Help in the header of all func tion windows ANA W ElectricField E To Menu __ Debug Help Const Browse w Python M syntax 2 The Help popup menu lists all predefined routines names and syntax elements that you can use The items are organized hierarchically It is easy to find an item by scanning all the different hierarchical menu items in this menu In addition to this the function window provides a popup menu called Const that provides a list of some of the most important constants of nature things like the speed of light the charge of an electron the mass of a neutron etc When you move the mouse over an entry in the menus Help and Const a help tag is shown giving a short description When you choose an entry and release the mouse its definition is pasted into the function window You can enable disable the help tags for these two menus by choosing the entry Sh
63. If Show function is checked during a fit the function is redrawn from time to time to show how it changes during the fit This lets you monitor how well the fit converges However drawing the func tion takes time You should close the preview window or uncheck Show Function to obtain the fastest fitting The same applies to the preview that is built into the fitting dialog box when that box is kept open The same thing happens when you use the Error Analysis feature To perform error analysis pro Fit generates random sets of synthetic data points and fits the function to them If Show Func tion is checked in the preview window you will see how the function curve varies in correspon dence to the fitted parameters See Chapter 8 Fitting for more details on the fitting process and the Error Analysis algorithm Choosing initial values of function parameters You can display the data you want to fit together with the fit function in the preview window You can then use the fitting tool to drag the function curve while varying some parameter values until it 45 follows the data points as closely as possible This is a kind of hand fitting that can be a very useful and fast way to set up a reasonable set of starting parameters for a fit For special applications you can also mark certain features of your data set using coordinate markers and write a small program that reads the coordinates of these markers and uses them to calculat
64. Info plist Y Utilities Y mc le CarbonStuff c CarbonStuff h Moduleutilities c ModuleuUtilities h StringSupport c ih StringSupport h E ICMEAGME zo ff amp amp amp 0O C C Programmin 4 Copy one of the example projects either the folder Example Plugin for a program plugin or In verse Erf example function for a function plugin Now modify the code as it suits you Then build the plug in and load it into pro Fit either by double clicking the bundle you have built or by loading it from pro Fit by choosing Load Plug in from the Customize menu The following gives some additional details about creating plug ins 157 Writing an a plug in with an external compiler Routines to be modified The following table lists the routines defined in ProgramTemplate c p and FunctionTemplate c p that can or should be modified by the user Functions or procedures that are only used by ad vanced programmers are marked with a T function name modify if defining a SetUp program or function CleanUp program or function InitializeProg t program Run program InitializeFunc t function Func function Derivatives function First t function Check t function Last t function In the following section we first describe the routines SetUp and CleanUp that are used for both types of plug ins Then we discuss the routines only used in external programs then the routines only used in external functions All the followin
65. Pascal definition language is that it produces very readable function defini tions that can be readily understood also by people who are not experienced in programming In addition it is very easy to learn making defining new functions very easy for everyone The advantage of Python is that it is a well known advanced programming language well adapted for scientific computing and with a wide availability of modules for scientific calculations and more In addition to these two built in languages pro Fit also supports Apple Script which means that you can use the AppleScript Editor to issue commands to pro Fit and to automate its functionality All commands that can be given to pro Fit using its menus can also be issued from scripts and Apple Script is useful for general purpose scripting However a scripting language like AppleScript is not necessarily the best choice for programming complex mathematical calculations That is why pro Fit has built in support for two programming languages like Pascal and Python pro Fit Pascal is the original pro Fit scripting dialect It is similar to the Pascal programming lan guage but does not implement all its features while adding some new functionalities You edit a Pascal function or program in a pro Fit function window and then compile it using pro Fit s built in compiler pro Fits debugger can be used to debug Pascal code e g by stepping through the individual instructions setting breakpoints
66. Preferences folder of the Library folder and again tries to load all plug ins it finds there If you are running pro Fit directly from a server the modules found in the pro Fit Plug ins folder in the application folder on the server will be available to all users the plug ins in the pro Fit plug ins folder of your system s Preferences folder will only be available to you You can also add plug ins to the pro Fit application directly To do so select the pro Fit application in the Finder and choose Get Info from the File menu In the window that appears go to the Plug ins section and click the button Add to select a plug in to add Note You can also place drawing data or function files into the pro Fit plug ins folders The names of these files will automatically appear in the Prog menu Choosing the name from that menu will open the file Loading a set of plug ins together with a new preferences file In multi user environments different users might want to use the multi preferences file mechanism provided by pro Fit The pro Fit preferences file holds the default settings and other information for many pro Fit s op tions Different users may want to use different preferences files pro Fit normally uses the prefer ences file found in the Preferences folder inside your Library folder It is possible however to start pro Fit by double clicking another preferences file or to switch to a new preferences fi
67. a Transforming data Defining a data set to work on Working with functions Editing in the parameter window Using functions Viewing function outputs Calculating output values Optimization of functions Analyzing functions The Spline function The Preview Window The basics Preview Window Appearance Preview Window Tools Managing coordinate markers Tips and tricks Using the preview window during a fit Choosing initial values of function parameters Drawing and Plotting The drawing window Drawing tools Coordinates accuracy and drawing info Drawing objects Shape properties Drawing windows in dialog mode 50 Editing drawings 51 General drawing commands 51 Objects created with the tools palette 53 Editing polygons and lines 55 Data Points 56 Control shapes 58 Editing drawing objects 59 Exporting pictures 60 File formats for graphics exports 60 Import graphics 61 Plotting 62 Plot types 62 Axis types 64 Plotting a function 65 Plotting a two dimensional data set 69 Plotting a three dimensional data set 72 Graphs and legends 72 Editing legends 73 Editing graphs 74 Panel Frame 83 Panel Grid 84 Panel Bar charts 85 Graph Styles 87 Graph coordinates and zooming 89 8 Fitting 90 Mathematical background 90 The mean square deviation Chi Squared 93 Zero X errors 94 The usual case Chi squared and zero x errors 94 Error analysis and confidence intervals 95 Fitting algorithms 95 The Monte Car
68. a series of commands in pro Fit While a function s role is clear it is part of the Func menu and has a parameters window a pro gram s role can be more multifaceted A program appears in the Prog menu but apart from that it can do almost anything It can perform a single calculation execute a data transformation opera tion using one or more windows create a new diagram in a drawing window or finally replay a set of instructions that manipulate pro Fit s windows its settings and executes commands like those found in pro Fit s menus one after the other It is the latter functionality that is often described as scripting Because functions and programs can perform calculations and access pro Fit data and built in mathematical functions they need to be defined using a definition language that is flexible enough and that is essentially a programming language Scripts that automate some lengthy sequence of operations on the other hand can also be defined by general purpose scripting commands such as Apple Script which pro Fit supports Here is a small list of what functions or programs can do e Calculate any kind of numerical value even if it cannot be expressed in a closed mathematical formula e Access the data in a data window write results into the results window use dialog boxes and alert boxes Execute any command from pro Fits menus open and save files create and close windows Run fitting operations and prede
69. about Apple Script consult the dedicated literature such as the Apple Script Language Guide distributed by Apple Examples Opening and closing a single file tell application pro Fit open file measured data open a file run program Analyze analyze it close window measured data close it end tell 164 The script starts with the statement tell application pro Fit which indicates that all subsequent statements until end tell are to be sent to pro Fit The following lines tell pro Fit to open a file called measured data run the program Analyze from the Prog menu and then close the file again To use this script you must enter it in a script editor such as Apple s Script Editor As mentioned above you can save the script as a Compiled Script and then load the compiled script from pro Fit by choosing Load Plug in from the Customize menu The script is added to the Prog menu from where it can be run Batch processing Imagine you have a large number of data files in a folder You want to open each of these files from pro Fit and analyze its data Without scripting you would have to open each file by hand run your analysis then close it again boring work if you have to do it often The following script does it all for you bring up a dialog for selecting the folder of the files to analyze set myFolder to choose folder with prompt Choose a folder with data files c
70. advancing one step in your code When you click Step Into and the next step is a local function or a procedure pro Fit steps into this procedure and stops at the first instruction there If you click Step Over and the next step is a function or procedure pro Fit will execute it and stop again at the next instruction right after it If the next step is not a function or procedure Step Into and Step Over just advance by one step Click Step Out if you are in the midst of a local function or procedure and you want pro Fit to stop once execution returns from this function or procedure i e you do not want to stop again until the function or procedure is done and execution would continue at the spot immediately after it Click Run to continue operation to the next breakpoint or if there are no more breakpoints to the end of your code Click Kill to abort execution of your function or program Click Show Evaluate to add additional edit field and result fields at the very right of the window They can be used for entering expressions or in Python scripts whole statements and viewing their results You can set breakpoints by clicking into the left margin of the source code in the debug window Red dots mark the breakpoints To remove a breakpoint click it again When you run a program or function and pro Fit encounters such a breakpoint execution is interrupted and the debug window comes up 146 Using pro Fit plug ins After you h
71. ameters of the function Up to 128 input values can be used The output variable the function s return value for single values functions It must be set by your function Y O y n An array holding the output values of a multi valued func tion y 0 is the function s default output value It is possible to define your own input and output names in the function header and to use your own names instead of the a 1 a n function foo ampl freq phase real var result begin result ampl cos freq x phase end If you do this input and output values retain their numbering defined by their sequence when you define them ampl freq phase The ali remain available as synonyms a 1 ampl a 2 freq a 3 phase and the indices can still be used in predefined function such as SetParamName Example 1 You want to define the function y al In a2 x2 Your definition looks like this function logSquare K Q real begin y K sqrt Q cosh x end This is a function in its most simple form If you work with it often you may want to assign default values to the parameters You will also see that Q should not be negative You might therefore im prove the above definition as follows function logSquare K Q real inputs K 1 active K amplitude Q 1 active Q multiplier of cosh 0 INF begin y K sqrt Q cosh x end 189 The first line after the keyword inputs de
72. an handle sqrt 1 The output of the above program is 0 000 i 1 000 Type conversion from real or other simple numeric types to complex is automatic For converting complex numbers to real use one of pro Fit s predefined functions such as abs phase re im see below To define complex numbers use the predefined function compl or the predefined constant ii which fulfills sqr ii 1 All predefined functions in pro Fit such as sin cos gamma erf etc automatically become complex valued functions if they notice that their argument is a complex number and return com plex numbers as a result 191 Expressions of the complex type can be used with all mathematical operators and with all mathe matical functions When the type of a parameter is complex the function will recognize it and re turn an appropriate complex or real result The following are the few special functions that only make sense for complex numbers Conj returns the complex conjugate of a complex number Re Im return the real and imaginary part respectively of a com plex number phase returns the phase of a complex number r ei abs returns the absolute value rof a complex number r ei compl used to define a complex number compl x y x i y Matrix and Vector types The Matrix and Vector data types are used to represent 2 dimensional and 1 dimensional ar rangements of complex floating point values program MatrixTest var m matrix 2 begin m
73. an import the following image formats PICT Quickdraw Picture PDF PNG TIFF GIF and JPEG There are three ways of importing pictures over the clipboard by choosing Paste in the Edit menu by dragging the file or image from the Finder or another application into pro Fit drawing window or by choosing the Import command from the File menu to open a supported image file type Note that pro Fit imports pictures as a whole and does not take them apart If you use a drawing application to create a line and a rectangle and paste these objects together into pro Fit they are interpreted as one picture not as a line and a rectangle An imported picture can be resized or rotated but it cannot be edited in any other way Rotating and resizing may not work with imported pictures if they contain any non standard information such as PostScript commands To obtain size and resolution information of an imported image double click it The dialog box that comes up also allows you to set or reset the size of the picture 61 Plotting One of the main jobs of pro Fit is to create high quality graphs which are graphical representations of mathematical functions and data sets A graph consists of two or more axes and one or more plots Each plot represents a plotted data set or a plotted function Plot types Pro Fit supports several plot types Two dimensional function plots consist of a single line They can be used for plotting one ou
74. apes allow you to add buttons check boxes Button radio buttons text fields popup menus and image wells s to a drawing window Gee is used to a auser y H Tape interface that can be accessed by a user defined pro Pe aO utton gram In essence you can transform any drawing win ae Field tatic Text Fie Popup Menu Well dow itno a sophisticated dialog box for your own addi tions to pro Fit Chapter 9 of the manual section Attaching scripts tells Arr you more about how to use control shapes The available control shapes are e Buttons These are simple objects that highlight when clicked Checkboxes They change their state when they are clicked e Radio buttons They are checked when they are clicked They usually come in groups The pro gram that manages the radio buttons is responsible for un checking all other radio buttons when one radio button is clicked e Text fields These are objects that contain text Generally text fields can be edited If you don t want the text field to be editable use a Static text field 58 e Popup menus These are objects that have several values which can be selected by choosing them from a popup menu Wells These shapes are usually used as background for other objects e g a graph They con sist of a white rectangle For a more detailed description on how to use control shapes see Chapter 9 of the manual sec tion Attaching scripts
75. aph and your paper A logarithmic scaling indicates that there is a logarithmic relationship between the coordinates of the graph and your paper this expands the lower end of an axis and compresses its upper end The min and max values for logarithmic axes must both be positive The 1 x scaling type can be used to plot a function whose y value is expected to be proportional to 1 x If you plot such a function on a 1 x scaled x axis the function is a straight line The min and max values for 1 x axes must both have the same sign The probability scaling type can be used for plotting normally distributed data or to be more accurate their integral If you have a sample of sand and you de termined the percentage of grains having a diameter smaller than x plot this per centage as a function of x using probability scaling for the y axis If the size of the grains is normally distributed your data points will lie on a straight line 64 pro Fit lets you plot on any one of the coordinate axes contained in a graph add new coordinate axes and change their characteristics The next section discusses the general options that are always available when plotting Then we discuss the procedures for plotting functions and data sets and finally we describe how to edit and use existing graphs Plotting a function To plot the output value of a function versus one of its input values 1 Choose the function you want to plot from the
76. as that contain calls like data pro Fit cannot anymore be sure that there is no recursion Avoid the definition of such recursive calculations because they can lead to unpredictable results Entering data You can type data in the data window copy and paste it or drag it and drop it everywhere you want Instead of entering a number directly you can also enter a mathematical expression like exp 1 or 6 sin m 4 or any predefined function or variable available in Pascal script You can also import data from text files 30 Transforming data pro Fit offers various methods for transforming data Numeri Cale Func Prog Customiz cal transformations data reduction sorting transposing and Simple Curve Fit HR Fourier transforms In addition you can write programs that Curve Fit XER edit manage or create data in any conceivable way for Multiple Fit ASR more information on writing such programs see Chapter 9 Optimize X30 Defining functions and programs Get Fitted Params ge All the commands for transforming data are found in the Calc Tabulate XT menu and they work on the data window which is in front of Analyze WK all other windows Data Transform HY The following commands are available Data Reduction Data Transform Simple data transformation such as sore N algebraic transformations on columns or cells Statistics dU Binning e Data Reduction Algorithms for
77. ave added a function or program to the menus that you have written in pro Fit Pascal you can save its compiled code as a separate file for later use This file is called a plug in You can also create plug ins in an external compiler pro Fit comes with a set of plug ins for differ ent tasks You can use them to add functionality to your copy of pro Fit in much the same way you do with the built in definition languages but using the flexibility and power of an external program ming environment to generate your code See Chapter 10 Working with plug ins for an explana tion on how to build plug ins This section explains how to use such plug ins Saving functions and programs as plug ins To save a function or program as a plug in choose Save as Plug In from the Customize menu to see a submenu with all the functions and programs that can be saved as plug ins Note this pres ently works for Pascal functions or programs only This sub menu has two sections divided by a horizontal line The first section lists the functions the second section lists the programs Choose the function or program you want to save as a plug in and pro Fit will ask you where you want to save it Note that you can only save functions and programs that you compiled in pro Fit you cannot save built in functions or plug ins The resulting file is a pro Fit document You can load it by using the Load Plug in command or by double clicking it in the Finder
78. bed stored in a retrieval system or translated into any language or computer language in any form or by any means electronic mechanical magnetic optical chemical manual biological or otherwise without prior written permission of the publisher The information in this user s guide is subject to change without notice Macintosh and LaserWriter Finder Mac OS Mac OS X PowerBook Quickdraw Quartz Power Macintosh Macintosh Programmers Workshop MPW and Xcode are registered or non registered trademarks of Apple Computer Inc PostScript is a regis tered trademark of Adobe Systems Incorporated Python and the Python logos are trademarks or registered trademarks of the Python Software Foundation www python org They are used with permission of the Founda tion pro Fit is a trademark of QuantumSoft Z rich The main resource for customer support is the QuantumSoft webpage at www quansoft com Additional customer support is provided on a discretionary basis by e mail at the address profit quansoft com Table of contents Introduction This handbook Key features System Requirements A note on updates Basic concepts Getting started A first session Our data Entering the data Plotting the data A function to fit our data Fitting Defining your own functions pro Fit Pascal and Python Writing programs Working with data Data editing The data window Data types Permanent transformations Entering dat
79. ble help panels and disable pasting templates Clicking a marker in the Preview window option to transform the clicked marker into the reference marker Moving a marker with the arrow keys in the Preview window option to let the marker go outside the ranges of the preview Using the left and right arrow keys in a data window option to move the insertion point by one character within the active data cell Clicking in the data window command to create a discontinuous selection Starting pro Fit option and shift in order not to load the standard preferences file Another commonly used shortcut is typing a period while holding down the command key This is equivalent to pressing the escape key and it interrupts most activities and calculations Use it to stop the plotting of a function to stop fitting to cancel printing or to interrupt lengthy calculations The combination Command key period is also interpreted as typing Cancel in dialog boxes The escape character is also interpreted as Cancel Return or Enter are always interpreted as clicking the outlined button 177 Appendix A About numbers Floating point numbers pro Fit can use two different formats for representing floating point numbers or float real or float This format has smallest accuracy but requires minimum size It is used in data windows if you set the range of a column to 1E30 1E30 e doub
80. box plot is an advanced plot that represents the statistical properties of one or more data sets Each plot symbol is a box that represents each data set and its statistical features The box indi cates the minimum maximum lower quartile upper quartile and median of the data set Optionally the box can also show some or all data points in the data set To generate a box plot choose Box Plot from the Plot menu 63 Axis types A N BOYS GIRLS The second important part of a graph are its coordinate axes A graph can have several axes All of pro Fit s built in graph types have x and y axes The x axes extend by definition horizon tally while the y axes extend vertically The z axes are not directly visible in the drawing plane They define the scaling color scheme and position of contour lines for z values in color plots and contour plots A graph always maintains one special coordinate axis for each dimension These special axes can never be deleted These are the main coordinate axes and are called X1 Y1 and Z1 The other axes are called X2 X3 Y2 Y3 Z2 Z3 and so on All axes can have linear scaling logarithmic scaling 1 x scaling or probability scaling 1 x logarithmic linear robability a 10 101 100 L 0 99 0 90 0 50 0 10 0 01 The linear scaling type is the standard scaling type It indicates that there is a linear relationship between the coordinates of the gr
81. calculate the mean deviation xr during fitting pro Fit calculates the function for each data point xi yi This may involve up to several thousand executions of the body of the function definition If your function definition contains expressions that do not depend on the value of x such as sin a 2 a 3 they will still be recalculated for each new value of x wasting a lot of time You can evaluate these expressions in the procedure First and store their values in vari ables used by the main part of the function The following example implements the function y sum_of_column a1 a2 x where sum_of_column a1 is the sum of the values in column a1 of the current data window As summing the values is a potentially slow operation the function only calculates the sum when the input values and therefore a1 change i e when First or xxx_first is called In addition it is calculated once when the function is compiled In Python function SampleFunc the defaults for the input values input 1 constant al 1 10 input 1 active a2 import numpy as np def sum_of column c return np nan_to_num pf GetData pf RowRange c sum def SampleFunc_first al a2 global s s sum_of_column al s sum_of_column 1 initialization at compile time def SampleFunc x al a2 global s 135 return x s a2 In Pascal function SampleFunc inputs Default values of the parameters optional a 1
82. cally 168 11 Printing There is a wide range of different printers that can be connected to a Mac OS machine and each of these printers have different capabilities resolutions and command languages pro Fit allows you to get the best out of most of the commonly used printers Basically there are two possibilities for printing pro Fit drawings You can print from pro Fit directly using the Print command from the File menu or you can export a drawing to another application such as a word processor using the Copy command or by dragging it to the other application and print it from there The next two sections discuss these two possibilities separately Printing from pro Fit Before printing you should choose Page Setup from the File menu You can print the active window by selecting the Print command from the File menu For drawing windows pro Fit offers two different modes of printing Quartz and Postscript You can select the desired method by choosing Preferences from the pro Fit menu In the dialog box that comes up click the icon Print in the list at the top If Quartz is checked pro Fit prints drawings using the Quartz engine introduced with Mac OS X This provides best results on all printers as well as for PDF generation If PostScript is checked pro Fit generates PostScript commands while printing a drawing window On a PostScript printer this provides good results However printing text with PostScript may s
83. can interrupt fitting by holding down the command key and the period key simultaneously While a fit is running you may want to place pro Fit s progress window in a corner of your screen to watch what is going on while you use other applications Inspecting the progress of a fit During lengthy fits you can inspect what is going on and see if the fitting algorithm is behaving cor rectly pro Fit displays information on the current fit in its progress window r pro Fit Fitting iteration 212351 chi squared 3 2301 best chi squared 0 6160 const 0 3714408 m 1 7225593 x0 0 8847926 type to interrupt This window lists the total number of iterations the current values of chi squared and the current values of the best parameter set You can see the progress of the fit graphically if you use the Preview Window to see the current data set and function During a fit pro Fit will periodically draw the function corresponding to the current best parameter set This allows you to see how the function as it approaches the data set during a successful fit Because of this previewing feature you will notice soon enough if the fit is not converging correctly and will then be able to interrupt it Note If your function performs many lengthy calculations redrawing the function periodically can slow down the fit Hide the Preview Window or uncheck Show Function if fitting speed matters Error analysis and confidence int
84. corre sponds to the ith data cell in the first column of the current data window The while loop The while loop takes the following form while condition do statement The statement of the while loop is executed as long as the expression in condition returns true If more than one statement should be executed in the loop the statements must be enclosed by be gin and end Example while i lt NrRows do begin if DataOK i 1 then if data i 1 lt 0 then begin DeleteRow i i i 1 end i i 1 end of while loop The repeat loop The last kind of loop is the repeat loop Its general form is repeat statement until condition 196 In contrast to the while loop the statement of a repeat loop is always executed at least once After the execution of the statement the condition is tested If the condition is true the loop is termi nated else the loop statement is executed again until the condition becomes true Loop control statements cycle and leave You can place the keyword leave into a for while or repeat loop to exit the loop even if its end condition is not yet reached Example for i 1 to NrRows do begin if not DataOK i 1 then begin Writeln Empty cell loop aborted leave exits the for loop end end The above example loops through the first column of a data window and does some calculations indicated by If however an empty cell is found the loop is aborted You can
85. ction curve to go through the current cursor coordinates You basically drag the function curve around to new positions under the constraint that only the value of the cur rently selected parameter can be varied to do so This is a great way to get a feeling of how a parameter influences a function When you select the tool and click and drag in the view port the curve of the function follows the position of the mouse while the selected function parameter is adjusted accordingly The parameter that is varied by this tool is set either by changing the selected parameter in the parameters window or by choosing a new parameter from the popup menu that ap pears below the tools palette in the preview window when this tool is selected You can only vary one parameter at a time When you select the fitting tool and click into the preview the selected parameter is varied until the function curve goes through the point indicated by the mouse pro Fit does this by numerically solving the function f a x y where a is the selected parameter and x y is the point indicated by the fitting tool If it is mathematically not possible for the function to go through that point no matter what the value of the selected parameter is then you will not be able to drag the function curve to that point The same applies if pro Fit fails to find nu merically the correct value for the parameter If you use the fitting tool with a slow function pro Fit will automatically r
