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TPrincipal.cxx
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1// @(#)root/hist:$Id$
2// Author: Christian Holm Christensen 1/8/2000
3
4/*************************************************************************
5 * Copyright (C) 1995-2004, Rene Brun and Fons Rademakers. *
6 * All rights reserved. *
7 * *
8 * For the licensing terms see $ROOTSYS/LICENSE. *
9 * For the list of contributors see $ROOTSYS/README/CREDITS. *
10 *************************************************************************/
11
12/** \class TPrincipal
13 \ingroup Hist
14Principal Components Analysis (PCA)
15
16The current implementation is based on the LINTRA package from CERNLIB
17by R. Brun, H. Hansroul, and J. Kubler.
18The class has been implemented by Christian Holm Christensen in August 2000.
19
20## Introduction
21
22In many applications of various fields of research, the treatment of
23large amounts of data requires powerful techniques capable of rapid
24data reduction and analysis. Usually, the quantities most
25conveniently measured by the experimentalist, are not necessarily the
26most significant for classification and analysis of the data. It is
27then useful to have a way of selecting an optimal set of variables
28necessary for the recognition process and reducing the dimensionality
29of the problem, resulting in an easier classification procedure.
30
31This paper describes the implementation of one such method of
32feature selection, namely the principal components analysis. This
33multidimensional technique is well known in the field of pattern
34recognition and and its use in Particle Physics has been documented
35elsewhere (cf. H. Wind, <I>Function Parameterization</I>, CERN
3672-21).
37
38## Overview
39Suppose we have prototypes which are trajectories of particles,
40passing through a spectrometer. If one measures the passage of the
41particle at say 8 fixed planes, the trajectory is described by an
428-component vector:
43\f[
44 \mathbf{x} = \left(x_0, x_1, \ldots, x_7\right)
45\f]
46in 8-dimensional pattern space.
47
48One proceeds by generating a a representative tracks sample and
49building up the covariance matrix \f$\mathsf{C}\f$. Its eigenvectors and
50eigenvalues are computed by standard methods, and thus a new basis is
51obtained for the original 8-dimensional space the expansion of the
52prototypes,
53\f[
54 \mathbf{x}_m = \sum^7_{i=0} a_{m_i} \mathbf{e}_i
55 \quad
56 \mbox{where}
57 \quad
58 a_{m_i} = \mathbf{x}^T\bullet\mathbf{e}_i
59\f]
60allows the study of the behavior of the coefficients \f$a_{m_i}\f$ for all
61the tracks of the sample. The eigenvectors which are insignificant for
62the trajectory description in the expansion will have their
63corresponding coefficients \f$a_{m_i}\f$ close to zero for all the
64prototypes.
65
66On one hand, a reduction of the dimensionality is then obtained by
67omitting these least significant vectors in the subsequent analysis.
68
69On the other hand, in the analysis of real data, these least
70significant variables(?) can be used for the pattern
71recognition problem of extracting the valid combinations of
72coordinates describing a true trajectory from the set of all possible
73wrong combinations.
74
75The program described here performs this principal components analysis
76on a sample of data provided by the user. It computes the covariance
77matrix, its eigenvalues ands corresponding eigenvectors and exhibits
78the behavior of the principal components \f$a_{m_i}\f$, thus providing
79to the user all the means of understanding their data.
80
81## Principal Components Method
82Let's consider a sample of \f$M\f$ prototypes each being characterized by
83\f$P\f$ variables \f$x_0, x_1, \ldots, x_{P-1}\f$. Each prototype is a point, or a
84column vector, in a \f$P\f$-dimensional *Pattern space*.
85\f[
86 \mathbf{x} = \left[\begin{array}{c}
87 x_0\\x_1\\\vdots\\x_{P-1}\end{array}\right]\,,
88\f]
89where each \f$x_n\f$ represents the particular value associated with the
90\f$n\f$-dimension.
91
92Those \f$P\f$ variables are the quantities accessible to the
93experimentalist, but are not necessarily the most significant for the
94classification purpose.
95
96The *Principal Components Method* consists of applying a
97*linear* transformation to the original variables. This
98transformation is described by an orthogonal matrix and is equivalent
99to a rotation of the original pattern space into a new set of
100coordinate vectors, which hopefully provide easier feature
101identification and dimensionality reduction.
102
103Let's define the covariance matrix:
104\f[
105 \mathsf{C} = \left\langle\mathbf{y}\mathbf{y}^T\right\rangle
106 \quad\mbox{where}\quad
107 \mathbf{y} = \mathbf{x} - \left\langle\mathbf{x}\right\rangle\,,
108\f]
109and the brackets indicate mean value over the sample of \f$M\f$
110prototypes.
111
112This matrix \f$\mathsf{C}\f$ is real, positive definite, symmetric, and will
113have all its eigenvalues greater then zero. It will now be show that
114among the family of all the complete orthonormal bases of the pattern
115space, the base formed by the eigenvectors of the covariance matrix
116and belonging to the largest eigenvalues, corresponds to the most
117significant features of the description of the original prototypes.
118
119let the prototypes be expanded on into a set of \f$N\f$ basis vectors
120\f$\mathbf{e}_n, n=0,\ldots,N,N+1, \ldots, P-1\f$
121\f[
122 \mathbf{y}_i = \sum^N_{i=0} a_{i_n} \mathbf{e}_n,
123 \quad
124 i = 1, \ldots, M,
125 \quad
126 N < P-1
127\f]
128The `best' feature coordinates \f$\mathbf{e}_n\f$, spanning a *feature
129space*, will be obtained by minimizing the error due to this
130truncated expansion, i.e.,
131\f[
132 \min\left(E_N\right) =
133 \min\left[\left\langle\left(\mathbf{y}_i - \sum^N_{i=0} a_{i_n} \mathbf{e}_n\right)^2\right\rangle\right]
134\f]
135with the conditions:
136\f[
137 \mathbf{e}_k\bullet\mathbf{e}_j = \delta_{jk} =
138 \left\{\begin{array}{rcl}
139 1 & \mbox{for} & k = j\\
140 0 & \mbox{for} & k \neq j
141 \end{array}\right.
