Fityk 0.8.5 - User's Manual
Contents
1. 3 Siegmund Brandt Data Analysis Springer Verlag 1999 4 PeakFit 4 0 for Windows User s Manual AISN Software 1997 5 Zbigniew Michalewicz Algorytmy genetyczne struktury danych programy ewolucyjne WNT 1996 282. ao 1 ag exp In 2 5 ro y 23 List of functions Pseudo Voigt is a name given to the sum of Gaussian and Lorentzian a3 parameters in Pearson VII and Pseudo Voigt are not related Equation A 10 Pseudo Voigt Area PseudoVoigtA 1 ag VIn2 eh in 2 04 M 13 y ao 2 Qay N a a V ur 1 se re e Equation A 11 Voigt 4 Too exp t QQ Jo Ee z d aj t 1 t pte exp t dt 5 00 a3 t7 The Voigt function is a convolution of Gaussian and Lorentzian functions ag heigth a center a is proportional to the Gaussian width and a3 is proportional to the ratio of Lorentzian and Gaussian widths Voigt is computed according to R J Wells Rapid approximation to the Voigt Faddeeva function and its derivatives Journal of Quantitative Spectroscopy amp Radiative Transfer 62 1999 29 48 See also http www atm ox ac uk user wells voigt html Is the approximation exact enough for all possible uses of fityk program Equation A 12 VoigtA ap too exp t y ZA 5 dt Vra J a3 t Equation A 13 Exponentially Modified Gaussian EMG NEK 2m xo b c c d gt bb c MT e BRE d 2d4 ld Mv2e vd Equation A 14 Doniach Sunjic DoniachSunjic d F x EP Equation A 15 Polynomial5 2 3 4 5 y ao x aor azt asx asr 24 Appendix B Command shortenings The pipe symbol I shows the
3. n peaks in n guess x range in n data expression in Qn n F n Z C n dF data expression der mathematic function version info der shows derivatives of given function info der sin a 3 exp b a f a b sin a 3 exp b a df d a cos a 3 exp b a b a 2 df d b 3 exp b a a commands dump sleep reset quit All commands given during program execution are stored in memory They can be listed using the command info commands n m or written to file info commands n m gt filename To put all commands executed so far during the session into the file foo fit type info commands gt foo fit With the plus sign i e info commands n m information about the exit status of each command will be added To log commands to a file when they are executed use commands gt filename or to log also the output commands gt filename To stop logging use commands gt dev null Scripts can be executed using the command commands lt filename There is also a command dump gt filename which writes the current state of the program together with all datasets to a single fit file Command sleep sec makes the program wait sec seconds before continuing The command quit works as expected If this command is found in a script it quits the program not only the script 19 Chapter 4 Using and extending Use cases TODO Extensions How to add your ow
4. is flexibility parameters of peaks can be arbitrarily bound to each other e g the width of a peak can be an independent variable the same as the width of another peak or can be given by complex and general for all peaks formula Fityk is free software you can redistribute and modify it under the terms of the GPL version 2 See Appendix C License for details You can download the latest version of fityk from http www unipress waw pl fityk or http fityk sf net To contact the author visit the same page How to read this manual After this introduction you may read the Chapter 2 Getting started If you are using the GUI version you can look at the screenshots based tutorial http www unipress waw pl fityk screenshots html in preparation and postpone reading Chapter 3 Reference until you need to write a script put constraints on variables add user defined function or understand better how the program works In case you are not familiar with the term weighted sum of squared residuals or you are not sure how it is weighted have a look at the section called Nonlinear optimization Remember that you must set correctly standard deviations of y s of points otherwise you will get wrong results GUI vs CLI The program comes in two versions the GUI Graphical User Interface version more comfortable for most users and the CLI Command Line Interface version named cfityk to differentiate Unix only If the CLI versio
5. Using and extending eicere pee 20 USO CASES wst toot et aeo t a I CREER TRE VOTO eee A Edo DELE RII ree tke ve Oto E eus 20 ExteNSIONS MESE E 20 How to add your own built in function sesssssse eee 20 As Last 0f FUNCIONS goi O O Ed 23 B Command Shortenin gs sacs te Pte t E ET PTT O e ap Rx RC re Pos O dE sa ENS 25 Ce LICENSE OPE 26 D About this manilall wsi PA O AT ORA nd sav W PATA 27 Bibliography AC 28 Chapter 1 Introduction What is the program for Fityk is a program for nonlinear fitting of analytical functions especially peak shaped to data usually experimental data The most concise description peak fitting software There are also people using it to remove the baseline from data or to display data only It is reportedly used in crystallography chromatography photoluminescence and photoelectron spectroscopy infrared and Raman spectroscopy to name but a few Although the author has a general understanding only of experimental methods other than powder diffraction he would like to make it useful to as many people as possible Fityk offers various nonlinear fitting methods simple background subtraction and other manipulations to the dataset easy placement of peaks and changing of peak parameters support for analysis of series of datasets automation of common tasks with scripts and much more The main advantage of the program
6. criterium nm convergence If the value of the expression 2 M m M m where M and m are the values of the worst and best vertices respectively values of objective functions of vertices to be precise is smaller then the value of nm convergence option fitting is stopped In other words fitting is stopped if all vertices are almost at the same level The remaining options are related to initialization of the simplex Before starting iterations we have to choose a set of points in space of the parameters called vertices Unless the option nm move all is set one of these points will be the current point values that parameters have at this moment All but this one are drawn as follows each parameter of each vertex is drawn separately It is drawn from a distribution that has its center in the center of the domain 11 of the parameter and a width proportional to both width of the domain and value of the nm move factor parameter Distribution shape can be set using the option nm distribut ion as one of uniform gaussian lorentzian and bound The last one causes the value of the parameter to be either the greatest or smallest value in the domain of the parameter one of two bounds of the domain assuming that nm move factor is equal 1 Genetic Algorithms TODO Settings This chapter is not about GUI settings things like colors fonts etc but about settings that are common for both CLI and GUI version Command info set shows t
7. errors with square root of reduced chi i e with sqrt WSSR DoF where DoF is the number of degrees of freedom i e the number of active data points minus the number of parameters Fityk is not doing this Fitting related commands To fit sum to data use command fit number of iterations im en The plus sign prevents initialization of the fitting method It is used to continue the previous fitting where it left off All non linear fitting methods are iterative number of iterationsisthe maximum number of iterations There are also other stopping criteria so that the number of executed iterations can be smaller fit in Q fits all datasets simultaneously Fitting methods can be set using the set command set fitting method method where method is one of Levenberg Marquardt Nelder Mead simplex Genetic Algorithms All non linear fitting methods are iterative and there are two common stopping criteria The first is the number of iterations and can be specified after the fit command The second is the number of evaluations of the objective function WSSR specified by the value of option max wssr evaluations 0 unlimited It is approximately proportional to time of computations because most of time in fitting process is taken by evaluating WSSR There are also other criteria different for each method If you give too small n to fit command and fit is stopped because of it not because of convergence it makes sens
