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User Manual of FitSuite 1.0.4

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1. MaxStep is the scaled maximum step length allowed in line searches in BFGS For further details see stpmx variable in the corresponding Numerical Recipes in C function Fortran subroutine Alpha ensures sufficient decrease in function value in line searches in BFGS For further details see ALF variable in the corresponding Numerical Recipes in C function Fortran subroutine TolxLnsrch gives convergence criterion on the location of the mini mum in line searches in BFGS For further details see TOLX variable in the corresponding Numerical Recipes in C function Fortran sub routine e Levenberg Marquardt method Numerical Recipes in C based section 15 5 may see also on wikipedia e Levenberg Marquardt method LMDER from the MINPACK 4 5 pack age available at http www netlib org minpack This was translated from For tran into C and modified a bit by us so it may work a bit differently than the original one e N2F N2FB N2G N2GB based on NL2SOL 6 7 from PORT package available with documentation at http www bell labs com project PORT for some reason this is often unreachable try it These were translated from Fortran into C and modified a bit by us so it may work a bit differently than the original ones Here are a lot of parameters which are not always easy to set and these methods were not tested in FitSuite extensively On 64 bit Linux it is not working at all If the result of the
2. The number 2000 is not graved in stone in literature we may find lower values This depends on the problem on the paper 13 USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil rd elements In case of such interdependency the transformation matrix technique may be useful 2 6 Fitting Without going into the details here we try to summarize the approaches used to fit experimental data sets using optimization methods enhancing the facts which may be important even for the users who sorrily do not know and maybe would not like to know the mathematical background scrupulously For further details we refer to the rich literature about this topic E g Numerical Recipes in C 15 available at http www nrbook com b Chapter 10 or as a good starting point see the optimization page on Wikipedia and the references available there We use optimization method as fitting parameters we want to find the param eters for which the fitted statistic assumes its minimum As the fitted statistics are positive definite such minimum should exist but there maybe several one and a lot of local minima Therefore finding the global minimum ma of a general function depending on a lot of parameters variables is not easy There is no per fect solution user interaction intuition is needed In case of fitting we usually have some preliminary knowledge about most of the fitted parameter values and we want just to have more accurate values and to
3. USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil rd their ordinal numbers and with different colours in the plot legend Clicking on the corresponding part of the legends the user may hide show the corresponding curves Right clicking on the plot window the user may zoom in out a selected first click on zoom in in the pop up menu and after that select with mouse rect angular region of the plot You may choose logarithmic scale for y axes but this 1s not always perfect as in Qwt it is done in a bit queer way this may be changed in next versions of FitSuite The colors can be set changed in pop up also click ing on Change Line Colors Here you can change the available colors and add new ones to the list Pushing the button Set default you save these settings on the computer in order to have the same colors when you start FitSuite next time For offspecular problems 8 we have spectrograms and contours which can be changed similarly But in these cases the levels should be set also these may be chosen to be elements of a geometric or arithmetic sequence by clicking in the menu of the corresponding dialog In case of spectrograms you may create line section graphs along the edges of a polygon by choosing from the pop up menu Polygon section pressing the right mouse button you add the first vertex moving the mouse to a new point and pressing spacebar a new vertex can be added Press ing enter or the right mouse button you fi
4. As we can examine the function only at finite number of points if the resolution of the method specified by proper options is not high enough we may even step over some minima if it is too high it may take much longer time to 14 USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil rd get there The methods were devised to make the best possible compromise but they are not perfect It is the task of the user to influence the fitting by setting the proper options of the method appropriately 2 6 1 Constraints Simple bounds As we mentioned in the beginning knowing approximately the domain in which the parameters may be found can be very helpful We can add to our optimization problem constraints of the form g P lt 0 To solve such optimization problems with constraints several methods were devised Here we do not dive into the nonlinear programming which tackles the most general problems We will show only three simple methods two of which are used currently in our program We have currently only simple bounds in which case g P P c S 0 or g P P cj 0 where c s are constants simplifying the problem further The first two methods the penalty and barrier methods have common features as in both cases the objective function is modified In both cases we replace the problem with a series of unconstrained optimization problems which should con verge to the original problem We will have a sequence o
5. Luki r 1 s T guru odis Todd A Tnu gt Tu Toda r 1 which correlates the parameters belonging to columns j 7 except of the u th column which becomes the sum of the s correlated columns The inverse of this operation is the decorrelation of parameters The latter generally is not unequivocal of course therefore user interaction may be needed thereafter in order to set the proper fit simulation parameter values and transformation matrix elements We may also split a matrix Split Mnxm Minima Mixs i112 sik j1 j2 Is 1 lt lt n l amp j amp m USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil rd E g in case of ti Tio Tis tia tis Tie tw Ta t22 lo3 124 To to 127 Split 731 t32 t33 134 135 t36 37 GE ta T Tag tas tas Tao tar Ts1 ts2 ts3 T54 Tos tse 157 2 15i 124 Tos 127 e Tis Tig Ta T34 135 T37 Tin dus Ta j Ts T54 155 Ts where the elements denoted by 7 will correspond to the matrix My s and the el ements denoted by 7 to the matrix M n k x m s The elements denoted by will be eliminated therefore information will be lost if they were not 0 Unification of two matrices can be conceived as the reverse of splitting but there we set the cross elements denoted by t to 0 Sometimes it is useful to insert a new simulation fit parameter this corre sponds to insertion of a new column in the transformation matrix E g
6. appears with an initial name Model i 0 1 in the window Problems on the left of the main window see Fig 8 You can change the name clicking on the text Model and typing the new name in this window Please do not use whitespaces space tabulator etc in names in FitSuite as it may have very queer consequences Currently only ASCII characters may be used in names Using other type of characters may have undesired side effects see model definition demo 21 USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil rd 7 Fitsuite OREP Eim Sip offspeku Model Editor m File Add Edit View Fit Simulate Plot Help D B e 90 Problems grid type ES 1 Omega theta scattered Models Data w fe1 down Default fe2 gt up Default Default fe2 gt down Default Default Object tree Correlation functions Figure 10 Model Editor the part used to choose the correlation functions 3 2 Building up the model structure On the right side of the main window Fig 8 should be seen a Model Editor now In this we can build up the hierarchical structure of the model On the top is always one object the experimental scheme We can add other objects to this and to every object with right clicking on it them and choosing Add or Insert from the arising pop up menu If there are more than one possibil ities you may choose by moving the mouse on Add In the curr
7. we know the value of the sum of some parameters but we do not know their value In such a case we may insert a new parameter which we keep constant set it to the value of the known sum and set the corresponding matrix elements properly like here 1 1 1 1 Pus pi 1 P p2 l P Ds 3 1 P Dn Thus we add a constraint for the corresponding parameters and we can eliminate a redundant fitting parameter and we do not increase the number of fitting param eters as we would using Lagrange multipliers Some parameters are integer numbers e g channel numbers switches etc These are never fitted but the transformation matrix technique is useful for them especially if some of them have the same value The integer and real number based parameters are separated in the program and on the user interface too to avoid the possibility of rounding errors because of the finite precision of the computer representation of the numbers This does not means that in some cases it would not be reasonable to mix the number types but it is safer than what we could gain allowing mixing Everything we said about the transformation matrix for real parameters is valid for integer parameters except of the fact that in this case we have integer matrix elements just because of the above mentioned considerations USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil rd 2 3 Parameter distribution In physics we often have not a single w
8. 8 Polak Ribiere method 29 30 PORT package 30 Powell s method 28 preprocessed data 9 23 24 Problems Window 21 projected gradient method 16 projection method 16 properties 3 Q quantile 11 13 Qwt library 31 R range distribution 6 reduced y 9 Reflection factor 29 repetition group ject number 22 Replace Data 24 27 replace data set 24 rescaling 18 26 Results Export 30 32 Report 26 see group physical ob Romberg s method 19 root mean square error 23 rounding 26 S Settings Editor Settings 25 26 Export Settings 31 32 Sound Settings 32 sfp file 21 simulation parameter 3 simultaneous fit project 2 fitting 3 4 8 10 problem 10 split matrix 4 26 statistic classical x 7 fitted 9 11 13 14 17 23 31 Gaussian MLE 8 Goodness Of Fit 8 9 24 Neyman s modified X 8 Pearson s y 8 Poisson MLE 8 reduced x 9 Statistical Properties 24 Stretch factor 29 submatrix 3 initial 4 subspectrum 20 synthetic dataset 12 13 31 T tolerance tolLine 29 TolxLnsrch 30 tooltip 26 transformation matrix 3 24 integer 5 28 29 48 USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil rd technique 3 Transformation Matrix Editor 26 U unify matrix 5 26 units physical 25 V View Show Open spec Files 23 49
9. Todo 34 6 Installation 36 Dl LIM 2209 oo guo ORB apon n a i gie ER DICAS a 36 02 WiINdOWS coe edos ptor be Sut OH ebe rra SM 36 7 License 36 USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil rd References Glossary Index II 38 41 45 USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil rd 1 Introduction antecedents FitSuite is an environment for simultaneous fitting and or simulation of experi mental data of vector scalar type such as parametric curves and surfaces typically collected in a physics experiment Simultaneous here means that several sets of the same type of experiment and even of different types of experiments can be simu lated fitted in a self consistent statistically correct framework with provisions for cross correlation of the theoretical and specimen parameters Experiments often provide raw data of measurements performed on the same sample by different methods and or using different experimental conditions like temperature pressure magnetic field and the like Data often partly depend on the same set of experimental and sample parameters therefore a simultaneous eval uation of all experimental data is prerequisite However data evaluation programs are dominantly organized around a single method therefore a simultaneous access to the data for a common fitting algorithm is not typical Lacking suitable pro grams some parameters are determined from one measuremen