142\f]
143Multiplying (3) by \f$\mathbf{e}^T_n\f$ using (5),
144we get
145\f[
146 a_{i_n} = \mathbf{y}_i^T\bullet\mathbf{e}_n\,,
147\f]
148so the error becomes
149\f{eqnarray*}{
150 E_N &=&
151 \left\langle\left[\sum_{n=N+1}^{P-1} a_{i_n}\mathbf{e}_n\right]^2\right\rangle\nonumber\\
152 &=&
153 \left\langle\left[\sum_{n=N+1}^{P-1} \mathbf{y}_i^T\bullet\mathbf{e}_n\mathbf{e}_n\right]^2\right\rangle\nonumber\\
154 &=&
155 \left\langle\sum_{n=N+1}^{P-1} \mathbf{e}_n^T\mathbf{y}_i\mathbf{y}_i^T\mathbf{e}_n\right\rangle\nonumber\\
156 &=&
157 \sum_{n=N+1}^{P-1} \mathbf{e}_n^T\mathsf{C}\mathbf{e}_n
158\f}
159The minimization of the sum in (7) is obtained when each
160term \f$\mathbf{e}_n^\mathsf{C}\mathbf{e}_n\f$ is minimum, since \f$\mathsf{C}\f$ is
161positive definite. By the method of Lagrange multipliers, and the
162condition (5), we get
163\f[
164 E_N = \sum^{P-1}_{n=N+1} \left(\mathbf{e}_n^T\mathsf{C}\mathbf{e}_n -
165 l_n\mathbf{e}_n^T\bullet\mathbf{e}_n + l_n\right)
166\f]
167The minimum condition \f$\frac{dE_N}{d\mathbf{e}^T_n} = 0\f$ leads to the
168equation
169\f[
170 \mathsf{C}\mathbf{e}_n = l_n\mathbf{e}_n\,,
171\f]
172which shows that \f$\mathbf{e}_n\f$ is an eigenvector of the covariance
173matrix \f$\mathsf{C}\f$ with eigenvalue \f$l_n\f$. The estimated minimum error is
174then given by
175\f[
176 E_N \sim \sum^{P-1}_{n=N+1} \mathbf{e}_n^T\bullet l_n\mathbf{e}_n
177 = \sum^{P-1}_{n=N+1} l_n\,,
178\f]
179where \f$l_n,\,n=N+1,\ldots,P\f$ \f$l_n,\,n=N+1,\ldots,P-1\f$ are the eigenvalues associated with the
180omitted eigenvectors in the expansion (3). Thus, by choosing
181the \f$N\f$ largest eigenvalues, and their associated eigenvectors, the
182error \f$E_N\f$ is minimized.
183
184The transformation matrix to go from the pattern space to the feature
185space consists of the ordered eigenvectors \f$\mathbf{e}_1,\ldots,\mathbf{e}_P\f$
186\f$\mathbf{e}_0,\ldots,\mathbf{e}_{P-1}\f$ for its columns
187\f[
188 \mathsf{T} = \left[
189 \begin{array}{cccc}
190 \mathbf{e}_0 &
191 \mathbf{e}_1 &
192 \vdots &
193 \mathbf{e}_{P-1}
194 \end{array}\right]
195 = \left[
196 \begin{array}{cccc}
197 \mathbf{e}_{0_0} & \mathbf{e}_{1_0} & \cdots & \mathbf{e}_{{P-1}_0}\\
198 \mathbf{e}_{0_1} & \mathbf{e}_{1_1} & \cdots & \mathbf{e}_{{P-1}_1}\\
199 \vdots & \vdots & \ddots & \vdots \\
200 \mathbf{e}_{0_{P-1}} & \mathbf{e}_{1_{P-1}} & \cdots & \mathbf{e}_{{P-1}_{P-1}}\\
201 \end{array}\right]
202\f]
203This is an orthogonal transformation, or rotation, of the pattern
204space and feature selection results in ignoring certain coordinates
205in the transformed space.
206
207Christian Holm August 2000, CERN
208*/
209
210#include "TPrincipal.h"
211
212#include "TVectorD.h"
213#include "TMatrixD.h"
214#include "TMatrixDSymEigen.h"
215#include "TMath.h"
216#include "TList.h"
217#include "TH2.h"
218#include "TDatime.h"
219#include "TBrowser.h"
220#include "TROOT.h"
221#include "Riostream.h"
222
223
225
226////////////////////////////////////////////////////////////////////////////////
227/// Empty constructor. Do not use.
228
230 : fMeanValues(0),
231 fSigmas(0),
232 fCovarianceMatrix(1,1),
233 fEigenVectors(1,1),
234 fEigenValues(0),
235 fOffDiagonal(0),
236 fStoreData(kFALSE)
237{
238 fTrace = 0;
239 fHistograms = 0;
243}
244
245////////////////////////////////////////////////////////////////////////////////
246/// Constructor. Argument is number of variables in the sample of data
247/// Options are:
248/// - N Normalize the covariance matrix (default)
249/// - D Store input data (default)
250///
251/// The created object is named "principal" by default.
252
254 : fMeanValues(nVariables),
255 fSigmas(nVariables),
256 fCovarianceMatrix(nVariables,nVariables),
257 fEigenVectors(nVariables,nVariables),
258 fEigenValues(nVariables),
259 fOffDiagonal(nVariables),
260 fStoreData(kFALSE)
261{
262 if (nVariables <= 1) {
263 Error("TPrincipal", "You can't be serious - nVariables == 1!!!");
264 return;
265 }
266
267 SetName("principal");
268
269 fTrace = 0;
270 fHistograms = 0;
273 fNumberOfVariables = nVariables;
274 while (strlen(opt) > 0) {
275 switch(*opt++) {
276 case 'N':
277 case 'n':
279 break;
280 case 'D':
281 case 'd':
283 break;
284 default:
285 break;
286 }
287 }
288
289 if (!fMeanValues.IsValid())
290 Error("TPrincipal","Couldn't create vector mean values");
291 if (!fSigmas.IsValid())
292 Error("TPrincipal","Couldn't create vector sigmas");
294 Error("TPrincipal","Couldn't create covariance matrix");
295 if (!fEigenVectors.IsValid())
296 Error("TPrincipal","Couldn't create eigenvector matrix");
297 if (!fEigenValues.IsValid())
298 Error("TPrincipal","Couldn't create eigenvalue vector");
299 if (!fOffDiagonal.IsValid())
300 Error("TPrincipal","Couldn't create offdiagonal vector");
301 if (fStoreData) {
302 fUserData.ResizeTo(nVariables*1000);
303 fUserData.Zero();
304 if (!fUserData.IsValid())
305 Error("TPrincipal","Couldn't create user data vector");
306 }
307}
308
309////////////////////////////////////////////////////////////////////////////////
310/// Copy constructor.
