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4b23a517 | 1 | |
2 | #include "AliTMinuitToolkit.h" | |
3 | #include <TNamed.h> | |
4 | #include <TVirtualFitter.h> | |
5 | #include <TH1F.h> | |
6 | #include <TH2F.h> | |
7 | #include <TF1.h> | |
8 | #include <TFormula.h> | |
dc697472 | 9 | #include <TVector.h> |
10 | #include <TMatrix.h> | |
4b23a517 | 11 | #include <TMath.h> |
12 | #include <TString.h> | |
13 | #include <TROOT.h> | |
14 | #include <TCanvas.h> | |
15 | #include <TRandom.h> | |
16 | ||
17 | ||
dc697472 | 18 | //-------------------------------------------------------------------------------------- |
4b23a517 | 19 | // |
20 | // The AliTMinuitToolkit serves as an easy to use interface for the TMinuit | |
21 | // package: | |
22 | // | |
23 | // - It allows to fit a curve to one and two dimensional histograms | |
24 | // (TH2F::Fit() only allows to fit a hyperplane). | |
25 | // - Or n points can be specified directly via a n x 2 matrix. | |
dc697472 | 26 | // - An option for robust fitting with non-linear functions is implemented. |
305a4a95 | 27 | // |
28 | // A small example illustrating the usage of AliTMinuitToolkit is given in the function | |
29 | // "AliTMinuitToolkit::Test()". | |
4b23a517 | 30 | // |
31 | // | |
32 | // 1. Setting the formula: | |
33 | // | |
34 | // The formula is simply set via "void SetFitFunction(TFormula * formula)". | |
35 | // | |
36 | // | |
37 | // 2. Adding the data points | |
38 | // | |
39 | // - In order to fit a histogram, use "void FitHistogram(TH1F * his)" or | |
40 | // "void FitHistogram(TH2F * his)". The fitter is started automatically | |
41 | // - Alternatively, the direct specification of the points is possible via | |
42 | // "void SetPoints(TMatrixD * points)". Note, that the each point | |
43 | // corresponds to one row in the matrix. The fitter is then started with | |
dc697472 | 44 | // the command "void Fit()". The weight of each point can be specified |
45 | // with an n-dimensional vector using "void SetWeights(TVectorD * weights)". | |
4b23a517 | 46 | // |
47 | // | |
48 | // 3. Accessing the fit results | |
49 | // | |
50 | // The N parameters of the formula are stored in a N-dimensional vector which | |
dc697472 | 51 | // is returned by "TVectorD * GetParameters()". In a similar way the covariance |
4b23a517 | 52 | // matrix of the fit is returned via "TMatrixD * GetCovarianceMatrix()" which |
53 | // is of the type N x N. | |
54 | // | |
55 | // | |
56 | // 4. Non-linear robust fitting: | |
57 | // | |
58 | // Even a few outliers can lead to wrong results of a least-squares fitting | |
59 | // procedure. In this case the use of robust(resistant) methods can be | |
60 | // helpful, but a stronger dependence on starting values or convergence to | |
61 | // local minima can occur. | |
62 | // | |
dc697472 | 63 | // The robust option becomes active if EnableRobust(true, sigma) is called. It is |
64 | // very much recommended that a normalization value (scale variable) corresponding | |