8. same x but y and sigma of a merged point is set as an average of components shirley bg Calculates Shirley background useful in X ray photoelectron spectroscopy rm shirley bg Calculates data with removed Shirley background Functions and variables in data transformation information in this section are not often used in practice Read it after reading the section called Sum of fitted functions Variables foo and functions bar can be used in data transformations and a current value of data expression can be assigned to the variable Values of the function parameters e g fun a0 and pseudo parameters Center Height FWHM and Area e g fun Area can also be used Pseudo parameters are supported only by functions which know how to calculate these properties Some properties of functions can be calculated using functions numarea findx and extremum numarea f xl x2 n gives area integrated numerically from x1 to x2 using trapezoidal rule with n equal steps findx f x1 x2 y findsxininterval x1 x2 such that f x y using bisection method combined with Newton Raphson method It is a requirement that f x1 lt y lt f x2 extremum f x1 x2 finds xin interval x1 x2 such that f x 0 using bisection method It is a requirement that f x1 and f x2 have different signs A few examples foo y 0 data expression can be used in variable assignment foo2 y 0 in 0 4 dataset can be given if ne
9. the range is omitted and the parameter center is given the peak is searched around the center value of the option guess at center pm Fityk offers only a primitive algorithm for peak detection It looks for the highest point in a given range and than tries to find the width of the peak If the highest point is found near the boundary of the given range it is very probable that it is not the peak top and if the option can cancel guess is set to true the guess is cancelled There are two real number options related to guess height correctionand width correction The default value of them is 1 The guessed height and width are multiplied by the values of these options respectively Displaying information If you are using the GUI most of the available information can be displayed with mouse clicks Alternatively you can use the info command Using info instead of info sometimes displays more verbose information Below is the list of arguments of info related to this chapter The full list is in the section called info show information info guess range shows where the guess command would find a peak info functions lists all defined functions info variables lists all defined variables info n F shows information about F info n Z shows information about Z info formula in n shows the mathematical formulae of the fitted functions info n dF x compares the symbolic and numerical derivatives in x useful for debug
10. 14 Reference parameters are determined by the data set In other words we need to know the likely errors of the best fit parameters Finally it is not uncommon in fitting data to discover that the merit function is not unimodal with a single minimum In some cases we may be interested in global rather than local questions Not how good is this fit but rather how sure am I that there is not a very much better fit in some corner of parameter space Our function of merit is WSSR the weighted sum of squared residuals also called chi square y yi y z a Y x a gt 23 w u v z a 2 Weights are based on standard deviations i 1 fi You can learn why squares of residuals are minimized e g from chapter 15 1 of Numerical Recipes So we are looking for a global minimum of chi This field of numerical research looking for a minimum or maximum is usually called optimization it is non linear and global optimization Fityk implements three very different optimization methods All are well known and described in many standard textbooks The standard deviations of the best fit parameters are given by the square root of the corresponding diagonal elements of the covariance matrix The covariance matrix is based on standard deviations of data points Formulae can be found e g in GSL Manual http www gnu org software gsl manual chapter Linear regression Overview weighted data version Some programs scale
11. Every data point has four properties x coordinate y coordinate standard deviation of y and active inactive flag Lower case letters x y s a stand for these properties before transformation and upper case X Y S A for the same properties after transformation M stands for the number of points Data can be transformed using assignments Command Y y will change the sign of the y coordinate of every point You can also apply transformation to selected points Y 3 21 2 will change point with index 3 which is 4th point because first has index 0 and Y 3 6 21 2 will do the same for points with indices 3 4 5 but not 6 Y 2 21 2 will apply the transformation to points with index 2 and above You can guess what Y 6 1 2 does Most of operations are executed sequentially for points from the first to the last one n stands for the index of currently transformed point The sequance of commands M 500 x n 100 y sin x will generate the sinusoid dataset with 500 points If you have more than one dataset you have to specify explicitly which dataset transformation applies to See the section called Working with multiple datasets for details Reference Points are kept sorted according to their x coordinate so changing x coordinate of points will also change the order and indices of points Expressions can contain real numbers in normal or scientific format e g 1 23e5 constant pi binary operators one argume
12. Fityk 0 8 5 User s Manual Te Tint OGUC HOM ness obs hE ID es os ee nes ng i ds EEOSE TEEN 1 What 18 the program LOE 2065 easet POPE RZ WA Ne RENE ban VER RR a Eve Pee O PAL et 1 How t read this manual err eere re Ire e PA Guan dees PA AAA 1 GULYSCH ez i O ies te e e rp i ED Umi 1 2 Getting Started z odii o W a z O tu eee tee ea we O Poda chess dE 2 The minimal example 5 pe oett tree denna PA OOOO W OT POWA 2 Inyvoking fityk 6 ssa eL eee epe eee ETOWE W Ce ENTE E AC 2 Graphical interface edia uei dee orm erre ER RETRO MEER wies De EUR 3 Pl ts and other windOWS ia di wore alas z Z Ed eee opu re TENE neue Y Nose e PE HUNE OC 3 MOUSE USAGE ass sis tette sans angen re Eee erm a mtb E Rr Se lea ERE OK RC ses yates 3 Se References 4 io ie uh ue dev rp oves tege des ea produ uode EA eR ER E EE 5 General Synt x se RO secu ET ER RE e T a EE erre PER RR 5 Data from Experiment i e eee gn ME Tee errore ide 5 Loading data iiir titer tete ste rete e te aber dees 5 Active and inactive porfits cosciente eee doe eee De e eee ce seb eee eere dr cep p re ree p vens 6 Standard deviation or weight 2 1 er cer RE RR CERES AREA ERREUR 6 Data transformations e eat ess epe ert REN soa er pe eR eee ARE OPERE IE covtevaceswpiuens 6 Functions and variables in data transformation e eee eee aaa aa aaa aaa aa e eene 8 Working with multiple datasets e eeeu eee aaa aaa meme 9 EX porting data c en e
13. Some fitting methods and functions such as randnormal in data expressions use a pseudo random number generator In some situations 17 Reference one may want to have repeatable and predictable results of the fitting e g to make a presentation Seed for a new sequence of pseudo random numbers can be set using the option pseudo random seed If it is set to 0 the seed is based on the current time and a sequence of pseudo random numbers is different each time variable domain percent See the section called Variables 11 verbosity Possible values quiet normal verbose debug width correction See the section called Guessing peak location Examples set fitting method show info set fitting method Nelder Mead simplex change default method set verbosity verbose with fitting method Levenberg Marquardt fit 10 with fitting method Levenberg Marquardt verbosity only warnings fit 10 Other commands plot viewing data In the GUI version there is hardly ever a need to use this command directly The command plot controls visualization of data and the sum It is used to plot a given area in GUI it is plotted in the program s main window in CLI the popular program gnuplot is used if available plot xrange yrange in n xrange and yrange have one of two following syntaxes D min max The second is just a dot and it implies that the appropriate range is not to be changed E