311
313 TNamed(pr),
314 fNumberOfDataPoints(pr.fNumberOfDataPoints),
315 fNumberOfVariables(pr.fNumberOfVariables),
316 fMeanValues(pr.fMeanValues),
317 fSigmas(pr.fSigmas),
318 fCovarianceMatrix(pr.fCovarianceMatrix),
319 fEigenVectors(pr.fEigenVectors),
320 fEigenValues(pr.fEigenValues),
321 fOffDiagonal(pr.fOffDiagonal),
322 fUserData(pr.fUserData),
323 fTrace(pr.fTrace),
324 fHistograms(pr.fHistograms),
325 fIsNormalised(pr.fIsNormalised),
326 fStoreData(pr.fStoreData)
327{
328}
329
330////////////////////////////////////////////////////////////////////////////////
331/// Assignment operator.
332
334{
335 if(this!=&pr) {
340 fSigmas=pr.fSigmas;
346 fTrace=pr.fTrace;
350 }
351 return *this;
352}
353
354////////////////////////////////////////////////////////////////////////////////
355/// Destructor.
356
358{
359 if (fHistograms) {
361 delete fHistograms;
362 }
363}
364
365////////////////////////////////////////////////////////////////////////////////
366/// Add a data point and update the covariance matrix. The input
367/// array must be <TT>fNumberOfVariables</TT> long.
368///
369///
370/// The Covariance matrix and mean values of the input data is calculated
371/// on the fly by the following equations:
372///
373/// \f[
374/// \left<x_i\right>^{(0)} = x_{i0}
375/// \f]
376///
377///
378/// \f[
379/// \left<x_i\right>^{(n)} = \left<x_i\right>^{(n-1)}
380/// + \frac1n \left(x_{in} - \left<x_i\right>^{(n-1)}\right)
381/// \f]
382///
383/// \f[
384/// C_{ij}^{(0)} = 0
385/// \f]
386///
387///
388///
389/// \f[
390/// C_{ij}^{(n)} = C_{ij}^{(n-1)}
391/// + \frac1{n-1}\left[\left(x_{in} - \left<x_i\right>^{(n)}\right)
392/// \left(x_{jn} - \left<x_j\right>^{(n)}\right)\right]
393/// - \frac1n C_{ij}^{(n-1)}
394/// \f]
395///
396/// since this is a really fast method, with no rounding errors (please
397/// refer to CERN 72-21 pp. 54-106).
398///
399///
400/// The data is stored internally in a <TT>TVectorD</TT>, in the following
401/// way:
402///
403/// \f[
404/// \mathbf{x} = \left[\left(x_{0_0},\ldots,x_{{P-1}_0}\right),\ldots,
405/// \left(x_{0_i},\ldots,x_{{P-1}_i}\right), \ldots\right]
406/// \f]
407///
408/// With \f$P\f$ as defined in the class description.
409
411{
412 if (!p)
413 return;
414
415 // Increment the data point counter
416 Int_t i,j;
417 if (++fNumberOfDataPoints == 1) {
418 for (i = 0; i < fNumberOfVariables; i++)
419 fMeanValues(i) = p[i];
420 }
421 else {
422
424 for (i = 0; i < fNumberOfVariables; i++) {
425
426 fMeanValues(i) *= cor;
428 Double_t t1 = (p[i] - fMeanValues(i)) / (fNumberOfDataPoints - 1);
429
430 // Setting Matrix (lower triangle) elements
431 for (j = 0; j < i + 1; j++) {
432 fCovarianceMatrix(i,j) *= cor;
433 fCovarianceMatrix(i,j) += t1 * (p[j] - fMeanValues(j));
434 }
435 }
436 }
437
438 // Store data point in internal vector
439 // If the vector isn't big enough to hold the new data, then
440 // expand the vector by half it's size.
441 if (!fStoreData)
442 return;
443 Int_t size = fUserData.GetNrows();
445 fUserData.ResizeTo(size + size/2);
446
447 for (i = 0; i < fNumberOfVariables; i++) {
449 fUserData(j) = p[i];
450 }
451
452}
453
454////////////////////////////////////////////////////////////////////////////////
455/// Browse the TPrincipal object in the TBrowser.
456
458{
459 if (fHistograms) {
460 TIter next(fHistograms);
461 TH1* h = 0;
462 while ((h = (TH1*)next()))
463 b->Add(h,h->GetName());
464 }
465
466 if (fStoreData)
467 b->Add(&fUserData,"User Data");
468 b->Add(&fCovarianceMatrix,"Covariance Matrix");
469 b->Add(&fMeanValues,"Mean value vector");
470 b->Add(&fSigmas,"Sigma value vector");
471 b->Add(&fEigenValues,"Eigenvalue vector");
472 b->Add(&fEigenVectors,"Eigenvector Matrix");
473
474}
475
476////////////////////////////////////////////////////////////////////////////////
477/// Clear the data in Object. Notice, that's not possible to change
478/// the dimension of the original data.
479
481{
482 if (fHistograms) {
483 fHistograms->Delete(opt);
484 }
485
487 fTrace = 0;
492 fSigmas.Zero();
494
495 if (fStoreData) {
497 fUserData.Zero();
498 }
499}
500
501////////////////////////////////////////////////////////////////////////////////
502/// Return a row of the user supplied data.
503/// If row is out of bounds, 0 is returned.
504/// It's up to the user to delete the returned array.