65 | // to an expected deviation (sigma) is specified via | |
66 | // "EnableRobust(Bool_t b, Double_t sigma)". | |
67 | // | |
68 | // Performing the fit without knowledge of sigma is also possible if only | |
69 | // "EnableRobust(true)" is activated, but this is NOT RECOMMENDED. | |
70 | // | |
71 | // The method is based on another estimator instead of chi^2. For small deviations | |
72 | // the function behaves like x^2 and for larger deviations like |x| - the so | |
73 | // called Huber estimator: | |
74 | // | |
75 | // h(x) = x^2 , for x < 2.5*sigma | |
76 | // h(x) = 2*(2.5*sigma)*x - (2.5*sigma)^2 , for x > 2.5*sigma | |
77 | // | |
78 | // If a weighting function is specified in addition, a second run with the ordinary | |
79 | // metric is started, but before entering the iteration every point is weighted | |
80 | // according to its distance to the outcoming function of the first run. The weighting | |
81 | // function w(x) must be defined on the intervall x in [0,1]. w(0) then | |
82 | // corresponds to the weight of the closest point and w(1) to the point with the | |
83 | // largest distance. | |
4b23a517 | 84 | // |
85 | // Some standard weighting functions are predefined in | |
86 | // "SetWeightFunction(Char_t * name, Float_t param1, Float_t param2 = 0)": | |
87 | // - "BOX" equals to 1 if x < param1 and to 0 if x > param1. | |
88 | // - "EXPONENTIAL" corresponds to "Math::Exp(-TMath::Log(param1)*x)" | |
89 | // - "ERRORFUNCTION" corresponds to "TMath::Erfc((x-param1)/param2)" | |
90 | // | |
dc697472 | 91 | // |
92 | // REFERENCE for non-linear robust fitting: | |
93 | // Ekblom H. and Madsen K. (1988), Alogrithms for non-linear Huber estimation, | |
94 | // BIT Numerical Mathematics 29 (1989) 60-76. | |
95 | // internet: http://www.springerlink.com/content/m277218542988344/ | |
96 | // | |
97 | // | |
98 | // 5. examples: | |
99 | // | |
100 | // A small example illustrating the working principles of AliTMinuitToolkit is given | |
101 | // in the function "AliTMinuitToolkit::Test()". | |
102 | // | |
103 | // | |
104 | // | |
105 | // Comments and questions are always welcome: A.Kalweit@gsi.de | |
106 | //-------------------------------------------------------------------------------------- | |
4b23a517 | 107 | |
108 | ||
109 | ClassImp(AliTMinuitToolkit) | |
110 | ||
111 | AliTMinuitToolkit::AliTMinuitToolkit() : | |
112 | TNamed(), | |
113 | fFormula(0), | |
114 | fWeightFunction(0), | |
115 | fFitAlgorithm(0), | |
116 | fPoints(0), | |
dc697472 | 117 | fWeights(0), |
4b23a517 | 118 | fParam(0), |
119 | fParamLimits(0), | |
120 | fCovar(0), | |
da7c9ae3 | 121 | fChi2(0), |
122 | fMaxCalls(0), | |
dc697472 | 123 | fPrecision(0), |
124 | fUseRobust(0), | |
125 | fExpectedSigma(0) | |
4b23a517 | 126 | { |
127 | // | |
128 | // standard constructor | |
129 | // | |
130 | fMaxCalls = 500; | |
131 | fPrecision = 1; | |
dc697472 | 132 | fUseRobust = false; |
133 | fExpectedSigma = 0; | |
4b23a517 | 134 | } |
135 | ||
136 | ||