10. determine only a few totally unknown parameters and even in that case we may have some conjecture about the range in which we should look after them Preliminary simulations may be very helpful at this stage We start the fitting from a point of the parameter space which according to expectation is not too far from the solution we are looking for If we have luck the method will find it or will get nearer to the solution The method may stuck in a local minimum or in more unlucky cases the method may become divergent or it may need further iteration to get closer to the min imum To understand these features we have to tell more about these methods It is common in all of them that they are some sort of iteration algorithms And that in each iteration step the value and or derivatives of the objective function whose minimum is to be found are calculated in discrete points of the parameter space determined by the method and it is tried to determine whether is there a minimum or where we have to take up the new points in which the objective function should be examined next If we look at the path of the iteration steps getting nearer and nearer to the current minimum we will get a curve similar ex cept of some jitter to a meandering river flowing always to lower levels It may be imagined that this path may be quite complex in higher dimensions Reaching the minimum can take a lot of time and we may stuck into local minima much easier
11. domains in off specular problem Model gt ParentObject gt LinkedObject gt Property Component E g FirstProblem gt ithLayer gt jthDomain gt size e Incase of correlation function parameters of two linked objects Model gt FirstParent gt FirstLinkedObject SecondParent gt SecondLinkedObject gt gt FunctionTypeName Property Component E g FirstProblem gt ithLayer gt jthDomain lthLayer gt mthDomain gt gt Gauss gt sigma In Parameter Editor Edit Fitting Parameters everything is included what is not integer independently on being constant or not The parameters with check 24 USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil rd IA E File Add Edit View Fit Simulate Plot Help x Problems Models Data w mms N kG kG Figure 12 Parameter Editor boxes change them by double click filled with x Linux or y Windows denote the free parameters The constants do not have check boxes thus they can not be freed There are calibration constants user defined constants and model defined constants Calibration constants have a label ca instead of check box They may be fitted right clicking on the check box or selecting the appropriate calibration constants SHIFT cursor right clicking and selecting Let Not Be Constant from the arising menu The user may set arbitrary parameters to be constant by choosing in this menu Let Be Constant If
12. for us and our funding agencies to keep track of the distribution Source code of the routines of the theories are already included in the binaries The program uses 3rdparty libraries Qwt5 LAPACK TSFIT these could be installed by the user They are included in the directory 3rdParty only in order to make the installation of FitSuite 1 0 4 easier Qwt has its own license a bit modified version of GNU LGPL given in the file 3rdParty qwt COPYING for further details see that TSFIT may be distributed under GNU GPL see 3rd Party TSFIT COPYING For LAPACK see 3rdParty lapack 3 1 0 COPYING it 1s not GPL but something like that 37 USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil rd References 1 H Spiering L De k L Botty n EF FINO Hyp Int 2000 125 197 204 doi 10 1023 A 1012637721433 2 K Kulcs r D L Nagy L P cs in Proc Conf on M ssbauer Spectrometry Dresden 1971 3 T Hauschild M Jentschel Comparison of maximum likelihood estimation and chi square statistics applied to counting experiments Nucl Instr and Meth A 2001 457 384 401 doi 10 1016 S0168 9002 00 00756 7 4 J More B Garbow K Hillstrom User Guide for MINPACK 1 Technical Report ANL 80 74 Argonne National Laboratory 1980 5 J More D Sorenson B Garbow K Hillstrom The MINPACK Project in Sources and Development of Mathematical Software editor W Cowell 1984 88 111 6
13. is made This is quite oversimplified just to explain the use of GLimit For further details see the routine mnbrak in Numerical Recipes in C Fortran e Nelder Mead method Numerical Recipes in C based section 10 4 A good description with animation is available on wikipedia Nelder Mead method is sensitive to scaling of the parameters It has the following pa rameters and options appearing on the interface if Details button is pressed down Maxlter as in Powell s method tolerance as in Powell s method Initial simplex size is a parameter determining the length of the vec tors pointing to the vertices of the simplex from the initial parameter vector consisted only the free components Reflection factor see Numerical Recipes in C Stretch factor see Numerical Recipes in C Contraction factor see Numerical Recipes in C 20 USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil rd e Polak Ribiere and Fletcher Reeves methods Numerical Recipes in C based section 10 6 and on wikipedia They have the same options and parameters as Powell s method e Broyden Fletcher Goldfarb Shanno on wikipedia variant of the David son Fletcher Powell method Numerical Recipes in C based section 10 7 may see also on wikipedia It has the following parameters and options appearing on the interface if Details button is pressed down Maxlter as in Powell s method
14. is pressed down They have the following parameters 28 USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil rd MaxlterLine is the maximum number of iteration steps in the 1 di mensional optimization method if a minimum was not found in so many steps the current line optimization is finished The main method may continue its own iteration The fitting is not finished just because MaxlterLine was reached tolLine is the fractional tolerance trine with which the minimum along the given direction should be found by brent or dbrent method Opti mization along a direction is finished if 2trinelz gt xo x where zo is the true local minimum and z is the calculated one GLimit In the optimization methods Powell Fletcher Reeves and Polak Ribiere the minimum of the function f which in case of fit ting of experimental data sets is the x is searched along directions specified by them The first step to find a minimum along a direc tion is to find three points a b and c where b is between a and c fur thermore f a and f c are both greater than f b The three points are searched by starting from an initial triplet moving similarly to an inchworm i e updating a b c by a b b c c u where u is the minima of the parabolic fit on a fla b f b oc f c This type of move is accepted only if the obtained u is between c and Uim b GLimit c b otherwise the a b b c c uj move
15. is the middle of the distribution range distribution range is the width of the histogram outside of which the approxi mated distribution is assumed to be negligible EFFI Environment For Fitting is the name of the program written by Hartmut Spiering experimental scheme contains the information necessary for description of the system consisted of the experimental apparatus es about the experimental method performed with them and of the system under study e g a measured sample see subsection 2 1 extraction type spec file is a structure which may be defined by the user used to collect the information needed to extract a scan from a spec file into a FitSuite data set feasible region in case of optimization problems with constraints is the region in the parameter space determinded by the constraints where the optimum should be found feasible set see feasible region 41 USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil rd fitted statistic In case of simultaneous fitting we can have experimental data with different distributions therefore the statistics used to fit for each fit ting problem may be different We fit a resulting statistic their weighted sum This is not a problem as if we start from the MLE from which all of them are derived we would obtain also such a resulting statistic In the case of the resulting statistic the names like classical x Pearson s x etc will not have any meaning ther
16. iteration is not what you like you can get back the state before the iteration by clicking Fit Simulation Revert Before a fit is started the project is saved in the file sfp which is loaded on the command Revert From version 1 0 4 the results are not written in files automatically They may be exported using the menu items Results Export You may choose the data 30 USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil rd which should be exported and set the format of the created files using Settings Export Settings 3 9 Error calculation Errors of free parameters may be calculated clicking Fit Simulation Calculate Errors There are currently two approaches used in the program for this purpose One is based on covariance matrix the other is named bootstrap method be aware that bootstrap method is very expensive in computation time Explanation of the principles of these method can be found in Numerical Recipes in C available at http www nrbook com b section 15 6 You may choose one of them or both by proper settings after clicking menu item Fit Simulation Select Method on the page with title Parameter Errors Here you may also change the required con fidence level Still in the same dialog on page with title Bootstrap method you can set the parameters used by bootstrap method namely the number of synthetic data sets and the convergence criterion According to the literature the number of synthetic data se
17. the parameter was not a calibration constant the check box will replaced by c these are the user defined constants There may be constants which are never fitted these are denoted by label cn these are the model defined constants see free fix make constant pa rameter demo To increase the transparency of the parameter list the user may hide parameters which s he thinks have the correct value and will not be fitted This can be done similarly to previous operations just a different menu item should be chosen For obvious reasons free parameters may not be hidden and hidden pa rameters may not freed The hidden parameters may be seen pressing the proper button or menu item appearing after parameters were hidden The hidden param eters will appear with a different background color This color may be changed in the Editor Settings Settings Editor Settings see hidding parameters demo From version 1 0 3 the parameters may have units In older project files they will appear only after the command Edit Regenerate Matrices was given for the program Sorrily with this the transformation matrices changed by the user will be lost and should be made again The units may be changed several ways Just 25 USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil rd double clicking on the unit in the parameter editor the unit may be changed If the button with an arrow is pressed down not only the unit is changed but the par