14. a from a file is of interest We should be able to exclude selected points from fitting and all computations Every point can be either active or inactive This can be done with the command A see the section called Data transformations for details or with a mouse click in the GUI The idea of active and inactive points is simple only the active ones are subject to fitting and peak finding inactive ones are neglected in these cases Standard deviation or weight When fitting data we assume that only the y coordinate is subject to statistical errors in measurement This is a common assumption To see how the y standard deviation influences fitting optimization look at the weighted sum of squared residuals formula in the section called Nonlinear optimization We can also think about weights of points every point has a weight assigned that is equal i l o Standard deviation of points can be read from file together with the x and y coordinates Otherwise it is set either to max sqrt y 1 0 or to 1 depending on the value of data default sigma option Setting std dev as a square root of the value is common and has theoretical ground when y is the number of independent events You can always change standard deviation e g make it equal for every point with command S 1 See the section called Data transformations for details You can not set data errors standard deviations as unknown Data transformations
15. an do it in method do_precomputations This possibility is provided for calculating expressions which do not depend on x Write the declaration here void do_precomputations std vector lt Variable gt const amp variables and provide a proper definition of this method in src bfunc cpp If you want to optimize the calculation of your function by neglecting its value outside of a given range see option cut unction level in the program you will need to use the method bool get nonzero range fp level fp amp left fp amp right const This method takes the level below which the value of the function can be approximated by zero and should set the left and right variables to proper values of x such that if x left or x gt right than If x l lt level If the function sets left and right it should return true If your function does not have a center parameter and there is a center like point where you want the peak top to be drawn write bool has center const return true fp center const return vv 1 In the second line between return and the semicolon there is an expression for the x coordinate of peak top vv 0 is the first parameter of function vv 1 is the second etc Finally close the definition of the class with Now go to file src bfunc cpp Write the function formula in this way const char FuncFoo formula Foo height center hwhm hei
16. cessary Y y foo and variables can be used in data transformation Reference D Y y f x substracts function f from data Y y Q0 F x substracts all functions in F Z Constant 0 fit constant x correction this can be caused fit by a shift in scale of the instrument collecting data X x QO Z x remove it from the dataset Z 0 and clear the x correction in the sum info numarea fun 0 100 10000 shows area of function fun info Sfun Area it is not always supported info _1 extremum _1 40 50 shows extremum value calculate FWHM numerically value 50 can be tuned Sc f Center i findx f c c 50 f Height 2 findx f c c 50 f Height 2 i f FWHM should give almost the same Working with multiple datasets Let us call a set of data that usually comes from one file a dataset All operations described above assume only one dataset If there are more datasets created it must be explicitly stated which dataset the command is being applied to e g M2500 in Q0 Datasets have numbers and are referenced by with the number e g Q3 means all datasets e g Y y 10 in To load dataset from file use one of commands Qn filename xcol ycol scol block filetype options lt filename xcol iycol scol block filetype options The first one uses existing data slot and the second one creates a new slot Using increases the number
17. e are also other parameters to the CLI and GUI versions of the program Option h gives the full listing wojdyr ubu fityk src fityk help Usage fityk h V c lt str gt I r script or data file h help show this help message V version output version information and exit ec cmd lt str gt Script passed in as string g config str choose GUI configuration qp Shon nL don t process SHOME fityk init file r reorder reorder data 50 xy before 100 xy The example of non interactive using CLI version on Linux wojdyr ubu foo cfityk h Usage cfityk h V c lt str gt script or data file h help show this help message V version output version information and exit c cmde str Script passed in as string I no init don t process HOME fityk init file q quit don t enter interactive shell wojdyr ubu foo ls rdf dat a rdf dat r rdf out rdf wojdyr ubu foo cfityk q I gt set verbosity quiet autoplot never VN gt rdf gt i min x if y gt 0 in in 00 dat a 1 8875 in 1 dat r 1 5105 in 02 out 1 8305 Graphical interface Plots and other windows The GUI window of fityk consists of from the top menu bar toolbar main plot auxiliary plot output window input field status bar and of sidebar at right hand side The input field allows you to type and execute commands in a similar way as is done in the CLI
18. e to use fit command to process further iterations TODO how to stop fit interactively Setting set autoplot on fit iteration will draw a plot after every iteration to visualize progress see autoplot 18 Information about goodness of fit can be displayed using info fit To see symmetric errors use info errors and info errors additionally shows the variance covariance matrix Available methods can be mixed together e g it is sensible to obtain initial parameter estimates using the Simplex method and then fit it using Levenberg Marquardt 15 Reference Values of all parameters are stored before and after fitting if they changed This enables simple undo redo functionality If in the meantime some functions or variables where added or removed the program can still load the old parameters but the result can be unexpected The following history related commands are provided fit undo move back to the previous parameters undo fitting fit redo move forward in the parameter history info fit history show number of items in the history fit history n load the n th set of parameters from history fit history clear clear the history Levenberg Marquardt This is a standard nonlinear least squares routine and involves computing the first derivatives of functions For a description of the L M method see Numerical Recipes chapter 15 5 or Siegmund Brandt Data Analysis chapter 10 15 Essentially it combines an inverse Hessian met