86. ction definition syntax If you want to define a function of your own to use it for fitting or plotting you must write a function definition The structure of a function definition is the same as the structure of a program definition but it can optionally contain additional information about the parameters and the contents of the parameters window This additional information is placed right at the beginning of the function defi nition A function definition starts with the keyword function instead of program Then follows op tional information on the parameters and the parameters window function myFunc the name of the function myFunc will appear in the Func menu 187 function myFunc ampl freq real var outl out2 real description parameters 4 inputs a 1l 1 2 active a 2 3 0 inactive name a 3 2 0 constant a 4 l active i 0 INF inputs ampl 1 2 active freq 3 0 constant outputs y 0 0 E absolute va lue Out1l 0 Ex x component Out2 0 Ey y component const c 2 997E8 var temp extended myVar t integer textl text2 optional definition of names that will be used to access input and or output values in the function code and as a default name in the parameters window Output values are preceded by the keyword var optional these two strings will appear in the parameters window optional the number of additional besides the standard x val
87. d String and char types Use the type Char for representing simple characters String for representing strings of up to 255 characters UnicodeString for representing strings of unicode characters of arbitrary length Example program StringAndCharTest var c Char s String u UnicodeString begin c g x s hi there writeln c writes c writeln s writes hi there s s Joe s now is hi there Joe c s 2 c now is i u Unicode strings can contain any characters such as gt or writeln u end Conversion between Strings UnicodeStrings and Chars is automatic For conversion between Char ASCII values and Integer use the functions Ord and Chr For conversions between Strings and numbers use NumberToString and StringToNumber To access the n th character in a string or UnicodeString s use s n In other words strings are ar rays of type char The following is a list of the most important functions for working with strings 194 Length Returns the length of a string Pos Delete Find delete a sub pattern in a string UpperString LowerString Convert between upper and lower case strings See pro Fit s on line help for a complete list Arrays pro Fit allows the definition of one dimensional arrays The following syntax is used var name array miniIndex maxIndex of type Where name is the name of the array minIndex is its minimum index maxIndex is its maximum
88. d for the first and last value of the table and its step width The input value that is varied to generate the table is set in the dialog box that appears and the table will contain all output values Optimization of functions The command Optimize from the Calc menu lets you find the maximum or minimum of a function by varying any of the input values To select which inputs must be varied to optimize the function set their fitting mode to active and use the check boxes in the dialog box that appears when choos ing the Optimize command Analyzing functions The Analyze in the Calc menu brings up a window to calculate the roots the extremes and the in tegral of a function 36 ANP Analysis of Polynomial x Parameters for basic analysis v Preview Status Analysis completed Type of analysis Calculate B atX fo 1 0 x range from to Number of Subintervals Use fitted parameters C Tabulate by varying x ce from to step Found xX Y Calculated 0 0000 0 0000 Calculate and copy values to Results Window New Data Window Done ee 4 N _ 7 With this tool you can do different types of analysis A preview area to the right helps you to find the answers you are looking for While the analysis window stays open you can modify the func tions parameters in the Parameters window or even change the current function and observe the effects on the results shown in the analysis window According the type
89. d then drag the mouse down to the first row to be selected To select all cells choose Select All from the Edit menu You can create a discontinuous selection e To extend or modify a current selection to a discontinuous selection click and drag into the cells to be selected or deselected while holding down the command key Note that selecting data in the Preview window can also create a discontinuous selection See also Chapter 6 Preview Window Data types By default each column of a data window contains numerical data i e real valued numbers There are two choices for the precision and range of these numbers e 1E 300 to 1E300 with approximately 12 significant digits double precision e 10E 38 to 10E38 with approximately 6 significant digits single precision By default a new data window opens with either single or double precision columns The default type can be selected by choosing the command Preferences from the File menu In the dialog box that comes up click the General icon 28 Acolumn can also contain text up to 255 arbitrary characters in each cell Finally a column can contain absolute dates or relative time values durations To change the format of the data in a column first select the column or columns you want to change and then choose Column Properties from the Calc menu Alternatively you can also double click the column number of a column you want to change 8 Column P
90. dditional ones specified as var parameters The parameter window uses the names of the var parameters for the additional output values and the hy 9d name y for the output value corresponding to the function name Instead of using the names defined in the function header it is also possible to use the predefined array for the output values function electricfield phi real var Ex Ey real real var E real begin E 1 x y 0 E y 1 E cos phi y 2 E sin phi end You can even mix the predefined output value array with the names you defined in the header even inside the outputs statement function electricfield phi real var Ex Ey real real outputs y 0 0 E absolute value of the field Ex 0 Ex x component of the field Ey 0 Ey y component of the field var E real begin E 1 x y 0 E Ex E cos phi Ey E sin phi end Selective calculation of output values In most cases it is not necessary to calculate all output values of a function The classic example is when only one of the output values is being plotted in a normal 2D plot In order not do unneces sary calculations and to speed up things pro Fit provides a predefined function Output i integer that you can call to find out if a given output value must be calculated Note that sometimes a function should calculate more than one output value at the same time as 128 an example when the output val
91. does this by extrapolating from the low resolution appearance in such a way that a high resolution view would give the same symmetry For example at the 100 view you can draw three overlapping lines with thicknesses of 0 25 0 5 and 1 0 pts All three lines have exactly the same appearance e g they appear 1 pt thick pro Fit sets up the floating point coordinates of the lines in such a way that the thinner lines are centered on the 1 pt thick line Thanks to this interpretation you get the same result at 100 view if you draw a 1 pt line and then make it 0 25 pt thick or if you draw a 0 25 thick line directly On the other hand if you draw a 0 25 pt thick line at 400 view go to 100 view and draw another 0 25 pt thick line on top of it the two lines will not necessarily overlap This is because the first line was positioned with a much larger precision than the first line Use the Align submenu in the Draw menu to make sure that such lines really overlap or look at their coordinates using the Coords window see later Likewise if you have two graphs or rectangles set their size to be exactly equal and overlap them by hand at 100 view one of their borders might be off by one pixel if their position is not exactly the same This is because round off errors must occur when calculating their rounded coordinates at 100 view If you set their position to be exactly equal using the Align command or using the Drawing Info window the ro
92. e Get Info from the File menu and check the option Display As Dialog The window now looks like a dialog box O O f Simple button example Multiply by 2 obviously you can do a better job with the appearance of the window and its contents 4 Attach the program To attach the program hold down the control key while clicking anywhere into the window and choose Show Program Window Then enter the following program import numpy if pf GetTagObject program tag msgWhy value control clicked msgShape pf GetTagObject program tag msgShape value if msgShape Button editField pf GetShapeObject shape Number x editField value if not numpy isnan x editField value x 2 Or if you prefer Pascal program attached var msgwWhy String msgShape String x real begin GetTag program tag msgWhy stringValue msgWhy if msgWhy control clicked then begin GetTag program tag msgShape stringValue msgShape if msgShape Button then begin x GetShapeProperty Number value if not Invalid x then if valid number SetShapeProperties shape Number value x 2 154 end end end Hit Command L to add the program to the window Now your dialog box is ready to use Enter a number in the edit field then hit Multiply by 2 Notes If you want to modify the items in the dialog window switch it back into
93. e See the beginning of this section for a discussion of scaling types Note that First can be larger than Last if you want to reverse the axis The Ticks field to the right of the Draw check boxes is used to edit the tick marks Enter the first major tick the distance between major ticks and the number of minor ticks between two consecu tive major ones The edit field 1st major gives the coordinate of the first major tick on the axis For a linear axis the Distance field defines the distance between the major ticks For a logarithmic axis this field changes its name to Decades and defines the number of decades between major ticks For a 1 x scaling the edit fields work in the same way as for linear scaling For probability scaling you can edit the list of tick marks directly using the Custom check box For a linear axis the minor field gives the number of minor ticks that are drawn between two ma jor ones For a logarithmic axis this field is replaced by a check box called small ticks which must be checked to draw the minor ticks If major ticks are drawn for each decade the minor ticks are drawn for each multiple of ten If there is more than one decade between major ticks the minor ticks are drawn at all the powers of ten between the positions of the major ticks T7 Instead of automatically calculating the positions of individual ticks you can set them manually Check the custom check box This changes the contents of the ticks fie
94. e Results window and then modifies it to first column As a Python script script ObjectExample obj pf GetPlotObject plot 1 print obj name get the name obj name first plot set the name As a Pascal script program ObjectExample var obj Object begin obj GetPlotObject plot 1 writeln GetPlotProperty obj name get the name 143 SetPlotProperties plot obj name first plot set the name end Automatic Macro Recording pro Fit can record most operations that you perform and generate Pascal instructions Python instructions or Apple Script instructions from them Open Apple s script editor to record your activ ity as an apple script See chapter 11 Apple Script for more information If you do not know how to program a certain action with pro Fit s definition language s switch on recording perform the action you want to program and look at the recorded commands Each text window has record play and stop buttons 000 M To Menu __ De function Lin_py The record button is the one with the circle in its center the stop button is the one with the square the play button is the one with the triangle To record your actions click the record button pro Fit will automatically generate a Pascal or Py thon instructions depending on your preferred language script for nearly everything you do When you have finished recording click the sto
95. e are the following modifier codes prede fined modButtonState modCommand modShift modAI phaLock modOption modControl You can set msgCharCode to an empty string to suppress the event A control has received keyboard input Called after the key is processed Same parameter as for message con trol keydown start The drawing window was opened The drawing window will be saved The drawing window will be closed A command added by the procedure AddCommand has been called The tag msgCommand contains the name of the command The script is being called because the value in its prop erty idleCallTime corresponds to the present value of TickCount In addition to the tag msgWhy attached scripts can always rely on the presence of the msgOwn erWindow The value of this tag is the ID of the window to which the script is attached The following code snippet retrieves the msgWhy tag using Pascal var msgWhy String GetTag program tag msgWhy stringValue msgWhy 150 And this retrieves the value of the msgWhy tag in Python msgWhy pf GetTagObject program tag msgWhy value if msgWhy check here for known tags The smallest skeleton of an attached program in Pascal looks like program attached var msgWhy String begin GetTag program tag msgWhy stringValue msgWhy if msgWhy then check here for known tags end It is also possible to attach a script
96. e data window If you select some rows in the data window the corresponding selection is shown in the preview window There is even the possibility of clicking and dragging data points in the preview window Doing so changes their coordinates in the data window 41 Preview Window Appearance You can get a more compact toolbar without the text under the buttons by commana clicking the button in the top right corner of the window s title bar Doing this cycles through different toolbar appearances You can set the color and appearance of data points function curve and markers by choosing Preferences from the pro Fit menu Preview Window Tools To the right of the preview window there is a palette of five different tools You can use them to se lect data points and change their coordinates graphically to change the ranges of the preview win dow view port to graphically change the value of the function parameters and to set coordinate markers k Use the arrow tool to select data points Simply click a data point to select it Click and drag to select a range of points with a selection rectangle Hold down the shift key to add points to the current selection or to remove points from the current selection If you hold down the shift key while dragging a selection rectangle the selection state of the data points contained in the rectangle toggles between selected and not selected Hold down both shift and option keys t
97. e the optimal initial values for the parameters of the current function and transfer them to the parameters window 46 7 Drawing and Plotting Drawing and plotting happen in a drawing window This window is essentially a small drawing edi tor for various standard objects such as rectangles lines ovals etc and one important new ob ject the graph that contains plots of functions and data The section The drawing window discusses standard drawing objects and editing techniques The section Plotting is devoted to the plotting commands used to produce graphical representa tions of your data and functions It discusses how to manage graphs and how to edit them The drawing window A drawing window always contains one single page or a set of pages arranged side by side You can select size and orientation of a page thorugh the Page Setup command in the File menu A dotted rectangle frames the printable area of the page Objects that lie outside this rectangle do not print See your printer s manual for more information on printers and paper sizes You can view the drawing in a drawing window using various zoom factors which you can set us ing a popup menu in the drawing window tools palette Drawing tools RIA pro Fit provides various tools for editing drawings These tools are collected in a tool O jr box which is either placed in the left margin of a drawing window or in a separate OIW d floating windo
98. e verti cal or horizontal 175 Clicking objects in drawing win dows shift to select an object without de selecting other already selected ob jects Clicking graphs in drawing win dows option amp command click to see the plot coordinates of the point you are indicating with the cursor option amp command click and then press shift to select an area of the graph to be enlarged command double click to make a graph the current graph command amp shift double click to remove the current graph setting Clicking nothing in drawing win dows command click to zoom in centering the clicked point in the new view option amp command click to zoom out Using the line style pop up menu in a drawing window to change the line styles in a legend shift to change the line styles of all the lines in the legend option to change the line style and set the attribute points connected for the data plot in the first row of the legend shift amp option to change the line styles of all the lines in the legend and set points connected for all data plots Using the point style pop up menu in a drawing window to change the point styles in a legend shift to change the point style of all the data plots in the legend Clicking a cell in a data window option to select the whole column above the clicked cell shift to enlarge a selection Clicking the column n
99. ecording operations Even for Pascal and Python you do not need to know exactly what to do in order to program an operation such as fitting plotting etc The code that corresponds to any user action can be gen erated automatically by switching on recording in a pro Fit function window Introduction to Python and Pascal programming in pro Fit In many situations pro Fit will try to guess what language you are using However before starting to define a function or program you should tell pro Fit what your preferred language is This will help pro Fit to make the right choice when determining the language used in a window To do so choose Preferences from the pro Fit menu go to the Scripting tab and select your preferred language from the popup menu in the top left corner 115 aac Preferences E era i _ Wri sk Oe L k General Print Copy Draw Plot Preview Date Scripting Interface Extension Meta Prefs Preferred macro language Python B M Try to guess the language Path to Python library Change Set to Default System Library Frameworks Python framework Versions 2 6 Python M Syntax coloring Set to Defaults gt Background Comments ll Global functions ill Normal text il Strings Global constants Keywords iii Global variables BEE Replace tabulators with spaces only in Python scripts If replacing tabulators with spaces use fa spaces If not
100. educe the resolu tion with which the function is drawn so the function will not appear to be smooth any more The resolution will be increased again once you are finished dragging Click the Re fresh button to achieve the maximum resolution 43 The last tool in the tools palette can be used to place coordinate markers on a given data point or on the function curve Select the marker tool and click the curve or a data point pro Fit creates a new marker at the indicated position While you move the marker tool around inside the view port of the drawing window the corresponding coordinates are displayed in the bottom left corner of the preview window When you create a new marker it becomes the active marker The active marker is always flashing on and off You can create any number of markers The first marker you create is the reference marker Subsequently created markers are auxiliary markers and are numbered starting from 1 Their number appears when they are active when they are flashing To set the color of the reference marker and of the auxiliary markers use the Preferences com mand If the reference marker cannot be distinguished by its color pro Fit automatically draws it larger Marker coordinates are displayed in the bottom left corner of the preview window If the marker is a data marker and the preview window is big the data window row number that corresponds to the marked data point is also displayed It is found abo
101. ering the chi squared that was obtained in the fitting For the standard deviations of the parameters this amounts to calculating them from Oa Cj X71 12b If you need to work with the covariance matrix and in general for any serious statistical analysis of a fit you must define errors for your data points The matrix A is also called curvature matrix and it is defined by the errors standard deviations of the data points and by the partial derivatives of the function with respect to the parameters When x errors are specified the derivative of the function with respect to x must also be calculated and the curvature matrix A is given by B 1 of x f x A T a a 13 O J k xk Ox If the x errors can be considered to be zero the curvature matrix A has the simpler form 1 f of A v of Of 14 k O a da Loosely speaking this matrix describes the propagation of the errors from the data points to the parameters We refer to the specialized literature for more details 98 If the x errors can be regarded as zero pro Fit lets you specify unknown y errors In this case the yi are assumed to be normally distributed all with the same standard deviation o For fitting Oyi is taken to be 1 for all i The real oy is then estimated from x2 v where vis the number of degrees of freedom i e the number of data points minus the number of parameters and daiis cal culated f
102. errors for each data point must be uncorrelated and described by probability distributions centered around the true values The above assumption might appear harmless but it is in fact more stringent than one would causally expect For example in most cases one tends to assume that the probability distribution is Gaussian but the actual probability distribution for the measurement errors might be different with a sizable probability of finding larger errors from time to time i e points that are clearly outside the expected trend outliers To allow for an analysis of such cases pro Fit provides a set of deviation functions R which corre spond to various error probability distributions The most common deviation function provided by pro Fit is the squared deviation R d 3 When using this deviation function Eq 2 becomes the mean square deviation between data points and function Eq 2 then corresponds to the negative logarithm of the probability of obtain ing the data set in the presence of normally distributed measurement errors The deviation function 3 corresponds to a Gaussian error distribution In this case the probability density that a certain error occurs when measuring x or y is given by a Gaussian distribution or normal distribution The deviation function R d idl 4 gives a mean absolute deviation instead of a mean square deviation It corresponds to a two sided exponential error distribu
103. ervals Check Error Analysis in the Fit dialog box to get more information on the confidence intervals of the parameters When Error analysis is checked two more edit fields appear in the Fitting Setup dialog box The Error Analysis algorithm simulates a number of data sets equal to the value specified in the iterations edit field For each iteration the corresponding parameter set will be determined by the fitting algorithm you selected either the Robust algorithm or the Levenberg Marquardat algorithm Note You should always use the Levenberg Marquardat algorithm when performing error analysis Using the Robust algorithm is not recommended because this algorithm is inherently slower than the Levenberg Marquardt algorithm Since error analysis can need thousands of iterations the convergence speed of the algorithm is very important 106 All parameter sets generated during error analysis will be collected and displayed in a data win dow You can then use them for a more complete analysis of the distribution of each parameter Based on the simulated parameter sets pro Fit estimates confidence intervals You must specify which confidence interval you want pro Fit to calculate by entering the corresponding probability in the confidence intervals field pro Fit calculates a confidence interval in such a way that the given percent of the simulated parameter values are contained inside it If you want to see what happens during error analysi
104. erwise the minimizing parameters found are in general not meaningful The goodness of fit which is the probability Q that a value of chi square should occur by chance is calculated by the incomplete Gamma function N M x O gan ia c 22 It depends on the degree of freedom defined as the difference between the number of measured points N and the number of varied parameters M 101 If Q is large e g gt 0 1 the fit seems reasonable If it is small e g lt 0 001 there might be some thing wrong Note that the Goodness of fit is only meaningful if you used meaningful errors for your data Literature and suggested reading W H Press B P Flannery S A Teukolsky W T Vetterling Numerical Recipes the Art of Scientific Computing Second Edition University Press Cambridge 1992 P L Jolivette least squares fits when there are errors in X Computer in Physics Vol 7 No 2 1993 The fitting process General features With pro Fit fitting is a highly interactive process You can decide which parameters have to be varied set their starting values estimates and choose a fitting method You can inspect the fitting process while it is running and interrupt it if you don t like it You can reiterate the process and change fitting algorithms The fitting process starts from the parameter values given in the parameters window You can change these values click the numbers and edit them The window also show
105. es for slight variations in a data set by simply loading a set of data files and looking at the preview window display GED la ss Le Q ne Fit a Undo Refresh Help Fitting Options b Selected Data Only i Print to Results Window after Fit Transfer Fitted Parameters after Fit Fitting Algorithm f Am ae ees At the edges of the rectangular view port that displays the function and the data set are four edit fields giving the coordinate range that is shown You can edit the values to change the x or y range Between these edit items there are check boxes labeled auto and log Check them to let pro Fit automatically recalculate the ranges based on the ranges of current function and data set or to use logarithmic scaling The preview window always displays an up to date representation of the current function and data set Change a coordinate in the data window or add a data point and the corresponding point will automatically appear in the preview window Change a function parameter and the representation of the function in the preview window will be updated automatically Modify a function definition and add it to the menu once again and the preview window will automatically display the new function When Live Fit is enabled the preview window automatically displays the best curve fit when any thing in the current data set changes If you select data points in the preview window the corresponding rows are selected in th
106. es of x Note that you can insert additional spaces or lines anywhere between keywords The new version of the function which now has the name LogSine shows how you can define names for the parameters B and D simply list them between brackets after the function name as shown and use the if statement for conditional execution The if statement takes the general form if condition then do this else do that do this is executed if the condition is met do that if it is not met If you work with your function more often you might want to make sure that the Parameter window shows reasonable default values for the parameters and a short description of what the function does Here is a final and more complex definition implementing this note that texts between curly brackets and are comments they are ignored 125 function Myfunction B D real description text to appear in parameters window x gt 0 y A sin x ln x B x lt 0 y B inputs names and defaults for the parameters B 1 active B the amplitude of this function D 0 inactive D a constant offset begin if x lt 0 B B sin x ln x D then y else z i end In this version of our sample function two additional elements have been added e A keyword description followed by two texts between quotes which appear in the header of the parameters window A keyword inputs in pr
107. es that are used by plotting applications for managing graphs and the data used to generate them e The first one consists in maintaining a permanent link between the data you plot and the result of the operation the graph In this approach whenever you edit the data you used for creating the plot the plot automatically changes to reflect the new values of the data set Since the link between data and plot needs to be maintained it is in generally not possible to save data and graphs separately and they must be saved in the same document In applications using this approach the graph is only a different view of the data but does not lead an independent life e In the second approach graphs and data are independent Although a graph can be created from data and data can be recovered from a graph the two documents lead separate lives Af ter it has been created the graph does not know anymore about the origin of the data used to create it and if you modify that data the graph remains untouched pro Fit uses the second approach while still providing a link between graphs and data that allows to update a graph easily when the data changes pro Fit has separate data documents and drawing documents From the data you can create graphs From the graphs you can recover the data used to plot them Drawing and Data documents can be stored and maintained separately and don t automatically affect each other In Chapter 7 Drawing and plottin
108. ets And much more Such customizable data file import and export services multi dimensional functions etc System Requirements pro Fit 6 2 has been developed for Mac OS 10 6 or better Compatibility with earlier systems has not been tested even though pro Fit 6 2 will probably also work mostly on MacOS 10 5 10 For users interested in running a version of pro Fit on older systems and machines pro Fit 5 1 through 6 1 are still available upon request pro Fit 5 1 is MC 680x0 based pro Fit 5 5 is Power PC based and allows to save most of its documents in pro Fit 5 1 format pro Fit 5 6 requires Mac OS X 10 0 4 or later or Mac OS 9 with CarbonLib 1 3 or later pro Fit 6 0 requires Mac OS X 10 3 or later pro Fit 6 1 requires MacOS X 10 4 or later pro Fit 6 2 requires MacOS X 10 6 or later A note on updates Development of pro Fit continues To check for updates to your current version visit Quantum Soft s web site at http Awww quansoft com You can also use pro Fit s built in mechanism for checking for updates Choose About pro Fit from the application menu and click the button Check for newer versions pro Fit will contact our servers and tell you about any updates that you may want to download 11 2 Basic concepts pro Fit works with data drawings functions and scripts You can enter data into spreadsheet windows Data can be analyzed plotted and transformed For transforming data y
109. evious pro Fit versions defaults was used which is followed by additional information for each input that appears in addition to x and takes the form ali value mode name lowLimit highLimit where value is the default value of a parameter mode its default fitting mode it can be active i e the parameter will be fitted inactive will not be fitted or constant cannot be fitted and name its name between quotes used in the pa rameters window Instead of using ali you can also directly use the name that you declared between brackets after the function name Using the keyword inputs you can also define a range of acceptable values for an input given in the optional parameters lowLimit and highLimit See the detailed description of the inputs keyword later in this chapter Once you have successfully defined a function and you have added it to the Func menu you can save it as a plug in A plug in is a file that contains the computer code for your function and that can be loaded by pro Fit at start up or at any other time Go to the last section of this chapter for more information on plug ins Functions with multiple outputs It is possible to define a function that has multiple output values The default output value can then be specified in the same way that you specify the default input value Using the corresponding check box or using the Func menu The case where a function has only one output is the