505/// Row 0 is the first row;
506
508{
509 if (row >= fNumberOfDataPoints)
510 return 0;
511
512 if (!fStoreData)
513 return 0;
514
515 Int_t index = row * fNumberOfVariables;
516 return &fUserData(index);
517}
518
519
520////////////////////////////////////////////////////////////////////////////////
521/// Generates the file `<filename>`, with `.C` appended if it does
522/// argument doesn't end in .cxx or .C.
523///
524/// The file contains the implementation of two functions
525/// ~~~ {.cpp}
526/// void X2P(Double_t *x, Double *p)
527/// void P2X(Double_t *p, Double *x, Int_t nTest)
528/// ~~~
529/// which does the same as `TPrincipal::X2P` and `TPrincipal::P2X`
530/// respectively. Please refer to these methods.
531///
532/// Further, the static variables:
533/// ~~~ {.cpp}
534/// Int_t gNVariables
535/// Double_t gEigenValues[]
536/// Double_t gEigenVectors[]
537/// Double_t gMeanValues[]
538/// Double_t gSigmaValues[]
539/// ~~~
540/// are initialized. The only ROOT header file needed is Rtypes.h
541///
542/// See TPrincipal::MakeRealCode for a list of options
543
544void TPrincipal::MakeCode(const char *filename, Option_t *opt)
545{
546 TString outName(filename);
547 if (!outName.EndsWith(".C") && !outName.EndsWith(".cxx"))
548 outName += ".C";
549
550 MakeRealCode(outName.Data(),"",opt);
551}
552
553////////////////////////////////////////////////////////////////////////////////
554/// Make histograms of the result of the analysis.
555/// The option string say which histograms to create
556/// - X Histogram original data
557/// - P Histogram principal components corresponding to
558/// original data
559/// - D Histogram the difference between the original data
560/// and the projection of principal unto a lower
561/// dimensional subspace (2D histograms)
562/// - E Histogram the eigenvalues
563/// - S Histogram the square of the residues
564/// (see `TPrincipal::SumOfSquareResiduals`)
565/// The histograms will be named `<name>_<type><number>`, where `<name>`
566/// is the first argument, `<type>` is one of X,P,D,E,S, and `<number>`
567/// is the variable.
568
570{
571 Bool_t makeX = kFALSE;
572 Bool_t makeD = kFALSE;
573 Bool_t makeP = kFALSE;
574 Bool_t makeE = kFALSE;
575 Bool_t makeS = kFALSE;
576
577 Int_t len = strlen(opt);
578 Int_t i,j,k;
579 for (i = 0; i < len; i++) {
580 switch (opt[i]) {
581 case 'X':
582 case 'x':
583 if (fStoreData)
584 makeX = kTRUE;
585 break;
586 case 'd':
587 case 'D':
588 if (fStoreData)
589 makeD = kTRUE;
590 break;
591 case 'P':
592 case 'p':
593 if (fStoreData)
594 makeP = kTRUE;
595 break;
596 case 'E':
597 case 'e':
598 makeE = kTRUE;
599 break;
600 case 's':
601 case 'S':
602 if (fStoreData)
603 makeS = kTRUE;
604 break;
605 default:
606 Warning("MakeHistograms","Unknown option: %c",opt[i]);
607 }
608 }
609
610 // If no option was given, then exit gracefully
611 if (!makeX && !makeD && !makeP && !makeE && !makeS)
612 return;
613
614 // If the list of histograms doesn't exist, create it.
615 if (!fHistograms)
616 fHistograms = new TList;
617
618 // Don't create the histograms if they are already in the TList.
619 if (makeX && fHistograms->FindObject(Form("%s_x000",name)))
620 makeX = kFALSE;
621 if (makeD && fHistograms->FindObject(Form("%s_d000",name)))
622 makeD = kFALSE;
623 if (makeP && fHistograms->FindObject(Form("%s_p000",name)))
624 makeP = kFALSE;
625 if (makeE && fHistograms->FindObject(Form("%s_e",name)))
626 makeE = kFALSE;
627 if (makeS && fHistograms->FindObject(Form("%s_s",name)))
628 makeS = kFALSE;
629
630 TH1F **hX = 0;
631 TH2F **hD = 0;
632 TH1F **hP = 0;
633 TH1F *hE = 0;
634 TH1F *hS = 0;
635
636 // Initialize the arrays of histograms needed
637 if (makeX)
638 hX = new TH1F * [fNumberOfVariables];
639
640 if (makeD)
641 hD = new TH2F * [fNumberOfVariables];
642
643 if (makeP)
644 hP = new TH1F * [fNumberOfVariables];
645
646 if (makeE){
647 hE = new TH1F(Form("%s_e",name), "Eigenvalues of Covariance matrix",
649 hE->SetXTitle("Eigenvalue");
650 fHistograms->Add(hE);
651 }
652
653 if (makeS) {
654 hS = new TH1F(Form("%s_s",name),"E_{N}",
656 hS->SetXTitle("N");
657 hS->SetYTitle("#sum_{i=1}^{M} (x_{i} - x'_{N,i})^{2}");
658 fHistograms->Add(hS);
659 }
660
661 // Initialize sub elements of the histogram arrays
662 for (i = 0; i < fNumberOfVariables; i++) {
663 if (makeX) {
664 // We allow 4 sigma spread in the original data in our
665 // histogram.
666 Double_t xlowb = fMeanValues(i) - 4 * fSigmas(i);
667 Double_t xhighb = fMeanValues(i) + 4 * fSigmas(i);
668 Int_t xbins = fNumberOfDataPoints/100;
669 hX[i] = new TH1F(Form("%s_x%03d", name, i),
670 Form("Pattern space, variable %d", i),
671 xbins,xlowb,xhighb);
672 hX[i]->SetXTitle(Form("x_{%d}",i));
673 fHistograms->Add(hX[i]);
674 }
675
676 if(makeD) {
677 // The upper limit below is arbitrary!!!