18. they are not independent we will have a TE components are given as Tis P R 4 5 0 5 A D i 1 n 5 In general case having distributed parameters we may fit R P and the his togram i e the set h These can be handled as additional fitting parame ters In order to have an appropriate result we have to take into account additional constraints for the histogram It is obvious that we may assume that R gt 0 Air in 2 0 and hj 1 There are several approaches applying fur ther constraints in order to get appropriate results for the parameter distributions 19 20 21 22 23 One of them 23 which is also used in FitSuite currently is the maximum entropy principle The entropy from information theory is defined as S So hilnh 6 h gt 0 6 USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil rd We try to fit our parameters with the constraint that S should assume its maxi mum For further details we will return to this topic in subsection 2 6 2 Another nontrivial question is how the objects corresponding to histogram ele ments contribute to the resulting spectrum how they should be taken into account The most simple approach which currently is also used by FitSuite is based on the assumption that the resulting sub spectrum y belonging to the lowest rank submodel which still contains the physical object objects if the distributed pa rameter is a correlated on
19. with m constraints B P 5 In g P 26 and B P Y cy r 1 2 3 27 i 1 In contrast with the penalty method the fitting program using the barrier method may not get out of the domain defined by constraints This is not quite the case working numerically This feature may be especially useful if the calculated spectrum as a mathematical function of parameters has a finite domain out of which we may get into unpredictable problems The third method handling the constraints is quite different We use our fit ting method for problems without constraints but we check in each iteration step whether we are out from the feasible region If we are out we continue with the nearest parameter vector on the boundary on the feasible set and if the method uses also gradient we continue with the component of the gradient projected on the boundary This method is almost the projected gradient method see e g 14 but may also be used in case of optimization methods without derivatives This projection method is much faster than the barrier or penalty method but it may have its own problems If some constraint cuts through a bend belonging to the path obtained connecting the steps of the optimization method we may get an artificial local minimum on the boundary where the method may stuck see Fig 4 In case of a very complex meandering path lot of constraints parameters to fit the number of such artificial local minima is
20. 07 19 10 2756 2779 doi 10 1162 neco 2007 19 10 2756 15 W H Press S A Teukolsky W T Vetterling B P Flannery Numerical Recipes in C 2 ed Cambridge University Press 1992 http www nrbook com b 16 W H Press S A Teukolsky W T Vetterling B P Flannery Numerical Recipes in C 2 ed Cambridge University Press 1992 683 http www nrbook com b 17 W H Press S A Teukolsky W T Vetterling B P Flannery Numerical Recipes in C 2 ed Cambridge University Press 1992 140 http www nrbook com b 18 W H Press S A Teukolsky W T Vetterling B P Flannery Numerical Recipes in C 2 ed Cambridge University Press 1992 689 699 http Iwww nrbook com b 19 B Window M ssbauer spectra of Invar alloys J Phys E Sci Instrum 4 401 402 20 J Hesse A Riibartsch Model independent evaluation of overlapped M ss bauer spectra J Phys E Sci Instrum 1974 7 526 532 21 G Le Ca r J M Dubois Evaluation of hyperfine parameter distributions from overlapped Mossbauer spectra of amorphous alloys J Phys E Sci Instrum 1979 12 1083 1090 22 I Vincze Fourier evaluation of broad Mossbauer spectra Nucl Instr and Meth 1982 199 247 262 23 R A Brand G Le Ca r Improving the validity of Mossbauer hyperfine pa rameter distributions The maximum entropy formalism and its applications Nucl Instr and Meth B 1988 34 272 284 24 E T J
21. J E Dennis Jr D M Gay and R E Welsch An Adaptive Nonlinear Least Squares Algorithm ACM Trans Math Software 1981 7 348 368 7 J E Dennis Jr D M Gay and R E Welsch Algorithm 573 NL2SOL An Adaptive Nonlinear Least Squares Algorithm ACM Trans Math Software 1981 7 369 383 8 L De k L Botty n D L Nagy H Spiering Yu N Khaidukov Y Yoda Perturbative theory of grazing incidence diffuse nuclear resonant scat tering of synchrotron radiation Phys Rev B 2007 76 224420 doi 10 1103 PhysRevB 76 224420 9 B Effron Bootstrap Methods Another Look at the Jackknife The Annals of Statistics 1979 7 1 1 26 10 B Effron The jackknife the bootstrap and other resampling plans Society of Industrial and Applied Mathematics 1982 CBMS NSF Monographs 38 11 B Effron RJ Tibishrani Bootstrap Metods for Standard Errors Con fidence Intervals and Other Measures of Statistical Accuracy Stat Sci 1986 1 54 77 12 B Effron R J Tibishrani An introduction to the bootstrap Chapman amp Hall CRC 1993 38 USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil rd 13 B Winkler Bootstrapping Goodness of Fit Statistics in Loglinear Pois son Models Collaborative Research Center 1996 386 Discussion Paper 53 http epub ub uni muenchen de 1449 14 C J Lin Projected Gradient Methods for Non negative Matrix Factorization Neural Computation 20
22. The vector statistics defined by 16 20 should be modified similarly to 28 i e we should replace in these equations K by 1 sim ste Ls S i 1 N GD 17 USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil rd The penalty or barrier function can be added the same way too In order to fulfill automatically the constraints R gt 0 h 2 0 and Y hj i 1 we fit instead of these parameters 77 me a j and calculate from these R and h using the formulae Ri Ary 32 h 2 T MEUM 33 gt Um o d which give the definition of 7 parameters as well This is working correctly but sometimes we may experience a bit different behaviour fitting these distribution parameters compared to the normal parameters 2 6 3 Rescaling parameters The expected order of magnitude may also be a helpful information during opti mization fitting as there are methods most of them which do not work properly they may become even divergent for parameters of different order of magnitude see fit using rescaling demo In these cases we can rescale the parameters P by dividing each Pj by the corresponding order of magnitude m and optimiz ing the modified objective function f P f P according to the rescaled parameters Pm P m The parameter bounds give the order of magnitude only if the signs of the upper and lower bounds are identical that is the main reason why the magnitude should be provi
23. User Manual of FitSuite 1 0 4 SAJTI Szil rd July 18 2009 USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil rd Contents 1 Introduction antecedents 1 2 Basic concepts of FitSuite 2 2 1 Experimental scheme and its Structure 2 2 2 Transformation matrix technique oss eo om mms 3 2 3 P rameter distribution lt A eS A L 6 AUI Statisties s uos 9 wn qox A ANA Roe RR ad oe E 2 5 Error estimation bootstrap method 11 26 EIUS cud s dedo a a as 14 2 6 1 Constraints Simple bounds 15 2 6 2 Distributed parameters using maximum entropy principle 17 2 6 3 Rescaling parameters ax ag x 490 RA 18 2 6 4 Numerical derivatives 18 2 7 SUBSPECHUIN ici AAA 20 3 Working with FitSuite 20 3 1 Starting anew Project auos s ae A C9 oed 21 3 2 Building up the model structure 22 2 9 Addn UNA ra A A EM kzt en RO 9 t 23 3 4 Changing parameters matrices ss eR o Ro Ro Rx s 24 35 Report generator 6 ck cca eh hehe eb eGR RS 26 20 Conine ses e eU b n BSN Sx HS A 27 24 Model groups lt ses ca 4 28 Sus debo REARS SE ca 27 3 8 Simulation Fit 28 3 9 EttorcGulculatli n 21 222220 x93 a A 31 3 10 Calculating statistics scenic UR OY s 31 OIL PHOEBE 2 222224 EGER e 9 9 vox 31 S12 SOUS AAA AAA AA RS 32 3 12 Peas sa 2d ok 9 30x oO Re RS CR a a 33 4 Sources documentation 34 5
24. a col ored map Usually the color ranges are not appropriately chosen but you change as it is written in the section 3 11 Under Windows and sometimes under some Linuces for some unknown reason there may be a segmentation fault if you calculate in too much points Therefore loading the off specular problems set n omega n theta sca etc to smaller values e g 200 and 200 or smaller 33 USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil rd STI C oo0TSimultana sip o File Add Edit View FivSimulate Plot Help D B OF OQA Problems x ModAsPrepared Plot ModHe11 Plot Models Data w cud type ModAsPrepared Data0 1 grazing 1e 06 ModHeti Datal 1 grazing ModHe74 Data2 1 grazing 700009 100000 n ModHe150 Data3 1 grazing i4 z 10000 o o 10000 1000 1000 ETTTTTTTTTTTTTTTTTTTTTTTTTTTI ETTTTTTTTTTTTTTTTTTTTTTTTTTTT 0 2 4 B 10 1214 0 2 4 6 B 10 12 14 Angle mrad Angle mrad ModHe74 Plot ModHe150 Plot theory gt data theory data 1e 06 100000 100000 10000 10000 1000 YA 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 Angle mrad Angle mrad Simulation time 0 08s ModHe74 Figure 16 Fitted X ray reflectometry spectra e OffspecularResonantXRay is a project simulating off specular problem for synchrotron radiation It is a recently added problem It takes a long time even to simulate fit has not been made tested e miscellaneous some simple funct
25. ameter value is also converted from the previous unit to the new one Editing a parameter value with units the unit may also be changed In this case pressing down the button with arrow the new unit and value is set there is no conversion If the button is not pressed down the unit remains the original one but the value will correspond to the selected value and unit converted to the original one see parameter values and units demo You may change the units of the minimum maximum and magnitude the latter is used to rescaling values as well If these units are identical with the unit of the parameter value they are represented by shortcut The user is able to specify a bit the behaviour of unit editor in Settings Editor Settings see editor settings demo You can get some information about the parameters by first clicking Help What is this or pressing SHIFT F1 on this the cursor icon should change to a question mark clicking thereafter on the parameter name in the editor a short help should appear Presently this type of help is not complete for some problems e g stroboscopic mode problems there is nothing available The displayed numbers in Parameter Editor and in Transformation Matrix Editor Edit T Matrices also are rounded to a few digits If a number is longer than that the rounded number is displayed in blue or other user set color and we can see the real not rounded value by pulling the mouse over that cell in the e
26. aynes Information Theory and Statistical Mechanics Phys Rev 1957 106 620 630 39 USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil rd 25 E T Jaynes Information Theory and Statistical Mechanics IT Phys Rev 1957 108 170 190 26 R Lieu Maximum entropy data analysis another derivation of S x J Phys A Math Gen 1988 21 L63 L65 27 R Lieu R B Hicks C J Bland Maximum entropy in data analysis with error carrying constraints J Phys A Math Gen 1987 20 2379 2388 40 USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil rd Glossary big O notation is used to give the order of magnitude e g O h should be read as order of h check box is a graphical user interface element widget that permits the user to make multiple selections from a number of options Normally check boxes are shown on the screen as a square box that can contain white space for false or a tick mark or X for true see wikipedia correlation is an operation transformation matrix used to eliminate the redundan cies arising because of common model parameters see eq 1 its inverse is the decorrelation see subsection 2 2 decorrelation is the inverse of correlation which is not unequivocal therefore further user interaction may be needed see subsection 2 2 degree of freedom DOF is the number of data points minus the number of fit ted parameters distribution midrange