110. f data points evaluation of 8 will require more or less ten times more function calls when x errors are defined The Linear Regression algorithms In this case we assume a straight line model for the measured data with normally distributed er rors y x a bx 15 If there are no x errors and the y errors are assumed to be known 0 is the uncertainty of yi equa tion 9 can easily be simplified At its minimum the derivatives after the two parameters a b van ish This leads to a set of linear equations that are solved analytically SoS SS S So SS ga A be a a A A 16 using the following definitions N 1 N es N y Se Se Sie i Qi izl Gi i l Qi N 2 N y Xi Xi yi Sx 2 Sx a 17 j Oi izi i A s5 s From these we are also able to calculate the variances of a and b and the correlation coefficient between them 02 Su A 05 S A S 18 Fab S S S If the measurement shows errors in the x the minimization of 8 becomes more difficult i e the set of equations derived for a b are not linear any more However they are solved with numerical means i e with a standard root finding algorithm Together with the fitting parameters and their variances the correlation coefficient r Pearson s r is calculated It assumes a value between 1 and 1 depending on how much the x values and the corresponding y values are correlated r 1 if there is a complete correlation with a p
111. fined numerical algorithms to retrieve their results Create graphs and other drawings in a drawing window using a precise floating point coordi nate system Access the properties of drawing objects in drawing windows and mange buttons check boxes popup menus or other interface elements that can be drawn there 113 Note All the above can also be done from a plug in a piece of code that you can generate yourself using your favorite programming environment If you are used to programming your own code for data or function analysis you can consider pro Fit as a big library offering routines for numeric analysis data input output and high resolution graphics Information on how to create such a plug in is found in Chapter 10 Working with plug i INS Built in languages for creating new functions and programs pro Fit supports two different built in languages for defining functions and programs pro Fit Pascal This language is loosely based on the Pascal programming language It is the traditional language for scripting pro Fit and pro Fit comes with numerous examples of written in this language Python Starting with version 6 2 pro Fit fully supports Python as an alternative to pro Fit Pas cal The Python programming language is an advanced object oriented dynamic programming language For more information see the www python org the website of the Python Software Foundation The advantage of the
112. fines the default value default mode active means that it will be varied in a fit and the name of K that will appear in the parameter window The second line defines the default value mode and name as well as the lower and upper limit of Q Example 2 You want to define the function y al sinc x x1 a2 sinc x x2 with sinc x sin x x The value of the function sinc is not defined for x 0 but it converges to 1 for x 0 When calculat ing sinc we must test if its argument is 0 to handle this special case Since the sinc function is used twice in our example it makes sense to put it into a local function function DoubleSinc inputs a l 1 active al a 2 20 active xl a 3 1 active a2 a 4 20 active x2 function Sinc u real real sin u u begin if u 0 then sinc 1 0 0 is illegal else sinc sin u u end begin body y a l sinc x a 2 a 3 sinc x a 4 end Types The following basic types are supported in pro Fit Pascal real real is the standard type for floating point numbers It has at least 64Bit accuracy optional but equivalent type extended integer integer is the standard type for integer numbers It has at least 32Bit accu racy optional but equivalent type longint string string is the type for strings It is at maximum 255 Bytes long char char is the type for single characters boolean boolean is the type for booleans It takes the
113. for each pa rameter The confidence interval thereby corresponds to the range enclosing a given percentage of the values When error analysis is complete the results are printed in the Results window and a list of the fit ted parameters for each synthetic data set appears in a new data window You can use the set of simulated parameters for further statistical analysis Fitting algorithms In the previous section we have given a short overview of the most important mathematical tools used to establish criteria distinguishing a good fit from a bad one Once these criteria are estab lished one can use them to analyze parameter sets and to find out in which direction the best pa rameter set can be found The search for the best parameter set is the responsibility of a fitting algorithm and pro Fit lets you choose between three different ones The Monte Carlo Levenberg Marquardt and Robust algo rithms The algorithms differ by the method they use to orient them in parameter space and to find the lo cation of the best parameter set e The Monte Carlo algorithm minimizes 2 with any definition of R by randomly varying the pa rameters and if the x errors are not zero the set of x coordinates and looking for the smallest 95 value of 2 This algorithm is often useful in difficult situations to scan parameter space and find initial values for a Levenberg Marquardt or Robust fit The Levenberg Marquardt algorithm minimizes the mea
114. foremost data window or the window you have selected yourself When a data window is used as the current data window by a function or by some commands four of its columns can have a special meaning They are the input and output columns You can de fine these columns using the pop up menu that appears when you click a column while holding down the control key For example the Spline function uses the data in the default input and output columns of the foremost of all the data windows other windows of a different kind e g a drawing window can be active A small in out appears in the head of a column marks the default columns that will be connected to input and output values of a function when performing calculations or used when plotting Note that the index column can also become one of the default columns 32 5 Working with functions Functions in pro Fit take a set of numerical values input values and convert them into one or more output values by applying a mathematical transformation Output and output values in pro Fit functions all have their individual names Many functions have only one output and in such a case its name is very often y But in general a function can have a number of different outputs Depending on how the function is used the inputs are often divided into two distinct populations The ones that are varied according to some pre defined rule in order to obtain new outp
115. g you will see how you can use the Draw menu to plot a function and a data set obtain graphical representations of your data and functions and edit the graphs to obtain the precise graph style you are looking for There is an ongoing discussion between the supporters of the first approach outlined above and the supporters of the second approach used by pro Fit A link between data and its graphical rep resentation is in fact also useful and pro Fit provides it as a convenience but a real time con stantly updated representation of data may not be what you want especially when a graph that has been carefully prepared for publication is unexpectedly updated while working with data files However pro Fit does provide a constantly update view of the current function and the current data set in a dedicated window the Preview Window The preview window gives you a graphical view of the function and the data set Any change in the data set or in the function is reflected in the preview window You can even use the preview window to graphically edit the function parameters or the data set Use the preview window to have a quick look at a function or a data set without actually plotting it For instance you can let the Preview Window be a floating window and keep it in front while you load many different data files The preview window will automatically display all data contained in the current x and y columns of the front window
116. g routines have a parameter called pb It is a pointer to a struct of type ExtModulesParamBlock Most users won t need the information stored in it Routines to be defined in functions and programs void SetUp short const moduleKind Str255 name long const requiredGlobals ExtModulesParamBlock pb This routine is called when your plug in is loaded by pro Fit It must return the following values e moduleKind must be set to the constant isProgram if your plug in is an external program and to isFunction if your plug in is an external function name must be set to the name of your plug in If you are programming in C you must make sure that you return a Pascal string For this purpose you can use the function SetPascalStr that is defined in proFit_interface c SetPascalStr name pmyName 255 The last parameter is the maximum length of the resulting string e requiredGlobals should usually be set to 0 Advanced programmers can set it to the size in bytes of a global data buffer they want to have allocated If requiredGlobals is returned with a 158 value gt 0 pro Fit allocates a block with the corresponding number of bytes and stores a pointer to it in pb gt globals pb is a pointer to a record called ExtModulesParamBlock and is passed to all routines called by pro Fit Note that memory allocated in this way is de allocated automatically when your plug in is unlinked from pro Fit you must not de allocate this memory
117. ge A singularity occurred This can be caused by badly chosen starting values for fitting However this error is of ten a consequence of poorly defined or redundant parameters For example consider the expo nential function y A exp x xo to const This function has four parameters A Xo i and const However the parameters A Xo and to are not independent When trying to fit them simultaneously the fit fails Note Another problem often encountered during fitting is caused by the poor definition of a parameter Example If you are trying to fit the data points x 1 y4 2 01 2 3 99 3 6 00 4 8 02 5 9 98 6 12 00 to a polynomial of second or higher degree y Got xt ax 83X you will get a very poor estimation of the parameters ae a3 because your data points are nearly on a straight line and are sufficiently described by the parameters ao and a The standard deviations of the coefficients az as will be accordingly large The errors of the data set When using errors standard deviations for your data it is useful to keep some points in mind e Multiplying all errors of your data points with a common factor does not affect the results of fit ting but changes the estimate of the standard deviations or the confidence intervals of the fitted parameters Changing the relative errors of your data points affects the numerical weight of the data points Example If you
118. ge returns char 13 or r Alternatively the linefeed character char 13 n can be used The first line with the column titles is optional These names are separated by tabs character code 9 here denoted as gt char 9 or t If pro Fit reads a file without column titles it sets the col umns names to Column 1 Column 2 etc Some applications produce data text files using other formats or read data text files only when they are in special formats pro Fit provides options to read and write text files in other formats as well The details are given in the next section Importing text files For reading text files choose Import from the File menu choose Text Files All Files or All Known Files from the Enable pop up and select the file to be read A dialog comes up that allows you to specify the format of the file That dialog contains a pop up titled Format having a first en try With title and a second entry Without title corresponding to the file formats described in the 179 previous section A third entry in the pop up menu is called Custom and allows you to specify a custom text file format ame Text file import options Import to text window Settings v Import to data window Format Custom re Header terminated by delimiter R CR LF LF CR or CR LF V Copy header to info M Column title
119. h Styles Saved Styles 6 5 6 temp UV Save style as Default This box shows a list of the styles that are already saved in the current preferences file You can delete one of these styles by selecting it and clicking Delete To save a new style enter its name and click Save To load a style select its name in the scrolling list and click Load The name of the button changes from Save to Load when you move from the Style name edit field to the Saved styles scrolling list 87 If you click the Default check box when saving a style or if you define a style with the name Nor mal this style becomes pro Fit s default style The next time you start up pro Fit the first graph you create will use this style When you load a style a dialog box appears asking you to choose which parts of the style you want to apply to your graph Graph Style Options To apply the style select the characteristics you want to use v Bounds Xmin Xmax v Axes and grids styles thickness color v Curve styles thickness color v Text styles font size vi Label styles number of digits v Tick styles number of minor ticks M Size of graph and position of labels and legend Cancel ox The characteristics of a style are Bounds The ranges of the graph i e the minimum and maximum of all the axes the positions of the first ticks the distance between major ticks the number of minor t
120. he curve represent ing the function is strongly bent in a given interval then the number of points that are required for drawing the function is large On the other hand if the function is a straight line the number of points needed is smaller The following figure illustrates this function Va caclulated points Notes The number of calculated points is optimized for the range a function is plotted in If you change the axes range of a graph later e g for zooming into a detail the number of calculated points may not be sufficient anymore for representing the curve accurately In this case you should redraw the function to create an optimized plot for the new range Plotting with the auto option results in the smallest number of data points stored to rep resent a function s curve In this way you can create a plot that uses a minimum amount of memory and that is redrawn at maximum speed To do so pro Fit may need to calculate a large number of points If you are working with a slow function you may prefer to use a fixed step to quickly obtain a rough plot and to go over to auto step only when you want to produce a final graph The function can be plotted over the whole given range Alternatively you can specify start point and end point manually Specify this by choosing one of the icons titled X range 66 6 4 2 0 2 4 6 6 4 2 0 2 4 6 The controls under the heading Magnitude define the settings for the
121. he function Plot Function M Use fitted parameters Coordinate X axis x W Scaling lin P from 1930 to 1990 X step auto X range dai Magnitude Y axis v W Scaling lin _ Auto range from 2000 to 5500 Options New Window Line style Plot into current graph Crit a Graph Styles 2 Cancel Cancel o We don t need to change any of these options 4 Click OK to draw the curve The curve is drawn in the graph You can now rearrange the items in the drawing window to obtain a representation of data and theory like the following 21 population data e population in millions o 5000 theory S E 4000 2 3000 w Q 8 2000 1940 1960 1980 year Defining your own functions In the previous session you have fitted the built in exponential function to your data Fine But what do you do if your model is described by some mathematical equation that does not appear among the built in functions in the Func menu Define your own function pro Fit can work with virtually all functions you can think of Let us look at an example Imagine you want to analyze a function of the form y a sin x x ln x b 3 3 with the parameters a and b To define this function Choose New Function from the File menu This opens a new empty function window Enter the definition of your function in the new window Just enter a l sin x ln x a 2 on the first li
122. he language pro Fit tries to guess the scripting language you are using For example if a script contains a large number of begin and end or it is likely to be Pascal Once you have settled for a preferred scripting language it is a good idea to select it as default in the above preferences pane Writing programs Besides defining functions for fitting and plotting you can also define any data generation data processing and data transformation algorithms using the same syntax 25 Let us have a quick look at a small program that fills the first column of a data window with the powers of two 2 4 8 16 etc To define this program again open a new function window choose New Function from the File menu and enter program User Program var i integer begin for i 1 to nrRows do data i 1l 2 i SetColumnName 1 Powers end If you prefer Python here is the same program in a Python version for r in pf RowRange pf SetData r 1 2 r pf GetColumnObject column 1 name Powers Once you have entered the program click the To Menu button Then run the program by choosing its name from the Prog menu In this chapter you have seen some of the most important features of pro Fit For in depth informa tion consult the following chapters of this manual as well as pro Fit s on line help 26 4 Working with data Data editing The data window The data window is organized in horiz
123. he right of the Algorithm popup menu Please refer to the beginning of this section for a basic discussion of the Fitting window Clicking Ranges presents another dialog box where you can define the ranges within which the parameters can be varied 108 Note The Monte Carlo fit slows down exponentially when the number of parameters to be fitted is increased Using the Linear Regression algorithm To run a Linear Regression fit choose Linear Regression in the algorithm popup menu of the fit dialog box This dialog box appears when selecting Fit from the Calc menu and it was de scribed above As the name indicates this algorithm forces you to select the Polynomial function of degree 1 with both parameters being fitted It assumes a Gaussian distribution of errors X errors and Y errors are possible When finished the parameter values and their standard deviations are printed to the results win dow Additionally the correlation coefficient r is calculated as well as its significance which is the probability that Irl should be larger than its observed value in the null hypothesis x and y being un correlated Using the Polynomial fitting algorithm To run a Polynomial fit choose Polynomial in the algorithm popup menu of the fit dialog box This dialog box appears when selecting Fit from the Calc menu and it was described above As the name indicates this algorithm forces you to select the Poly
124. here The native data format If you want to exchange binary data with pro Fit you can use pro Fit s native file format A descrip tion of this format is given in the technical note pro Fit file formats in the folder Notes that comes with the pro Fit package Drawings There are various ways to export pro Fit drawings to other applications and pro Fit provides sev eral image formats to do so You can export data through the clipboard by drag and drop as well as by means of files Exporting through the clipboard by means of copy and paste is the most traditional way When you copy a drawing to the clipboard pro Fit provides picture and PDF data for the target appli cation to use see below Exporting by drag and drop may be more convenient in some situations When you export by means of drag and drop pro Fit also provides picture and PDF data for the target application to use see below Exporting by files requires you to save the drawing window using a custom format This is more cumbersome but allows you to use a variety of standard file formats such as PDF TIFF GIF PNG and jpg To export a drawing in a file choose Export from the File menu select the desired file format from the Format pop up and select the appropriate options by clicking the button Op tions 181 Image formats The image formats supported by pro Fit are Pictures PICT format This is the classic format for images under Mac OS
125. hickness color and dash pattern of the connecting line Check Draw points to draw a plot symbol for each point Check Use line color to force the color of the data points to be equal to the color of the connecting line between the data points Check Draw error bars to draw error bars for each point using the default x error and y error columns of the selected data window To specify those columns click into each one while hold ing the control key down and choose Default X error or Default Y error Then choose the Plot XY Data commana The controls titled Fill allow you to specify if and how the area below the curve is to be filled 70 e Use Plot type to select if you want a scatter plot bar chart or skyline plot scatter plot bar charts skyline plot 4 4 4 0 0 0 2 2 2 123 4 5 6 7 123 45 6 7 123 4 5 6 7 When using scatter plots use the Point style pop up menu to select a plot symbol If you are plot ting multiple data sets only the first set will be drawn with this symbol The symbols of subsequent sets are chosen according to the current graph style See section Graph Styles later in this chap ter for further information about graph styles Bar charts and skyline plots can also start from the vertical axis iO N A O CO 1O N A O 3 2 1012345 3 2 1012345 The main differences between bar charts and skyline plots are In bar charts the bars have always the same width The width is either derived from
126. ialog boxes where you can choose the data to be plotted In the XYZ Columns command you can choose the X Y and Z Column In the Table of Z Values command you can choose the data window and the region in the window that holds your data Note The Table of Z Values command assumes that the rows and columns indices corre spond to the current range of the X and Y axes of the graph In other words a plot created with this command will always cover the complete graph even if you change the range and scaling of an axis In practice you determine how the row and column numbers correspond to coordi nates by editing the x and y axes of the graph The z values are represented by contour lines and or by a color encoding The color encoding is derived from a color scheme and its range corresponds to the range of one of the z axes of the graph When plotting you can specify which z axis should be used Graphs and legends graph When you plot data or functions you create a i function graph object and a legend object 20 e data Graphs and legends are the most important draw 10 A ing objects 0 legend 0 1 2 3 4 5 12 Editing legends A legend contains a description for each curve or data set of its graph The description consists of a symbol identifying the plot and a text You can change the line and point style of a plot as well as the text by double clicking the respective items in the legend e Double click the text of
127. ible to create things such as check boxes and buttons in a drawing window Once you have created these con trols you can set up the window as a dialog box and the controls will start to respond to user ac tions A program attached to a window like this can interpret what happened to these controls and perform any appropriate action In this way the window becomes a container for both the program and for the controls that steer what the program does To attach a program to a drawing window or to modify an attached program bring the drawing window to the front choose Get Info from the File menu and check Show program window Then click OK Alternatively click into the drawing window while holding down the control key and choose Show program window from the contextual menu A window with the source of the attached pro gram appears Once you have defined the program choose Attach to drawing window from the Customize menu or simply click the Attach button The program is compiled and its code is attached to the window A program attached to a window an attached program communicates with pro Fit using tags see below An attached program should always check its tag msgWhy to find out why it has been called If this tag contains an unknown stringValue the script should do nothing Otherwise it should take some action according to its needs The tag msgWhy can have the following stringValues clicked The drawing
128. ical description of deviation functions and fitting algorithms the second part shows you how to select these options in pro Fit and how to run a successful fit Mathematical background First it is necessary to establish a quantitative method to measure the distance between a data set and the function that should describe it in order to find the best parameter set describing a given measurement This requires the introduction of weights for the data points and of probability distribution functions They are described in the next sections Distribution functions and data weights Consider a function f x faz a2 x also written as f x we won t write explicitly the function pa rameters every time and a measured data set x7 yz x2 y2 XN yn Let s assume that the function with its true parameter set correctly describes the quantity that was measured We further assume that when the data point x yi was determined the true system the one described by the function f x was at the coordinates x f x When the x coordinate was determined an inevitable experimental error oc curred and x was measured instead of x When the y coordinate was determined another inevitable experimental error occurred and the measurement gave y instead of f x x fx Xi Yi 90 In real life the true parameter set is not known One has to measure it by measuring many data
129. icks Axes and grid styles The line thickness dash and color of the axes the frame and the grid the distance of the labels from the axes the location of the ticks inside or outside Curve styles The line style of all plots i e curves and data points Text styles The font size and text style of the labels Label styles The number format of the labels and the number of digits after the decimal point and the representation exponential auto decimal of the labels Tick styles The number of minor ticks the axes scaling logarithmic or linear and whether the labels are visible Graph size The horizontal and vertical size of the graph length of the coordinate axes and the relative position of its labels and legend Note You can apply a style to several graphs in a single step This is particularly useful if you have a series of graphs that should look the same To apply a style to several graphs first select the graphs and then choose Styles from the Graph Options submenu in the Plot menu 88 Graph coordinates and zooming Normally you can look at coordinates and analyze data sets and functions using the Preview win dow However options similar to the ones available in the preview window can be used when edit ing graphs Hold down the command and option key simultaneously and click and drag over a graph object pro Fit displays the mouse location in the main axes coordinate system The coordinates are displayed to
130. indow using the command set myTable to make new table open new data window The symbol myTable becomes a reference to the new data window Subsequently the files of the designated folder are opened one by one After a file is opened the ScriptProgram is run and the file is closed again Then the script retrieves the result of the program from globalData 1 The values in the array globalData can be accessed from scripts by using the object globalData and an index such as set sum to globalData 1 get result The result retrieved in this way is transferred to the k th row of column 1 of the data window myTa ble set cell k of column 1 of myTable to sum store it in the table As you can see scripts can exchange data with pro Fit either through globalData or by accessing values in a data window There are other ways of interaction between scripts and pro Fit They are explained in the last sec tion of this chapter which lists all Apple Script commands and objects supported by pro Fit 167 When to use Apple Script As you may have realized there are various things you can do through Apple Scripts as well as from a program defined within pro Fit For example you could define the following program for writ ing the sum of the two first cells of a data window into the results window program Sum begin Writeln data 1 1 tdata 1 2 end A Python script would even be shorter Alternatively you could do the same from an A
131. ing multiple bars in side by side mode In this case each group of bars bars for the same value has the given group width which can again be a percentage of the available space or an absolute value in pixels centimeters or inches The Base line is the line the bar charts start from There is a horizontal base line for vertical bar charts and a vertical base line for horizontal bar charts You can set the position and line style of each base line Check Frame bars with rectangles to select how bar charts are framed 86 framed with rectangles standard framing 0 2 0 2 01234567 0 123 45 6 7 Note You must check the option With line in the Panel Curves for the framing to appear Graph Styles The appearance of a graph is defined by many parameters such as its size the ranges of its axes the number of minor ticks the symbols used for plotting etc These settings are called the style of a graph You can save the style of a graph to use it or parts of it later for another graph Styles are saved in the preferences file By using styles you can create graphs with equal formats e g graphs having the same size the same length of the ticks the same fonts etc To save the style of a graph you can either double click the graph and click the button Organize in the dialog box that comes up or you can choose Styles from the Graph submenu in the Draw menu after having selected your graph Grap
132. istance between xi x fice yi and x f x The term is minimized when the line connecting the data point to the function curve is perpendicular to the function curve A fit algorithm must thus adjust the function until the sum 2 of the squared Xi Yi perpendicular distances between data points and function curve reaches a minimum We refer to the literature for more detailed discussions of the above deviation functions A short description is also found in the classical book by W H Press B P Flannery S A Teukolsky W T Vetterling Numerical Recipes the Art of Scientific Computing The mean square deviation Chi Squared When squared deviation functions are used 2 gives the mean square deviation which is often called or chi squared 93 pana y2 fR y O Oi The mean square deviation chi squared should be used when the measurement errors are de scribed by a Gaussian probability distribution and in this case the errors Oxi and Oyi correspond to the standard deviations of the Gaussian distributions The denomination chi squared has become so common that it is often used to indicate the result of 2 and not only to indicate the particular case 8 Note For the sake of simplicity pro Fit follows this somewhat dirty convention and uses the denomination chi squared when referring to the result of 2 even if deviation func tions other than square deviations are used The same is true