678 Double_t dlowb = 0;
679 Double_t dhighb = 20;
680 Int_t dbins = fNumberOfDataPoints/100;
681 hD[i] = new TH2F(Form("%s_d%03d", name, i),
682 Form("Distance from pattern to "
683 "feature space, variable %d", i),
684 dbins,dlowb,dhighb,
686 1,
688 hD[i]->SetXTitle(Form("|x_{%d} - x'_{%d,N}|/#sigma_{%d}",i,i,i));
689 hD[i]->SetYTitle("N");
690 fHistograms->Add(hD[i]);
691 }
692
693 if(makeP) {
694 // For some reason, the trace of the none-scaled matrix
695 // (see TPrincipal::MakeNormalised) should enter here. Taken
696 // from LINTRA code.
698 Double_t plowb = -10 * TMath::Sqrt(et);
699 Double_t phighb = -plowb;
700 Int_t pbins = 100;
701 hP[i] = new TH1F(Form("%s_p%03d", name, i),
702 Form("Feature space, variable %d", i),
703 pbins,plowb,phighb);
704 hP[i]->SetXTitle(Form("p_{%d}",i));
705 fHistograms->Add(hP[i]);
706 }
707
708 if (makeE)
709 // The Eigenvector histogram is easy
710 hE->Fill(i,fEigenValues(i));
711
712 }
713 if (!makeX && !makeP && !makeD && !makeS) {
714 if (hX)
715 delete[] hX;
716 if (hD)
717 delete[] hD;
718 if (hP)
719 delete[] hP;
720 return;
721 }
722
723 Double_t *x = 0;
726 for (i = 0; i < fNumberOfDataPoints; i++) {
727
728 // Zero arrays
729 for (j = 0; j < fNumberOfVariables; j++)
730 p[j] = d[j] = 0;
731
732 // update the original data histogram
733 x = (Double_t*)(GetRow(i));
734 R__ASSERT(x);
735
736 if (makeP||makeD||makeS)
737 // calculate the corresponding principal component
738 X2P(x,p);
739
740 if (makeD || makeS) {
741 // Calculate the difference between the original data, and the
742 // same project onto principal components, and then to a lower
743 // dimensional sub-space
744 for (j = fNumberOfVariables; j > 0; j--) {
745 P2X(p,d,j);
746
747 for (k = 0; k < fNumberOfVariables; k++) {
748 // We use the absolute value of the difference!
749 d[k] = x[k] - d[k];
750
751 if (makeS)
752 hS->Fill(j,d[k]*d[k]);
753
754 if (makeD) {
755 d[k] = TMath::Abs(d[k]) / (fIsNormalised ? fSigmas(k) : 1);
756 (hD[k])->Fill(d[k],j);
757 }
758 }
759 }
760 }
761
762 if (makeX||makeP) {
763 // If we are asked to make any of these histograms, we have to
764 // go here
765 for (j = 0; j < fNumberOfVariables; j++) {
766 if (makeX)
767 (hX[j])->Fill(x[j]);
768
769 if (makeP)
770 (hP[j])->Fill(p[j]);
771 }
772 }
773 }
774 // Clean up
775 if (hX)
776 delete [] hX;
777 if (hD)
778 delete [] hD;
779 if (hP)
780 delete [] hP;
781 if (d)
782 delete [] d;
783 if (p)
784 delete [] p;
785
786 // Normalize the residues
787 if (makeS)
789}
790
791////////////////////////////////////////////////////////////////////////////////
792/// Normalize the covariance matrix
793
795{
796 Int_t i,j;
797 for (i = 0; i < fNumberOfVariables; i++) {
799 if (fIsNormalised)
800 for (j = 0; j <= i; j++)
801 fCovarianceMatrix(i,j) /= (fSigmas(i) * fSigmas(j));
802
804 }
805
806 // Fill remaining parts of matrix, and scale.
807 for (i = 0; i < fNumberOfVariables; i++)
808 for (j = 0; j <= i; j++) {
811 }
812
813}
814
815////////////////////////////////////////////////////////////////////////////////
816/// Generate the file <classname>PCA.cxx which contains the
817/// implementation of two methods:
818/// ~~~ {.cpp}
819/// void <classname>::X2P(Double_t *x, Double *p)
820/// void <classname>::P2X(Double_t *p, Double *x, Int_t nTest)
821/// ~~~
822/// which does the same as TPrincipal::X2P and TPrincipal::P2X
823/// respectively. Please refer to these methods.
824///
825/// Further, the public static members:
826/// ~~~ {.cpp}
827/// Int_t <classname>::fgNVariables
828/// Double_t <classname>::fgEigenValues[]
829/// Double_t <classname>::fgEigenVectors[]
830/// Double_t <classname>::fgMeanValues[]
831/// Double_t <classname>::fgSigmaValues[]
832/// ~~~
833/// are initialized, and assumed to exist. The class declaration is
834/// assumed to be in <classname>.h and assumed to be provided by the
835/// user.
836///
837/// See TPrincipal::MakeRealCode for a list of options
838///
839/// The minimal class definition is:
840/// ~~~ {.cpp}
841/// class <classname> {
842/// public:
843/// static Int_t fgNVariables;
844/// static Double_t fgEigenVectors[];
845/// static Double_t fgEigenValues[];
846/// static Double_t fgMeanValues[];
847/// static Double_t fgSigmaValues[];
848///
849/// void X2P(Double_t *x, Double_t *p);
850/// void P2X(Double_t *p, Double_t *x, Int_t nTest);
851/// };
852/// ~~~
853/// Whether the methods <classname>::X2P and <classname>::P2X should
854/// be static or not, is up to the user.
855
856void TPrincipal::MakeMethods(const char *classname, Option_t *opt)
857{
858
859 MakeRealCode(Form("%sPCA.cxx", classname), classname, opt);
860}
861
862
863////////////////////////////////////////////////////////////////////////////////
864/// Perform the principal components analysis.
865/// This is done in several stages in the TMatrix::EigenVectors method:
866/// - Transform the covariance matrix into a tridiagonal matrix.
867/// - Find the eigenvalues and vectors of the tridiagonal matrix.