27. cal system in cases different from M ssbauer spectroscopy even if the whole spectrum cannot be obtained as weighted sum of single contributions subspectra see subspectrum demo 3 Working with FitSuite In the following we try to show the features the usage of the program in an or der as a new user should go step by step through the different interfaces of the 20 USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil rd SOS Ofisp_sim stp offspeku Model Editor y File Add Edit View FilSimulate Plot Help DAB 07 Problems Models Data w 5 E Motel ye E sel nchRefiOfs Synch Offspec refi ze m e scattered E Domain A su ite defaultSite n e ny_n e i Layer_group 10 Layer wavelength Poincare v Figure 8 Model Editor the part used to build up the structure represented by a tree program 3 1 Starting a new project Starting the program the user can start a new project File New Project or load File Open a previously saved one The extension of the project files is sfp simultaneous fit project We can save our project anytime clicking File Save If we have a new empty project we have to add models theories in EFFI terminology This can be done in two ways The user can load it from a mod file if there is such available e g such files may be created by exporting or you can click Add New Model and then can choose from a list The model
28. change it if it is possible The iteration fitting can be started by clicking Fit Simulation Fit It is pos sible to fit only a single model fitting problem or models of a model group choosing the proper menu items of Fit Simulation Fit Only Choosing the menu item Fit Simulation Select Method you can select the fitting method and set their parameters At present we have the following methods e Powell s method which is a slightly modified version of the code available in Numerical Recipes in C available at http www nrbook com b section 10 5 The method has parameters options which appear if the Details button is pressed down These options are the following Maxlter is the maximum number of iteration steps if a minimum was not found in so many steps the optimization is finished tolerance is the tolerance fij with which the minima of the function f is determined The result f of the n iteration step is accepted if foi fn 1 MS gt i fa Powell s method uses line minimization methods as Golden section search see in Numerical Recipes in C available at http www nrbook com b section 10 1 and or on on Wikipedia Brent s method brent Brent s method using derivatives dbrent see in Numerical Recipes in C section 10 2 and 10 3 respectively The user may be choose one of them They and their pa rameters appear on the interface if the second 1 dim optimization method Details button
29. dat 4k5t dat in directory adatok etc We can give the line with which the reading begins First line to read in the numbering starts with 0 the number of lines we want to read in if it is O then it will read all and we may give a string this of course could be a number until whose first occurrence after the line which was given in First line to read in we want to read in the lines Presently the program does not read parameters constants from data file In compact format we can add the number of data columns in the above mentioned example this is 5 as the first column 0 5 10 gives the number of data in former lines see reading ASCII data file demo Scans from Certified Scientific Software s spec M X Ray Diffraction and Data Acquisition software files can be obtained in another way from version 1 4 1 on Open spec file clicking File Open spec File wereafter a window should appear where you may filter select the scans and extract the chosen data sets If the required spec file has already been opened it is enough to choose the menu item View Show Open spec Files to have this window Extraction types may be defined changed For further details see this spec file demo The experimental data have some distribution E g data obtained using parti cle counters usually have Poisson distribution Ordinary experimental data are expected to have Gaussian distribution Fitting the experimental data we have to kno
30. ded by the user separately 2 6 4 Numerical derivatives Usage of methods using or not using derivatives is another question we should ad dress a bit Mathematicians usually assume that we have objective functions whose derivatives are known therefore most of the optimization method uses them as well The problem is that in practice we usually do not have the deriva tives in case of complex problems as we do not have an analytical function but we have algorithms which may use a lot of numerical iterative algorithms al ready e g determining eigenvalues in quantum mechanical problems as M ss bauer line positions and line strengths calculated from Hamiltonian and we have a lot of parameters Therefore calculating the derivatives requires tremendous additional work for which we usually do not have time and it may not be worth ei ther Still we should provide the derivatives for the methods requiring it therefore we should do it numerically The problem is that because of the finite computer representation of the numbers and because it needs dividing of a number with a 18 USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil rd small number numerical differentiation is dangerous therefore it is avoided in numerical computations whenever possible The method implemented by us uses the same trick as the Romberg s method for integration see 17 http www nrbook com b section 4 3 or wikipedia to get high precision
31. derivatives We know that f xo h f xo hi af f xo 4 1 therefore using the notation h f ro h d TT iiis Di flay EE ug 3 Trin d 2 FF 0 OP where we use the big O notation O h should be read as order of h We can eliminate the higher order terms as in Romberg integration E g AJ to h f xo h _ f xo 2h f xo 2h 2h Ah D f zo 3 35 _ AD f o Dof ro _ d a sg HS FOU Generally we may use c k D f 2 4D f a Don f xo d f xo O h 36 4k 1 du In order to calculate with accuracy O h we need 2k function evaluation simu lation f zo 2h f xo 2 1h f xo 2 h f zo 2 h for deriva tives and one to get f xp which may be slow the fitting very much If we fit sev eral parameters and therefore we have to calculate several partial derivatives the program will slow even further And even then in general case we cannot guaran tee that the function has not a very great high order derivative which deteriorates everything We may check the convergence comparing different order of approxi mations E g we may accept and stop calculating approximations of higher order if Dg f zo Dk f zo lt Dzf zo ID f zo 37 where 0 lt C lt 1 is a small number giving the user required precision It is useful to have a maximum for the order of approximations as numerically we cannot take h arbi
32. ditor and waiting until it appears in a tooltip The user is able to specify the number of the displayed digits choose the precision what he needs and a lot of other options in Settings Editor Settings In current version the matrices can be united split and the parameters can be correlated decorrelated and user defined parameters may be inserted see subsec tion 2 2 and the demos of parameter correlation decorrelationl decorrelation2 and matrix split unite The handling of integer parameters and the related integer transformation ma trices may be handled quite similarly The main difference is that there we have to use the Integer Parameter Editor Edit Integer Parameters and similarly the Integer Transformation Matrix Editor Edit Integer T Matrices 3 5 Report generator On clicking Results Create Report appears a window with a report of the current project containing the model structure and the model parameter values The report can be saved in an html file This feature is still in a very early development stage 26 USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil rd Pisae threespectra_FIt Stp Edit Transformation Matrices ON File Add Edit View Fit Simulate Plot Help DgBgeu oo Problems x Change Models qP iData0 1 drive EFG H_external Model CLONE2 Data2 1 drive H internal UnCategorized Datal 1 drive baseline broadening calibration geomet
33. e can be obtained as Yres h ny Po in p x 7 Jis n Of course other expression can be conceived and put into the program if some physical reason can be attributed to it yes may be and is used as an intermediate result if it belongs to a submodel a part of a model which itself is a model too The number of histogram elements will be II therefore it is not too ad visable to increase the number of distributed parameters too much As fitting too much parameter is always a danger but fitting too less may also be dangerous in case of a complex distribution where the spectrum depends on the distributed parameter strongly If some of them are independent we may gain a lot as e g for independently distributed parameters we will have only gt gt V additional parame 2 ters to fit because of the histogram elements Before showing how the maximum entropy principle is used we will see the minimized functions during fitting in absence of distributed parameters 2 4 Statistics We usually mean by fitting parameters finding the parameters for which the clas sical x statistic is minimal This function is given as exp _ theo V 2 x p gt y Xi p 8 i where y and yiheo are the experimental and theoretical values for the i th data point and o is the standard deviation error of measurement of th data point X is the independent variable it may be a vector Fitting simultaneously w
34. e minimize the weighted sum of the x s of the different fitting problems The x is not the single statistic which may be used for this purpose e g see absolute deviation on wikipedia Furthermore its use is justified only for normally Gauss distributed experimental data In the problems handled by FitSuite currently we have data obtained by particle counters which have usually Poisson distribution 7 USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil rd For large count rates there is no problem the Poisson distribution can be approx imated well with Gaussian distributions but in other cases we have to use other statistics in order to fit parameters There are several approaches to tackle this problem 3 There are approximations based on 8 and on the fact that the vari ance and the expectation value of the Poisson distribution is the same These are of the form of classical x namely Pearson s x exp theo 2 2 u yp xo p X earson s p eo 9 Pearson s P yte xi p modified Neyman s x ys y x p XNeyman s D 10 max y 1 In unmodified Neymann s x statistic y appears instead of max y 1 Furthermore there are other statistics based on Maximum Likelihood Estimation namely Poisson MLE 2 exp theo exp ga Xi p XPMLE 2 25 y y Xi P M Y ine ee y gt 11 i ys 0 obtained by using MLE for Poisson distributed data And Gaussian MLE e
35. e values are badly measured The exclusion is not working correctly for all the models at the moment do not use it in case of M ssbauer and stroboscopy spectra From version 1 0 3 on you may exclude data points according to their values as well The user may also add his er notes to the data after clicking on button Notes at the bottom of the data window The above mentioned statistical properties of the data set can be changed click ing on button Statistical Properties at the bottom of the data window Here you can choose the distribution of the data set Set the normalization factor if there is such one preprocessed data Besides the user may choose the statistic whose minimum has to be found by fitting the parameters and the GOF goodness of fit statistic For further details see 2 4 3 4 Changing parameters matrices From version 1 0 3 the parameters and transformation matrices are generated au tomatically With Edit Regenerate Matrices you may generate them if there is some problem or you want to set the transformation matrices starting from the initial ones The parameters and the matrices may be changed with Edit For the program generated parameter variable names the following name convention is used e For the parameters of simple physical objects ModelName gt ObjectName gt PropertyName ComponentName E g FirstProblem gt SecondIncoherentFraction gt ExternalMagneticField x e Incase a linked objects e g