19. ed Data from experiment Loading data The basic file format is ascii text file with every line corresponding to one data point If there are more than two columns of numbers it can be specified which columns corresponds to x and y and optionally also sigma Numbers in line can be separated by whitespace commas or semicolons Lines that can t be read as numbers are ignored The modified version of xy1ib library is used to read data from file New formats can be easily added Points are loaded from files using the command dataslot lt filename xcol ycol scol block filetype options where dataslot should be replaced with 0 unless many datasets are to be used simultaneously for details see the section called Working with multiple datasets filetype and options usually can be omitted in most of the cases the filetype can be detected automatically all supported filetypes are listed at the end of this section xcol ycol scol supported only in text file are columns corresponding to x y and std dev of y A column number of 0 generates a number increasing from zero with each point block is supported by formats with multiple blocks of data If the filename contains blank characters a semicolon or comma it should be put inside single quotation marks together with colon separated indices if any Multiple y columns and or blocks can be specified see the examples below Q0 lt foo vms Q0 foo fii
20. es of the functions The sum is constructed by specifying which functions are in F and which in Z In many cases x correction Z can safely be ignored The fitted curve is thus the sum of all functions in F These functions can be listed with info F Command F function puts function into F command Z function puts function into Z To remove function from F or Z either do F function or delete function del function If there is more than one dataset F and Z must be prefixed with the dataset number e g 1 F function The following syntax is also valid create and add funtion to F g Gaussian height 66254 hwhm 0 264 center 24 7 0 F g ole eo create automatically named funtion and add it to F G0 F Gaussian height 66254 hwhm 0 264 center 24 7 clear F O F 0 clear F and put three functions in it O F a Sb c show info about the first and the last function in 0 F info Q0 F 0 0 F 1 the same as bcp copy b bcp copy 0 F 1 make 1 F the exact shallow copy of 0 F G1 F QO F make 1 F a deep copy of 0 F all functions and variables are duplicated 1 F copy 0 F The sum can be exported as data points using the syntax described in the section called Exporting data or as mathematical formulae using the command info formula in n gt filename Some primitive simplifications are applied to the formula To prevent it put plu
21. foo 5 3 or c 3 1 or bar 5 sin foo foo is here a so called simple variable It is created by assigning to it real number prefixed with The means that the value assigned to the variable can be changed when fitting the sum of the functions to the data For people familiar with optimization techniques the number of defined simple variables is the number of dimensions of space we are looking for the optimum in In the above example the variable c is actually a constant bar depends on the value of foo When foo changes the value of bar also changes Compound variables can be build using operators and the functions sqrt exp log10 ln sin cos tan sinh cosh tanh atan asin acos erf erfc lgamma voigt This is a subset of the functions used in data transformations Every simple parameter has a value and optionally domain The domain is used only by the fitting algorithms which need to randomly initialize or change variables Genetic Algorithms are a good example Variables can be used in data tranformations e g Y y a The value of the data expression can be used in the variable definition but it must be inside braces e g bleh M or to create a simple variable bleh M Sometimes it is useful to freeze a variable i e to prevent it from changing while fitting There is no special syntax for it but it can be done using data expressions in this way a 12 3 a is fittable a a Sa
22. ght 1 x center hwhm 2 The syntax of the formula is the similar as that of the UDF but for built in functions only the left hand side of the formula is parsed The right hand side is for documentation only Write how to calculate the value of the function FUNC_CALCULATE_VALUE_BEGIN Foo fp xala2 x vv 1 vv 2 fp inv denomin 1 1 xala2 xala2 FUNC CALCULATE VALUE END vv 0 inv denomin The expression at the end i e vv 0 inv denomin is the calculated value xalxa2 and inv denomin are variables introduced to simplify the expression Note the fp you can also use double at the beginning and semicolon at the end of both lines The meaning of vv has already been explained Usually it is more difficult to calculate derivatives FUNC CALCULATE VALUE DERIV BEGIN Foo fp xala2 x vv 1 vvI2 fp inv denomin 1 1 xala2 xala2 21 Using and extending dy_dv 0 inv_denomin fp dcenter 2 vv 0 xala2 vv 2 inv denomin inv denomin dy dv 1 dcenter dy dv 2 dcenter xala2 dy dx dcenter FUNC CALCULATE VALUE DERIV END vv 0 inv denomin You must set derivatives dy dv n for n 0 1 number of parameters of your function 1 and dy dx In the last brackets there is a value of the function again If you declared do precomputations or get nonzero range methods do not forget to write definitions for them After compila
23. ging Fitting Nonlinear optimization This is the core We have a set of observations data points to which we want to fit a model or sum of functions that depends on adjustable parameters Let me quote Numerical Recipes chapter 15 0 page 656 f you do not know the book visit http www nr com The basic approach in all cases is usually the same You choose or design a figure of merit function merit function for short that measures the agreement between the data and the model with a particular choice of parameters The merit function is conventionally arranged so that small values represent close agreement The parameters of the model are then adjusted to achieve a minimum in the merit function yielding best fit parameters The adjustment process is thus a problem in minimization in many dimensions however there exist special more efficient methods that are specific to modeling and we will discuss these in this chapter There are important issues that go beyond the mere finding of best fit parameters Data are generally not exact They are subject to measurement errors called noise in the context of signal processing Thus typical data never exactly fit the model that is being used even when that model is correct We need the means to assess whether or not the model is appropriate that is we need to test the goodness of fit against some useful statistical standard We usually also need to know the accuracy with which
24. he syntax of the set command and lists all possible options set option shows the current value of the option and set option value changes it It is also possible to change the value of the option for one command only by prepending the command with with option value The examples at the end of this chapter should clarify this autoplot can cancel guess cut function level data default sigma epsilon exit on warning fitting method formula export style guess at center pm height correction lm max wssr evaluations nm pseudo random seed See the section called plot viewing data 18 See the section called Guessing peak location See the section called Speed of computations See the section called Standard deviation or weight It is used for floating point comparison a and b are considered equal when la bl lt epsi lon You may want to decrease it when you work with very small values like 10 10 If the option exit on warningis set any warning will also close the program This ensures that no warnings can be overlooked See the section called Fitting related commands See the section called Sum F and Z 13 See the section called Guessing peak location See the section called Guessing peak location Setting to tune Levenberg Marquardt fitting method See the section called Fitting related commands Setting to tune Nelder Mead downhill simplex fitting method