868
870{
871 // Normalize covariance matrix
873
875 TMatrixDSymEigen eigen(sym);
878 //make sure that eigenvalues are positive
879 for (Int_t i = 0; i < fNumberOfVariables; i++) {
880 if (fEigenValues[i] < 0) fEigenValues[i] = -fEigenValues[i];
881 }
882}
883
884////////////////////////////////////////////////////////////////////////////////
885/// This is the method that actually generates the code for the
886/// transformations to and from feature space and pattern space
887/// It's called by TPrincipal::MakeCode and TPrincipal::MakeMethods.
888///
889/// The options are: NONE so far
890
891void TPrincipal::MakeRealCode(const char *filename, const char *classname,
892 Option_t *)
893{
894 Bool_t isMethod = (classname[0] == '\0' ? kFALSE : kTRUE);
895 const char *prefix = (isMethod ? Form("%s::", classname) : "");
896 const char *cv_qual = (isMethod ? "" : "static ");
897
898 std::ofstream outFile(filename,std::ios::out|std::ios::trunc);
899 if (!outFile) {
900 Error("MakeRealCode","couldn't open output file '%s'",filename);
901 return;
902 }
903
904 std::cout << "Writing on file \"" << filename << "\" ... " << std::flush;
905 //
906 // Write header of file
907 //
908 // Emacs mode line ;-)
909 outFile << "// -*- mode: c++ -*-" << std::endl;
910 // Info about creator
911 outFile << "// " << std::endl
912 << "// File " << filename
913 << " generated by TPrincipal::MakeCode" << std::endl;
914 // Time stamp
915 TDatime date;
916 outFile << "// on " << date.AsString() << std::endl;
917 // ROOT version info
918 outFile << "// ROOT version " << gROOT->GetVersion()
919 << std::endl << "//" << std::endl;
920 // General information on the code
921 outFile << "// This file contains the functions " << std::endl
922 << "//" << std::endl
923 << "// void " << prefix
924 << "X2P(Double_t *x, Double_t *p); " << std::endl
925 << "// void " << prefix
926 << "P2X(Double_t *p, Double_t *x, Int_t nTest);"
927 << std::endl << "//" << std::endl
928 << "// The first for transforming original data x in " << std::endl
929 << "// pattern space, to principal components p in " << std::endl
930 << "// feature space. The second function is for the" << std::endl
931 << "// inverse transformation, but using only nTest" << std::endl
932 << "// of the principal components in the expansion" << std::endl
933 << "// " << std::endl
934 << "// See TPrincipal class documentation for more "
935 << "information " << std::endl << "// " << std::endl;
936 // Header files
937 outFile << "#ifndef __CINT__" << std::endl;
938 if (isMethod)
939 // If these are methods, we need the class header
940 outFile << "#include \"" << classname << ".h\"" << std::endl;
941 else
942 // otherwise, we need the typedefs of Int_t and Double_t
943 outFile << "#include <Rtypes.h> // needed for Double_t etc" << std::endl;
944 // Finish the preprocessor block
945 outFile << "#endif" << std::endl << std::endl;
946
947 //
948 // Now for the data
949 //
950 // We make the Eigenvector matrix, Eigenvalue vector, Sigma vector,
951 // and Mean value vector static, since all are needed in both
952 // functions. Also ,the number of variables are stored in a static
953 // variable.
954 outFile << "//" << std::endl
955 << "// Static data variables" << std::endl
956 << "//" << std::endl;
957 outFile << cv_qual << "Int_t " << prefix << "gNVariables = "
958 << fNumberOfVariables << ";" << std::endl;
959
960 // Assign the values to the Eigenvector matrix. The elements are
961 // stored row-wise, that is element
962 // M[i][j] = e[i * nVariables + j]
963 // where i and j are zero-based.
964 outFile << std::endl << "// Assignment of eigenvector matrix." << std::endl
965 << "// Elements are stored row-wise, that is" << std::endl
966 << "// M[i][j] = e[i * nVariables + j] " << std::endl
967 << "// where i and j are zero-based" << std::endl;
968 outFile << cv_qual << "Double_t " << prefix
969 << "gEigenVectors[] = {" << std::flush;
970 Int_t i,j;
971 for (i = 0; i < fNumberOfVariables; i++) {
972 for (j = 0; j < fNumberOfVariables; j++) {
973 Int_t index = i * fNumberOfVariables + j;
974 outFile << (index != 0 ? "," : "" ) << std::endl
975 << " " << fEigenVectors(i,j) << std::flush;
976 }
977 }
978 outFile << "};" << std::endl << std::endl;
979
980 // Assignment to eigenvalue vector. Zero-based.
981 outFile << "// Assignment to eigen value vector. Zero-based." << std::endl;
982 outFile << cv_qual << "Double_t " << prefix
983 << "gEigenValues[] = {" << std::flush;
984 for (i = 0; i < fNumberOfVariables; i++)
985 outFile << (i != 0 ? "," : "") << std::endl
986 << " " << fEigenValues(i) << std::flush;
987 outFile << std::endl << "};" << std::endl << std::endl;
988
989 // Assignment to mean Values vector. Zero-based.
990 outFile << "// Assignment to mean value vector. Zero-based." << std::endl;
991 outFile << cv_qual << "Double_t " << prefix
992 << "gMeanValues[] = {" << std::flush;
993 for (i = 0; i < fNumberOfVariables; i++)
994 outFile << (i != 0 ? "," : "") << std::endl
995 << " " << fMeanValues(i) << std::flush;
996 outFile << std::endl << "};" << std::endl << std::endl;
997
998 // Assignment to mean Values vector. Zero-based.