36. echnicque are needed to calculate the spectra Optimally the number of simulation fit parameters is less than the number of model parameters as already some redundancy is eliminated by proper choice of the transforma tion matrix see subsection 2 2 spec file is a file format of Certified Scientific Software s spec X Ray Diffrac tion and Data Acquisition software used for experimental data in ESRF and many other places submodel is a model which is part of another model it represents a physical system to which we can relate intermediate results sub spectra which are used calculating the spectra measured in the experiment subspectrum see subsection 2 7 43 USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil rd tooltip The tooltip is a common graphical user interface element It is used in conjunction with a cursor usually a mouse pointer The user hovers the cursor over an item without clicking it and a small hover box appears with supplementary information regarding the item being hovered over see wikipedia transformation matrix is a matrix technique used to eliminate the redundancy arising because of common parameters and or to take into account linear interdependencies of the parameters see subsection 2 2 transformation matrix split is an operation transformation matrix used to split a submatrix into two in order to have smaller more transparent submatrices which build up the sparse transformation mat
37. efore in the program it is referred to just as the Fitted Statistic goodness of fit statistic is a statistic measuring the goodness of a fit see sub section 2 4 GUI Graphical User Interface integer parameter Some parameters are integer numbers these are not fitted and are handled separately from real number based parameters in order to avoid rounding errors model is an object to which belong algorithms for calculation of characteristic spectra see subsection 2 1 model parameters are all the parameters which are needed by the models in or der to calculate the spectra without using transformation matrix technicque Usually there are a lot of common parameters wherefore transformation matrix technicque is used see subsection 2 2 object The word object is used in several sense in this text including its everyday meaning too It may mean e in most of the cases means a physical object or concept see the corre sponding item in the glossary and subsection 2 1 e a program language concept used in Object Oriented Programming objective function is the function whose location of minimum and or maximum is to be found by the optimization method optimization method is an algorithm used to find the location s where a given function assumes its minimum or maximum parameter distribution see subsection 2 3 parameter insertion inserts a new simulation fit parameter inserting a new col umn in the transformati
38. ell defined physical object but rather an ensemble of them Even in this case it may be useful to represent them with a single object in our computer model but we have to know that which parame ters are the same for the members of the ensemble and which are different We can group the objects according to these parameters The parameter distribution fa p will give the probability that the ensemble has objects which can be differ entiated from other members according to the parameter set pes which is part of the set of all the parameters p As usually this distribution is unknown we have to determine it by fitting too There is no sense in defining the distributions on the level of model parameters as it would be to complex and would require an enormous administration therefore we will have Pf from which p can be calculated using the transformation matrix and f p T P P F P P Practically as we usually do not know the analytical shape e g Lorentzian Gaus sian etc of the distribution either and even if we know it in general case it may not be easy to use it for our calculations we fit histograms with finite resolution and finite range which is divided up equidistantly around the midrange E g for a single distributed parameter P with resolution M range R and midrange 7 we will have histogram values h for the parameter values P R 2 i j 0 N 1 4 If we have n distributed parameters and
39. ent built in types of models only the layers may be grouped Selecting them with SHIFT cursors or SHIFT mouse and right clicking on selection you can choose Group from the pop up menu In case of the objects representing the groups the column labelled with Nrep contains the repetition number telling us how many times the elements of the group are repeated in the real physical system This could be changed with right clicking on it in previous versions Now you can change it setting the corresponding integer parameter see later In case of off specular synchrotron Mossbauer reflection problems we have to give the correlation functions between the domains belonging to different lay ers The domains available at the level of scheme can be linked to the layers The correlation functions can be chosen with the help of graphical user interface which appears after clicking on tab with label Correlation function on the bot tom of the model editor see Fig 10 22 USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil rd 3 3 Adding data We can import from files the experimental data selecting the menu item Add Data after this we can choose the format of the file e g one column two columns three columns compact I name so as I could not find better name the format 0 68348 68699 68315 68375 68253 5 68198 68508 67983 68114 68041 10 67436 67776 68123 68143 68480 E g see the files 4kOt dat 4k3t
40. er space which contains the parameter vectors providing fits with the same quality for some pos sible outputs of our experiment s according to the experimental results and our knowledge about the precision of the measurement Therefore using the set of fitted parameter vectors of synthetic data sets we may get a probability density histogram f p corresponding to possible experimental errors Knowing f p we can calculate the expectation value and the standard deviation of fitted param eters This standard deviation can be used as an error estimate The bootstrap method needs a lot of 2 2000 synthetic data sets therefore it may be very expensive in computation time It is advisable to calculate errors only when we have a quite good fit For usability conditions and further details of the bootstrap method see the cited works It is inappropriate to give just the errors for a given confidence level as we may never know when somebody will need our data with higher our lower confidence levels In that case it is very useful to have the errors for quantile 1 which is the sample standard deviation as thereafter the errors knowing the degree of freedom can be calculated for arbitrary confidence level easily The covariance matrix has other uses than the error estimation We can dis cover interdependencies between the parameters looking for off diagonal ele ments with large gt 0 absolute values compared to the corresponding diagonal
41. f objective functions of form Ay P f P a2Z P qx dk 17 0 qk oo 21 where P is the original objective function Z P is the penalty function and q is the monotonically increasing divergent sequence of penalty coefficients If M is the feasible region in the parameter space given by the constraints the penalty function should be of the form 0 PeM zP 4 2 P M Pu Two often used examples for such penalty functions with m constraints Z P max 0 91 P m P 1 1 1 1 2 3 Q3 and m Z P Y max 0 g P 1 m r 1 2 3 24 i 1 Solving the modified optimization problems consecutively each time starting from the solution of the previous modified problem we may get close to the real solu tion of the original constrained problem With barrier method we have a sequence of objective functions of form H P FP w B P 0 lt Wk lt Wri wj 0 25 15 USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil rd where f P is the original objective function wx is the sequence of monotonically decreasing barrier coefficients converging to 0 and B P is the barrier function growing to oo on the boundary of M Because of this property of the barrier function it is useful in case of g P lt 0 constraints but numerically as we have always a finite precision using computers there is not much difference between g P lt 0 and g P 0 Two often used examples for such barrier functions
42. freedom which is the number of data points minus the number of fitted parameters usually is called the goodness of fit As a rule of thumb we may say that models with Q lt 0 001 are likely wrong To be honest we never saw such a good fit for spectra fitted by the program X ray reflectometry is very far from that The M ssbauer spectra are nearer but still far below this value so for them it should be taken more seriously The probable problem with X ray reflectometry is that our theoretical models still neglect some properties of the physical system e g the material inhomogeneities For data with Poisson distributions this value can be much lower Models with small Q values may be accepted only if we know the reason The X ray reflectometry is a good example for this Very good fits Q z 1 are also suspicious as they usually arise if the experimenter overestimated her is error or made something we would never assume of anyone As another measure of goodness of fit often the reduced or relative x statistic is used X a 2 A 14 Xreduced x2 V 9 USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil rd As a rule of thumb x2 ace should be close to 1 for good models for M ssbauer spectra this is usually true for X ray reflectograms not This depends strongly on the degrees of freedom This rule is based on the fact that the x statistic has a mean v and standard deviation 2v and for la
43. hat physical characteristics the method in question is sensitive to such as domains atomic groups lattice sites molecules etc E g in M ssbauer related problems each layer may contain several sites with their own hyperfine theories to calculate the corresponding subspectra The original idea of cross correlation and of hierarchy of theories as well as a number of subroutines of FitSuite were inherited from EFFI Environment For USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil rd Fitting 1 an originally M ssbauer spectroscopy related Fortran program de veloped over the years by Prof Hartmut Spiering from University of Mainz In view of the friendly and fruitful collaboration between the Budapest and Mainz groups over the last decades Prof Spiering kindly agreed to build in the theo ries written by him and tested within EFFI into FitSuite in order to promote both projects Enlightening discussions with him greatly promoted the FitSuite project Although in the last three years the two projects developed in different directions we are very grateful for Prof Spiering s essential contribution to FitSuite In the following we go through the basic concepts used by FitSuite first Thereafter we give a short description of the GUI Graphical User Interface how FitSuite should be used from start to fit and present shortly some examples which can be found in the directory examples 2 Basic concepts of FitSuite In
44. have the results of experiments performed in a bit different environment external field tempera ture etc and or different type of experiments using the same sample Therefore there are a lot of common parameters To eliminate this type of redundancy and as it is also convenient for the user to use as few parameters as possible as it is more transparent for human and easier to fit in a parameter space with lower dimension at least if we want to get correct results transformation matrix tech nique is used 2 For this we need also parameter vector array Because of these considerations we have to generate the parameter vector and the initial transfor mation matrix from the object tree structure mirroring the real physical system The model parameters which still contain all the redundancy can be collected in a vector p P1 P2 Pn Where p is the vector containing all the parameters belonging to the 7 th model in the current simultaneous fit project Let denote the vector of the fitting or if you like simulation parameters with P and the transfor mation matrix with T The transformation matrix technique uses the expression p TP where dim P lt dim p Above it was mentioned that this technique is used in order to eliminate the redundancy arising because of the common param eters but this is not the unique reason We can take into account some possible linear relations between the parameters also which is a redundancy to