133. itialize xxx_init Input value checking Check xxx_check Input value pre calculations First xxx_first Calculating derivatives Derivatives xxx_derivatives Finishing up Last xxx_last 111 111 113 113 114 115 117 118 120 120 120 122 124 124 125 125 125 129 129 129 130 130 131 131 131 131 133 135 136 137 Determining how multiple output values are rendered in the preview window 138 Programming examples Accessing data Drawing pro Fit Objects Automatic Macro Recording External functions and programs Debugging window Using pro Fit plug ins Saving functions and programs as plug ins Loading Plug ins Removing functions and programs from the menus Loading plug ins automatically on startup Loading a set of plug ins together with a new preferences file Attaching scripts Working with controls in drawing windows Working with plug ins Creating a plug in with a compiler Writing an a plug in with an external compiler Routines to be modified Routines to be defined in functions and programs Routines to be modified in external programs only not used for functions Routines to be modified in external functions only not used in programs 140 140 142 143 144 144 145 147 147 147 147 147 148 148 151 156 156 158 158 158 159 159 Predefined constants and types Global variables Procedures provided by pro Fit 10 Apple Script Batch processing When to use Apple Script pro Fit s Apple Scri
134. ld A list appears that contains all the ticks of the axis To add a tick click a free space in the list there is always a free space at the bottom of the list and enter the desired coordinate Ticks 0 75 0 Standard 0 5 v 0 25 0 w 4 j L nos T To remove a tick select it in the list and press the delete key To change the position of a tick click it and enter a new value Check major to create a major tick Major ticks are written in bold face in the ticks list Click the Standard button to automatically re calculate the tick positions according to the present axis settings To set the label of a tick mark to some general text instead of a number double click the label in the drawing window The text edit dialog box appears and you can then enter any kind of text you want Click Labels in the axis dialog box to edit the format of the labels The the dialog box now looks like this Graph Settings General 1 m F Y axes Z axis ERS LE Lal Frame Grid Bar charts at Y 0 2 Range amp Ticks Labels Prefix Lines Format Auto ea Digits fi _ Delete trailing O s Font Helvetica m B Size 10 a syle pain Color Location below axis B by 3 pt Rotation 0 H 10 05 00 05 1 0 Apply Cancel 0K 78 Use the Format field to set the format of the numbers Choose Decimal to suppress exponential representation Auto exponent t
135. ld outputGroup 1 SetOutputProperties output 3 name r distance from the origin outputGroup 2 SetOutputProperties output 4 name h horizontal displacement 138 outputGroup 2 SetOutputProperties output 5 name v vertical displacement outputGroup 2 end begin E 1 x y 0 5 Ex E cos phi Ey E sin phi r x h x cos phi v x sSin phi end Here the SetFunctionProperties call is used by the function to assign to itself the property that its outputs are shown in the preview window whenever they belong to the same group as the default output The remaining SetOutputProperties calls give the outputs names that are used in the parameter window and assign the outputs to two different groups Outputs from the same group are shown at the same time in the preview window In Python the same function definition looks as follows function electricfield output 0 E absolute value of the field output 0 Ex x component of the field output 0 Ey y component of the field output 0 r horizontal displacement output 0 h horizontal displacement output 0 v vertical displacement import numpy as np def electricfield_init phi pf SetFunctionProperties function preview pf SetOutputProperties output 0 outputGroup pf SetOutputProperties output 1 outputGroup pf SetOutputProperties output 2 outputGroup pf SetOutputProperties output 3 ou
136. le This format has better accuracy but requires more size It is used in data windows if you set the range of a column to 1E300 1E300 The following list summarizes the features of each data type for the Power Macintosh and the 68k version of pro Fit minimum negative number maximum negative number minimum positive number maximum positive number decimal digits size bytes Apart from the values in the list above pro Fit knows four other numbers 0 INF infinity INF infinity NAN The first three of them will do what you expect them to do E g 1 0 INF INF 3 INF etc NAN Not A Number is the result of any computation that cannot be carried out such as sqrt 1 The occurrence of NAN values in computations is reported as a run time error Date and Time data pro Fit understands and works with time data i e absolute calendar dates and relative time The Mac OS stores dates as the number of seconds since January 1 1904 for the technically minded the date is stored as a long integer number 8 byte long pro Fit uses the same convention as the Mac OS to store dates but uses double floating point values instead of integers With this number representation pro Fit can store and recognize dates with second precisions until up to 1015 this corresponds more or less to a 6 byte long integer sec onds after January 1 1904 This means that pro Fit can store dates with second precision up to 31 million
137. le while pro Fit is in use by choosing Preferences from the File menu This allows each user to use an own set of preferences See Chapter 13 Preferences to learn how to use preferences files pro Fit provides a mechanism that allows users to load their favorite plug ins together with their preferences file whenever a preferences file is opened pro Fit looks for a folder named pro Fit plug ins in the same folder as the preferences file and loads all the plug ins it contains To take advantage of this mechanism simply put your preferences file and pro Fit plug ins folder inside a common folder Whenever pro Fit opens the preferences file it also loads all the plug ins found in the pro Fit plug ins folder Attaching scripts Scripts can be attached to drawing windows Such script are called whenever there is a user interaction with drawing windows e g when they are clicked opened closed etc This feature is 148 allows to use the drawing window to design an interface for a user defined program The attached script can then read the results of the actions a user performed on the drawing window that is act ing as a dialog box interface and interpret them One example of a useful program that can be attached to a drawing window is a program that automatically exports the contents of the drawing window as PDF file every time the window is saved Another example is the use of a drawing window as a dialog box It is poss
138. le with nearly all Mac OS applications but it often does not yield optimum results when rendering a picture on screen or for printing PDF data provides much better quality and portability but may not yet be accepted by all applications Some other applications may accept PDF via clipboard but not via drag and drop In pro Fits preferences dialog which you can bring up by choosing Preferences command there is a section titled Copy In this section you can define what image data format PICT and or PDF pro Fit is to generate when you copy items form a drawing menu You can also define the depth and resolution of the PICT data to be exported Note Starting from pro Fit 6 1 PICT data is exported as bitmap only because pro Fit does not natively use Quickdraw anymore File formats for graphics exports To save a drawing for exporting to other programs choose Export from the File menu and select the desired file format from the Format popup 60 Save As Untitled Drawing 1 a aa h O C Handbook files 4 Q DEVICES Name te Date Modified L Macinto ata window explanation plot 2 OR 04 E iDisk ia __ Han amp EPS file 4 PICT file SHARED GIF file JPEG file Forma PNG file TIFF file Options i aand Cria d aa M Hide extension New Folder Cancel Save Depending on the file type you choose you can specify various formatting options through the Op tions button Import graphics pro Fit c
139. line 172 O Results Ll sqrtt2 7871 x 100 78 7187 The language used for evaluating these expressions is the one selected as preferred macro lan guage in the Scripting section of the preferences window Note If you select Python as your preferred macro language each line that you end by hitting the Enter key or cmd return is evaluated as a complete Python statement Hence this not only allows you to evaluate expressions but also to execute complete Python commands For example you can enter import numpy then hit the enter key at the end of the line This imports the numpy library into the environment of the results window Then in the next line enter dir numpy and hit Enter This lists the symbols defined in numpy This mechanism provides a great way to test and explore code snippets interactively The following is an example session that shows some of the things you can do by entering Python commands in the results window Note The character denotes the enter key pf NewDataWindow name data open a new data window w pf GetWindowObject window data create a window object print w nrRowse print the number of rows 200 w nrRows 5 set the number of rows pf SetData 1 1 range 0 5 set the first column print pf GetData pf RowRange 1 print the first column 0 0 1 0 2 0 3 0 4 0 You can also use mathematical expressions in all pro Fit dialog boxes As an example if you
140. ll picture clipping which you will be able to use later on either by dragging it back to a pro Fit window or by using it in another applica tion e You can drag objects into the Trash to delete them To change the size of an object 1 Select the object 2 Click into one of the four black selection handles at its corners and drag 51 While dragging the new outline of the object is shown If you hold down the shift key when resizing the proportions of the object are maintained or the height or width remains constant If you hold down the option key when resizing the horizontal and vertical dimensions of the object become equal If the object is a group of different objects hold down the command key to tell pro Fit to resize all of the objects of the group regardless of their type normally pro Fit would not automatically resize texts or data points To rotate an object 1 Select the object 2 Choose the desired rotation from the Rotate submenu in the Draw menu All objects except graphs and legends can be rotated by angles multiple of 90 Lines Polygons and Rectangles can be rotated by any angle not just multiples of 90 degrees To flip an object i e to exchange its left and right sides or its top and bottom 1 Select the objects to be flipped 2 Choose the desired operation from the Flip submenu in the Draw menu Flip Horizontal exchanges the left and right side of the objects Flip Vertical turns it u
141. lo algorithm 96 The Levenberg Marquardt algorithm 96 The Robust minimization algorithm 99 The Linear Regression algorithms 100 The Polynomial fitting algorithm 101 Goodness of fit 101 Literature and suggested reading 102 The fitting process 102 General features 102 Parameter limits 103 Running a fit 103 Inspecting the progress of a fit 106 Error analysis and confidence intervals 106 Fitting results 107 Using the various fitting algorithms 107 Using the Levenberg Marquarat algorithm 108 Using the Robust minimization algorithm 108 Using the Monte Carlo algorithm 108 Using the Linear Regression algorithm 109 Using the Polynomial fitting algorithm 109 Fitting multiple functions and x values 109 General hints for fitting 111 Starting parameters Redundancy of parameters Defining functions and programs Introduction Built in languages for creating new functions and programs Introduction to Python and Pascal programming in pro Fit A simple program A simple function Using the Python language Introduction Python Defining programs Python Defining functions The module pf Compatibility notes Using the pro Fit Pascal language Introduction Pascal Defining functions Pascal Defining programs pro Fit Pascal in depth On line help for programming The help menus Browsing functions and programs Finding the definition of a symbol The compiler Advanced function and program definitions Initializing and shutting down programs and functions In
142. ltiply them with a given factor or to offset them by a given value 80 Graph Settings Wem EE ba Y axes Z axis Plots Frame Grid Bar charts X axes General x W at Y 0 2 Range amp Ticks Labels Prefix Lines Offset 0 Prefix Postfix 3 j Lt i ts 10 05 00 05 1 0 Apply Cancel 0K Click Prefix or Postfix to prepend or append a string to each label The value in field Factor is multiplied with the value of each label before its string is generated You may e g enter 100 here to display values between 0 and 1 in percent The value in the field Offset is added before the string of the label is generated Panel Plots You can change the appearance of the individual plots in a graph in many ways Choose Plots from the Graph submenu Draw menu or click the Plots icon in the Graph Settings box You can also double click a plot symbol in the legend The Graph Settings dialog box now displays the panel used to edit data and function plots 81 Graph Settings CE Lal Frame Grid Bar charts C General X axes Y axes Z axis Curve Plot Sequence Plot type Function Plot H4 e y Value Polynomial x Mine E Q Fill i MM toaxis X1 Plots Points E width Auto Use line color Contour plot Color plot Coordinate axes x Wd y z1 M Appears in legend Pa Update Tabula
143. masked cell is invisible hidden during calculations plotting fitting etc 31 To mask or unmask a cell or a set of cells select it and then choose the command Masked from the Calc menu Defining a data set to work on Some of pro Fits commands access data in the data windows If you have several data windows open at the same time pro Fit uses some rules for selecting the data window it works on The transformation commands in the Calc menu work on the active window the window in front of all other windows If the active window is not a data window these commands cannot be used Example the Data Transformation command is only enabled when the front most win dow is a data window Some other commands use the front most of all data windows In this case it does not matter if windows of other kinds are in front If the active window is not a data window these commands look at all the windows behind the active window and work on the first data window they find This will be the data window that is closest to the front Example the Spline function uses the data window that is closest to the front The commands for curve fitting and for plotting data display dialog boxes where you can choose the data window from a pop up menu Examples Plot Data and Curve Fit The data window containing the data used in a particular operation is called the current data win dow in this manual The current data window is either the
144. min max with 122 initialValue the default value of the parameter as it will appear in the Parameters window mode inactive active or constant depending on if the parameter will be used for fitting not be used for fitting unless switched to active or not fittable name the name of the parameter in quotes min the minimum value of the parameter max the maximum value of the parameter Note the min and max values are optional Example function myFunc input 1 active inl 0 10 input 2 inactive in2 Note The input directive overrides the names and defaults used in the arguments list of the func tion Functions with multiple outputs To define a function with multiple outputs Several y values you have to add several output di rectives and the function must return a tuple with the y values Each output directive is a single text line starting with followed by the keyword output The keyword is followed by the default value for the output a comma and the name of the output Example function complexExp output 2 re output 1 im from cmath import def complexExp x c0 0 f 10 a cool python function y c0 exp f x 135 return y real y imag In most cases it is not necessary to calculate all output values of a function The classic example is when only one of the output values is being plotted in a normal 2D plot In order not do unneces sary calculati
145. mn 3 inititialization end of initialize 132 begin main part of program ask for an input column default is the one that was set in initialize Input which column inputColumn transform data end of main part Input value checking Check xxx_check If you define a procedure called Check Pascal or a function called xxx_check Python where xxx is the name of the function in the function directive it is called each time the user changes an input value in the parameters window It can check the input value that was changed and act accordingly For example it can refuse a value if it is not acceptable It can also recalculate some other input values and cause the parameters window to be redrawn In the following example this mechanism is used for preventing the user from entering a value for input a2 that is larger than input a1 In Python function SampleFunc def SampleFunc_check pNumber al a2 if pNumber 2 and al lt a2 pNumber is the changed parameter return 2 return 2 if not acceptable return 1 return 1 if the parameter is fine def SammpleFunc al a2 In Python xxx_check has a first argument pNumber corresponding to the index of the input value that has been changed The following arguments are the input values xxx_check returns 0 if the new input value is fine and the parameter window needs to be updated 1 if the parameter is fine and the parameter window does no
146. mplex are interpreted as floating point numbers Boolean expres sions are evaluated as floating point numbers a 0 0 representing false any non zero value rep resenting true No records structures or pointers are supported Arrays are one dimensional Case statements are not supported Nested declarations of functions or procedures are not supported Optional parameter predefined procedures and functions are supported A general purpose complex type is supported General purpose matrix and vector types are supported Pointer types are not supported 201
147. n if your function has many parameters that you do not fit and you do not want all the values of inac tive or constant parameters to clutter the results window Check Print full description to get for each fit a header that describes the settings that were used for fitting Check Error Analysis if you want to obtain more information on the accuracy of the fitted parame ters When this option is checked additional statistical calculations will be performed after a fit is completed Confidence intervals for each fitted parameter will be determined by a Monte Carlo method that simulates a large number of fits with a series of synthetic data sets More about this in the Error Analysis section below To start fitting click Fit Fitting can run for fractions of a second or for hours depending on the execution speed of the function you selected the number of parameters to fit and the number of data points The results of the fit appear in the result window You might want to choose its name from the windows menu and position it in a comfortable place before running a fit You can let pro Fit always bring the results window to the front after a fit by setting this option using the Prefer ences command To speed up fitting when you are using one of your functions you can define the function s partial derivatives with respect to its parameters The section Calculating derivatives below gives more information on this topic 105 You
148. n square deviations using 8 It finds at the same time the set of x coordinates and the function parameters that minimize the mean square deviations between the data points and the function values When the x errors are zero the Levenberg Marquardt algorithm minimizes 9 The Robust fitting algorithm minimizes 2 with any definition of R by continually moving down hill in parameter space until the bottom of a valley is found The Linear Regression and the Polynomial fitting algorithms are specialized for polynomials of 1st and nt degree While the Linear Regression allows for x errors the Polynomial fitting algo rithm is restricted to y errors only The mathematics used by the various algorithms to perform their job is outlined in the next sec tions The Monte Carlo algorithm This method randomly varies the parameters of a function within given intervals When x errors are defined the algorithm also varies randomly the set of xi coordinates while observing the given errors and error distributions For each random guess xr is calculated according to Eq 2 and the parameter sets correspond ing to the smallest values of xr are remembered The strength of this method is also its biggest disadvantage It looks for the best parameter set by shooting blindly inside the given region of parameter space Although there is an option of letting this parameter space region follow the position of the currently best parameter set this
149. nd begin Writeln SumSelection end Note Calling Exit from the main body of a function or program has the same effect as calling Halt Predefined constants functions procedures and operators This section lists the operators and the most important predefined constants that are available in the Pascal syntax An full list of all predefined functions procedures and constants is found in pro Fit s on line help The following are the most important predefined constants m or pi equals 3 141592 TRUE equals 1 FALSE equals 0 inf infinity 1 inf 0 The operators are identical to those that are defined in standard Pascal In addition the power op erator or has been added The operators in ascending order of precedence are lt gt lt lt gt gt comparison returning true 1 or false 0 or add subtract logical or 199 and multiply divide logical and xx power x y X y XY not logical not You can change the order of precedence of the operators in the above list by using brackets and Note that there are two ways for using the power operator x y and x y They are equivalent Use whichever you prefer Notes On some machines x y x4y is calculated as exp y In x As a consequence of this the x4y may not work for negative x and may be slow Therefore you should not use this notation for calculating small integer powers
150. nd the single default input and output set in the parameter window are not used When the standard x value is used as the default input it is possible to use some additional optimizations that can make calculations of some functions faster such as special code used in user defined functions to initialize calculations based on parameter values Note that whenever the default input is used for any operation outside of the parameter window its value is varied in that operation and the value entered in the parameter window is ignored An entry in the Func menu makes it possible to save the current inputs under various names to be used later or as the default set for a function The Parameter Sets submenu in the Func menu al lows to save this data for the current session permanently in the pro Fit preferences file or even as an attachment to a function definition file Whenever pro Fit finds Parameter Sets that can be used for a function it makes them available in the Parameter Sets submenu 34 pro Fit has a set of built in functions that you can use as is But this set of built in functions is meant to be extended by the individual users in their own way This is a very important part of what pro Fit is about and there are therefore a few different ways to add additional functions and gives you the possibility of defining your own functions there are also many external functions that are distributed with pro Fit and that are ready to u
151. ndard deviations are printed to the results win dow If you need to access the complete covariance matrix you can define a program that uses the predefined function CovarMatrix See pro Fit s on line help for more details on how to use this function Using the Robust minimization algorithm To run a Robust fit choose Robust in the algorithm popup menu of the fit dialog box This dialog box appears when selecting Fit from the Calc menu and it was described above Using the Distribution popup menu which appears when you define errors you can select the er ror distribution that best describes your experiment Robust fitting will deserve its name if you se lect a distribution that diminishes the importance of outliers like Andrew s sine or Tuckey s bi weight When finished the resulting parameter values are printed to the results window This algorithm does not determine a standard deviation for each parameter like the Levenberg Marquardt algo rithm does To obtain error estimations you have to run a Levenberg Marquardt fit after the Robust fit converged or you have to check the Error Analysis check box and perform a Monte Carlo analysis See the corresponding section for more details Using the Monte Carlo algorithm To run a Monte Carlo fit choose Monte Carlo Fit from the Calc menu or chose Monte Carlo in the Algorithm popup menu of the Fitting Setup dialog box When you do this two more items appear to t
152. ne Click the To Menu button in the function window or choose Compile amp Add To Menu from the Customize menu This translates your function into computer code pro Fit looks at what you wrote and sees that you used the standard input x and the standard pa rameter array input values a 1 a 2 It therefore assumes that you want to define a new function and interprets your text accordingly 22 The new function is added to the Func menu and the parameter window shows its default parame ters Your simple expression is replaced by a complete syntactically correct function definition function User_Function begin y a l sin x 1ln x a 2 end The first line defines the name of the function as it appears in the Func menu User_Function is the default proposed by pro Fit You can change it to something like LogSine Then enclosed between begin and end there follows the definition of the function In the third line the function is calculated from the variable x and the parameters a 1 and a 2 and it is assigned to the variable y Note An alternative way to define the same function is function logSine ampl offset real begin y ampl sin x ln x offset end In this definition the parameters input values of the function are defined in the function header The names used in the header are then used in the function body This is the syntax used for stan dard Pascal functions pro
153. ne constants and variables The definition of constants is preceded by the keyword const which is followed by the name of each constant the operator not and the value of the constant Example const c 3e8 startValue 22 185 Once you have defined the value of a constant you cannot change it anymore The definition of variables is preceded by the keyword var which is followed by a list of variables var u v real done boolean m matrix 3 c complex Note that you can specify the type of each variable such as real boolean complex If you omit the type specification it is assumed that the variable is of type real All statements within the program and the program s procedures and functions can access vari ables and constants that you define in the head of a program You can use any name you like for a constant or variable as long as it is not yet used for any other purpose It can contain letters and digits but must start with a letter Examples for names are myFunc xx J0 legal names 2ToOX illegal starts with a digit then illegal reserved keyword The same rules apply to the names of procedures and functions See below Following the definition of constants and variables you can optionally define local procedures and functions The general form of their definition is for a procedure procedure MyProc m n real i integer variable and constant definitions begin
154. ng these symmetry settings and using only one or two definition points 57 Hold down the shift key while dragging a polygon handle to constrain the dragging along radial di rections Hold down the command key while dragging a polygon handle to resize and rotate the whole polygon in one single move Choose a Symmetry of 1 and no mirror symmetries to draw a polygon free hand Note that if you do this you can draw a polygon that is not centered inside the preview area This means that if you use such symbols for plotting the symbols will not be centered on the mathematical coordinates of the data points Click the Copy to Menu button to add the point symbol you just defined to the point symbols menu for later use The data point symbols you define are normally used when plotting see the section Plotting be low for more details on this However you might also want to use them to achieve some special effect in a drawing For example you can use a symmetric triangle to define a point and you can rotate it by any amount You can t do this that easily using the standard drawing tools You can also define closed polygons with any special symmetry The data point symbols you define can then be used as drawing objects in the drawing window their size can be quite big You will be able to re size them as usual by dragging a selection handle and you can always modify them by double clicking them Control shapes Control sh
155. ngs from the Draw menu You can also double click most shapes for getting into the corresponding dialog box When a window is in dialog mode command double click the shape The most important setting you can access through this box is the shape s name This is a unique string attributed to each shape and used by scripts for identifying the shape Drawing windows in dialog mode Drawing windows can be put into dialog mode In this mode the window obtains the same back ground as dialog boxes This is required when you want to create a complex dialog box using con trol shapes You create and edit the control shapes while the drawing window is in its normal state When you have finished you switch the dialog window to dialog mode In this mode the drawing window cannot be edited anymore normal mode A Simple button example 7 dfgsdfg Multiply by 2 v gt dialog mode O fA Simple button example Multiply by 2 To switch a drawing window into dialog mode hold down the control key while clicking anywhere into the window and choose Display As Dialog Alternatively choose Get Info from the File menu and check the option Display As Dialog For more information on creating dialog boxes see Chapter 9 section Working with control shapes 50 Editing drawings This section describes the general drawing commands and the use of the tools palette General drawing
156. nomial function of any degree It assumes a Gaussian distribution of errors Only Y errors are possible When finished the parameter values and their standard deviations are printed to the results win dow Fitting multiple functions and x values You may sometimes need to fit simultaneously several functions fi fg with one or more common parameters Or in other words you may have a function with many outputs and use it to fit a multi dimensional set of data Or you may want to fit a function that does not depend on a single x value but on a set Xx Xe Xp of x values Or you might even encounter a combination of these two cases In the most general case you have q functions or q different outputs ff fg each of them de pending on one or more inputs Among the inputs some are used as independent variables corre sponding to the quantities that were varied during a measurement others as parameters Each output is calculated using one or more of the various inputs You can see this as a set of functions that can share one or more parameters and that depend on any subset of the independent vari ables 109 y fi X1 X2 Xp Y2 fe X1 X2 Xp Yq fo X1 X2 Xp For each function or output value you have a set of data points that should be described by it Now you want to fit all these functions simultaneously pro Fit provides a dedicated tool for this type of multidimensional fitting To invoke it ch
157. ns a set of standard pre E E mms oe dA defined point symbols The last line contains user defined sym TOUrY en rrvy bols and the Other field lets you define new point symbols a cOOmetK After having selected the desired point style click into the draw ash bad o Other ing window to create the point I To change the plot symbol of a point object select it and choose the desired symbol from the point style pop up menu in the tools palette or double click it to go directly to the custom points dialog box If the selected object is a graph or a legend the new point style is applied to the data plots con tained in the graph See the section Plotting later in this chapter for details As you can see from the screenshot above pro Fit comes with a variety of different point styles In addition to this pro Fit provides a tool for hand crafting any other custom point style To create such a point style either double click an existing data point or choose Other in the pop up menu shown above The following dialog box appears 56 Custom Points Background View 800 l X HXO Size 112 5 CTA E k Fill black F Pts 1 1 Symm 6 vi Frame Mirror V L Foreground XH Size i45 5 olLAlV sS Pts 1 J Fill white B p 1 W Line thickness Auto H Symm 3 H M Frame a Mirror V _ H Copy to Menu C Cancel GOK pro Fit defines a data point symbol