999 outFile << "// Assignment to sigma value vector. Zero-based." << std::endl;
1000 outFile << cv_qual << "Double_t " << prefix
1001 << "gSigmaValues[] = {" << std::flush;
1002 for (i = 0; i < fNumberOfVariables; i++)
1003 outFile << (i != 0 ? "," : "") << std::endl
1004 << " " << (fIsNormalised ? fSigmas(i) : 1) << std::flush;
1005 // << " " << fSigmas(i) << std::flush;
1006 outFile << std::endl << "};" << std::endl << std::endl;
1007
1008 //
1009 // Finally we reach the functions themselves
1010 //
1011 // First: void x2p(Double_t *x, Double_t *p);
1012 //
1013 outFile << "// " << std::endl
1014 << "// The "
1015 << (isMethod ? "method " : "function ")
1016 << " void " << prefix
1017 << "X2P(Double_t *x, Double_t *p)"
1018 << std::endl << "// " << std::endl;
1019 outFile << "void " << prefix
1020 << "X2P(Double_t *x, Double_t *p) {" << std::endl
1021 << " for (Int_t i = 0; i < gNVariables; i++) {" << std::endl
1022 << " p[i] = 0;" << std::endl
1023 << " for (Int_t j = 0; j < gNVariables; j++)" << std::endl
1024 << " p[i] += (x[j] - gMeanValues[j]) " << std::endl
1025 << " * gEigenVectors[j * gNVariables + i] "
1026 << "/ gSigmaValues[j];" << std::endl << std::endl << " }"
1027 << std::endl << "}" << std::endl << std::endl;
1028 //
1029 // Now: void p2x(Double_t *p, Double_t *x, Int_t nTest);
1030 //
1031 outFile << "// " << std::endl << "// The "
1032 << (isMethod ? "method " : "function ")
1033 << " void " << prefix
1034 << "P2X(Double_t *p, Double_t *x, Int_t nTest)"
1035 << std::endl << "// " << std::endl;
1036 outFile << "void " << prefix
1037 << "P2X(Double_t *p, Double_t *x, Int_t nTest) {" << std::endl
1038 << " for (Int_t i = 0; i < gNVariables; i++) {" << std::endl
1039 << " x[i] = gMeanValues[i];" << std::endl
1040 << " for (Int_t j = 0; j < nTest; j++)" << std::endl
1041 << " x[i] += p[j] * gSigmaValues[i] " << std::endl
1042 << " * gEigenVectors[i * gNVariables + j];" << std::endl
1043 << " }" << std::endl << "}" << std::endl << std::endl;
1044
1045 // EOF
1046 outFile << "// EOF for " << filename << std::endl;
1047
1048 // Close the file
1049 outFile.close();
1050
1051 std::cout << "done" << std::endl;
1052}
1053
1054////////////////////////////////////////////////////////////////////////////////
1055/// Calculate x as a function of nTest of the most significant
1056/// principal components p, and return it in x.
1057/// It's the users responsibility to make sure that both x and p are
1058/// of the right size (i.e., memory must be allocated for x).
1059
1060void TPrincipal::P2X(const Double_t *p, Double_t *x, Int_t nTest)
1061{
1062 for (Int_t i = 0; i < fNumberOfVariables; i++){
1063 x[i] = fMeanValues(i);
1064 for (Int_t j = 0; j < nTest; j++)
1065 x[i] += p[j] * (fIsNormalised ? fSigmas(i) : 1)
1066 * fEigenVectors(i,j);
1067 }
1068
1069}
1070
1071////////////////////////////////////////////////////////////////////////////////
1072/// Print the statistics
1073/// Options are
1074/// - M Print mean values of original data
1075/// - S Print sigma values of original data
1076/// - E Print eigenvalues of covariance matrix
1077/// - V Print eigenvectors of covariance matrix
1078/// Default is MSE
1079
1081{
1082 Bool_t printV = kFALSE;
1083 Bool_t printM = kFALSE;
1084 Bool_t printS = kFALSE;
1085 Bool_t printE = kFALSE;
1086
1087 Int_t len = strlen(opt);
1088 for (Int_t i = 0; i < len; i++) {
1089 switch (opt[i]) {
1090 case 'V':
1091 case 'v':
1092 printV = kTRUE;
1093 break;
1094 case 'M':
1095 case 'm':
1096 printM = kTRUE;
1097 break;
1098 case 'S':
1099 case 's':
1100 printS = kTRUE;
1101 break;
1102 case 'E':
1103 case 'e':
1104 printE = kTRUE;
1105 break;
1106 default:
1107 Warning("Print", "Unknown option '%c'",opt[i]);
1108 break;
1109 }
1110 }
1111
1112 if (printM||printS||printE) {
1113 std::cout << " Variable # " << std::flush;
1114 if (printM)
1115 std::cout << "| Mean Value " << std::flush;
1116 if (printS)
1117 std::cout << "| Sigma " << std::flush;
1118 if (printE)
1119 std::cout << "| Eigenvalue" << std::flush;
1120 std::cout << std::endl;
1121
1122 std::cout << "-------------" << std::flush;
1123 if (printM)
1124 std::cout << "+------------" << std::flush;
1125 if (printS)
1126 std::cout << "+------------" << std::flush;
1127 if (printE)
1128 std::cout << "+------------" << std::flush;
1129 std::cout << std::endl;
1130
1131 for (Int_t i = 0; i < fNumberOfVariables; i++) {
1132 std::cout << std::setw(12) << i << " " << std::flush;
1133 if (printM)
1134 std::cout << "| " << std::setw(10) << std::setprecision(4)
1135 << fMeanValues(i) << " " << std::flush;
1136 if (printS)
1137 std::cout << "| " << std::setw(10) << std::setprecision(4)
1138 << fSigmas(i) << " " << std::flush;
1139 if (printE)
1140 std::cout << "| " << std::setw(10) << std::setprecision(4)
1141 << fEigenValues(i) << " " << std::flush;
1142 std::cout << std::endl;
1143 }
1144 std::cout << std::endl;
1145 }
1146
1147 if(printV) {
1148 for (Int_t i = 0; i < fNumberOfVariables; i++) {
1149 std::cout << "Eigenvector # " << i << std::flush;
1152 v.Print();
1153 }
1154 }
1155}
1156
1157////////////////////////////////////////////////////////////////////////////////
1158/// Calculates the sum of the square residuals, that is
1159///
1160/// \f[
1161/// E_N = \sum_{i=0}^{P-1} \left(x_i - x^\prime_i\right)^2
1162/// \f]
1163///
1164/// where \f$x^\prime_i = \sum_{j=i}^N p_i e_{n_j}\f$
1165/// is the \f$i^{\mbox{th}}\f$ component of the principal vector, corresponding to
1166/// \f$x_i\f$, the original data; I.e., the square distance to the space
1167/// spanned by \f$N\f$ eigenvectors.