25. hod with a steepest descent method by introducing a lambda factor When lambda is equal to O the method is equivalent to the inverse Hessian method When lambda increases the shift vector is rotated toward the direction of steepest descent and the length of the shift vector decreases The shift vector is a vector that is added to the parameter vector If a better fit is found on iteration lambda is decreased it is divided by the value of 2m 1ambda down factor option default 10 Otherwise lambda is multiplied by the value of 2m 1ambda up factor default 10 The initial lambda value is equal to 2m 1ambda start default 0 0001 The Marquardt method has two stopping criteria other than the common criteria If it happens twice in sequence that the relative change of the value of the objective function WSSR is smaller then the value of the 2m stop rel change option the fit is considered to have converged and is stopped Additionally if lambda is greater than the value of the 2m max 1ambda option default 10 15 usually when due to limited numerical precision WSSR is no longer changing the fitting is also stopped Nelder Mead downhill simplex method To quote chapter 4 8 3 p 86 of Peter Gans Data Fitting in the Chemical Sciences by the Method of Least Squares A simplex is a geometrical entity that has n 1 vertices corresponding to variations in n parameters For two parameters the simplex is a triangle for three parameters the s
26. implex is a tetrahedron and so forth The value of the objective function is calculated at each of the vertices An iteration consists of the following process Locate the vertex with the highest value of the objective function and replace this vertex by one lying on the line between it and the centroid of the other vertices Four possible replacements can be considered which I call contraction short reflection reflection and expansion It starts with an arbitrary simplex Neither the shape nor position of this are critically important except insofar as it may determine which one of a set of multiple minima will be reached The simplex than expands and contracts as required in order to locate a valley if one exists Then the size and shape of the simplex is adjusted so that progress may be made towards the minimum Note particularly that if a pair of parameters are highly correlated both will be simultaneously adjusted in about the correct proportion as the shape of the simplex is adapted to the local contours Unfortunately it does not provide estimates of the parameter errors etc It is therefore to be recommended as a method for obtaining initial parameter estimates that can be used in the standard least squares method This method is also described in previously mentioned Numerical Recipes chapter 10 4 and Data Analysis chapter 10 8 Reference There are a few options for tuning this method One of these is a stopping
27. ined either by giving a full formula or as a sum of already defined functions with possible re parametrization see GaussianArea and GLSum below for the example of the latter When giving a full formula right hand side of the equality sign is similar to the definiton of variable but the formula can also depend on x Hopefully the examples below will make the syntax clear How it works you can skip this paragraph the formula is parsed derivatives of the formula are calculated symbolically all expressions are simplified but there is a lot of space for optimization here bytecode is created for a kind of virtual machine and when fitting the VM calculates the value of the function and derivatives for every point Common Subexpression Elimination is not implemented yet I suppose it will noticeably speed up UDFs Hint use the init file for often used definitions See the section called Invoking fityk for details Examples first how some built in functions could be defined define MyGaussian height center hwhm height exp ln 2 x center hwhm 2 define MyLorentzian height center hwhm height 1 x center hwhm 2 define MyCubic a0 height al 0 a2 0 a3 0 a0 al x a2 x 2 a3 x 3 supersonic beam arrival time distribution define SuBeArTiDi c s v0 dv c s x 3 exp s x v0 dv 2 x area based Gaussian can be defined as modification of built in Gaussian it is the same as built in GaussianA fu
28. is not fittable a f a a is fittable again 10 Reference It is also possible to define a variable as e g bleh 9 1 exp 2 In this case two simple variables with values 9 1 and 2 will be created automatically Automatically created variables are named _1 _2 __3 and so on Variables can be deleted using the command delete variable Some fitting algorithms need to randomize the parameters of the fitted function i e simple variables For this purpose the simple variable can have a specified domain Note that the domain does not imply any constraints on the value the variable can have it is only a hint for fitting methods such as the Nelder Mead simplex or Genetic Algorithms Further information on how the domain is used in these methods is contained in the appropriate fitting description The syntax is as follows a 12 3 11 5 center and width of the domain is given Sb 12 3 5 if the center of the domain is not specified current value of the variable is used If the domain is not specified the value of variable domain percent option is used domain is value of variable value of the option 100 Function types and functions Let us go back to functions Function types have names that start with upper case letter e g Linear or Voigt Functions i e function instances have names prefixed with a percent symbol e g func Every function has a type and variables b
29. ll expressions that can be used on the right hand side of data transformations can also be used in the column list Additionally F and Z can be used with dataset prefix e g info QO n l x y F x y F x Z x foo x a sin pi x ty 2 gt bar tsv Sum of fitted functions Sum Introduction The sum of functions S the curve that is fitted to the data is itself a function The value of the whole sum is computed as a sum of the functions like Gaussians or polynomials and can be given by the formula S f i where f is a function of x and depends on a vector of parameters a This vector contains all fitted parameters Because we often have the situation that the error in the x coordinate of data points can be modeled with function z x a we introduce this term to sum 5 z a gt f z 2 x a a where z a 25j zj z a Note that the same x correction z x is used in all functions fj Now we will have a closer look at f functions Every function f has a type chosen from the function types available in the program The same is true about functions z One of these types is the Gaussian It has the following formula sfr E ao exp In 2 p There are three parameters of Gaussian These parameters do not depend on x There must be one variable bound to each parameter Variables Variables in Fityk have names prefixed with the dollar symbol A variable is created by assigning a value to it e g
30. minimum length of the command defline means that the shortest version is def but defi defin and define are also valid and mean exactly the same Arguments of info command can not be shortened i e you must write i fit not i f Commands which cannot be shortened are not listed here clommands defline flit gluess ilnfo pllot slet undefline wlith 25 Appendix C License Fityk is free software you can redistribute and modify it under terms of GNU General Public License version 2 There is no warranty Text of the license is distributed with the program in the file COPYING 26 Appendix D About this manual This manual is written in DocBook XML and converted to other formats The fitykhelp xml file is distributed with the program sources and can be modified with any text editor All changes improvements corrections etc are welcome Following people have contributed to this manual in chronological order Marcin Wojdyr maintainer Stan Gierlotka Jaap Folmer Michael Richardson This version of the manual is produced from fitykhelp xml Revision 403 last modification Date 2008 03 10 16 02 28 0100 Mon 10 Mar 2008 27 Bibliography 1 William Press Saul Teukolsky William Vetterling and Brian Flannery Numerical Recipes in C http www nr com 2 Peter Gans Data Fitting in the Chemical Sciences by the Method of Least Squares John Wiley amp Sons 1992