158. ns and programs See www python org for more information Support for Apple Script Apple Script is the high level scripting language of Mac OS used to automatize tasks and control applications Interactive parameter modeling and curve fitting A key feature of pro Fit is its intuitive and flexi ble interface for modeling and fitting data offering the choice of several fitting algorithms and optional restriction of parameter ranges Fitting supports y as well as x errors and allows a Monte Carlo error analysis for fitted parameters A parameter can also be fitted manually by dragging the function s curve with the mouse Professional plotting Data and functions can be plotted in a multitude of ways in high quality publication ready graphs Plots can have multiple coordinate axes using linear logarithmic 1 x and normal probability scalings including reverse scaling Plotting types include scatter plots line plots skyline plots histograms contour plots color plots and box plots Interactive 3D models of function and data can be generated and edited through the 3DplotterGL plug in Built in drawing editor A complete range of drawing tools allows flexible editing and annotation of plots and presentations Specific drawing objects such as buttons check boxes and pop up menus are supported to be used as interfaces for user defined programs Extensive graphical output possibilities Pro Fit supports output and in part input for the
159. nt function is to be drawn in the preview window and if it should display the results of the last fit in stead of using the values in the parameter window and the current data set Otherwise the window displays the x and y columns of the current data window You can use the Data button to change the data set that is displayed or else you can directly set the default x and y columns in the data window using a contextual menu there 40 Click the Refresh button if you want to let pro Fit redraw the complete function at maximum resolu tion pro Fit automatically decreases the resolution at which it draws the function if it notices that the function is too slow You can override this with this button The Undo button allows you to undo the last operation This is an alternative to choosing Undo from the Edit menu Clicking the Fit Button is equivalent to choosing the Simple Curve Fit command from the Calc menu and allows you to run a fit without being prompted for the settings that are otherwise acces sible through the Curve Fit command A limited set of options for this feature can be specified by clicking the Settings button The Settings button also allows you to enable Live Fit When this feature is enabled pro Fit keeps the fit displayed in the preview window up to date A new curve fit operation will be performed automatically whenever the current dataset is changed which can be useful to rapidly observe how a Fit chang
160. o always add the points inside the selection rectangle to the current selection You can set the color of the data points and the color used to mark selected data points using the Preferences command in the File menu If you have a monochrome monitor pro Fit will use a dithered pattern to mark the selected points Whenever you select a data point in the preview window the corresponding row is se lected in the data window If you then choose Data Transform from the Calc menu you can perform calculations on the data in the selected rows only Note Selecting a data point in the preview window always selects the whole correspond ing row in the data window If you select a range of data points in the preview and then delete them you will delete all data in the selected rows and not only in the current x and y columns Use the dragging tool to change the x or y ranges of the preview Click in the view port area with the drag tool and drag the area of the data set or function curve displayed by the preview The ranges of the preview will change accordingly You start dragging inside the view port but you can go on dragging also outside thus changing the coordinates by a large amount Click the view port area with the zoom tool to zoom in and magnify the clicked area Hold down the option key while clicking to zoom out 42 With this fitting tool you can click and drag to change a function parameter by constrain ing the fun
161. o have all labels in exponential format with varying exponent Fixed exponent to have all labels in exponential format with a common exponent The Digits field defines the number of digits to be shown after the decimal point Check Delete trailing O s to cut off any trailing O digits after the decimal point which is particularly useful for loga rithmic axes Use the Font field to specify the text font size and style to be used for the labels of the current axis The Location popup menu defines where the labels of an axis are drawn The edit field to its right defines the distance between the labels and the axis or the frame of the graph The value in this field is in points 1 72 inch or 0 35 mm Note that it can also be negative The pop up titled Rotation allows rotating the labels by multiples of 90 Click Lines to change the appearance of the lines used for drawing the axis and its tick marks and to set the position where the tick marks are drawn The the dialog box now looks like this Graph Settings mea F J ETE tan Y axes Z axis Plots Frame Grid Bar charts General X axes atY 0 2 Range amp Ticks Labels Prefix Lines Axis line style E Ticks location top Major ticks Minor ticks length is pt length 3 pt a Apply Cancel oK Use the Ticks location popup menu to set the position of the tick marks In the Major ticks
162. o parameter value selected the text on the clipboard is assumed to contain several values separated by spaces tabs or carriage returns which are then used to change all parameter values You can also paste a list of tab delimited value while a given input is selected to replace its value and the following ones Limits All inputs but the standard one x can have upper and lower limits which are displayed in the information area at the bottom of the Parameters window These limits are used to constrain the parameter during fitting and function optimization To change a limit select the input and enter the limit in the corresponding field To remove a limit select the input and clear its limit Fitting mode To change the fitting mode of a parameter check or uncheck the option Use for fit ting in the lower right part of the window If you check this option the parameter will be varied dur ing fitting and optimization otherwise it will be kept fixed The fitting mode of a parameter can also be recognized by the typeface used for its name Parameters with names displayed in bold face will be varied during a fit Parameters displayed in normal type face are kept fixed during a fit As an alternative to using the Use for fitting checkbox simply click the name of the parameter to toggle its fitting mode Note If you define your own function you must keep the predefined input x as the default input if you want to use the procedure firs
163. of analysis you are performing the result of the analysis is immediately shown in the preview area i e the right half of the window in the function display as well as in a results list If your function is too slow you can switch off the preview check box Click the help button of the Analysis window for further information The Spline function There is one special function in the list of predefined functions the Spline function The Spline function is defined as a smooth cubic Spline curve going through all your data points The Spline function can be useful for interpolation especially when you do not have a mathematical model for your data This is a simple data set together with its Spline function A Spline x O y Value 1 0 0 0 1 0 x Value 37 The Spline function calculates its return value using a set of data points xi yi This set of points can be defined by the inputs in the parameter window of the Spline function or it can be defined by the data in the current data window You can set how the Spline function operates in the parameter window itself When using a set of inputs as the set of data points used to calculate its return value it is even possible to use the Curve Fitting command to adapt the position of those points by e g adapting their y value so that the Spline function optimally approximate a noisy data set 38 6 The Preview Window The basics There are generally two different approach
164. of the data win dow or you can write a small macro a pro Fit program that transfers the fitted parameters directly to their data window See chapter 9 Defining functions and programs to see how to do this An example program for transferring parameter values to a data window is distributed together with pro Fit Using the various fitting algorithms pro Fit provides three different fitting algorithms The Monte Carlo Robust and Levenberg Marquardt algorithms They are described in the preceding section The following sections describe how each of these fitting algorithms is used and what particular options you can set for each algorithm 107 Using the Levenberg Marquardt algorithm To start a fit with the Levenberg Marquardt algorithm choose Fit from the Calc menu after having selected the appropriate function from the Func menu The Fitting Setup dialog box appears with Levenberg Marquardt pre selected in the Algorithm popup menu See the preceding section for a description of this dialog box When you define errors the Distribution popup menu is dimmed and set to a Gaussian distribu tion The Levenberg Marquarat algorithm can only work if the errors of the data set are normally distributed The Levenberg Marquardt fit stops running when the chi squared determined from the current pa rameter set doesn t decrease appreciably anymore from one iteration to the next When finished the parameter values and their sta
165. ome of the more popular modules ___builtins__ since pro Fit is not a console application raw_input prompt and input prompt bring up a dialog that lets you enter a single line string sys sys argv is always empty since pro Fit is not a console application sys stdout and sys stderr are redirected to the Results window As to sys stdin pro Fit will bring up a dialog box for line based input sys exit is overwritten to exit the python script but not the pro Fit application pylab matplotlib These modules kind of work However spurious error messages about a CGContextRef being NULL may be dumped to the results window A simple example import matplotlib pyplot as plt plt plot 1 2 3 plt ylabel some numbers plt show Note matplotlib and pylab are not included by default in MacOS X matplotlib can be downloaded from drawinports com or www finkproject org or directly from 183 Tkinter The version of Tkinter installed on MacOS 10 5 is incompatible with pro Fit as it tries to take control of the whole menu bar thereby destroying pro Fit s menus wx This module works with pro Fit to some degree Some guidelines for best compatibility Do not tamper with the menu bar Windows brought up by your script should be modal dialogs An example of a reasonably well behaved wx based dialog can be found in the file wx example dialog func 184 Appendix D pro Fit Pascal Syntax Introduction
166. ometimes lead to problems on some printer drivers Do not use Postscript when printing to a non PostScript printer Note 1 Bitmap based patterns appear as gray levels under PostScript Note 2 If you experience postscript problems when drawing plots containing a large number of points try to increase the number of points drawn with one stroke in the Plotting panel of the Pref erences command In the printing section of the Preferences dialog you can also choose the default page setup to be used for the various window types within pro Fit Printing a pro Fit drawing from another application When drag or copy pictures to other applications pro Fit can encode these pictures in two different ways PICT and PDF PICT is a legacy format dating from the times prior to Mac OS X and has limited capabilities but it has the advantage that it is understood by older applications PDF is a more modern format with much improved capabilities but not all applications are able to handle it For example Microsoft Word up to version 2003 did not support importing PDF images through the clipboard 169 In the Copy section of pro Fit s Preferences command you can specify if pro Fit is to export one or both of these formats A Je Preferences a RY Sto BO i amp d L AeIE bad b Draw Plot Preview A Scripting Interface Export Extension Meta Prefs Add the following data types for exporting M PDF A m Quickdraw PICT Color depth Millions
167. ompati bilities with the present or future versions of pro Fit General remarks Strings passed between pro Fit and a plug in are always Pascal strings and not C strings If you are programming in Pascal you won t have any problems with this If you are programming in C you must remember that a Pascal string must be introduced by p example pMyS tring For assignments you can use the function SetPascalString described earlier in this chapter When compiling for PowerPC processors records passed between pro Fit and a plug in always use 68k alignment Therefore for compatibility with Power Macintosh compilers definitions for C structs are always preceded by if defined powerc defined __powerc pragma options align mac68k endif and followed by if defined powerc defined __powerc pragma options align reset endif e Parameter indices under Pascal always run from 1 to maxNrParams in C they run from 0 to maxNrParams 1 The following lists the most important constants and types 162 Global variables Global variables or static variables as they are often called by C programmers are variables that remain statically in memory Their values are preserved between individual calls to your plug in In Xcode you can define global variables in the way you are used to In C you declare them out side your functions or if you declare them inside a function you declare them as static Procedure
168. omputer means that the Software can be executed on said com puter either because it is stored locally on said computer or accessible by said computer through a network 3 Competent court and law This Agreement shall be governed by the laws of Switzerland and the competent court is in Z rich Switzer land 4 If for any reason any provision of this Agreement or portion thereof is found to be unenforceable under the applicable law that provision of the Agreement shall be enforced to the maximum extent permissible so as to effect the intent of the parties and the remainder of this Agreement shall continue in full force and ef fect 5 QuantumSoft reserves the right to change this agreement at any time Any changes will be announced at www guansoft com 6 Pre release software By using any alpha or beta version pre release version of the Software you ac knowledge that you are aware that such versions are for testing purposes only and that they often contain substantial errors that affect their functionality or may damage data on or functionality of a computer Pre release versions of the Software should never be used on computers containing data critical for your work QuantumSoft Buhistr 18 CH 8707 Uetikon am See Switzerland Copyright Trademarks Customer Support pro Fit QuantumSoft 1990 2010 All rights reserved No part of this publication or the program pro Fit may be repro duced transmitted transcri
169. on Debug at the top of the function win dow that contains the code you want to debug When you run the program or function its debug window will automatically show up with execution stopped at the first instruction in your code iO 08 Debugger Step into Step Over Step Out Run Kill Show Evaluate Show Pointer a fa Ss 9 9 9 9 2 Calling Sequence Variables Values PrOCES Directory func gt globals gt locals Source python script This is an example showing how to process all files in a directory from a python script import os ask the user for the directory if the script has never been run and mydir is therefore undefined or if the user presses the shift key if pf KeyPressed pf shiftKey or not mydir in locals mydir pf ChooseDirectory i i sum 0 os listdir returns a list of the files in a directory alternatively you could use the functions in module glob for f in os listdir mydir we only process the files ending on data Use any other criterion if you want if str endswith f data try we open each file read its cell 1 1 and add it to sum b H pf OpenFile file mydir f type pf dataType a if pf DataOK 1 1 gt i i Li q 4 i 4 4 i 4 4 i 4 l i d 4 ij 4 i 4 4 j a v a e a X X X X X X X Now you can step through your program
170. one _ Print full description O a m Print active parameters only _ Done Cri Using this dialog box you can set a number of fitting options and observe the set up for the fit us ing the preview of function and data set this preview can be hidden or shown by clicking its disclo sure triangle Once you are satisfied with your set up click OK and fitting will start If you don t want this settings window to stick around during and after a fit check the Close window when done check box You can switch from one fitting algorithm to the other using the Algorithm popup menu More de tails about particular options for each algorithm are given below The Data Window menu lets you select a data window by default the foremost data window If Selected rows only is checked only rows intersecting the current selection are used for fitting Otherwise all data in the X and Y columns will be used The Data Column menus define the data set coordinates xi input data and yi output data 104 The popup menus Error type let you specify the errors of your data For the input data choose zero to use zero x errors the usual case In the output data choose unknown if you don t want to specify y errors in this case a value of 1 0 will be used as the error for all data points re gardless of the order of magnitude of the y values and all points will have the same weight in the calculation of xR which is calculated with
171. only one graph When you double click a graph or choose General from the Graph Options submenu the fol lowing dialog box appears Graph Settings et J a sd L t General X axes Y axes Z axis Plots Frame Grid Bar charts Draw M Legend M Current graph Frame C Grid Drawing sequence Grid Curves Axes Frame req Graph height 2 893 cm Range calculation M Autorange after updates Styles Organize Apply Cancel The icons in the toolbar of this box correspond to the items in the Graph Options submenu Click the icons to access and edit the various parts of a graph Click the Apply button to see the effects of your changes Panel General Check Current graph to make this graph the currently active graph This is the graph where plot ting takes place per default The three Draw check boxes indicate if legend frame and grid should be drawn or not If you un check the box named legend the legend is deleted If you check it again the legend reappears to the right of the graph 74 The Drawing Sequence popup menu defines the order in which the various parts of a graph curves axes grid are drawn This is especially important if you use color to highlight your curves or if you use very large data points A grid in front of a curve can then look quite different from a grid behind a curve The Graph width and Graph height edit fields let you enter precise dimensions for the graph
172. ons and to speed up things pro Fit provides a predefined function pf Output i that you can call to find out if a given output value must be calculated Note that sometimes a function should calculate more than one output value at the same time as an example when the output values are displayed in the parameters window or when a function is tabulated The prede 123 fined function Output accepts the index of the output value as a parameter and returns true if it must be calculated function electricfield output 0 E output 0 Ex output 0 Ey import numpy as np def electricfield x phi E 1 x Ex 0 Ey 0 if pf Output 1 Ex E np cos phi if pf Output 2 Ey return E Ex Ey E np sin phi The module pf The functions for controlling pro Fit from Python are defined in a module called pf This module is automatically imported into all Python scripts compiled within pro Fit For example the script print pf NrRows calls pro Fit s built in function NrRows which returns the number of rows of the current data window pf NewDataWindow calls pro Fit s function NewDataWindow to create a new data window Some of pro Fit s built in functions have named optional parameters which can be defined using Python s standard syntax For example pf NewDrawingWindow name My Python Drawing calls pro Fit s built in function NewDrawingWindow passing it the title of the new
173. ont win dow instead The pro Fit s on line help states which of pro Fit s predefined procedures use optional parameter lists Aborting procedures functions and programs Use the keyword Halt to immediately end the execution of a function or program Use the keyword Exit for exiting from a local function or procedure to the caller The following is an example of a program calculating the sum of the presently selected cells in a data window The program aborts when the selection contains empty data cells Note that it uses the predefined variables selectLeft selectRight selectTop selectBottom which return the enclosing rectangle of the currently selected data cells program CalcSum var row col integer sum real begin sum 0 for col selectLeft to selectRight do for row selectTop to selectBottom do begin if not DataOK row col then Halt sum sum data row col end writeln sum end The following program does basically the same as the one above but the sum is calculated in a local function which is aborted by Exit program CalcSum 198 function SumSelection real sums the selected data returns 1 if a selected cell is empty var row col integer sum real begin sum 0 for col selectLeft to selectRight do for row selectTop to selectBottom do begin if not DataOK row col then begin SumSelection 1 Exit end sum sum data row col end SumSelection sum e
174. ontal rows and vertical columns It can hold up to 16 millions columns with up to 16 millions rows if enough memory is available home field r D i 1 8 amp population data eee resize field 1 1940 00000 2200 00000 2 1950 00000 2500 00000 rag fiel T drag field_____ 3 1960 00000 3000 00000 4 1969 00000 3600 00000 5 1975 00000 4000 00000 6 _1981 00000 4400 000007 7 1987 00000 5000 000000 f s rm S700 oOnnMnM 8 2 v pair 4 To change the size of a data window i e the number of rows and columns click the resize field in the top left corner of the window To bring the first cell of the first column into view click the home field to the right of the resize field To insert or delete empty rows or columns click one of the drag fields and drag the mouse There are also commands to add and delete columns and rows in the Calc menu To change the width of a column click and drag the separation line between column titles Dragging down the info hook opens an empty area at the top of the data window In this area you can enter general information or comments about your data O ___population data The world s population GEO 1 1990 lilly 2 out population in millions 2200 00000 2500 00000 27 When editing numbers in a data window the arrow keys move the selection mark to neighboring data cells If you hold down the option key
175. oose Multi ple Fit from the Calc menu 8 00 Multidimensional Fitti Algorithm Levenberg Marquardt 7 Vl Auto search Ranges z Guess initial parameters Function Log Experiment 1 ay Data Window Table 1 B _ Selected data only x col x Value input x y col y Value output Simple Y Plot Show last fit 4 0 F Show function O log W m u F Show data auto 8 0 1 0 Mawo Clog 1 0 Error analysis _ Close window when done m Print full description Coone 2 Print active parameters only This window is similar to the Curve Fit dialog but offers a dedicated area for setting up a com plex multi dimensional fit pro Fit s on line help has an exhaustive description of how to use it Click the Help button to the left of the Cancel button to see these instructions 110 General hints for fitting Starting parameters As already pointed out the success of a fit often depends critically on the choice of a good set of starting parameters Bad starting parameters can cause convergence to a false i e local mini mum of the mean deviation xR It is good practice to always try to figure out reasonable values for starting parameters Some of the functions are able to take their own guess at a good set of start ing parameters but in many cases a human is still the best judge Redundancy of parameters Sometimes a fit converges slowly or is even stopped with the cryptic error messa
176. or of the file must be ftCD and NLft or it should have one of the following ex tensions code fitcode plugin fitplugin proFitCode or proFitPlugin Note If you have loaded a plug in and you subsequently change it e g by recompiling it you must remove the loaded plug in from pro Fit before loading its new version To load your plug ins automatically at start up put them into a folder called pro Fit plug ins lo cated in the same folder as the application itself or in the Preferences folder of the Library folder See the end of Chapter 9 Defining functions and programs for a more detailed discussion of how to work with pro Fit plug ins The rest of this chapter explains how you can write plug ins using your own compiler Creating a plug in with a compiler To write a plug in you need some experience in programming and a programming environment a compiler such as Apple s XCode To create plug in locate the pro Fit development files in the pro Fit distribution files You will find them in the folder Example Files Programming 7 plug ins 156 25 items 31 2 GB available Vv Interfaces v c le proFit_interface c h proFit_interface h h proFit_paramBlk h gt Pascal lt read me rtf v E Sources v mc Y oD Example Plugin e main c F PB bundle plugin xcodeproj plugin Info plist Y Inverse Erf example function le Inverse erf c as InverseErf xcodeproj _ plugin
177. or by clicking one of the question marks that you find in pro Fit windows and dialog boxes Balloon help is also sup ported When defining functions and programs there is a special feature based on a dedicated help menu that is always present in the header of function windows See chapter 9 Defining functions and programs for more information on this help menu The on line help system is embedded into the pro Fit application bundle and goes wherever the pro Fit application goes Help tags pro Fit supports help tags for providing information on individual user elements Help tags on Mac OS X are displayed automatically when you keep the mouse over an item of in terest for at least half a second On line evaluation of mathematical expressions Wherever pro Fit expects a numerical input such as in spreadsheets or dialog boxes you can en ter a mathematical expression For example instead of typing a number directly you can use a mathematical expression like exp 1 or 6 sin pi 4 pro Fit reads the mathematical expression you typed or pasted and calculates the numerical result Note These expressions are Pascal expressions not Python expressions Text windows such as the result window can be used as a calculator by typing an expression on a new line positioning the insertion point on that line and hitting the Enter key or the return key while holding down the command key The result is displayed on the next
178. ositive slope r 1 if there is a complete correlation with a negative slope and r 0 if there is no correla tion at all 100 The significance of the correlation is the probability that Irl should be larger than its observed value in the null hypothesis x and y being uncorrelated It ranges from 0 good correlation to 100 bad correlation We refer to the specialized literature for more details The Polynomial fitting algorithm Our model is the general linear combination of arbitrary functions M y x X ax Fx x 19 k 1 The functions Fx can be wildly nonlinear functions of x Linear refers only to the model s depend ence on its parameters ax Once again we assume that the measurement errors oj of the data point are known By defining the matrix A and the vectors b and a as A Fe pet di 20 Oi Oi it is possible to describe the minimization equations in matrix form A A a A b 21 The variances of the parameters can be found as the square root of the diagonal elements of the inverse matrix A To solve equation 21 we use the method of Singular Value Decomposition SVD It is a very ro bust algorithm for over determined as well as for underdetermined systems although it is a little slower and needs more memory resources than solving the normal equations For further details see the literature listed below Goodness of fit It is very important to know the quality of a fit oth
179. ost ele ments of the Pascal programming language for defining functions The pro Fit package comes with more examples of function definitions Look them up pro Fit Pascal and Python In the example above we have used pro Fit s built in Pascal like scripting language for defining the function pro Fit supports two other scripting languages namely Python and Apple Script While Apple Script cannot be used for defining functions Python scripts can The following is an example of the function logSine when defined using Python function logSine import numpy as np def logSine x ampl offset if x lt 0 y 0 else y ampl np sin x np log x return y offset As you can see the function starts with the comment function logSine which is ignored by the Python interpreter but tells pro Fit to look for the definition of logSine and use it as a pro Fit func tion logSine itself has three arguments the first one is the independent variable x the following arguments are the input values or parameters of the function The y value is returned in a Python return statement The choice of scripting language Pascal or Python in pro Fit is primarily a matter of preference Pascal as a compiled language pro Fit has a built in compiler for it may be faster in some situa tions but hardly ever in a significant way while Python is more widely supported and comes with a large number of powerful modules in particular fo
180. ot will be extended if necessary If you uncheck Auto range you can enter the ranges manually Note If you do not use Auto range but define your own ranges in min and max data points outside these ranges are ignored only data points within the ranges of the graph are plotted and stored together with the graph If you always want the complete data set to be stored with the graph check Auto range and resize your graph after plotting The controls under the heading Options define the graph to be used for plotting and the plotting style Check New window if you want to create a new graph in a new drawing window Uncheck this if you want to use the front most drawing window Check Plot into current graph to plot into the current graph The current graph is usually the one where the last plotting took place However you can define any graph to be the current graph by double clicking it and checking Current Graph in the dialog box that appears Read more about this dialog box later in this chapter As a shortcut you can hold down the com mand key and double click the graph If both Plot into current graph and New window are unchecked a new graph is drawn in the front most drawing window Use the button Style to select the graph style for the new graph A more detailed explanation of graph styles is given below Use the controls titled Connect points to specify if the data points are to be connected and if yes the t
181. ou can use built in transformation algorithms e g sort transpose filter Fourier transform or mathematical operations or user defined ones Data can be text numbers and dates You can define your own data transforms by writing scripts programs which can e g access the data in the spreadsheet windows pro Fit supports the following scripting languages Apple Script e g in conjunction with AppleScript Editor Pascal more accurately a Pascal like scripting language executed within pro Fit Python scripts executed within pro Fit Functions can be used for plotting analysis and curve fitting There are a number of built in func tions such as log cos exp etc You can define your own functions using Pascal or Python Functions and scripts can also be created using an external compiler plug ins Functions can be analyzed and they can be fitted to data pro Fit provides powerful fitting tools based on various algorithms and methods Fits can be multidimensional parameters can be con strained and their starting values can be selected manually or depending on the function and al gorithm to be used automatically You can plot your functions and data sets in a drawing window pro Fit offers most standard fea tures of a drawing program and the appearance of all graphical elements is customizable These various elements of pro Fit are described in the following sections 12 3 Getting started A first session Thi