45. here the data usually are preprocessed E g in case of neutron reflectometry the experimentalists normalize the results as they prefer to plot reflection and not count rate The problem with this approach is that cal culating the 9 12 statistics their value will not be the correct one In case of the classical x statistic defined by 8 this is not a problem as it is enough just to normalize o as well We are able even to fit as for that it is enough to know the location of the minimum The value of the statistic gives some information about the quality of the fit If we just fit the value of the statistic this question is not as interesting as in case of parameter error estimation or hypothesis test Without knowing the correct value the error estimations hypothesis test will be incorrect Therefore we need the raw unpreprocessed data set or at least the parameters used during preprocessing in order to be able to calculate the correct statistic e g in the above mentioned example we need the normalization factor With the above mentioned Goodness Of Fit statistic Sgor we may check whether the used model is correct or not For x statistic this can be done by calculating the probability Q x2 v that the observed x exceeds the fitted value x2 even for a correct model is the incomplete gamma function oo Qxzis V m f etiz udi 13 NIY laa 2 Xmin where I is the gamma function The number v is the degrees of
46. ions added just to test the fitting routines 4 Sources documentation The sources containing the Fortran subroutines Cfunctions used for simulations with the libraries created from them can be found in directory Repositories Their documentation and the files used as help in the Parameter Editor are also there These are still not complete The experimental data files are collected in adatok and the project files in examples 5 Todo e complete program documentation User manual 34 USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil rd EGE sim sip offspeku Plot 57 o E s E E g a E Figure 18 Off specular synchrotron Mossbauer reflection simulation write a separate GUI for creation of new or modification of model types let be possible to copy models and their parts the same way as we copy directories and files on GUI automatic names GUI extend plot features GUI selection of parameters GUI extend report generator log file when the independent variable is generated from a parameter the plot of the data set should be replotted also test methods adjust default parameters values write some usage notes More validators needed to do not allow illegal names parameter name ob ject model matrix name Have a directory for log file outputs and temporary work files 35 USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil rd 6 Ins
47. is test 8 9 I import data set 23 incomplete gamma function 9 Initial simplex size 29 initial submatrix 4 Insert 22 inserting parameter 5 integer parameter 5 transformation matrix 5 Integer Parameter Editor 26 midrange distribution 6 MINPACK 30 MLE see maximum likelihood estima tion Gaussian 8 Poisson 8 model 3 group 27 28 32 parameter 3 structure 22 26 Model Editor 22 multiplier Lagrange 5 N name convention 24 Nelder Mead method 29 neutron reflectometry 9 Integer Transformation Matrix Editor 26Neyman s modified y 8 L Lagrange multiplier 5 Let Be Constant 25 Let Not Be Constant 25 Levenberg Marquardt method 30 linked object 24 M machine precision 19 magnitude 18 26 matrix operation 26 split 4 unification 5 maximum entropy 6 7 likelihood estimation 8 Maxlter 28 30 MaxlterLine 29 MaxStep 30 mean square error 23 NL2SOL 30 normalization factor 9 23 24 normally distributed 10 Notes for dataset 24 Nrep 22 O objective function 15 17 18 optimization method 3 P parameter common 3 4 10 correlation 4 27 decorrelation 4 distribution 6 7 17 fitting 3 hidden 25 insertion 5 26 integer 5 magnitude model 3 18 26 47 USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil rd name convention 24 rescaling 18 26 simulation 3 Parameter Editor 24 Pearson s y 8 penalty coefficient 15 function 15 16 18 physical object 2 group 22 units 25 Poisson MLE
48. ivatives of the fitted statistic In this case the derivatives 11 USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil rd Figure 2 Error estimation for two free parameters As it can be seen the parabolic approximation is valid only in domain U where the parabolas I 2 are good ap proximations of the sections of the XP P gt with planes parallel to P and P respectively The true error for the given confidence level is given the bounding rectangle ABC D of V which is the corresponding elevation line The confi dence intervals are P PP and PP PF should be calculated numerically which may have quite unacceptable errors in some regions of the parameter space see subsection 2 6 4 Another approach is the bootstrap method 9 10 11 12 13 see section 15 6 of 18 using synthetic data sets generated by Monte Carlo methods We may generate synthetic data sets from the measured data sets e randomly erasing some data points or e randomly replacing some data points this can be used only if we have sev eral measurements for the same independent variable with another mea surement value or e replacing the measured values y with y generated as it would have been drawn randomly from a bootstrap sample whose mean is y and its elements are distributed according to the distribution assumed for the 12 USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil rd experime
49. multiplied Therefore we should be cautious using constraints Although the usage of the penalty and barrier meth ods is a bit safer in this regard but this problem may also arise there To use the penalty or barrier function method in case of vector statistics defined by 16 20 should be modified during fitting E g in case of penalty function we should replace in these equations x by 1 k alk 4 EZ P i 1 n 28 n 16 USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil rd I B Figure 4 Demonstration of artificial local minimum arising because of constraint on a contourplot where the contours denote decreasing levels approaching the real minimum C Without the bound represented by line I for parameter P the fitting started from A would be finished in C but because of bounds it will be finished in B 2 6 2 Distributed parameters using maximum entropy principle To fitting problems with distributed parameters we minimize the fitted statistic X and maximize the entropy S of the distribution therefore we minimize x S As we prefer positive definite objective function f remember on vector statistics defined in 16 20 instead of this we minimize N J x Sa S XHn N Y hilnh 29 i 1 h gt 0 as S is maximal if h x If we have distributed parameters we sum the en tropies of the m independent distributions m Nj p ye In Ny gt hji ln hji 30 E
50. nish the polygon In the profile window you may need to choose logarithmic scale The created line section profile may not be appropriate if the axes belonging to the two independent variable have very different scale or you have a polygon with too much vertices When the user would like to have an image file of the data and the fitted curve we recommend gnuplot or Origin etc From version 1 0 4 the results are not written in files automatically if it is not set so in Settings Export Settings They may be exported using the menu items Results Export You may choose the data which should be exported and set the format of the created files using Set tings Export Settings Clicking on the menu items Plot Plot you can choose the model or model group whose simulation fit result s you would like to plot Plot Plot All replots all the simulation fit results if you closed the plot windows before but this works only if there was a fresh simulation fit Plot Close All closes all the currently open plot windows 3 12 Sounds Currently after the fit was finished a wav sound file is played If you do not like this just select another file or switch it off using Sound Settings Settings Sound Settings On Linux systems no sound is played 32 USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil rd 3 13 Examples Presently there are only a few example project files They can be found in the directory examples whe
51. ntal data Therefore we may have to know besides y all the other parameters on which the distribution depends E g in case of normally distributed data we have to now the deviation i e the errors of the data but y is enough in case of Poisson distributed data as that already determines unequivocally the distribution The synthetic data sets which in principle could also be the results of the exper iments performed with the known errors are fitted The results of the fit of a synthetic data set is stored only if it was convergent and the corresponding fit ted statistic differs from the fitted statistic of the experimental data set less than a user defined small positive number We filter out this way the synthetic data sets which give a very different fitted statistic because they miss already too much information compared to the real experimental data or because the data points synth 2 were not changed really randomly e g C y becomes unaccept ably large because of a synthetic systematic error It may also happen that the fit has gone wrong unexpectedly and we do not want to throw out everything just because a few such bad fits If the fitted statistic of a synthetic data is less than the value used in the filtering criterion that value is updated with it Therefore the bootstrap method may also be used as some sort of fitting method Applying this algorithm we survey the basin in the paramet
52. o It is advisable to take into account that there are parameters which according to expectations will not have interdependencies and therefore the T matrix can be block diagonalized It is more transparent to handle submatrices with lower dimensions than one extended sparse matrix Therefore we have to categorize the parameters according to our expectation whether the subspace stretched by a subset of them may have interdependencies or this is very unlikely If the user USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil rd finds a case where our expectations are not met s he is able to unite or split the submatrices thus our choice here is not a constraint The initial submatrices generally are identity matrices but not always E g the thickness of a multilayer sample will be the sum of the layer thicknesses in M ssbauer spectroscopy in a doublet site the line positions and the measure of the splitting and the isomer shift will not be independent etc Before going further we have to mention a few concepts related to this trans formation matrix technique used for simultaneous fitting and simulation In or der to eliminate redundancy arising because common parameters we often use the operation Cor Maxm Mnx m s41 1 amp ji amp m UELI1 92 ys where M sm denotes n by m matrices defined by Corr T uE ida Js s Ti aii Tic Ty deii sa Ti u 1 2 Tij Ti u r s 1 Ti EDS Tod T det E Trui gt Tij
53. on matrix see subsection 2 2 42 USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil rd parameter vector is a vector array consisted of the simulation fit or model pa rameters as components penalty function method is a method used for optimization problems with con straints modifying the objective function by adding penalty functions out sied of the feasible region see subsection 2 6 1 pop up menu is a menu in a graphical user interface GUI that appears upon user interaction such as a right mouse click see wikipedia property Properties in FitSuite represent the physical quantities thickness rough ness hyperfine field susceptibility EFG effective thickness etc and some numbers characterizing the experimental scheme e g number of channels symmetries of the sites etc see subsection 2 1 reduced x is the x divided by degree of freedom see eq 14 repetition group of physical objects is a group of physical objects e g layers which more accurately the same sort of objects are repeated in the same order several times repetition group number shows how many times the elements of the repetition group are repeated in the real physical system root object is at the root of the object tree structure It is analogous to the root main in some operating systems directory in the filesystems used in com puting simulation fit parameters are all the parameters which using transformation matrix t