31. n built in function Add built in function only if user defined function UDF is too slow or too limited To add a built in function you have to change the source of the program and then recompile it Users who want to do this should be able to compile the program from source and know the basics of C C or another programming language The description that follows is not complete If something is not clear you can always send me e mail etc fp you can see in fityk source means a real floating point number typedef double fp The name of your function should start with uppercase letter and contain only letters and digits Let us add function Foo with the formula Foo height center hwhm height 1 x center hwhm 2 C class representing Foo will be named FuncFoo In src func cpp you will find a list of functions FACTORY FUNC Polynomial6 FACTORY FUNC Gaussian Now add FACTORY FUNC Foo Then find another list FuncPolynomial6 formula FuncGaussian formula and add the line FuncFoo formula Note that in the second list all items but the last are followed by comma In the file src bfunc h you can now begin writing the definition of your class class FuncFoo public Function DECLARE_FUNC_OBLIGATORY_METHODS Foo 20 Using and extending If you want to make some calculations every time parameters of the function are changed you c
32. n was compiled with the GNU Readline Library command line editing and command history as per bash will be available Especially useful is TAB expanding Data and curves fitted to data are visualized with gnuplot if it is installed The GUI version is written using the wxWidgets http www wxwidgets org library and can be run on Unix species with GTK and on MS Windows There are also people using it on MacOS X have a look at the fityk users mailing list archives for details Chapter 2 Getting started The minimal example Let us analyze a diffraction pattern of NaCl Our goal is to determine the position of the center of the highest peak It is needed for calculating the pressure under which the sample was measured but this later detail in the processing is irrelevent for the time being The data file used in this example is distributed with the program and can be found in the samples directory First load data from file nac101 dat You can do this by typing G0 lt nac101 dat in the CLI version or in the GUI version in the input box at the bottom just above the status bar In the GUI you can also select Data Load File from the menu and choose the appropriate file If you use the GUI you can zoom in to the biggest peak using left mouse button on the auxiliary plot the plot below the main plot To zoom out press the View whole toolbar button Other ways of zooming are described in the section called Mouse usage If
33. nction define GaussianArea area center hwhm Gaussian area fwhm sqrt pi ln 2 center hw sum of Gaussian and Lorentzian a k a PseudoVoigt should be in one line define GLSum height center hwhm shape Gaussian height 1 shape center hwhm Lorentzian height shape center hwhm to change definition of UDF first undefine previous definition undefine GaussianArea Speed of computations With default settings the value of every function is calculated at every point Functions such as Gaussian often have non neglectible values only in a small fraction of all points To speed up the calculation set the option cut function level to a non zero value For each function the range with values greater than cut function level will be estimated and all values outside of this range are considered to be equal zero Note that not all functions support this optimization If you have a number of loaded dataset and the functions in different datasets do not share parameters it is faster to fit the datasets sequentially fit Q90 fit 1 thenparallelly fit 12 Reference Each defined simple variable slows down the fitting although this is often negligible Sum F and Z As already discussed each dataset has a separate sum i e a model that can be fitted to the data As can be seen from the formula above the sum consists of functions f and z Each dataset has two sets named F and Z containing the nam
34. nt functions sqrt exp log10 1n sin cos tan sinh cosh tanh atan asin acos erf erfc gamma 1gamma ln lgammal abs round rounds to the nearest integer two argument functions min2 max2 e g max2 3 5 will give 5 randuniform a b random number from interval a b randnormal mu sigma random number from normal distribution voigt a b see below and ternary operator condition expressionl expression2 which performs expressionl if condition is true and expression2 otherwise Conditions can be built using boolean operators and comparisions AND OR NOT gt gt lt lt or lt gt TRUE FALSE voigt a b fr ECT at The voigt function above has formula m4 eo b a t The value of a data expression can be shown using the command info see examples at the end of this section t x expression where t x y s a X Y S A gives a linear interpolation of t between two points or the value of first last point if the given x is outside the current data range All operations are performed on real numbers Two numbers that differ less than epsilonie abs a b epsilon are considered equal Indices are also computed in real number domain and then rounded to the nearest integer Transformations can be joined with comma e g Xzy Y x swaps axes Before and after executing transformations points are always sorted according to their x coordinate You can change the orde
35. of datasets and command delete n decreases it The syntax Qn dataset transformation m k dataset_transformation m k can be used to duplicate a dataset n to create new dataset as a sum of two or more existing sets n m to perform dataset transformations 8 On dataset transformation n etc A sum of datasets contains all points from all component datasets If you want to merge points with the same x value use one of dataset transformations sum same x n m Each dataset has a separate sum i e a model that can be fitted to the data This is explained in the next chapter Each dataset also has a title it does not have to be unique however When loading file a title is automatically created either using the filename or by reading it from the file depending on the format of the file Titles can be changed using the command set n title new tit le To see the current title of the dataset use info title in n It is possible to show values of a data expression calculated for each dataset Example i avg y in Q Exporting data Command info dataslot expression gt filename can export data to an ASCII TSV tab separated values file To export data in a 3 column x y and standard deviation format use info n x y s gt file Ifa is not listed in the list of columns such as in this example only the active points are exported Reference A
36. ound to its parameters To see a list of available function types use the command info types You can also use the command info typename e g info Pearson7 to see the names of the parameters default values and formulae Functions can be created by giving the type and the correct number of comma separated variables in brackets eg f Gaussian 66254 24 7 0 264 or Gaussian 6e4 ctr b c Every expression which is valid on the right hand side of a variable assignment can also be used as a variable If it is not simply a name of a variable an automatic variable is created In the last example two variables are created value 60000 and the sum The second way is to give named parameters of a function in any order e g f Gaussian height 66254 hwhm 0 264 center 24 7 Function types can can have specified default values for some parameters so this assignment is also valid sf Pearson7 height 66254 center 24 7 fwhm 0 264 although the shape parameter of Pearson7 is not given A deep copy of function i e all variables that it depends on are also copied can be made using the command function copy anotherfunction Functions can be also created with the command guess as described in the section called Guessing peak location You can change a variable bound to any of the function parameters in this manner f Pearson7 height 66254 center 24 7 fwhm 0 264 New function f was crea