182. ow Help Tags from the Help popup menu Browsing functions and programs Navigating a lengthy function or program definition can be difficult To get a quick overview of your definition click the popup menu Browse in the toolbar of the function window ANA W lin_py func W To Menu __ Debug Help Const Browse y Python 7 Mi syntax 7 function lin_py lin_py_check i lin_py import scipy interpolate as interpolate import numpy The menu shows a list of all functions procedures and programs defined in the file Choosing an entry from this list takes you there 130 Finding the definition of a symbol If you want to find the definition of a symbol variable or command that appears in a function win dow double click it while holding down the option key This brings up a window of pro Fit s extensive on line help which helps in using pro Fit as well as in programming functions or data transform algorithms This help is permanently accessible through pro Fit s Help menu and you can browse it at will or use its search functionality to learn about how to use pro Fit and about the details of various commands The compiler When adding a definition to the list of functions or programs of pro Fit the definition text is trans lated into code that can be executed by your computer directly Pascal scripts or by Python s run time engine Python scripts This results in a very fast e
183. p button Then you can replay what you did by clicking the play button Alternatively you can use the commands Start Recording Stop Recording and Run or Run Selection from the Customize menu If you only want to run a part of a script first select it and then click the play button or choose Run Selection from the Customize menu If you don t select a part of the script before clicking the play button then the whole script is run If pro Fit can find function or program definitions in the midst of the script it will add them to their proper place in the Func or Prog menus The recorded commands appear at the current insertion point in the text window You cannot edit the text window while it is being recorded You can record new commands at any place inside an existing program definition Simply position the cursor where you want the new commands to appear and click the record button External functions and programs In addition to scripting through Pascal Python and Apple Scripts you can write your definition in an external programming environment of your choice such as Xcode and add the generated code 144 to pro Fit This process is called writing a plug in See Chapter 10 Working with plug ins for de tails Debugging window pro Fit provides a powerful debugging environment for the development of your programs and functions To use this environment simply check the opti
184. place the keyword cycle into a for while or repeat loop to immediately start a new it eration of the loop Example for i 1 to NrRows do begin if not DataOK i 1 then begin Writeln Empty cell skipped cycle goes to next value of i end end The above example loops through the first column of a data window and does some calculations indicated by If an empty cell is found the calculations are skipped and the loop is continued with the next value of i Optional parameter lists Usually you pass parameters to procedures and functions using the standard Pascal syntax For example you write DrawRect 10 10 50 100 In other words you pass a value for each parameter and separate the parameters by commas 197 However some of pro Fit s predefined procedures use an optional parameter list for passing val ues for instance CloseWindow window Data 1 saveOption dontSave In the above example window and saveOption are the names and Data 1 and dontSave the values of the parameters that are passed to the procedure CloseWindow In other words each pa rameter has a name that must be passed in front of its value The advantage of this calling convention is that you can omit some parameters if you want to use their default values For example you can call CloseWindow saveOption ask In this example we have omitted the parameter window and use its default value the fr
185. points at different coordinates and fit f x to the complete data set This is the way we usually find a parameter set which best describes the measurement The parameter set obtained in this way is not the true unknown parameter set but it should be a good approximation for it See the section on Error Analysis to find out how to estimate the errors of the fitted parameters The fitted parameter set corresponds to a function f x which maximizes the probability that the measured data set came from the system described by f x To maximize this probability we have to minimize the deviations between the measured data points and the function curve This devia tion can be defined in different ways depending on the way in which the experimental errors are distributed but it is usually a function of the weighted distances CAG 1a eo oe a tas Oxi Joa f x yj 1b yl Oyj Oxi and Oy give the magnitude of the errors expected when measuring x and yi respectively The role of these x and y errors is to define the correct scaling of the x and y deviations between a measured data point and the function that should describe it The errors normalize the deviations introducing dimension less numbers dxi and dyi Data points are weighted differently given more or less importance depending on their errors A small error will magnify the importance of a given difference a large error will make the normalized difference less important The dis
186. ppears beside the radio button Text fields These are shapes that contain editable text Generally text fields can be edited active Set to true if the field can be edited Set to false if it cannot be edited value The numeric equivalent of the text appearing in the field Use the function Invalid to check if the text corresponds to a value number text The text that appears in the edit field Static text fields These are shapes that contain non editable text Properties same as for text fields except that active has no influence on the shape s edit ability Popup menus Popup menu shapes have several val ues which can be selected by choosing them from a pop up menu Properties active Set to true if the pop up can be clicked Set to false if it cannot be clicked and is grayed value The currently selected item in the pop up menu 1 is the first item 2 the second item etc text The text that appears to the left of the pop up menultems The menu items separated by semico lons Wells These shapes are usually used as background for other objects e g a graph They consist of a white rec tangle 152 For further properties that you can use for controlling these shapes see the description of Get ShapeProperty and SetShapeProperties in pro Fit s on line help To use control shapes you first must create them in a drawing window Then you write a program that manages them and attach it to the
187. pple Script tell application pro Fit set sum to value of cell 1 of column 1 value of cell 1 of column 2 write line sum end tell Even though the above examples do the same you will prefer the program because defining pro grams is usually more convenient and faster In practice you probably use programs more often than Apple Scripts Programs can be defined within pro Fit they are much faster and they are better suited for numerical applications However there are some things that you simply cannot do from a program such as exchanging data with other applications communicating with the Finder batch processing a large number of files etc For these tasks you can use Apple Scripts You can combine the advantages of Apple Scripts and programs To call an Apple Script from a program first add it to the Prog menu choose Load Plug in from the Customize menu then call it with CallProgram To call a program from an Apple Script compile it and use the command run program pro Fit s Apple Script dictionary pro Fit provides a rich Apple Script dictionary with numerous methods and classes You can open this dictionary from within Script Editor or whatever tool you use for scripting to learn more In addition pro Fit is recordable i e you can e g enable recording from within Script Editor then execute the desired commands within pro Fit manually and Script editor will protocol their Apple Script equivalents automati
188. pro Fit 6 2 User Manual http Awww quansoft com 1990 2010 by QuantumSoft All rights in this product are reserved End User License Agreement License terms 1 The Software and its related documentation are provided AS IS and without warranty of any kind Quan tumSoft disclaims all other warranties expressed or implied including but not limited to the implied warran ties of merchantability and fitness for a particular purpose Under no circumstances shall QuantumSoft be liable for any incidental special or consequential damages that result from the use or inability to use the Software or related documentation even if QuantumSoft has been advised of the possibility of such dam ages In no event shall QuantumSoft s liability exceed the license fee paid if any 2 Rights granted a Use of the trial version of the Software is granted within the limits built into the trial version application b Use of the full version of the Software is granted for n users where n is the number of users indicated in the purchase acknowledgement for the registration key bought from QuantumSoft or from one of its re sellers An n user license is deemed to be exceeded if the Software is installed on more than n computers Exception Right is granted within the single user license to install the Software on two computers simultane ously if both said computers are regularly used by one person only Within this agreement installed on a c
189. pside down Note that you can only flip lines and polygons It is not possible to flip graphs legends imported pictures or text To change the order in which several objects overlap 1 Select the appropriate objects 2 Choose the desired operation from the Send submenu in the Draw menu You can move objects one position forward or backward commands Forward and Backward or you can bring them to the front or to the back of all other objects in the window To Front To Back To align objects 1 Select the objects to be aligned 2 Choose the desired operation from the Align submenu Using this menu you can align objects to each other or distribute them regularly If the objects are a group of text objects then every object retains its alignment when you edit it To group objects 1 Select all objects to be grouped 52 2 Choose Group from the Draw menu Objects that can be double clicked to change them e g text objects a graph or its legend can also be double clicked and changed while they are part of a group You don t have to ungroup them If the objects are text objects and you aligned them with the Align command before grouping them their alignment will be maintained when they are edited Ungroup from the Draw menu to ungroup a group Note If you resize a group containing text objects or data point symbols the proportions and size of the text and data points remain the same If yo
190. pt dictionary 11 Printing Printing from pro Fit Printing a pro Fit drawing from another application 12 Preferences 13 General features Help Help tags On line evaluation of mathematical expressions File info Shortcuts and other options Appendix A About numbers Floating point numbers Date and Time data Appendix B File formats Data The default text format Importing text files Saving text files The native data format Drawings Image formats Appendix C Python compatibility notes Appendix D pro Fit Pascal Syntax Introduction Program definition syntax Function definition syntax Types Simple numeric types Complex type Matrix and Vector types String and char types Arrays Loop statements The for loop The while loop The repeat loop Optional parameter lists Aborting procedures functions and programs Predefined constants functions procedures and operators Functions and procedures provided by pro Fit 162 163 163 164 165 168 168 169 169 169 171 172 172 172 172 174 175 178 178 178 179 179 179 179 180 181 181 182 183 185 185 185 187 190 191 191 192 194 195 195 195 196 196 197 198 199 200 Comparison to standard Pascal 200 1 Introduction This handbook This handbook is a conceptual introduction that describes key features and concepts and provides some step by step recipes It is neither a reference to all of pro Fits commands nor to its scripting en
191. r a pNumber in the function check so that the user can see these changes As an example assume that your function can have two input values that represent the same value in two different units of measurement Check Pascal or xxx_check Python can be used to update the value of one input value when the other input value is changed Note Check or xxx_check is not called during fitting It is called once when fitting is com plete Don t use it for calculating intermediate results for later use in the evaluation of the function You won t notice anything wrong as long as you modify the values in the parameters window but your function will not work when fitting Always use the proce dure First or xxx_first see below for calculating intermediate results 134 Input value pre calculations First xxx_first If you define a procedure called First Pascal or a function called xxx_first Python where xxx is the name of the function in the function directive it is called whenever the input val ues of a function have been changed before the body main part of the function is called The body of a function will never be called without first having been called beforehand First or xxx_first is mainly used for accelerating calculations that do not depend on the input value x This can make a fit considerably faster First or xxx_first should calculate all expres sions that appear in a function but that do not depend on x To
192. r scientific computing such as NumPy and SciPy see www scipy org with NumPy being installed by default on MacOS 10 6 To use a Python or Pascal script in pro Fit it has to be entered into a function window and then it needs to be compiled by choosing Compile amp Add to Menu from the Customize menu or by hitting the To Menu button in the toolbox of the function window The scripting language used in a function window is selected using a pop up menu the toolbar of the window 24 W logsin func v Const Browse M Synta default Pascal Python This pop up has an item called default If that item is selected pro Fit uses the preferred language specified in the pro Fit preferences under the tab Scripting ano Preferences Orie me on Ce me oy Export Extension Meta Prefs Preferred macro language Python req M Try to guess the language H Path to Python library _ Change Set to Default System Library Frameworks Python framework Versions 2 6 Python SS E v Syntax coloring Set to Defaults O Samras itt Background Ol Comments Global functions Normal text Strings A Global constants l Keywords LJ Global variables L Replace tabulators with spaces only in Python scripts E If replacing tabulators with spaces use a spaces If not replacing tabulators with spaces the tab spacing is 2 5 x font size If you check the checkbox Try to guess t
193. reate a list with all files in the folder set myFiles to list folder myFolder a list of files in myFolder set myFileCount to count myFiles the number of files in myFolder now start working with pro Fit tell application pro Fit set oldErrorAlerts to error alerts save error alert status set error alerts to false pro Fit should not show alerts activate bring pro Fit to front repeat with i from 1 to myFileCount go through all files set theFile to item i of myFiles get the i th file try open the file for processing as data file open file myFolder as string amp theFile as table write line found amp theFile write comment to Results window close window theFile saving no close without saving on error errText write line cannot open amp theFile amp amp errText amp end try end repeat set error alerts to oldErrorAlerts restore end tell This script first brings up a dialog box for selecting a folder by using the Apple Script extension choose folder with prompt Then it goes through all the files in this folder and uses the command open file name as table for opening the file as a data window It also uses the command write line text for writing a text into the results window Then it closes the file 165 The open file and close window commands are enclosed by the statements try and on error If any of these commands fails and returns an error the write line statement between on error
194. red position of one corner of the rectangle or the enclos ing rectangle for an ellipse and then drag the mouse to the opposite corner Note that you can draw partially closed ellipses or circles To do so create a Co regular ellipse or circle and double click it or choose Shape Settings from the Draw menu In the dialog box that appears you can select the starting angle and arc length of the section to be drawn Crating lines and polygons is easy as well Select the appropriate tool click the start of the line and drag to its end For polygons click at the positions corresponding to the cor ner points of the polygon release the mouse when moving from one point to the next Double click when finished To change between open and closed polygons holding down the mouse button for two seconds while you select the polygon tool Hold down the shift key to constrain lines or polygon sections to horizontal vertical or diagonal directions When drawing a polygon hold down the command key and double click to create a corner that remains a corner even when the polygon is smoothed Editing polygons and lines Lines and polygons can have arrows To define at which ends of the line polygon arrows must be drawn and to select the type and size of the arrow s use the arrow pop up menu in the tools pal ette To smooth polygons select the polygon to smooth and choose one of the options in the Smooth submenu of the Draw menu To
195. reducing the number of Gridding f data values e g by skipping smoothing averaging etc Transforms p Sort Sorting data numerically or alphabetically Insert Rows Columns Delete Rows Columns Transpose Swap rows and columns Resize Table Statistics Perform statistical analysis on a dataset Column Properties ked Binning Put data into bins You define a data range that oe encloses all your data values This data range is divided into consecutive intervals the bins For each bin the number of data values that lie in its inter val are counted This is useful for producing histograms Gridding Interpolating 3D data Imagine you have a series of x and y values xi and yi and a Z value zi for each pair of values xi yi The coordinates xi yi are not necessarily on a grid You now want to interpolate the z values and calculate them at a number of x and y values lying on a rectangular grid This interpolation process is called gridding Transforms Fourier Transforms FFT or inverse FFT for real and complex valued data Transforms Convolution and Correlation Perform convolutions and correlations Insert Delete Rows Columns Resize Table Commands for resizing data windows and inserting removing data More information regarding the various commands can be found in pro Fit s online help The bottom most entry in the Calc menu is called Masked The cells of a data window can be masked individually Any
196. replacing tabulators with spaces the tab spacing is 2 5 x font size Also generally you should check the option Try to guess the language which allows pro Fit to guess the language of a function or program definition entered in a Function window and will allow you to open any function or program definition and add it to pro Fit s menus irrespective of the lan guage in which they were defined and of you language preference The same dialog box also allows to choose the version of Python to be used in case you have several Python frameworks installed on your System By default pro Fit uses the version of Python that is installed in System Library Frameworks Python framework on MacOS 10 6 and later but you can change that to any other Python version pro Fit has been tested with reasonably recent builds of Python 2 5 as well as Python 2 6 Once you have selected your preferred scripting language you can open a new window for enter ing a new function or program Function Window by choosing New Function from the File menu 116 AOO Untitled Function 2 e Lal To Menu _ Debug Help y Const y Browse Python 7 y M Syntax ai gt The rightmost popup button in the toolbar of this window specifies the scripting language In the example above it is set to Python The question mark indicates that automatic language guessing is enabled and that the present guess is that the contents of the window will be
197. rically Set hasDerivatives to false in InitializeFunc if you are sure that you will never want to calculate any derivatives yourself Note that a call of Derivatives with a given x value is always preceded by a call of Func with the same x value therefore you might save a temporary result in Func for later use in Derivatives See also Chapter 9 Defining functions and Programs Parameters x The function s independent variable a The function s parameters ali Note that the index i ranges from 1 in Pascal but from 0 in C dydali The partial derivatives to be returned void First ParamArray a ExtModulesParamBlock pb This routine is called whenever the parameters a have changed before Func is called In most cases you can leave it empty Advanced programmers can use First for speeding up your function by evaluating temporary results that only depend on your function s parameters but not on its x value for more information see the description of First in Chapter 9 Defining functions and Pro grams Parameters a The function s parameters a i The index i ranges from 1 in Pascal but from 0 in C short Check short paramNo DefaultParamInfo const a0 ExtModulesParamBlock pb Check is called whenever the user has entered a value in the Parameters window In most cases you can leave Check empty returning the value good Advanced programmers can use it for im proving the parameters window s
198. rner gives the position of the selected axes in the main axes coordi nate system Use this field to change the position of a horizontal or vertical axis with respect to the vertical horizontal main axis coordinates The position is set by default to the minimum and maximum bounds of a plot when it is first created If the dialog box does not show the main axis X1 and Y1 are the main axes an additional check box is present It is called Same as X1 or Same as Y1 If it is checked most settings of the se lected axis such as the range scaling color line thickness tick positions are taken from the main X1 axis Note If you want to use two different axes for the top and for the bottom of your plot you have to uncheck this box before making any changes The tabs buttons Range amp Ticks Labels Prefix and Lines let you switch between different sub panels that are used to edit the general appearance of an axis the appearance of its labels and the kind of lines that are used to draw the axis and its tick marks 76 If you select Range amp Ticks you can set the following options l Range amp Ticks Labels Prefix Lines First 1 1 Last 1 lin Had Draw Ticks v axis line custom M ticks Ist major 1 v labels Distance 0 5 _ minor minor 1 The Draw check boxes determine which parts of an axis are drawn The First Last fields and the popup menu to their right are used to edit the range of the axis and its scaling typ
199. rom the expressions given above It is interesting to consider the case where a parameter reaches one of its limits during a fit As you know pro Fit lets you specify for each function parameter an interval of allowed parameter values If a parameter is at one of the boundaries of this interval after a fit its standard deviation cannot be calculated The parameter is then considered to be constant i e it is not a free parameter any more The standard deviations of the other parameters and x are calculated using the effective number of active parameters at the end of the fit The results obtained are the same as those that would have been obtained by fitting with the parameter fixed at its limit from the start Note The standard deviations of the parameters and the covariance matrix that are ob tained in a Levenberg Marquardt fit have a clear quantitative interpretation only if the errors of the data are normally distributed If the data errors are not given the calcula tions for evaluating the standard deviations of the parameters assume that the y are normally distributed and that the function is the correct description of reality Interpret the results carefully An alternative more general way to estimate the errors of the fitted parameters is described in the section Error analysis and confidence intervals The Robust minimization algorithm This method minimizes xr 2 with any definition of R by continually moving
200. roperties Format Calculations Role and Errors Format settings for Column 2 Width Number req Range 1e30 1e30 Format Decimals auto Q 1e300 1e300 C Scientific Example 1234 1235 To learn more about this dialog box click its help button About dates The Mac OS stores dates as the number of seconds since January 1 1904 For the technically minded the date is stored as an integer number 8 Byte long pro Fit uses the same convention as the Mac OS to store dates but uses double floating point values instead of integers With this number representation pro Fit can store and recognize dates with seconds precisions until up to 1015 seconds this corresponds more or less to a 6 byte long integer after January 1 1904 This means that pro Fit can store dates with second precision up to 31 million years in the future and it can store dates with day of the week precision up to 3 1 billion years 3 x 1012 in the future Up to 29 940 AD pro Fit can store dates with a precision of milliseconds while it can store dates in the present with a precision of approximately a microsecond If you choose Rel Time as the column format you can display a time difference Note You choose to use the units of years or months for a time duration but remember that month duration can vary and one year is not just 365 days Use the Date amp Time panel of the Preferences command to define how
201. s terminated by delimiter w spaces or tabs HA Lines M ends with delimiter R CR LF LF CR or CR LF eq _ maximum number of values 4 values per line Values common format Values can be enclosed in double quotes Excel CSV files Values can contain dates Format yyyy MM dd hh mm ss custom auto B terminated by delimiter B w spaces or tabs B line format string w v Show contents Input viewas Text Output Update x Value y Value x Value y Value 1 0 5 0 25 A 1 00000 0 5 0 25000 oO gt Selection Offset 0 Line 1 Cancel ox Click the help button of this dialog box for more information about the options you can specify here Saving text files You can also save data into text files in a custom format To do this choose Export from the File menu for your data window You are again prompted to specify the file format and you can choose between With Titles Without Titles and Custom the first two of which correspond to the de fault file format specified above while the third one allows to specify other file formats 180 Text file export options Format Custom a4 v Optimize size _ Write header lines Copy text from Info field Single line v Write column titles Column delimiter Line terminator Tab cRIsoD ie O t O r Click the help button of this dialog box for more information about the options you can specify
202. s and your function draws itself fast enough use the Preview Window pro Fit will redraw the function periodically during error analysis and you will be able to see how the fitted function changes depending on the simulated data sets which are generated randomly However doing so will waste time for drawing the function and slow down the error analysis procedure Hide the Preview Window or tell it to not draw the current function to make the error analysis procedure as fast as possible Fitting results When fitting is completed a summary of the results of the fit is displayed in the results window Depending on which fitting algorithm you used the data printed to the results window can vary slightly You may often want to transfer the values of the fitted parameters to the parameters window to use them as starting values for a further fit Choosing Get Fitted Params from the Calc menu transfers the best set of parameters to the parameters window The results of a fit are made available to custom functions and programs through a set of prede fined functions used for accessing the fitted parameters the confidence intervals the value of chi squared and for the Levenberg Marquardt algorithm the full covariance matrix See pro Fit s on line help for more details If you want to save the parameter sets obtained after a fit store them in a dedicated data window You can copy them from the parameters window and paste them into a single row
203. s chapter describes a typical pro Fit session It shows how to enter new data plot it and how to fit a mathematical function to it Our data The world s human population is growing rapidly The following table shows the number of inhabi tants of this planet for the period after 1940 year population in millions Let us plot and analyze these figures Entering the data First you must import your data into pro Fit To do this open a new data window 1 Choose New Data from the File menu An empty data window appears 13 00O Untitled Data 1 Data are arranged in horizontal rows and vertical columns The topmost cell of each column shows the name of the column by default Column 1 Column 2 etc The cells below contain the data of each column 2 Click into the first empty cell of column 1 and enter the first year 1940 We fill the first column with the years and the second column with the population 3 Click into the first cell of column 2 and start entering the population data Enter the values given in the table above Note that you can use the arrow keys the tab and the return or enter key to move from one cell to another 4 Enter the column titles year and population in millions Click into the titles Column 1 or Column 2 and enter the new names Move the mouse to the ver tical separation line to the right of the
204. s provided above This is described in the following sections Using the Python language Introduction This section describes in detail how to use Python within pro Fit It is not intended as a tutorial to Python There are numerous Python resources on the web e g at http docs python org tutorial where the interested reader can learn about Python programming Python Defining programs There are two ways to define a program with Python within pro Fit a Adding a python script as a program The first obvious way is to write a simple Python script such as a 5 print a a at2 print a When you compile this script by clicking the To menu button or by choosing Compile amp Add to Menu from the Customize menu the script is compiled but not executed and added to the Prog menu The name of the script in the Prog menu will be the title of the window that the script was written in When you choose the name of the script from the Prog menu it is executed The title of the script is derived from the name of the window that the script is written in To override this behavior you can add a single script directive to the beginning of the script A script directive is a single text line starting with followed by the keyword script and the name the script should have in the Prog menu Example script my script print hello world from my script 120 b Adding a python function definition as a program