54. ponent of this vector should give the contribution of the th data point to the corresponding statistic i e X M KF Xo DKF Qu p amp x p amp x p 15 E g in case of classical X statistic defined by 8 we will have exp theo Yi Y Xi P ix p SUD 16 0 in case of Pearson s x statistic defined by 9 exp theo green p Y a xi p 17 yi Xi p in case of modified Neyman s x statistic defined by 10 exp theo coe x p Yi Yi xi p 18 max y 1 10 USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil rd in case of Poisson MLE statistic defined by 11 PMLE J 2 Vui ye xi p if yj 0 Ki Xi p v2 ex ex meg Xi ex ya ur ln g an p if v 0 19 and in case of Gaussian MLE statistic defined by 12 K GMLE 2 exp theo 2 theo exp 2 MEN Xi i5 i E yO iP y p OPP e eo I m 20 y xn p ci ci 1 1 where c a 2 5 Error estimation bootstrap method Error estimation of the fitted parameters pg is also a complex issue The usual procedure to obtain the errors is based on the fact that for x statistics the er ror of parameters with confidence level c and degree of freedom M can be ob tained looking for the minimal hyperrectangle with edges parallel to the unit vectors belonging to parameter components containing the hypervolume V PpIX p x pa y where y is the c quantile belonging to chi s
55. problem with this If the program does not find some of the libraries available here set using the command export LD_LIBRARY_PATH prefix FitSuite 1 0 4 lib SLD LIBRARY PATH use full path If other li braries are missed by the program please check whether they are installed on your system etc The binary created for X86 64 of course will not run under 32 bit systems do not try to run binaries created for later Suse Linux versions in earlier distributions as probably the required system gcc libraries may be too old 6 2 Windows Download the setup file fitsuite 1 0 4 Win32 exe from this ftp site and start it 7 License FitSuite is a scientific software provided free as it is under the terms of GNU GPL license except for one additional condition should you use FitSuite to any ex 36 USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil rd tent for your publication you should cite the article on FitSuite currently available at http arxiv org abs 0907 2805v1 by mentioning also the name of the program and the version number e g FitSuite 1 0 4 The above link will be updated when the regular journal article will be out Please use to Forum to disseminate the bibliographic data of your published papers that make use of FitSuite Fitsuite uses the open source version of Qt4 Qt4 licensing Source code of the main program is available on personal request to the author szilard Ormki kfk1 hu since it is essential
56. quare dis tribution with degree of freedom M A c quantile 0 lt c lt 1 of a cumulative distribution function F x is y if F y c As the fitted parameters ps are ob tained with minimization of x therefore their vector will be part of V For further details see chapter 15 in 15 This approach is also valid for statistics 9 12 for reasons see section 2 4 in 3 To simplify the procedure further it is often assumed that the fitted statistic as a function of fitted parameters has a parabolic profile in V and therefore knowing the second derivatives of the fitted statistic the parameter errors dp can be ob tained by solving a second order equation gt gt ij AijPi0p Y Sorrily in case of nonlinear problems the assumption about parabolic profile is usually correct only in a small part of V in neighborhood of ppt and not in the whole V see Fig 2 Therefore we may estimate the errors quite inaccurately using this method If it is applicable we may also use the fact that A is with good approximation the inverse of the covariance matrix 16 This may have advantages in case of some methods where this matrix is calculated in each iteration step There is another possible source of error using this approach if we do not have as is the case mostly the analytical derivatives according to the fitted parameters of the theoretical function used to model the problem which is needed to calcu late the second order der
57. rd Gaussian 7 8 13 Poisson 8 13 midrange 6 range 6 DOF see degree of freedom Dynasync 1 E Edit Fitting Parameters 24 Integer Parameters 26 Integer T Matrices 26 Regenerate Matrices 24 25 T Matrices 26 EFFI 1 entropy 6 7 17 maximum 6 7 error estimation 9 11 13 experimental data 7 9 13 23 31 type 23 excluding bad data points 24 experimental data 2 23 31 error 7 9 13 23 31 preprocessed 9 23 24 scheme 2 3 22 extraction type 23 F factor Contraction 29 Reflection 29 Stretch 29 file extension mod 21 sfp 2 First line to readin 23 Fit Simulation Calculate Errors 31 Fit 28 Fit Only 28 Force Simulation 28 Force Simulation of 28 Revert 30 Select Method 28 31 Simulate 28 Simulate Only 28 FitSuite 1 Fitted Statistic 9 11 13 14 17 23 31 fitting parameter 3 problem 2 7 8 10 17 28 Fletcher Reeves method 29 30 Fortran 2 29 30 34 free parameters 25 31 function correlation 22 24 incomplete gamma 9 objective 14 15 17 18 penalty 15 16 18 G Gaussian MLE 8 GLimit 29 gnuplot 31 32 GOF statistic see Goodness Of Fit statistic Golden section search 28 Goodness Of Fit statistic grid type 28 group model 27 28 32 physical object 22 repetition see group physical ob ject number 22 Group Model s 28 GUI 2 8 9 24 46 USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil rd H Help What is this 26 hide curve 32 parameter 25 hypothes
58. re we can find several subdirectories In each one we can find project files saved at different phase of the project definition process The files ending with e Simulation sfp usually contain simulations no data e Fit sfp contain fits with data sets e Simultan sfp contain several data sets fitted simultaneously In the directory e XrayReflectionFePd we can find projects for non resonant X ray reflectom etry experiments ResonantXraySelfDiffusionFePd and XrayRes_NoResRelectionFeCr we can find projects for resonant and non resonant X ray reflectometry experiments MossbauerSpectraFemtz we can find projects for a M ssbauer experiment The project containing simultaneous fitting problems contains three spectra for which the internal magnetic field Hint and the effective thicknesses of sites 52 and s3 and a few other parameters were chosen to not to be corre lated during cloning SMRStroboscopicMode we can find a project for Stroboscopic M ssbauer simulations SynchrotronTransmission we can find projects for synchrotron transmission experiments SynchrotronTimeDifferential we can find projects for a time differential re flection experiment NeutronReflection there are two specular simulation projects CrFeSimu lation sfp and FePdSelfDiffusionSimulation sfp and two off specular sim ulations CrFeOffspecularSimulation sfp CrFeOffspecularSimulationModi fied sfp In off specular case the 3dimensional results are plotted in
59. rge v becomes normally distributed As the above mentioned statistics also have asymptotically x distribution these rules of thumb may be useful for them and their weighted sum but we should be cautious If we start from the maximum likelihood estimation in case of a simultane ous fitting problem we will get that the weights of the statistics with which they are summed up in order to get the common resulting statistic s should be 1 But sometimes it may be useful to have the possibility to change these weights This may be useful especially when we are still far from the minimum Usually it is not a good idea to start fitting all the problems at once if we are far from the true minimum Initially it is worth to fit the most simple and most error free prob lem which can be simulated fast and promises to get good preliminary results for the common parameters easier And only if we reached using this simpler fit ting problem an acceptable result should we continue the fitting adding the other fitting problems This way we can progress faster still having the simultaneous fitting which is the single correct way for evaluation of spectra where we have measured the same sample with different methods and or under different condi tions and so forth Some fitting methods do not require these statistics or their sums directly In stead they require a vector k p and maybe its derivatives according to the param eters The square of i th com
60. rix see eq 2 and the second paragraph of subsection 2 2 transformation matrix unification is the reverse of split the cross elements are set yo zero whitespace characters used to represent white space in text see wikipedia 44 Index mod 21 sfp 2 A Add Data 23 New Model add data to model model 21 Alpha 30 21 23 B bootstrap method bound 15 18 brent 28 29 Brent s method 28 29 using derivatives 28 29 Broyden Fletcher Goldfarb Shanno BFGS method 30 12 13 31 C C 14 28 30 34 c 25 C 30 ca 25 y classical 7 modified Neyman s 8 Pearson s 8 clone 27 cn 25 coefficient penalty 15 common parameter 3 4 10 compact format 23 confidence interval 12 level 11 13 31 constant 24 25 calibration 25 model defined 25 user defined 25 constraint 5 7 15 16 18 Contraction factor 29 convergence criterion 30 31 correlate 26 parameters correlation function 22 24 covariance matrix 11 13 31 4 27 D data file formats compact 23 one column 23 spec 23 three column 23 twocolumn 23 data set replace 24 synthetic 12 13 31 Davidson Fletcher Powell DFP method 30 dbrent 28 29 decorrelate 26 parameters 4 degree of freedom 8 11 13 derivative 11 12 14 16 18 19 of fitted statistic 11 displayed number distributed parameter distribution 23 data set 23 26 6 7 17 45 USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil
61. ry intensity nonlinearity orientationEFG position source thickness gt set gt channel_vO ModelO set v max ModelO set channel vO ModelO set channels vmax Modelo CLONE1 Modelo CLONE2 set channel vO Model 0 set v max Model0 gt set gt channel_v0 Model0 gt set gt channels_vmax Model CLONE1 set v max Model0_CLONE1 gt set gt channel_v0 Model0_CLONE1 gt set gt channels_vmax Model0_CLONE2 gt set gt v_max Model0_CLONE2 gt set gt channel_v0 Model0_CLONE2 gt set gt channels_vmax e i 00000 Figure 14 Transformation Matrix Editor 3 6 Cloning It happens frequently that the user wants to fit the same type of experiment in a bit different environment or a bit different sample etc It would be inconvenient to build up almost the same model several times and then to correlate almost all the parameters Therefore in FitSuite we can clone the models This can be done by just right clicking on the model name in the window Problems and selecting Clone from the pop up menu Thereafter a dialog arises in which we can choose the number of clones and the parameters and or matrices which are not to to be correlated After this we will have copies of the chosen model and of the data belonging to it These data may be replaced by right clicking on them in the window Problems and choosing Replace Data from ari
62. sing pop up menu 3 7 Model groups The user may group models from version 1 0 3 The model groups are used just to select a few models from the available ones in the current project in order to simulate fit plot only them To create model groups just select them in the window Problems with SHIFT cursor right click with mouse and in the arising 27 USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil rd pop up menu select Group Model s In the dialog showing up thereafter the user may choose the models to be grouped and the name of the group according to which we can use them later on see modelgroup demo 3 8 Simulation Fit The simulation can be started by clicking Fit Simulation Simulate The program checks whether there was a former simulation and the parameter values were changed or not and calculates only when it is necessary It may happen that this program decision was not appropriate In such cases you may force simulation by clicking Fit Simulation Force Simulation It is possible to simulate only a single model fitting problem or models of a model group see subsubsection 3 7 choosing the proper menu items of Fit Simulation Simulate Only and Fit Simulation Force Simulation of The independent variable of the simulation fit may be specified by setting properly in the Problems window on the left side in the column with name grid type if there is a possibility Just click on the proper cell and