37. p menu with a range of options while the left allows you to select the range to be displayed on the x axis Clicking with the middle button or with left button and Shift pressed simultaneously will zoom out to display all data On the main plot the meaning of the left and right mouse button depends on current mode selected using either the toolbar or menu There are hints on the status bar In normal mode the left button is used for zooming and the right invokes the pop up menu The same behaviour can be obtained in any mode by pressing Ctrl or Alt The middle button can be used to select a rectangle that you want to zoom in to If an operation has two steps such as rectangle zooming i e first you press a button to select the first corner then move the mouse and release the button to select the second corner of the rectangle this can be cancelled by pressing another button when the first one is pressed Chapter 3 Reference General syntax Basically there is one command per line If for some reason it is more comfortable to place more than one command on one line they can be separated with a semicolon Most of the commands can have arguments separated by a comma e g delete a b c Most of the commands can be shortened e g you can type inf or in or i instead of info See Appendix B Command shortenings for details The symbol starts a comment everything from the hash to the end of the line is ignor
38. r of points using order t where t is one of x y s a X y s a Clearly this only makes sense for a sequence of transformations joined with comma as after finishing each transformation points will be reordered again Points can be deleted using the following syntax delete index or range or delete condition and created simply by increasing value of M There are two parametrized functions spline and interpolate The general syntax is parametrizedfunc paraml param2 expression e g spline 22 1 37 9 48 1 17 2 93 0 20 7 x will give the value of a cubic spline interpolation through points 22 1 37 9 48 1 17 2 in x Function interpolation is similar but gives a polyline interpolation Spline function is used for manual background substraction via the GUI There are also aggragate functions min max sum avg stddev darea They have two forms In the simpler one aggragatefunc expression the value of expression in brackets is calculated for all points min gives the smallest value max the largest sum avg and stddev give the sum of all values arithmetic mean and standard deviation respectively True value in data expression is represented numerically by 1 and false by 0 so sum can be also used to count points that fulfil given criteria darea gives the sum of expressions calculated using formulae t x n 1 x n 1 2 where t is the value of the expression in brackets darea y gives the area under interpola
39. rtet RR ves decd PR dhe P EHE ER cis SER ven orbs te E etai 9 Sum of fitted functions eee HII EI e EIE eU EE a 10 Sum Introducti N s iere rette tete eto Pare e eR epe re eee et ev Pieve e E ERR 10 Xariables wsi zaa eee esie AAA Z AEO T E CERE R ep Y ies 10 Function types and functions iced rrr REESE EE EP MER ERRARE A REESE RREE 11 User defined functions UDF sss nemen eme emere eese eere 12 Speed Of computations css 2 eter Rete POPU RE re ORGA rte aes restare OG tees 12 Sum Band Zi iiec R EA CE eer E vssreveg rete Wor dure i wey eese ba d cres 13 Guessing peak location 24 53 iicet PO EE REESE PPE EG ERR SERE E spy RN 13 Displaying informati n oi 5 eret terere vere Exe ere TO epe E ee PEE Ee YER 14 EINE AM IE 14 Nonlinear optimization mesan esi eee eee aaa aaa anawa aaa nhe nenne ee hne ee nhe ee rhe re hne nennen nene 14 Fitting related commands 1 terrre pez ASEOS PESEE EEPE Pp eR ERR 15 Levenberg Marqu rdt uper Ie urere eel RUDI RR REIR er rr zc asd 16 Nelder Mead downhill simplex method sss 16 Genetic Algorithms iii dette Gees ves lanes Secr euge diee Rege dee dka EES 17 kun EE 17 Oth r commands cede t ERG w MUEVE EM 18 plot viewing data oce ce t be e et pb e e tege PORT ats sites ss D ZOT 18 info show information s w s GA ces Ee HERE e prex W ge ER eave don eee gees oy 18 commands dump sleep reset quit esesta neer A Eas EE meme meme men rennen 19 4
40. s sign after info The style of the formula output governed by the formula export style option can be either normal exp x 2 or gnuplot exp x 2 Peak parameters can be exported using the command info peaks in n gt filename Put the plus sign after info to also export symmetric errors of the parameters will export formulae or parameters used in all datasets to the same file It is often required to keep the width or shape of peaks constant for all peaks in the dataset To change the variables bound to parameters with a given name for all functions in F use the command F param variable Examples F hwhm foo hwhm s of all functions in F that have parameter hwhm will be equal to foo hwhm here means half width at half maximum F shape _1 shape variable bound to shape of peak _1 is bound also to shapes of all functions in F F hwhm 0 2 For every function in F a new variable is created and bound to parameter hwhm All parameters are independent Guessing peak location It is possible to guess peak location and add it to F with the command name guess PeakType x1 x2 in n e g guess Gaussian 22 1 30 5 in QO If the range is omitted the whole dataset will be searched Name of the function is optional Some of parameters can be specified with 13 Reference syntax parameter variable e g guess PseudoVoigt 22 1 30 5 center ctr shape 0 3 in 00 As an exception if
41. ted Sf center 24 8 gt h 66254 f height h gt info f f Pearson7 h _5 _3 _4 gt h 60000 variables are kept by name so this also changes f gt pl center p2 center 3 keep fixed distance between pl and p2 Functions can be deleted using the command delete function 11 Reference User defined functions UDF User defined function types can be created using command define and then used in the same way as built in functions The name of new type must start with an upper case letter contain only letters and digits have at least two characters and must not be the same as the name of built in function Defined functions can be undefined using command undefine The name of a UDF should be followed by parameters in brackets see examples Names of parameters should contain only lower case alphanumeric characters and the underscore _ and start with lowercase letter The name x is reserved do not put it into parameter list just use it on the right hand side of the definition Each parameter can have a specified default value To allow adding a peak with the command guess the default value is given as an expression which can then be calculated for a known height center fwhm and area If the name itself is one of the following height center fwhm area or hwhm default value is deduced in case of hwhm it is fwhm 2 UDFs can be def
42. ted data points and can be used to normalize the area The second form aggragatefunc expression if condition takes into account only points for which the condition is true A few examples Y 1 Y n 1 y n integrate x 1 x n x n 1 2 reduces Reference yl 1 n y n 1 two times delete n 2 yl 1 number of points delete not a delete inactive points X 4 pi sin x 2 pi 180 1 54051 changes x scale 2theta gt Q make equal step keep the number of points the same X x 0 n x M 1 x 0 M 1 Y y x X S s x X A a x X take the first 2000 points average them and substract as background Y y avg y if n lt 2000 fityk can also be used as a simple calculator i 242 44 sin pi 4 cos pi 4 1 41421 i gamma 10 362880 p examples of aggregate functions max y the largest y value sum y avg y the number of points which have y value greater than arithmetic mean y darea y normalize data area darea y F x if 20 x 25 H K H He ak There is also another kind of transformations dataset tranformations which operate on a whole dataset not single points The syntax for one dataset is 0 dstransformation 90 where dstransformation can be one of sum same x Merges points which distance in x is smaller than epsilon x ofa merged point is the average and y and sigma are sums of components avg same x The same as sum