205. s provided by pro Fit pro Fit offers a list of functions and procedures that can be called by your plug ins If you are pro gramming in Pascal they are defined in the interface of the file proFit_interface p If you are pro gramming in C they are defined in the header file pro Fit_interface h Their implementation can be found in the files pro Fit_interface p or proFit_interface c respectively Most of the functions and procedures provided by pro Fit for plug ins are the 1 1 equivalents of the ones that can be used when defining a function or program with pro Fit s definition language Refer to pro Fit s on line help for more information on the individual routines 163 10 Apple Script Apple Script is a language for scripting applications on the Macintosh It provides a common tech nique for automating tasks exchanging data and process remote control You can use Apple Script with pro Fit Note however that pro Fit cannot create i e compile an Apple Script To use Apple Script with pro Fit you need an Apple Script compiler such as Apple s Script Editor you can find it in the folder Apple Script of your Application s folder You enter the script in the script editor and compile it there Once the script is compiled you can either run it from your script editor or you can save it in its compiled form When using Apple s Script Editor choose Save As from the File menu choose the type Compiled scrip
206. s useful to optimize the functions for execution speed Because the standard x value is indeed most often used as an independent variable while the rest of the inputs most often play the role of parameters we will often use the word parameters to loosely describe all named input values that usually play the role of parameters as explained above This reflects the most common usage However keep in mind that any input can be in prin ciple the default input and play the role of the independent variable Functions can be plotted used for curve fitting and analyzed in various ways The functions that are available in pro Fit are listed in the Func menu and the name of the function that is currently in use the current function is checked The inputs and outputs of the current function are always visible in a dedicated window that carries the name of the current Function We call this window the Parameters Window It can be closed but the name Parameters always appears in the Window menu and can be selected to make the parameters window visible or bring it to the front Here is the parameters window of the polynomial function 33 Kok Polynomial f x const al x a3 xA3 deg degree of the polynomial 7 Inputs x 0 0 deg 3 0 const 0 0 al 0 0 a2 1 0 a3 0 0 Outputs default y Selected Value const y offset Limits from none to none Default input M Use for fitting This window determines the behavior of the
207. s which parameters are to be fitted Only those whose name is shown in bold face will be fitted for these parameters the check box Use for fitting which appears when you select a parameter is checked The following is the parameters window for the built in Polynomial function Among all the inputs the one named x is also the default input value that is the independent variable for the function All other inputs are parameters that influence how the function generates its return value from the de fault input value x But for the parameter named deg they can all be varied during a fit They can be used as fitting parameters Ane Polynomial f x const al x 4 a3 xA3 deg degree of the polynomial Inputs default x x 0 0 deg 3 0 const al 0 0 a2 1 0 a3 0 0 Outputs default y Selected Value const y offset Limits from none to none O Default input Use for fitting To change the fitting mode of a parameter e g from fit to not fit click its name It will switch from bold to normal or from normal to bold Alternatively you can click the check box Use for fit ting in the selected parameter field 102 Some parameters can never be fitted For example it does not make sense to fit the degree of a polynomial The name of such a permanently fixed parameter cannot be made bold by clicking it and the Use for fitting check box is disabled Parameters that can never be fitted are called con
208. s window or to alert the user of some event For example you can let your ma chine beep when fitting is finished In Python def xxx_last pf Beep In Pascal procedure Last begin beep end Determining how multiple output values are rendered in the preview window Input and output values of a function have various properties that can be set by the dedicated func tions SetParameterProperties and SetOutputProperties These functions can be called by any pro Fit program or function and also by a function that wants to set some advanced proper ties of its outputs One of these properties determines if an output appears in the preview window even when it is not defined as the default output To get an idea of what this means look at one of the predefined Peaks functions in the preview window By using these calls in its initialization routine a function can determine if and how its outputs must appear in the preview window and even set their names there instead of using the outputs state ment Here is an example in Pascal function electricfield phi real var Ex Ey r h v real real var E real procedure Initialize begin SetFunctionProperties function preview groupOutputValues SetOutputProperties output 0 name E absolute value of the field outputGroup 1 SetOutputProperties output 1 name Ex x component of the field outputGroup 1 SetOutputProperties output 2 name Ey y component of the fie
209. se It is possible to create a simple custom func tion very simply and rapidly and it is also possible to create very complex custom functions that can perform all kinds of complex calculations including numerical approximations To define a new function you can use two different programming languages pro Fit pascal and Python or you can write your functions in your own compiler and import them as plug ins see Chapter 9 Defining functions and programs and Chapter 10 Working with plug ins Editing in the parameter window Input values Click any value and edit it Use tab or enter to move to the next value use shift tab or shift enter to move back When an input is selected the area at the bottom of the window dis plays the full name of the input and some other properties The interval of allowed values limits if it is the default input and its fitting mode Output values You cannot edit output values but you can select them and move from one to the other in the same way as you do for inputs When an output is selected the area at the bottom of the window displays its full name and if it is the default output Copy paste You can copy and paste input values between the parameters window and data or text windows using the Copy Cut and Paste commands from the Edit menu If you choose Copy with no parameter value selected all parameters are copied to the clipboard separated by tabula tors If you choose Paste with n
210. second column title click and drag the separation line a little bit to the right so that you see the complete title It s also possible to customize the appearance and formatting of numbers and text 5 Save the data by choosing Save As from the File menu You are prompted to enter a name for your file Your window should now look like this 14 LaKa 1 2 3 4 5 6 7 8 9 10 11 Plotting the data Now that we have entered the data we can display it graphically 1 Choose Data y x from the Plot menu A dialog box appears where you can enter the ranges of the plot the columns to be plotted and more In this introductory session we can use the settings as they are 2 Click OK A drawing window appears showing a graph of the data 15 population data oO Pie e population in mill a A a population in millions 1940 1960 1980 Page 1 x 14923m Oe E You can edit a drawing easily For example you can change most parts of the graph just by double clicking 3 Double click the vertical axis to change its range Double click the vertical axis itself not the numbers to the left of it A dialog box called Graph Settings appears presenting the settings of the left y axis 16 Graph Settings a E T oa Z axis Plots Frame Grid Bar charts General X axes i Labels Prefix Lines
211. sed which was patented by Unisys PNG Portable Network Graphics This is an extensible file format for the lossless portable well compressed storage of raster images supported by a large number of applications PNG is the most modern format with excellent compression PNG provides a patent free replacement for GIF and can also replace many common uses of TIFF Indexed color grayscale and true color images are supported plus an optional alpha channel Sample depths range from 1 to 16 bits TIFF Tag Image File Format This is a no loss bitmap based file format supported by a large number of applications TIFF describes image data that typically comes from scanners frame grabbers and paint and photo retouching programs 182 Appendix C Python compatibility notes Python scripts within pro Fit can import and use most of the available python modules However in view of the sheer number of such modules it has been impossible to test all of them Therefore if you find any issues or if you can confirm interoperability of pro Fit with any important or useful py thon modules please do let us know But we cannot provide support and interoperability with all of them The type of modules that are most likely to cause problems are those that create their own GUI elements such as windows and dialogs Such modules were often not written to be called from within an app that runs its own GUI The following are some compatibility notes on s
212. set To create a scatter plot skyline plot or histogram of a set of x and y values Open a data window with the data you want to plot 1 Choose Scatter Plot from the Draw menu 2 The following dialog box appears Plot XY Data Selected data only Data window Table 1 Cooordinate X axis X1 req Scaling lin E Data x Value eq m Auto range Magnitude Data Row Index x Value y Value Y axis Y1 B Scaling lin mM Auto range Options Use line color Sj o EE OC E B 0O 2 a c SE es oO Oot ODO a s ira zg vis g5 4 V o S as arr tes ne E B x a Draw error bars Cancel Graph Styles 69 Select the window containing the data set an x column in the section Coordinates and at least one y column in the section Magnitude Check selected rows only if you only want to plot those rows of the data set that are currently selected in the data window Then you can define the axes their ranges and their scaling The two popup menus titled X axis and Y axis are used to choose the axis to be used for plot ting and to determine its range The popups titled Scaling define the scaling type of the axis Check Auto range to let pro Fit automatically calculate the ranges of the axis based on the val ues of the selected data set If you plot into an existing graph the ranges of the axes you use for the pl
213. set of parameters is chosen This procedure is reiterated with this new set of parameters When the minimum is near the algorithm goes over to a more deterministic guessing at the position of the minimum and solves some equations to find it The fitting stops when the value of x does not decrease anymore between successive steps The algorithm is described in W H Press B P Flannery S A Teukolsky W T Vetterling Numerical Recipes the Art of Scientific Computing Second Edition University Press Cambridge 1992 When x errors are specified the algorithm is modified in such a way that it minimizes 8 It finds at the same time the set of x coordinates x and the function parameters that minimize the mean square deviations between the data points and the function values The extensions to the Levenberg Marquardt algorithm that allow the interpretation of x errors are described in P L Jolivette least squares fits when there are errors in X Computer in Physics Vol 7 No 2 1993 Partial derivatives To fit a function of the type y a ae x the Levenberg Marquarat algorithm needs the partial derivatives of the function with respect to its parameters It uses the partial derivatives when it es timates the influence of the parameter set ai on x2 The partial derivatives fi are given by f a 3X a l f x 11 and they are calculated for all x coordinates during every iteration Note When you
214. sful fit you automatically obtain a table containing the parameter values before the fit the values after the fit their difference and the resulting standard deviations File info pro Fit lets you save a comment with every one of its files You can edit this comment with the Get Info command from the File menu Choosing Get Info presents a dialog box with a large field for editing text You can add an info comment to data windows drawing windows and functions or programs The data windows let you view and edit this information directly without using the Get Info com mand For this you drag down the info hook a black area on top of the right scroll bar of a data window to create an info field of the desired size See Chapter 4 Working with data for more in formation on data windows 174 Note that the info comments are in general only saved in files that have pro Fit s standard formats If you save a function definition as normal text files TEXT format or if you save a drawing window as a picture or EPS file PICT format EPSF format the info comments are not saved If you save a data file as TEXT you have the option of placing the info comments right at the beginning of the text file as a header To set this option you have to choose Custom format in the dialog box that comes up when saving text files Shortcuts and other options Although most of pro Fit s features and commands are readily accessed thro
215. simplest Then the single output value can be set by assigning a value to the name of the function or to a predefined variable called y As an example function electricfield real var 126 E real Begin E 1 x electricfield E end or function electricfield real var E real begin E 1 x y E end The predefined variable y to which the output value is assigned becomes a predefined array for multi valued functions To tell pro Fit that a function has multiple output values there are two possi bilities 1 Use the outputs statement to declare and initialize the output values array Example function vectorfield phi real real outputs y 0 0 E absolute value of the field y 1 0 Ex x component of the field y 2 0 Ey y component of the field var E real begin E 1 x y 0 E y 1 E cos phi y 2 E sin phi end This allows to define for all output values the names with which they must be identified in the pa rameter window when output values are displayed or in menus 2 Use var parameters following the standard Pascal syntax to declare additional output values function electricfield phi real var Ex Ey real real var E real begin E 1 x electricfield E Ex E cos phi Ey E sin phi end Note that following the Pascal convention the above function has three output values The usual one specified by the function name plus the two a
216. sitioning precision that is much more accurate than any output device printer or monitor This is important because all drawing objects can also be created by a user program If you write a program that produces graphical output then you are likely to need a high precision coordinate system pro Fit gives you just this Any drawing that you generate from a program is produced at very high resolution and it will give optimal results when printed on any output device or when exported to other applications as a picture or a PDF shape The precise coordinates of any drawing objects can also be viewed after it has been created using the pro Fit Coords window which will be described later in this chapter Although all coordinates are precise floating point numbers apparent accuracy will obviously suffer when drawing on a low resolution device such as a normal monitor In order to represent your drawing at a certain resolution determined by the zoom popup menu pro Fit must round the floating point coordinates describing a drawing object In addition anti aliasing effects can repre sent fractional line widths and coordinates As an example a line that is half a pixel wide will be drawn with a light gray These effects are produced by the operating system s drawing engine but some customization is available through the Preferences command When you draw something at a low resolution pro Fit must figure out reasonable floating point co ordinates It
217. ssions that are also required for the calculation of the derivatives you can optimize your code to evaluate these expressions only once In Python xxx_derivatives has the same arguments as xxx itself and it returns a list of the partial derivatives for all inputs not including x though Example function SampleFunc import numpy as np def SampleFunc_derivatives x al a2 global t return t 1 return dy dal dy da2 def SampleFunc x al a2 global t t np sinh x store t for re use in derivatives return al t a2 In Pascal Derivatives can use the values x a l a 2 and returns the results in the array dyda 1 dyda 2 function MySinh var t real procedure Derivatives begin dyda 1 t use t calculated below dyda 2 1 end begin the function s body t sinh x save sinh for Derivatives y a 1 t a 2 end Finishing up Last xxx_last If you define a procedure called Last Pascal or a function called xxx_last Python where xxx is the name of the function in the function directive its is called when all calculations fitting etc are completed It is the last piece of function code called by pro Fit before returning control to 137 the user Last xxx_last can be used to clean up to make final calculations or to re initialize some variables to their starting values Last can also be used to print some special messages or results in the result
218. stant parameters those that are currently not fitted are called inactive parameters and those that are currently fitted are called active parame ters Parameter limits The value of a parameter can be limited to any specified interval by entering the boundaries of the allowed interval in the corresponding edit fields The edit fields appear in the active parameter field once you select a parameter A parameter is not allowed to leave the specified interval during fitting optimization of the function or when you enter a new value in the parameters window See Chapter 5 Working with functions and the chapter Defining functions and programs for more information on how to set parameter limits in user defined functions If no limits are specified the default values are and infinite to infinite During fitting each parameter is constrained to the interval specified by the parameter limits Running a fit Running a fit consists basically of three simple steps 1 Choose the function to fit in the Func menu Add your own function to the Func menu if it is not already there 2 Define which parameters you want to fit and their starting values You can do this in the parameters window as described above Look at the function and the data set in the Preview Window to see how good your starting values are Use the Fitting tool in the Preview Window to push the function towards the data points The impor
219. statements separated by semicolons end for a function function MyFunc m n real i integer real variable and constant definitions begin statements separated by semicolons myFunc return value end In this case MyProc or MyFunc is the name of the procedure function The name is followed by a list of arguments in brackets If the procedure or function has no arguments this list including the brackets is omitted In our examples we have three arguments m n and i together with their 186 type definitions If you define a function the declaration of its return type follows after the argument list Then follows a semicolon After the line defining the name of the function or procedure you can define constants or variables using the same syntax as described for the program see above These items are only known within this procedure or function The statements of the procedure or function follow enclosed by begin and end You can call a procedure or function anywhere after its declaration like this MyProc 1 72 3 13 20 r MyFunc 1 71 3 14 10 Local functions and procedures can also have var parameters When you change a var paramter you change the value of the corresponding variable of the calling function Example program Test procedure Increase var a Real increase value of a by 1 begin a atl end begin k 1 Increase k increases k by 1 Writeln k writes 2 end Fun
220. stomize its features You can choose the format for exporting pictures the preferred method for printing you can save your preferred user interface options etc All of these settings are saved in pro Fit s preferences file During start up pro Fit looks for a folder pro Fit Preferences in Libraries Preferences for its preferences file If the file is there pro Fit reads its standard settings from it If the file is not there pro Fit creates a new preferences file You can switch to another preferences file or create a new preferences file anytime later If you do not want to load the standard preferences file hold down the option and the shift key while starting up pro Fit Choosing Preferences from the pro Fit menu controls most of pro Fit s settings Doing so brings up a dialog box with several panels Each panel controls a set of options To choose a panel se lect its icon from the list at the top of the dialog box The panels Printing and Copy are discussed in Chapter 12 Printing This handbook does not provide a detailed discussion of the other panels because each G of them comes with its own help button Clicking the help button brings up a help screen that explains the functions of all the settings 171 13 General features Help pro Fit offers a powerful on line help based on Apple s Help Viewer application The pro Fit on line Help system can be accessed by choosing pro Fit Help from the help menu
221. t and save the script Such a compiled script can be loaded into pro Fit Choose Load Plug in from the Customize menu and select the compiled script It is added to the Prog menu In the following we give some examples for scripting pro Fit through Apple Script Then we discuss the differences between programs and scripts For a list of all Apple Script classes and methods supported by pro Fit read pro Fit s dictionary from your Apple Script compiler e g by choosing Open Dictionary from the File menu of Apple s Script Editor There is an Apple Script equivalent for most commands of pro Fit s built in compiler Appendix C of this manual provides a cross reference between the commands of pro Fit s compiler and the corre sponding operations in Apple Script Note Apple Script is a very powerful programming language However it may be confusing for the beginner The easiest way to get started is using Apple Script s recording ca pabilities Just open the Script Editor and click the Record button Now go into pro Fit and do by hand what your script is supposed to do Script Editor records all your ac tions as Apple Script commands Once you are through go back to Script Editor and click the Stop button Your script is now complete Note that this chapter is not intended to give a beginner s introduction to the Apple Script language We will however explain some its aspects as we use them To learn more
222. t to start fitting Fitting is very fast When it is completed the fitted function is shown in the plot preview in the lower half of the fitting window a0 Fitting Algorithm Levenberg Marquardt V Auto search Ranges Function Exp m Guess initial parameters Data Data Window population d B Dout data l Capen ONA Data Column year E Data Column population i Error type Zero EB Error type Unknown Selected data only Plot Show last fit 5500 0 M Show function O leg i Show data auto 1500 0 1930 0 auto Clog 1990 0 Error analysis _ Close window when done _ Print full description M Print active parameters only 2 Done Fit gt The fitted parameters appear in the results window e020 Results l Fit Algorithm Levenberg Marquardt Chi squared 1 6958e 4 Fitted parometers Standard deviations A 2113 2935 SA 30 2618 t 54 8120 At 1 0812 The fit yields 54 years for t0 and 2113 millions for A Note that above we wanted to show you the fitting dialog box For simple fits you can also execute them directly by using the Simple Curve Fit command in the Calc menu which simply reads its information from the current data window and immediately executes a fit without asking for any additional details 20 We can plot function 3 2 using the fitted parameters 3 Choose Plot Function f x from the Plot menu A dialog box appears displaying options for plotting t
223. t a predefined procedure used for speed op 35 timization when programming a function When another input is used as the default input its value would be undefined in first See Chapter 9 Defining functions and programs for more details If you plan to use other inputs as the default input do not use first Using functions The following explains what you can do with functions It does however not describe how to plot or fit functions these topics are covered in Chapter 8 Fitting and Chapter 7 Drawing and plot ting or how to define a new function a topic covered in Chapter 9 Viewing function outputs To see all the outputs of a function click the triangle to the left of the word Outputs in the parame ter window The list of output values will be updated automatically whenever an input value is modified Calculating output values The first direct way to observe how a function s output values change in response to a change in its input values is to display all the outputs in the parameter window by clicking the disclosure tri angle to the left of the word Outputs in the parameters window You can also calculate the default output of the currently selected function for a given value of the default input by choosing Analyze from the Calc menu Choosing Tabulate Function x allows you to create a table of the function s output values in a data window You are prompte
224. t need to be updated and 2 if it is not acceptable In Pascal function SampleFunc function Check begin 133 check OK return OK if the new parameter is fine if pNumber 2 and a 1 lt a 2 then pNumber is the changed parameter check BAD return BAD if not acceptable end begin end In Pascal Check can use the following predefined variables and constants pNumber The number of the modified parameter a l a n The input values as they appear in the parameters win dow They can be checked and or changed mode 1 The mode of each input which can be active inactive or mode n constant You can check and or change the modes active inactive constant These three constants can be used to be compared to or assigned to mode i check The function must store its return value in this variable ok bad update One of these three constants must be returned in the variable check Check must return one of the values ok bad or update in the variable check to tell pro Fit if it should accept the new input value and what it should do with the parameters window If check is ok pro Fit accepts the new parameter If check bad pro Fit refuses the new parameter and shows the old one in the parameters window If check update pro Fit accepts the new parameter and redraws updates the whole pa rameters window Use this feature whenever you have changed a parameter other than the one with index pNumbe
225. tance of good starting values depends on the function to be fitted Some functions like the Gauss function are more difficult to fit A polynomial can be fitted with almost any starting pa rameter set Note Some of pro Fit s built in functions such as Exp Log and Power provide algorithms for automatically choosing good starting values For these functions you can check the option Guess Initial Parameters in the fitting dialog box mentioned below and you don t have to set appropriate initial parameters before fitting 3 Choose Simple Curve Fit or Curve Fit from the Calc menu If you choose the Simple Curve Fit command the current function is used to fit the current data set defined by the default x and y columns and the corresponding error columns if any in the 103 current data window and the fit is immediately executed This is the simpler and most direct way to perform a fit If you choose the Curve Fit command you can set a number of additional fitting options in a dia log box K k Fitting Algorithm Polynomial B Auto search Function Polynomial Guess initial parameters Data Data Window Table 1 m concn Output data Z Selected data only Data Column x Value E Data Column y Value B a Error type Zero Error type Unknown Plot C Show last fit 1 0 A Show function O leg Show data auto 0 0 Tio lauto log 1 0 Error analysis x _ Close window when d
226. tances dxi and dyi give the difference between measured coordinates and true coordi nates Obviously we don t know the true coordinates otherwise there wouldn t be any need for a fitting program in the first place But we can estimate the true coordinates by minimizing some function of the distances dxi and dyi This function describes the difference between the model function and the set of data points and it is chosen in such a way that its minimization corresponds to the situation with the highest probability of producing the measured data set If the x and y errors are independent a fitting algorithm must generally minimize a mean deviation XR of the type Xp gt Ri dx Ry dy 2 l where the functions Rx are deviation functions R d that tell us in a quantitative way how bad it is that a certain normalized distance d is found for a data point They are normally related to the er 91 ror probability distribution This is the function that gives the probability that a certain measurement error occurs For example Rx can be the negative logarithm of the corresponding probability dis tribution for the distances dxi and dyi Minimization of yr as defined in Eq 2 adjusts the function f x in order to maximize the probabil ity that the measured data set corresponds to an underlying reality described by the adjusted f x This is true as long as the following assumption is fulfilled the measurement
227. te 38T Forward Backward ToFront ToBack Edit Text Delete 3D Se ee Here you can select and change or delete all curves and data sets of a graph To change the drawing order of the plots select a plot by clicking it in the list and click Forward To Front Backward or To Back to move it one position backward or forward or to move it to the back or front of all plots The first plot symbol at the top of the list is drawn first so back means top of the list and front means bottom of the list To change the text describing a curve or a data set select the curve in the list and click Edit Text Instead of doing this you can also double click the item in the list of curves and data sets To specify if a data set is to be show as scatter plot bar chart or skyline plot use the menu Plot type The pop up menus titled Line let you edit the line that draws a curve or connects the data points The pop up menu Points lets you select the symbol for data points Check Line to draw lines be tween successive data points or for a skyline plot The menu Thick defines the line thickness used to draw the data point symbols It can be set to auto in which case the line thickness will be cho sen depending on the size of the data points Click the E Bar Style button to define the style of any error bars that could be defined for a set of data points 82 You can fill the region between a curve and one
228. tion exp ldl The deviation function og R d log 7d 5 corresponds to a Lorentzian error distribution 1 1 d2 2 Two other deviation functions available in pro Fit are 92 c l cos d c d lt cx R d 6 2 Wiser with c 2 1 and 2 3 v i 4 ld lt c R d 4 s 7 ld gt c 6 with c 6 These deviation functions are called Andrew s sine the derivative of 6 is sin z c and Tuckey s biweight respectively They don t correspond to a particular probability distribution for the errors They are designed to decrease the weighting of data points with very big errors outliers in order to allow a robust fitting through the more reasonable data points It should be obvious that this procedure should only be used if you know your experiment and data set well enough and we repeat the usual calls for caution Note that using the deviation functions 6 and 7 with another constant c is equivalent to changing all errors of the data points and the resulting mean deviation value by a constant factor Each term in the sum 2 describes a deviation between the measured data point xi yi and the nearest point on the function curve x f x For each data point coordinate x must be chosen in such a way that each term in the sum is minimized When the deviation function R is the squared deviation R d d2 then each term in 2 gives the square of the Euclidean d