da7c9ae3 | 137 | AliTMinuitToolkit::AliTMinuitToolkit(const AliTMinuitToolkit&) : |
138 | TNamed(), | |
139 | fFormula(0), | |
140 | fWeightFunction(0), | |
141 | fFitAlgorithm(0), | |
142 | fPoints(0), | |
dc697472 | 143 | fWeights(0), |
da7c9ae3 | 144 | fParam(0), |
145 | fParamLimits(0), | |
146 | fCovar(0), | |
147 | fChi2(0), | |
148 | fMaxCalls(0), | |
dc697472 | 149 | fPrecision(0), |
150 | fUseRobust(0), | |
151 | fExpectedSigma(0) | |
da7c9ae3 | 152 | { |
153 | ||
154 | ||
155 | } | |
156 | ||
157 | ||
158 | AliTMinuitToolkit& AliTMinuitToolkit::operator=(const AliTMinuitToolkit&) { | |
159 | ||
160 | return *this; | |
161 | } | |
162 | ||
163 | ||
4b23a517 | 164 | |
165 | AliTMinuitToolkit::~AliTMinuitToolkit(){ | |
166 | // | |
167 | // destructor | |
168 | // | |
169 | delete fPoints; | |
dc697472 | 170 | delete fWeights; |
4b23a517 | 171 | delete fWeightFunction; |
172 | delete fParamLimits; | |
173 | delete fFormula; | |
174 | delete fParam; | |
175 | delete fCovar; | |
176 | delete fChi2; | |
177 | } | |
178 | ||
179 | void AliTMinuitToolkit::FitHistogram(TH1F * his) { | |
180 | // | |
181 | // Fit a one dimensional histogram | |
182 | // | |
183 | fPoints = new TMatrixD(his->GetNbinsX(), 2); | |
184 | ||
185 | for(Int_t ibin=0; ibin < his->GetNbinsX(); ibin++) { | |
186 | Double_t x = his->GetXaxis()->GetBinCenter(ibin+1); | |
187 | Double_t y = his->GetBinContent(ibin+1); | |
188 | ||
189 | (*fPoints)(ibin, 0) = x; | |
190 | (*fPoints)(ibin, 1) = y; | |
191 | } | |
192 | ||
193 | Fit(); | |
194 | } | |
195 | ||
196 | ||
197 | void AliTMinuitToolkit::FitHistogram(TH2F * his) { | |
198 | // | |
dc697472 | 199 | // Fit a curve to a two dimensional histogram |
4b23a517 | 200 | // |
201 | fPoints = new TMatrixD((Long64_t)his->GetEntries(), 2); | |
202 | Long64_t entry = 0; | |
203 | ||
204 | for(Int_t ibin=0; ibin < his->GetNbinsX(); ibin++) { | |
205 | Double_t x = his->GetXaxis()->GetBinCenter(ibin); | |
206 | for(Int_t jbin=0; jbin < his->GetNbinsY(); jbin++) { | |
207 | Long64_t n = his->GetBin(ibin, jbin); | |
208 | Double_t y = his->GetYaxis()->GetBinCenter(jbin); | |
209 | for(Int_t ientries=0; ientries < his->GetBinContent(n); ientries++) { | |
210 | (*fPoints)(entry,0) = x; | |
211 | (*fPoints)(entry,1) = y; | |
212 | entry++; | |
213 | } | |
214 | ||
215 | } | |
216 | } | |
217 | ||
218 | Fit(); | |
219 | } | |
220 | ||
221 | ||
222 | void AliTMinuitToolkit::SetWeightFunction(Char_t * name, Float_t param1, Float_t param2) { | |
223 | // | |
224 | // Set the weight function which must be defined on the interval [0,1]. | |
225 | // | |
226 | TString FuncType(name); | |
227 | FuncType.ToUpper(); | |
228 | ||
229 | if (FuncType == "EXPONENTIAL") fWeightFunction = new TFormula("exp", Form("TMath::Exp(-TMath::Log(%f)*x)", param1)); | |
230 | if (FuncType == "BOX") fWeightFunction = new TFormula("box", Form("TMath::Erfc((x-%f)/0.0001)", param1)); | |
dc697472 | 231 | if (FuncType == "ERRORFUNCTION") fWeightFunction = new TFormula("err", Form("TMath::Erfc((x-%f)/%f)", param1, param2)); |