43. text first line header Q0 lt foo dat 1 4 x y lst and 4th columns Q0 lt foo dat 1 3 4 load two dataset with y in columns 3 4 Q0 lt foo dat 1 3 5 load three dataset with y in columns 3 4 5 Q0 foo dat 1 4 6 2 load four dataset y 4 5 6 2 Q0 foo dat 1 2 3 read std dev of y from 3rd column eO lt foosdat Ost dox 0 452 Sy SELES Column Q0 lt foo raw 0 1 load two first blocks of data as one dataset Supported filetypes text ASCII format If option first line header is given the first line is read as title dbws format used by DBWS program for Rietveld analysis and DMPLOT cpi Sietronics Sieray CPI format uxd Siemens Bruker UXD format powder diffraction data Reference bruker_raw Simens Bruker RAW format version 1 2 3 canberra_mca Spectral data stored by Canberra MCA systems rigaku_dat Rigaku dat format powder diffraction data vamas VAMAS ISO 14976 only experiment modes SEM or MAPSV or MAPSVDP and only REGULAR scan mode are supported philips_udf Philips UDF powder diffraction data philips_rd Philips RD raw scan format V3 powder diffraction data spe Princeton Instruments WinSpec SPE format only 1 D data is supported pdcif CIF for powder diffraction what else would you like to have here Information about loaded data can be obtained with info data in dataslot Active and inactive points We often have the situation that only a part of the dat
44. timization is being minimized The default weights of points are not equal To see the peak parameters type info p or in the GUI move the cursor to the top of the peak and try out the context menu right button or use the right hand panel That s it To do the same a second time for example to a similar data set you can write all the commands to file you can do it now using command commands gt filename and use it as script commands lt nacl01 fit or select Session Execute script from menu or run program with the name of the script bash fityk naclO1 fit Invoking fityk On startup the program executes a script from the HOME fityk init file on MS Windows XP C Documents and Settings USERNAME fityk init Following this the program executes command passed with cmd option if given and processes command line arguments if the argument starts with gt string following gt is regarded as a command and executed otherwise it is regarded as a filename e if the filename has extension fit or the file begins with a t Fityk string it is assumed to be a script and is executed Getting started otherwise it is assumed to be a data file and is loaded It is possible to specify columns in data file in this way file xy 1 4 Multiple y columns can be specified file xy 1 3 4 5 or file xy 1 3 5 it will load each y column as a separate dataset with the same values of x Ther
45. tion of the program check if the derivatives are calculated correctly using command info dF x e g i dF 30 1 You can also use numarea findx and extremum see the section called Functions and variables in data transformation for details to verify center area height and FWHM properties Hope this helps Do not hesistate to change this description or ask questions if you have any Consider sharing your function with other users using FitykWiki or mailing list 22 Appendix A List of functions The list of all functions can be obtained using i types Some formulae here have long parameter names like height center and hwhm replaced with a Equation A 1 Gaussian 9 ca y agexp In 2 ag Equation A 2 SplitGaussian Gaussian zc ag a1 09 lt ay Y T Ag 44 do 43 Gaussian zc ag 01 03 gt a4 Equation A 3 GaussianA in ZE do Equation A 4 Lorentzian In 2 ag y y exp T do s ag 1 ex Equation A 5 LorentzianA ao T Equation A 6 Pearson VII Pearson7 y ag y 9 2 08 T Equation A 7 Split Pearson VII SplitPearson7 PearsonT zr ag a41 d9 a4 x a Y T a0 41 amp 2 43 44 45 Pearson7 x ag a1 a3 a5 gt a4 Equation A 8 Pearson VII Area Pearson7A _ aol az y 2 s 1 aal aa 3 VT 2 a i Equation A 9 Pseudo Voigt PseudoVoigt f M T aj 2 a3 y
46. version The output window which is configurable through a pop up menu shows the results Incidentally all GUI commands are converted into text and are visible in the output window providing a simple way to learn the syntax The main plot can display data points functions and or the sum of all functions Use the pop up menu click right button on the plot to configure it Some properties of the plot e g colors of data points can be changed using the sidebar One of the most useful things which can be displayed by the auxiliary plot is the difference between the data and the sum of functions also controlled by a pop up menu Hopefully a quick look at this menu and a minute or two s worth of experiments will show the potential of this auxiliary plot Configuration of the GUI visible windows colors etc can be saved using GUI Save current config Two different configurations can be saved which allows easy changing of colors for printing On Unix platforms these configurations are stored in a file in the user s home directory On Windows they are stored in the registry perhaps in the future they will also be stored in a file Mouse usage The usage of the mouse on menu dialog windows input field and output window is hopefully intuitive so the only remaining topic to be discussed here is how to effectively use the mouse on plots Getting started Let us start with the auxiliary plot The right button displays a pop u
47. xamples plot 20 4 50 10 20 show x from 20 4 to 50 and y from 10 to 20 plot 20 4 x from 20 4 to the end y range will be adjusted to encompass all data plot 10 x range will not be changed y from the lowest point to 10 prot Cal all data will be shown plot all data will be shown plot nothing changes The value of the option autoplot changes the automatic plotting behaviour By default the plot is refreshed automatically after changing the data or the sum of functions It is also possible to visualize each iteration of the fitting method by replotting the peaks after every iteration info show information First there is an option verbosity not related to command info which sets the amount of messages displayed when executing commands If you are using the GUI most information can be displayed with mouse clicks Alternatively you can use the info command Using the info instead of info sometimes displays more detailed information 18 Reference The output of info can be redirected to file using info args gt filename syntax to truncate the file or info args gt gt filename to append to the file The following arguments are recognized variables variable_name types TypeName functions function name datasets data in n title in n filename in n commands commands n m view set fit in n fit history errors in Q n formula in
48. you want the data to be drawn with larger points or a line or if you want to change the color of the line or background press right mouse button on the main plot and use Data point size or Color menu from the pop up menu To change the color of data points use the right hand panel Now all data points are active Because only the biggest peak is of interest for the sake of this example the remaining points can be deactivated Type a 23 0 lt x lt 26 0 or change to range mode press Data Range Mode button on toolbar and select range to be deactivated with right mouse button As our example data has no background to worry about our next step is to define a peak with reasonable initial values and fit it to the data We will use Gaussian To see its formula type info Gaussian or look for it in the documentation in Appendix A List of functions Incidentally most of the commands can be abbreviated e g you can type i Gaussian To define peak type 3 Gaussian 60000 24 6 0 2 F p or p guess Gaussian or select Gaussian from the list of functions on the toolbar and press the auto add toolbar button There are also other ways to add peak in GUI such as add peak mode These mouse driven ways give function a name like _1 96 2 etc Now let us fit the function Type fit or select Fit Run from the menu or press the toolbar button When fitting the weighted sum of squared residuals see the section called Nonlinear op
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