229. tput value of a function versus one input value Plot a new function by choosing the Function f x command from the Plot menu 1 0 1 0 0 5 0 0 0 5 1 0 Contour plots consist of a series of contour lines and or a color encoded pixels representing a function s output value versus two of the function s input values or a set of data points having x y and z coordinates To plot a function contour plot choose Function f x y from the Plot menu 6 4 2 0 2 4 6 Scatter plots represent sets of data points having x and y coordinates The data points can be marked by plot symbols and or be connected by lines To draw a scatter plot choose Data y x from the Plot menu and select the option Scatter Plot from the Plot Type pop up 62 2 012345 6 7 In a bar chart horizontal or vertical bars represent two dimensional data points To generate a bar chart choose Data x y from the Plot menu and select the option Horizontal Bar Chart or Verti cal Bar Chart from the Plot Type pop up 2 012345 6 7 In a skyline plot two dimensional data points are represented by horizontal or vertical lines inter connected by vertical or horizontal connecting lines giving the impression of a skyline To gener ate a skyline plot choose Data x y from the Plot menu and select the option Horizontal Skyline or Vertical Skyline from the Plot Type pop up 2 012345 6 7 A
230. tputGroup pf SetOutputProperties output 4 outputGroup pf SetOutputProperties output 5 outputGroup def electricfield r phi E 1 r Ex E np cos phi Ey E np sin phi pf groupOutputValues 1 1 1 2 2 2 139 h r np cos phi v r np sin phi return E Ex Ey r v h Programming examples Accessing data Imagine you have a data window with some data in the first column You want to write a program that fills the second column with the square root of the values of the first column You want to take some special cases into account e Ifa cell in the first column is negative the corresponding cell in the second column should be 0 Ifa cell in the first column is empty the corresponding cell in the second column should be empty too If any cell in the first column was empty the program should give the user a warning when it has finished A Pascal program that does this task is program MakeRoot var i integer the row counter doAlert boolean true if a cell was empty begin doAlert false for i 1 to nrRows do if DataOK i 1 then if cell not empty if data i 1 gt 0 then data i 2 sqrt data i 1 else begin data i 2 0 doAlert true end if doAlert then Alert Some data was negative end The same program writtin in Python is import numpy doAlert False for i in pf RowRange 140 if pf DataOK i 1 if pf GetData i 1 gt 0
231. try flag or country specific character towards the right end of the menu bar usually represents this menu If no such icon appears in you menu bar choose System Preferences from the Apple menu and go to the International pref erences pane There select the Input Menu tab and activate besides your customary keyboard layout e g the Greek keyboard or the Character Palette After having done so you can switch between keyboard layouts by hitting the spacebar while holding down the command key 54 OU a Note 1 pro Fit uses Unicode for character representation If you enter a Unicode character that cannot be handled by a given font pro Fit uses fallback algorithms to find a font that does handle this character This will usually give the expected results when displaying and printing through the default operating system rendering engine If however you are gen erating Quickdraw or Postscript output we recommend that you always explicitly switch to a font that carries the desired characters In particular when using symbol characters such asa B or y you may want to switch to the Symbol font Note 2 If your text object is part of a group object it is not resized when the group is re sized If you want to resize the text objects within a group you must hold down the com mand key while resizing the group Rectangles and ellipses are created using the corresponding tools of the palette Select the appropriate tool click the desi
232. ts to false pro Fit should not show alerts activate bring pro Fit to front compile scriptProgram add the above program to Prog menu set myTable to make new table open new data window set k to 1 a counter for opened files repeat with i from 1 to myFileCount set theFile to item i of myFiles get the i th file try open file myFolder as string amp theFile as table open the file write line processing amp theFile run program ScriptProgram run the program in pro Fit 166 close window theFile saving no close without saving set sum to globalData 1 get result set cell k of column 1 of myTable to sum store it in the table set k tok 1 on error errText write line cannot process amp theFile amp amp errText amp end try end repeat delete program ScriptProgram remove the program from Prog menu set error alerts to oldErrorAlerts restore end tell The above script starts with the definition of the program to be run by pro Fit set scriptProgram to This statement sets the symbol scriptProgram to the text following it The symbol at the end of the line tells the script editor that more lines follow to generate this symbol type the return key while holding the shift key down The statement compile scriptProgram add the above program to Prog menu sends this text to pro Fit and tells pro Fit to compile it i e to add it to the Prog menu Then the script creates a new data w
233. tting from the Pen and Dash pop up menus in the drawing tools palette To change the line style of the first item in a legend that is drawn using a line either connected data points or a function curve select the legend and choose the line style in the Pen and or Dash pop up menus To change the line style of all items in a legend select the legend and choose the line style in the Pen and or Dash pop up menus while holding down the shift key To add a connecting line to the first data point in a legend select the legend and choose a line style from the Pen or Dash menu while holding down the option key To add a connecting line to all data points in a legend select the legend and choose a line style the Pen or Dash menu while holding down the shift key 13 By default a legend lists every plot of the related graph You can however hide one or more plots from a legend by unchecking Appears in legend in the dialog box for editing curve styles This is explained later in this chapter Editing graphs The possibilities for changing and editing a graph are nearly unlimited A whole set of specialized options lets you create the graph you need These options are accessed either by double clicking the graph or its legend or by using the Graph Options submenu in the Plot menu This submenu is only available if a single graph is selected or if a drawing window contains
234. u want to resize them proportionally with the group hold down the command key while resizing the group Objects created with the tools palette The upper part of the tools palette contains the drawing tools needed to create some of the more simple drawing objects Ze AN k The lower part contains pop up menus to select background patterns line widths and dashing and arrows Their use is explained in the section Editing drawing objects So F r 53 A Use the text tool to create text objects When you select this tool and click into a drawing window a dialog box for editing the text appears Edit Text Electric gradient E Font Helvetica Neue M Use Greek keyboard for Symbol font ne Position 7 qS ee ees CD Bold 8 M fi TE M italic 31 Extended Size 29 gt 8 E A C Underline _ Condensed E z 2 Cancel OK _ S O Here you can enter your text and specify font font size text styles and the vertical position of each character pro Fit uses the shift command key equivalents H T and S for the fonts Helvetica Times and Symbol respectively Instead of clicking the OK button hit the Enter key or hold down the command key and hit the Return key To enter greek characters such as a R or y switch the keyboard to a layout that allows entering such characters For this purpose it is easiest to enable the Input Menu in the menu bar An icon carrying a coun
235. ue input values max 128 optional the default values for the parameters their de fault mode parameter window name lower and upper limit see the Chapter 8 Fitting If you do not define the defaults for a parameter it will be O inactive and limited by INF and INF If you do not define a parameter window name for a parameter its default name will be used The default name is either the name you define in the function header e g ampl or afi optional the default values for the outputs and their name to appear in the parameter window If you do not define the defaults for an output value its value will by default be 0 Alternatively if you do not want to define default values and names for all outputs of a function but you merely want to state that your function has n outputs simply as sign the constant n to the outputs keyword e g outputs 3 optional the definition of constants as in standard Pascal optional variable declarations as in standard Pascal After this you can optionally define your own local procedures and functions Then follows the body of the function definition between begin and end In this body you must calculate the function s yvalue from its x value and its parameters For this you can use the follow ing variables 188 var optional variable declarations as in standard Pascal temp extended myVar t integer a l a n The remaining input values par
236. ues are displayed in the parameters window or when a function is tabulated The predefined function output accepts the index of the output value as a parameter and returns true if it must be calculated function electricfield phi real var Ex Ey real real outputs y 0 0 E absolute value of the field Ex 0 Ex x component of the field Ey 0 Ey y component of the field var E real begin E 1 x if Output 0 then y 0 E if Output 1 then Ex E cos phi if Output 2 then Ey E sin phi end Pascal Defining programs Programs are defined in the same way as you would define a procedure in standard Pascal program PowersOf2 var i integer begin NewDataWindow for i 1 to nrRows do data i 1l 2 i end The keyword program at the very beginning means that this code will be added to the Prog menu Programs do not have a return value and do not use the spcialized statements that are used for functions such as those to define default input and output values Otherwise the defini tion language follows the same rules explained above for functions pro Fit Pascal in depth More information about pro Fit s Pascal syntax can be found in Appendix D of this manual On line help for programming The introduction given above only scratches the surface and does not describe many of the possi bilities that you have to determine how a function or program behaves and can be used within pro
237. ugh its menus there are some more advanced or rarely used features that require the use of modifier keys like the op tion key the command key or the shift key The following is a short list of these features action modifier keys Selecting a tool in the tools palette of drawing windows option to keep the tool selected after drawing the corresponding object Dragging objects in drawing win dows command to constrain the movement along 45 lines shift to constrain the movement to horizontal and vertical directions option to duplicate an object instead of simply moving it Drawings objects in drawing win dows option or shift to get a square bounding box Drawing lines in drawing windows shift to make the line horizontal vertical or diagonal at 45 option to make a diagonal line Drawing polygons in drawing win dows option shift same as for lines command double click to produce a corner that remains a corner even when the polygon is smoothed Resizing objects in drawing win dows option to keep the bounding box of the object square height wicth shift to maintain the horizontal and vertical proportions of the object its height or its width command to resize the size of texts in a group Resizing lines in drawing windows option to get a line constrained to 45 directions shift to maintain the direction of the original line or to make the lin
238. umber cell in command a data window to set the default columns x y Ax Ay using a pop up menu Clicking on the larger font size controls in the text edit dialog box option to increase the font size by 1 pt only Clicking on the subscript superscript position controls in the text edit dialog box option to change the vertical position of the selected text by 1 pt only Choosing New Function from the file menu option to open a new definition window containing a sample function defini tion option shift to open a new definition window containing a sample program defini tion Importing text files option to tell pro Fit not to ask for information and to open the text files as data files using the current settings 176 Saving a drawing as an EPS file option to create a TEXT file containing the PostScript information Using lists in dialog boxes e g the ycolumn list in the plot data dialog box shift click shift and drag to select more than one item shift click to de select an item Clicking with the lens tool in the Preview Window command to drag a selection rectangle specifying the region to en large option to zoom out instead of zooming in Selecting an item from the Help menu in a Function window option to paste the template with a and a carriage return command to enable pasting templates and disable help panels shift to ena
239. und off errors are exactly the same for the two objects and they do overlap exactly in the 100 view too If you are concerned with precise positioning e g when drawing overlapping lines or placing ar rows on the axes of a graph always go to a larger zoom factor e g 400 or have a look at the underlying floating point coordinates You can do this using pro Fit s Coords window 48 Choose Coords from the Windows menu to see this floating window Whenever a single drawing shape is selected the Coords window shows its Coords floating point coordinates i e its size and its position in coordinates that Left Top pt make sense for the particular shape which is currently selected 182 127 Width Height The first row of the Drawing Info window gives the absolute coordinates on 498 138 the paper the second row gives the dimensions of the selected shape and Angle Diagonal the third row gives the angle of its diagonal and its length The last row 0 0 1175 24 shows the current coordinates of the mouse The units used to display the Page 1 coordinates can be chosen using the Preferences command Mouse Position All the coordinate fields are editable Simply click a coordinate and enter a different number to change the size or the position of the selected shape C Lens For example you can set the precise length and orientation of a line by en tering the corresponding coordinates in the edit fields in the third ro
240. ut values and the ones that are only changed rarely to influence how the function calculates its outputs or that are determined by some optimization procedure like curve fitting One can describe in general the first set as the independent variables of a function and the second set as the parameters of a function Which group of inputs must be considered as independent variables and which one must be considered as parameters depends on the kind of operation that is performed using the function and can be changed But in the simplest most usual case there is a single default input called x and a single default output called y and there is an additional set of inputs a1 a2 that are treated as parameters y f x a1 a2 an For the polynomial function one of the predefined functions available in pro Fit the input called deg sets the degree of the polynomial while const and a7 a2 etc are parameters that can be determined e g by matching the polynomial function to an existing set of x and y values curve fitting Note that even though any input can be made the default input and it can therefore be transformed into the default independent variable in the above sense the standard x value cannot be used as a parameter in curve fitting for instance This limitation ensures that all pro Fit func tions have a standard x value that plays the same role consistently and it allows for a special treatment of the standard x value that i
241. ve the x coordinates and is labeled i If there is more than one marker there can be two other coordinates displayed They correspond to the distance between the reference marker and one of the other markers If a marker is active its coordinates are displayed in editable fields Edit any of these fields to set the coordinate of the marker Note The behavior when changing the text in the edit fields containing the marked coordi nates varies depending if the marker is on a data point or if it is on a function curve If the marker is on a data point the coordinates displayed in the edit field correspond to the coordinates of that data point in the data window Editing them changes the values in the data window If the marker is on a function curve editing the coordinates sets the position of the marker If you edit the y coordinates pro Fit numerically inverts the function to find the corresponding x value You can use this feature also as a shortcut to calculate the in verse of a function Coordinate markers can be accessed from pro Fit programs using the predefined functions GetMarkedX GetMarkedY and GetMarkedCoords Managing coordinate markers We already saw above how to create markers and look at their coordinates There are a few other simple operations that can be applied to markers Click a marker to make it active 44 Click a marker while holding down the option key to transform it into the reference marker
242. vironment such reference material is included in pro Fit s online help Key features pro Fit is an interactive tool for the investigation analysis and representation of functions and data It is designed for users in science research engineering and education The key features of pro Fit are Spreadsheet and data management Numerical and alphanumerical data can be stored trans formed and analyzed Predefined and user defined algorithms can be used for data transforma tion Analysis of mathematical functions The values passed to mathematical functions and those re turned from them are managed in a parameter window that allows to easily determine which input values must be used to calculate the output values of the function Input values can be easily edited and managed and the effects of any change in their values are previewed interac tively and in real time Customized functions and algorithms pro Fit provides very powerful and simple methods for defining mathematical functions programs data transformation algorithms drawings and gen eral macros Simple built in Pascal like compiler for user defined functions and programs This is a version of Pascal adapted to the needs of the pro Fit environment and extended to support floating point numbers complex numbers vectors strings and up to 4x4 matrices as general data types to be used in any mathematical expressions Built in support for Python for user defined functio
243. w The Lens check box lets you open a small view port with an enlarged larger pixels version of the region around the mouse Drawing objects A drawing contains different objects There are three different classes of objects in a drawing win dow Objects that are created by choosing Plot Function or Plot Data from the Draw menu such as a graph and its associated legend Objects that are created using the tools in the upper part of the tools palette such as texts lines and rectangles Objects that were created in another application and that are imported as pictures to pro Fit by pasting them by importing a picture file or by dragging and dropping them in a drawing window The first class graphs and legends is discussed in the section Plotting The other classes ob jects created by using the tools palette and imported pictures are discussed in the following sec tion Shape properties All shapes objects in a drawing window have properties such as their position or size These properties can be read and set from a program so it can manipulate shapes in a drawing window For more information see the documentation on GetShapeProperty and SetShapeProperties in pro Fit s on line help Most of these properties can also be set and changed manually For instance when you move a shape to a new location you change its position properties Some of the properties are accessible 49 by choosing Shape Setti
244. w j Pe To place the tools in a separate drawing window choose Drawing Tools from the Win 89 dows menu The floating tools palette appears To move the tools back to the drawing Fill window simply close the floating window If you will never want to have drawing tools N E inside the drawing windows you can disable this option Choose Preferences from A the Files menu and check Always use floating toolbox in the Drawing panel r gt Dash The upper part of the tools palette contains tools that are used to create simple objects gt such as rectangles and text Then there is a tool that can be used to pick up a color and _Arrow apply it to another graphic object and a tool that lets you draw the individual data points such as those used in graphs A further tool is provided for generating buttons and _ Zoom other controls used when creating the interface for user defined programs The rest of 100 the tools palette contains popup menus for setting line styles and fill patterns and for _Pages choosing the zooming factor of the current view in the drawing window The drawing Ixl gt window can be viewed at zoom factors from 25 to 400 To learn more about these 7 tools refer to the section Editing drawings later in this chapter at 47 Coordinates accuracy and drawing info pro Fit uses floating point numbers to store the size and position of the various drawing objects This provides a po
245. w much have you lost 0 print You now have myAccount cents 121 Python Defining functions Simple functions To define a function that pro Fit can use for plotting or fitting you define the function as a python function and use a function directive to tell pro Fit that it should be added to the Func menu A function directive is a single text line starting with followed by the keyword function and the name of a function defined within the script Example function func from math import def func x c0 kk if x 0 return c0 kk else return c0 sin kk x x The first argument of the function is assumed to be the default input x of the function the remain ing arguments are treated as additional inputs parameters You can specify default values for the parameters by assigning default values to the function s ar guments using regular Python syntax e g function sinc from math import def sinc x c0 0 kk 100 the classic sinc function if x 0 return c0 kk else return c0 sin kk x x Note The comment the classic sinc function is optional If it is provided it is added to the header of the parameter window As an alternative to defining the names and default values by means of the names and defaults of the function s arguments the input directive can be used This directive follows the function direc tive and has the form input initialValue mode name
246. window Finally you must switch the window to dialog mode The following is a simple example that shows this procedure 1 Open a new drawing window and create a button named Multiply 0 To do this choose the button tool from the windows toolbox Then click into the drawing window A dialog box appears where you can define the text that appears on the button You can also define a name for the button Ea that we will later use for accessing the button a a program In this ex button tool ample set the button text to Multiply by 2 and its name to Button A Control Shape Settings Shape name Button Title Multiply by 2 m enabled C default button Cancel Co 2 Create an edit field named Number Now click the button tool again and hold the mouse down until a popup menu appears Choose Text Field Now click into the drawing window and enter Number for the shape s name Then click OK You now should have a drawing window with an edit field and a button Arrange these items as you wish then save the file The window might e g look like this fP Simple button ex J a ae dfgsdfg oS Multiply by 2 i ee dete eae N A Pe I v x 196 pt y 64 pt IRI 3 Switch the window to dialog mode 153 To do this hold down the control key while clicking anywhere into the window and choose Display As Dialog Alternatively choos
247. window to be created My Python Drawing The functions available through the module pf are the same as those available to Pascal scripts with some few exceptions These exceptions are listed in pro Fit s on line help Compatibility notes Python scripts within pro Fit can import and use most of the available python modules with few exceptions A list of some known exceptions can be found in Appendix C of this handbook 124 Using the pro Fit Pascal language Introduction pro Fits Pascal scripting language is loosely based on the Pascal programming language but with various restrictions and a few additions The following describes how to define pro Fit functions and programs using Pascal and the major syntax elements of the language Pascal Defining functions Simple functions A simple example of a function definition in Pascal has been given above A less minimalistic defi nition for our function could be function LogSine B D real begin if x lt 0 then y D else B sin x 1ln x D KK l end which uses two names for the two additional inputs B and D and provides a mechanism to take into account the possibility that the x value can be negative Our sample function above was not defined for x lt 0 It would generate a run time error if used in calculations with negative x values But since the function converges to y D for x 0 we can just decide that the output value is D for all negative valu
248. window was clicked In this case the tag msgShape will have a stringValue set to the name of the clicked shape if a shape was clicked or will have an empty stringValue if no shape was clicked The tags msgClickedX and msgClickedY contain the clicked co ordinates control clicked A control shape was clicked successfully In this case the tag msgShape has a stringValue set to the name of the clicked shape The tags msgClickedX and msgClickedY contain the clicked coordinates 149 control keydown start control keydown end opened save close command idle A control receives keyboard input This tag message is sent before the key is processed In this case the tag msgShape has a stringValue set to the name of the shape The tag msgCharCode has a stringValue of length 1 giving the char code of the pressed key For easier comparison there are the following charCodes constants predefined charHome charEnter charEnd charBackspace charTab charLf charPageUp charP ageDown charCr charEsc charArrowLeft charArrow Right charArrowUp charArrowDown charDelete The tag msgKeyCode has a stringValue of length 1 giving the key code of the pressed key The tag msgModifiers has a value set to the keyboard modifiers it tells e g if the option key was pressed You can change the msgChar Code msgKeyCode and msgModifiers tag to change the keyboard event before it is processed For easier com parison ther
249. xecution speed of programs and functions The translation of your definitions into machine code is carried out when you choose Add to Menu from the Prog menu or if you click the button To Menu in the toolbar of the function window Any changes that you make to your definition after compilation will not affect the function or pro gram as it was added to pro Fit s menus To update your changes you must choose Add to Menu again Advanced function and program definitions pro Fit provides some mechanisms to improve the capabilities and performance of user defined functions and programs Many of these mechanisms are implemented by defining procedures with special names In Pascal these procedures have the names Initialize Check First Derivatives Last In Python scripts they are called xxx_init xxx_check xxx first xxx_derivatives xxx_last with xxx being the name of your function Initializing and shutting down programs and functions Initialize xxx_init When writing a program or function you can defined a piece of code that is called whenever the program or function has been added to pro Fit s menus Within that code you may want to initialize any variables or print some information into the Results window The simplest way to do this in a Python program is by adding that code to the top level of the script e g program DoMyStuff print This program converts a data column print into normalized units inputColumn
250. y axes O The popups titled Y axis and Scaling are used to define the y axis of the graph to be used as well as its scaling type Check Auto range to let pro Fit automatically calculate the ranges of the y axis start ing from the y values returned by your function If you plot into an existing graph the ranges of the axes you use for the plot will be extended if necessary If you uncheck Auto range you can enter the ranges manually The controls under the heading Options let you choose the following options O Check New window if you want to create a new graph in a new drawing window Un check this if you want to use the front most drawing window Check Plot into current graph to plot into the current graph The current graph is usu ally the one where the last plotting took place However you can define any graph to be the current graph by double clicking it and checking Current Graph in the dialog box that appears Read more about this dialog box later in this chapter As a shortcut you can hold down the command key and double click the graph If both Plot into current graph and New window are unchecked a new graph is drawn in the front most draw ing window Use the controls titled Line style to specify the style of the curve representing your function The controls titled Fill allow you to specify if and how the area below the curve is to be filled e Use the button Styles to select the graph style
251. years in the future and it can store dates with day of the week precision up to 3 1 billion years 3 x 109 years in the future Up to about 29000 years into the future pro Fit can store dates with a precision of milliseconds while it can store dates in the present with a precision of approximately a microsecond 178 Appendix B File formats This appendix describes the file formats used by pro Fit for transferring data or drawings to and from other applications Data The default text format To exchange data between pro Fit and other applications text files are used Usually such files hold one or more lines of text Each line contains all values of a row separated by tabs gt The lines are separated by carriage returns It is possible to use other characters instead of tabs and carriage returns See below There are two standard formats of data text files produced by pro Fit The standard format with titles is defined as follows lst line namel gt name2 gt name3 1 2nd line 0 123 gt 1 732 gt 1 122 f 3rd line 2 233 gt 2 125 gt 2 126 f The standard format without titles is very similar but without the column titles line There is an interesting exception for data text files to be loaded into pro Fit If the first line is a sin gle star pro Fit reads the second line as the column titles even if the file is loaded as being standard format without titles Lines are separated by carria
252. yourself Generally you should use static variables for global memory instead of using pb gt globals void CleanUp ExtModulesParamBlock pb CleanUp is called when pro Fit is quitting or when your plug in is removed from pro Fit In most cases you won t have to do anything here Advanced programmers may wish to de allocate some special memory to close a port or to clean up other stuff here Routines to be modified in external programs only not used for functions void InitializeProg ExtModulesParamBlock pb This routine is called before a program is run for the first time Most users can leave it empty Ad vanced programmers may wish to allocate some memory open a port initialize global static vari ables etc here void Run ExtModulesParamBlock pb This routine is called when your program is executed It should hold your program s main code Routines to be modified in external functions only not used in programs Note An important note about parameter indices When accessing arrays that hold values names etc of the parameters such as al a0 names mode dydal the index i ranges from 0 to 127 inC void InitializeFunc Boolean const hasDerivatives Str255 descristLine Str255 descr2ndLine short const numberOfParams DefaultParamInfo const a0 ExtModulesParamBlock pb This routine is called once after your external function has been linked to pro Fit It must return some default values and information

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