4b23a517 | 232 | |
233 | } | |
234 | ||
235 | ||
236 | void AliTMinuitToolkit::FitterFCN(int &npar, double *dummy, double &fchisq, double *gin, int iflag){ | |
237 | // | |
238 | // internal function which gives the specified function to the TMinuit function | |
239 | // | |
240 | ||
241 | // suppress warnings for unused variables: | |
242 | dummy = dummy; | |
243 | iflag = iflag; | |
244 | npar = npar; | |
245 | // | |
246 | AliTMinuitToolkit * fitter = (AliTMinuitToolkit*)TVirtualFitter::GetFitter()->GetObjectFit(); | |
247 | fchisq = 0; | |
248 | Int_t nvar = fitter->GetPoints()->GetNcols()-1; | |
249 | Int_t npoints = fitter->GetPoints()->GetNrows(); | |
250 | ||
dc697472 | 251 | // calculate mean deviation for normalization or use user-defined sigma |
252 | Double_t dev = 0.; | |
253 | if (fitter->GetExpectedSigma() == 0 && fitter->GetStatus() == true) { | |
254 | for (Int_t ipoint=0; ipoint<npoints; ipoint++){ | |
4b23a517 | 255 | Double_t x[100]; |
dc697472 | 256 | for (Int_t ivar=0; ivar<nvar; ivar++){ |
4b23a517 | 257 | x[ivar] = (*fitter->GetPoints())(ipoint, ivar); |
dc697472 | 258 | } |
259 | Float_t funx = fitter->GetFormula()->EvalPar(x,gin); | |
260 | Double_t delta = (*fitter->GetPoints())(ipoint, nvar) - funx; | |
261 | dev += TMath::Sqrt(TMath::Abs(delta)); | |
262 | } | |
263 | dev = dev/npoints; | |
264 | } else { | |
265 | dev = fitter->GetExpectedSigma(); | |
4b23a517 | 266 | } |
dc697472 | 267 | // calculate chisquare |
4b23a517 | 268 | for (Int_t ipoint=0; ipoint<npoints; ipoint++){ |
269 | Double_t x[100]; | |
270 | for (Int_t ivar=0; ivar<nvar; ivar++){ | |
271 | x[ivar] = (*fitter->GetPoints())(ipoint, ivar); | |
272 | } | |
dc697472 | 273 | Float_t funx = fitter->GetFormula()->EvalPar(x,gin); |
274 | Double_t delta = TMath::Abs((*fitter->GetPoints())(ipoint, nvar) - funx); | |
275 | if (fitter->GetStatus() == true) { | |
276 | delta = delta/dev; // normalization | |
277 | if (delta <= 2.5) fchisq+= delta*delta; // new metric: Huber-k-estimator | |
278 | if (delta > 2.5) fchisq+= 2*(2.5)*delta - (2.5*2.5); | |
279 | } else { | |
280 | Double_t weight = (*fitter->GetWeights())(ipoint); | |
281 | fchisq+= delta*delta*weight; //old metric | |
282 | } | |
4b23a517 | 283 | } |
dc697472 | 284 | } |
4b23a517 | 285 | |
286 | ||
287 | void AliTMinuitToolkit::Fit() { | |
288 | // | |
289 | // internal function that calls the fitter | |
290 | // | |
dc697472 | 291 | Int_t nparam = fParam->GetNrows(); |
292 | Int_t npoints = fPoints->GetNrows(); | |
293 | Int_t nvar = fPoints->GetNcols()-1; | |
4b23a517 | 294 | |
295 | // set all paramter limits to infinity as default | |
296 | if (fParamLimits == 0) { | |
297 | fParamLimits = new TMatrixD(nparam ,2); | |
298 | for (Int_t iparam=0; iparam<nparam; iparam++){ | |
299 | (*fParamLimits)(iparam, 0) = 0; | |
300 | (*fParamLimits)(iparam, 1) = 0; | |
301 | } | |
302 | } | |
303 | ||
304 | // set all weights to 1 as default | |
dc697472 | 305 | Bool_t weightFlag = false; |
4b23a517 | 306 | if (fWeightFunction == 0) { |