63. site for fitting for checking theory or measurement etc In FitSuite the experimental scheme is built up of objects which correspond to physical objects or concepts e g source sample detector layer site etc The experimental scheme is also an object It has a simple tree structure with one root USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil rd object the experimental scheme To these objects belong properties which repre sent the physical quantities thickness roughness hyperfine field susceptibility EFG effective thickness etc and some numbers characterizing the experimental scheme e g number of channels symmetries of the sites etc Besides objects and properties to each object may belong algorithms for cal culation of the characteristic spectra In the current built in problems only the experimental schemes and the sites have algorithms These type of objects are called model on the program interface as it is an almost perfect name for them This type of structure is needed because simulating our problem writing the simulation algorithms this structure mirrors the real physical system and therefore it is a practical logical choice 2 2 Transformation matrix technique In contrast to the last remark of the previous section the optimization methods used for fitting require a parameter vector and not an object tree structure with properties Furthermore in case of simultaneous fitting we usually
64. t assumed error free and kept constant when evaluating other experiments an obviously incor rect approach Besides for different methods different programs are used which makes it very difficult to tune parameters of such theories and their errors and cor relations to each other and to extend or modify the theories to describe different experimental data Therefore starting in 2004 as part of the Dynasync FP6 project sponsored by the European Commission and since 2006 within the NAP_VENEUS project sponsored by the Hungarian National Office for Research and Technology we developed FitSuite for Windows and Linux a code that consistently handles by now data of over ten spectroscopic methods with over twenty theories together with a large number of sample structures in a common inter related framework FitSuite is an environment in which besides the possibility of adding brand new user written theories the user can build new theories based on the com bination of existing ones subroutines which call each other To our opinion this feature of FitSuite is really essential since a complex physical system can in general be divided into subsystems and even this subsystem can be divided into further subsystems groups of which can be described with the same parameters and physical equations To provide an example assume we have a thin film built up from layers and the layers may be built up from further object depending on to w
65. tallation The current version was tested only under openSuse Linux 11 1 32 and 64 bit and under Windows XP The program is mainly written and tested under Linux therefore the Windows version sorrily is still much more error prone In principle the program can also be compiled for other Linux Unix distributions earlier Linux Windows versions and Mac 6 1 Linux Download the setup file fitsuite 1 0 4 LinuxDistribution architecture sh from this ftp site This is a Self Extracting Tar GZip compressed packages needs bin sh tar and gunzip for extracting and you may need to set the file permis sion in order to be allowed to execute This shell script may have the follow ing arguments prefix full path to directory were FitSuite should be installed help After starting the script from a terminal will ask whether you accept the license or not and that Do you want to include the subdirectory fitsuite 1 0 4 LinuxDistribution architecture For the last question answer boldly n o as the program will be in prefix FitSuite 1 0 4 If the keyboard repeat rate is too fast it may happen that the program does not react properly and you cannot install it in that case change your personal keyboard configuration Start the program fitsuite 1 0 4 available in this directory I tried to include all the non stan dard libraries needed by FitSuite these are installed in prefix FitSuite 1 0 4 lib hopefully there will no
66. this section we try to make the reader acquainted with the basic concepts used in the program which are necessary to know in order to be able to use it Here we summarize the principles The description of the user interface is given in the next section there we will see how these concepts principles are used in practice To be able to simulate fit we need to give the program our knowledge of the problems In FitSuite we should create simultaneous fit projects first which have file extension sfp which contain the fitting problems consisted of experimen tal data and of experimental scheme 2 1 Experimental scheme and its structure The experimental schemecontains the information necessary for description of the system consisted of the experimental apparatus es about the experimental method itself and of the system under study e g a measured sample We know that the meaning of the word experimental scheme is a bit different from the present usage as it usually contains only the apparatuses but we did not found a better one In our thoughts we usually separate the apparatus used for observation and the subject of observation Here we do not want to do this as it would lead a more complex structure and the experimental scheme selects the characteristic properties features of the studied system and of the apparatus which are essen tial to be able to calculate the theoretical pair of the experimental data set which is prerequi
67. trarily small at least not without extra work and computation time which is the cost of using a library using numbers with arbitrary precision The most plausible choice for h would be xo e for zo 4 0 which is defined as 1 e being the smallest number which may be differentiated from 1 for a given machine precision but the user may have other choice 19 USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil rd counts 10 5 0 1 5 velocity mm s Figure 6 Two subspectra s 52 the total calculated M ssbauer spectrum and the corresponding data set 2 7 Subspectrum Sometimes it is useful to see what is the contribution of the parts of the studied system to the whole spectrum Therefore M ssbauer spectroscopists devised the concept of subspectrum In M ssbauer spectroscopy the contributions of different sites atomic environments of the M ssbauer isotope of the sample are added up weighted by their concentration calculating the absorption coefficient used to determine the spectrum of the system Therefore plotting spectrum and subspectra Fig 6 i e the spectra calculated taking into account only one or a few but not all of the sites we may see which site is corresponding to a specific peak etc The subspectrum in FitSuite is a spectrum where some of the physical ob jects of the studied system are not taken into account This concept may also be helpful in better understanding of the studied physi
68. ts should be at least about a few thousands the current de fault value is 200 as we used most only for testing and because of the slowness of this method The convergence criterion gives the criterion to stop the fitting of a synthetic data sets if the difference of fitted statistics belonging to the real experimental and the current synthetic data sets is smaller than this Bootstrap method needs errors of the experimental data without that will not work ap propriately If the experimental data has Poisson distribution it is enough just to set the distribution properly 3 10 Calculating statistics to be written 3 11 Plotting Originally it was planned to use gnuplot for plotting of the results That way we could have more beautiful and appropriate press ready graphs The problem is that it is a bit circumstantial to get control over the plot windows created by gnuplot There is a solution but only for X systems and that would not be portable Therefore we use the open source plot library Qwt which is based on Qt This can be integrated in FitSuite and developed further without problems It is not as beautiful as gnuplot but it should be enough during fitting or simulations Presently the Qwt based plot windows are created if they are not available already when an iteration is started The data and the results of simulations be fore and after each iteration are plotted The theoretical results are represented by 31
69. w which distribution should be used as the fitted statistics should be cho sen accordingly The type of the distribution can be chosen here or later If the imported data contains errors too we may set its type root mean square or mean square as well Sometimes the raw data is already preprocessed E g neutron reflectometrists usually normalize their data as they measure reflection which has a maximal value of 1 The problem with this preprocessing is that in case of Poisson distribution we throw out this way information see subsubsection 2 4 Therefore here you can tackle this problem by providing the normalization factor if there was such one This piece of information may be provided later as well Right clicking in the window Problems on the name of the new dataset we can add the data to the chosen model At present the user should know which format 23 USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil rd is required accepted by a given type of model If you choose a wrong one it will warn you only with malfunctioning or with segmentation fault error message Data set may be replaced right clicking on the name of data set and clicking on Replace Data in the arising menu This is useful if there was a mistake made by the user choosing reading the data set or after cloning see later Some data points may be ex in cluded from the fit selecting with SHIFT cursor and right clicking This can be used if you are sure that som
70. x eo 2 eo ex 2 E i y x p In y xa p i in ci 12 XGMLE x i y xi p 5 7 exp 1 1 where c y MEC obtained by using MLE for normally Gauss distributed data and used for Poisson distributed data applying the substitution 0 y x p also used for 9 The detailed considerations leading to these statistics and their usage can be found in 3 Here we restrict ourselves to mention a few additional facts about them These statistics all follow asymptotically as the number of data points more accurately the degree of freedom DOF goes to oo a x distribution For Poisson data zy should be used during fitting But according to the tests avail able in 3 this is not the appropriate Goodness Of Fit statistic hypothesis test for that Pearson s Should be used In case of simultaneous fitting we can have experimental data with different distributions therefore the statistics used to fit for each fitting problem may be different We fit a resulting statistic their weighted sum This is not a problem 8 USER MANUAL OF FITSUITE 1 0 4 July 18 2009 SAJTI Szil rd as if we start from the MLE from which all of them are derived we would obtain also such a resulting statistic In the case of the resulting statistic the above mentioned names generally will not have any meaning therefore in the program it is referred to just as the Fitted Statistic There are experiments w

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