307 | fWeightFunction = new TFormula("constant", "1"); | |
dc697472 | 308 | } else { |
309 | weightFlag = true; | |
4b23a517 | 310 | } |
311 | ||
312 | // migrad fit algorithm as default | |
313 | if (fFitAlgorithm == 0) { | |
314 | fFitAlgorithm = "migrad"; | |
315 | } | |
316 | ||
dc697472 | 317 | // assign weights |
318 | if (fWeights == 0) { | |
319 | fWeights = new TVectorD(npoints); | |
320 | for (Int_t ipoint=0; ipoint<npoints; ipoint++) (*fWeights)(ipoint) = 1; | |
321 | } | |
322 | ||
4b23a517 | 323 | // set up the fitter |
324 | TVirtualFitter *minuit = TVirtualFitter::Fitter(0, nparam); | |
325 | minuit->SetObjectFit(this); | |
dc697472 | 326 | minuit->SetFCN(AliTMinuitToolkit::FitterFCN); |
4b23a517 | 327 | |
328 | // initialize paramters (step size???) | |
329 | for (Int_t iparam=0; iparam<nparam; iparam++){ | |
330 | minuit->SetParameter(iparam, Form("p[%d]",iparam), (*fParam)(iparam), (*fParam)(iparam)/10, (*fParamLimits)(iparam, 0), (*fParamLimits)(iparam, 1)); | |
331 | } | |
dc697472 | 332 | |
333 | // | |
4b23a517 | 334 | Double_t argList[2]; |
335 | argList[0] = fMaxCalls; //maximal number of calls | |
336 | argList[1] = fPrecision; //tolerance normalized to 0.001 | |
337 | if (fMaxCalls == 500 && fPrecision == 1) minuit->ExecuteCommand(fFitAlgorithm, 0, 0); | |
338 | if (fMaxCalls != 500 || fPrecision != 1) minuit->ExecuteCommand(fFitAlgorithm, argList, 2); | |
339 | // two additional arguments can be specified ExecuteCommand("migrad", argList, 2) - use 0,0 for default | |
dc697472 | 340 | |
341 | // fill parameter vector | |
342 | for (Int_t ivar=0; ivar<nparam; ivar++){ | |
343 | (*fParam)(ivar) = minuit->GetParameter(ivar); | |
344 | fFormula->SetParameter(ivar, minuit->GetParameter(ivar)); | |
345 | } | |
4b23a517 | 346 | |
dc697472 | 347 | // if a weight function is specified -> enter 2nd run with weights |
348 | if (weightFlag == true && fUseRobust == true) { | |
349 | // sort points for weighting | |
350 | Double_t *sortList = new Double_t[npoints]; | |
351 | Int_t *indexList = new Int_t[npoints]; | |
352 | for (Int_t ipoint=0; ipoint<npoints; ipoint++){ | |
353 | Double_t funx = fFormula->Eval((*fPoints)(ipoint, 0)); | |
354 | Double_t delta = TMath::Abs((*fPoints)[ipoint][nvar] - funx); | |
355 | sortList[ipoint] = delta; | |
356 | } | |
357 | TMath::Sort(npoints, sortList, indexList, false); | |
358 | for (Int_t ip=0; ip<npoints; ip++){ | |
359 | Double_t t = ip/(Double_t)npoints; | |
360 | (*fWeights)(indexList[ip]) = fWeightFunction->Eval(t); | |
361 | } | |
362 | ||
363 | // set up the fitter | |
364 | fUseRobust = false; | |
365 | for (Int_t iparam=0; iparam<nparam; iparam++){ | |
366 | minuit->SetParameter(iparam, Form("p[%d]",iparam), (*fParam)(iparam), (*fParam)(iparam)/10, (*fParamLimits)(iparam, 0), (*fParamLimits)(iparam, 1)); | |
367 | } | |
368 | // start fitting | |
369 | if (fMaxCalls == 500 && fPrecision == 1) minuit->ExecuteCommand(fFitAlgorithm, 0, 0); | |
370 | if (fMaxCalls != 500 || fPrecision != 1) minuit->ExecuteCommand(fFitAlgorithm, argList, 2); | |
371 | fUseRobust = true; | |
372 | ||
373 | delete sortList; | |
374 | delete indexList; | |
375 | } | |
376 | ||
4b23a517 | 377 | // fill parameter vector |
378 | for (Int_t ivar=0; ivar<nparam; ivar++){ | |
379 | (*fParam)(ivar) = minuit->GetParameter(ivar); | |
dc697472 | 380 | fFormula->SetParameter(ivar, minuit->GetParameter(ivar)); |
4b23a517 | 381 | } |
382 | ||
383 | // fill covariance matrix | |
384 | fCovar = new TMatrixD(nparam, nparam); | |
4b23a517 | 385 | for(Int_t i=0; i < nparam; i++) { |
386 | for(Int_t j=0; j < nparam; j++) { | |
387 | (*fCovar)(i,j) = minuit->GetCovarianceMatrixElement(i,j); | |
388 | } | |
389 | } | |
dc697472 | 390 | |
391 | if (weightFlag == false) fWeightFunction = 0; | |
4b23a517 | 392 | } |
393 | ||
394 | ||
395 | ||
396 | void AliTMinuitToolkit::Test() { | |
397 | // | |
398 | // This test function shows the basic working principles of this class | |
399 | // and illustrates how a robust fit can improve the results | |
400 | // | |
305a4a95 | 401 | |
402 | // 1. provide some example histogram | |
dc697472 | 403 | TH1F * hist = new TH1F("test", "with (red) and without (black) robust option", 20,0,4); |
4b23a517 | 404 | TRandom * rand = new TRandom(); |
405 | for (Int_t i = 0; i < 10000; i++) { | |
406 | hist->Fill(rand->Exp(1)); | |
407 | if (i < 1000) hist->Fill(3); //"outliers" | |
408 | if (i < 1070) hist->Fill(3.5); | |
409 | if (i < 670) hist->Fill(2); | |
410 | if (i < 770) hist->Fill(1.5);//"outliers" | |
411 | if (i < 740) hist->Fill(1); | |
412 | } | |
413 | TCanvas * canv = new TCanvas(); | |
414 | canv->cd(1); | |
415 | hist->Draw(); | |
dc697472 | 416 | |
305a4a95 | 417 | // 2. example fit without robust option |
418 | AliTMinuitToolkit * tool = new AliTMinuitToolkit(); | |
419 | TFormula *FormExp = new TFormula("formExp", "[0]*TMath::Exp(-[1]*x)"); | |
420 | tool->SetFitFunction(FormExp); | |
421 | TVectorD *vec1 = new TVectorD(2); // Set initial values | |
422 | (*vec1)(0) = 1800; | |
423 | (*vec1)(1) = 1; | |
424 | tool->SetInitialParam(vec1); | |
425 | tool->FitHistogram(hist); | |
426 | ||
4b23a517 | 427 | // draw fit function |
428 | TF1 *func = new TF1("test", "[0]*TMath::Exp(-[1]*x)", 0, 6); | |
305a4a95 | 429 | func->SetParameters((*tool->GetParameters())(0), (*tool->GetParameters())(1)); |
4b23a517 | 430 | func->Draw("same"); |
dc697472 | 431 | |
305a4a95 | 432 | // 3 . robust fit |
4b23a517 | 433 | TVectorD *vec2 = new TVectorD(2); |
434 | (*vec2)(0) = 1800; | |
435 | (*vec2)(1) = 1; | |
305a4a95 | 436 | tool->SetInitialParam(vec2); |
437 | tool->EnableRobust(true, 10); | |
438 | tool->SetWeightFunction("box", 0.75); | |
439 | tool->FitHistogram(hist); | |
4b23a517 | 440 | TF1 *func2 = new TF1("test2", "[0]*TMath::Exp(-[1]*x)", 0, 6); |
305a4a95 | 441 | func2->SetParameter(0, (*tool->GetParameters())(0)); |
442 | func2->SetParameter(1, (*tool->GetParameters())(1)); | |
4b23a517 | 443 | func2->SetLineColor(kRed); |
444 | func2->Draw("same"); | |
445 | ||
446 | } | |
447 |