1168
1170{
1171
1172 if (!x)
1173 return;
1174
1175 Double_t p[100];
1176 Double_t xp[100];
1177
1178 X2P(x,p);
1179 for (Int_t i = fNumberOfVariables-1; i >= 0; i--) {
1180 P2X(p,xp,i);
1181 for (Int_t j = 0; j < fNumberOfVariables; j++) {
1182 s[i] += (x[j] - xp[j])*(x[j] - xp[j]);
1183 }
1184 }
1185}
1186
1187////////////////////////////////////////////////////////////////////////////////
1188/// Test the PCA, bye calculating the sum square of residuals
1189/// (see method SumOfSquareResiduals), and display the histogram
1190
1192{
1193 MakeHistograms("pca","S");
1194
1195 if (!fStoreData)
1196 return;
1197
1198 TH1 *pca_s = 0;
1199 if (fHistograms) pca_s = (TH1*)fHistograms->FindObject("pca_s");
1200 if (!pca_s) {
1201 Warning("Test", "Couldn't get histogram of square residuals");
1202 return;
1203 }
1204
1205 pca_s->Draw();
1206}
1207
1208////////////////////////////////////////////////////////////////////////////////
1209/// Calculate the principal components from the original data vector
1210/// x, and return it in p.
1211///
1212/// It's the users responsibility to make sure that both x and p are
1213/// of the right size (i.e., memory must be allocated for p).
1214
1216{
1217 for (Int_t i = 0; i < fNumberOfVariables; i++){
1218 p[i] = 0;
1219 for (Int_t j = 0; j < fNumberOfVariables; j++)
1220 p[i] += (x[j] - fMeanValues(j)) * fEigenVectors(j,i) /
1221 (fIsNormalised ? fSigmas(j) : 1);
1222 }
1223
1224}
PyObject * fTrace
#define d(i)
Definition: RSha256.hxx:102
#define b(i)
Definition: RSha256.hxx:100
#define h(i)
Definition: RSha256.hxx:106
int Int_t
Definition: RtypesCore.h:41
const Bool_t kFALSE
Definition: RtypesCore.h:88
bool Bool_t
Definition: RtypesCore.h:59
double Double_t
Definition: RtypesCore.h:55
const Bool_t kTRUE
Definition: RtypesCore.h:87
const char Option_t
Definition: RtypesCore.h:62
#define ClassImp(name)
Definition: Rtypes.h:365
#define R__ASSERT(e)
Definition: TError.h:96
char name[80]
Definition: TGX11.cxx:109
TMatrixTColumn_const< Double_t > TMatrixDColumn_const
#define gROOT
Definition: TROOT.h:415
char * Form(const char *fmt,...)
Using a TBrowser one can browse all ROOT objects.
Definition: TBrowser.h:37
This class stores the date and time with a precision of one second in an unsigned 32 bit word (950130...
Definition: TDatime.h:37
const char * AsString() const
Return the date & time as a string (ctime() format).
Definition: TDatime.cxx:101
1-D histogram with a float per channel (see TH1 documentation)}
Definition: TH1.h:571
The TH1 histogram class.
Definition: TH1.h:56
virtual void SetXTitle(const char *title)
Definition: TH1.h:409
virtual Int_t Fill(Double_t x)
Increment bin with abscissa X by 1.
Definition: TH1.cxx:3275
virtual void Draw(Option_t *option="")
Draw this histogram with options.
Definition: TH1.cxx:2998
virtual void SetYTitle(const char *title)
Definition: TH1.h:410
virtual void Scale(Double_t c1=1, Option_t *option="")
Multiply this histogram by a constant c1.
Definition: TH1.cxx:6234
2-D histogram with a float per channel (see TH1 documentation)}
Definition: TH2.h:251
A doubly linked list.
Definition: TList.h:44
virtual void Add(TObject *obj)
Definition: TList.h:87
virtual TObject * FindObject(const char *name) const
Find an object in this list using its name.
Definition: TList.cxx:575
virtual void Delete(Option_t *option="")
Remove all objects from the list AND delete all heap based objects.
Definition: TList.cxx:467
TMatrixDSymEigen.
const TVectorD & GetEigenValues() const
const TMatrixD & GetEigenVectors() const
Int_t GetNrows() const
Definition: TMatrixTBase.h:124
virtual TMatrixTBase< Element > & Zero()
Set matrix elements to zero.
Bool_t IsValid() const
Definition: TMatrixTBase.h:147
virtual const Element * GetMatrixArray() const
Definition: TMatrixT.h:222
The TNamed class is the base class for all named ROOT classes.
Definition: TNamed.h:29
virtual void SetName(const char *name)
Set the name of the TNamed.
Definition: TNamed.cxx:140
TNamed & operator=(const TNamed &rhs)
TNamed assignment operator.
Definition: TNamed.cxx:51
virtual void Warning(const char *method, const char *msgfmt,...) const
Issue warning message.
Definition: TObject.cxx:866
virtual void Error(const char *method, const char *msgfmt,...) const
Issue error message.
Definition: TObject.cxx:880
Principal Components Analysis (PCA)
Definition: TPrincipal.h:20
virtual void MakeMethods(const char *classname="PCA", Option_t *option="")
Generate the file <classname>PCA.cxx which contains the implementation of two methods:
Definition: TPrincipal.cxx:856
virtual void AddRow(const Double_t *x)
Add a data point and update the covariance matrix.
Definition: TPrincipal.cxx:410
virtual void X2P(const Double_t *x, Double_t *p)
Calculate the principal components from the original data vector x, and return it in p.
Double_t fTrace
Definition: TPrincipal.h:37
virtual void Print(Option_t *opt="MSE") const
Print the statistics Options are.
const Double_t * GetRow(Int_t row)
Return a row of the user supplied data.
Definition: TPrincipal.cxx:507
TPrincipal()
Empty constructor. Do not use.
Definition: TPrincipal.cxx:229
virtual void Browse(TBrowser *b)
Browse the TPrincipal object in the TBrowser.
Definition: TPrincipal.cxx:457
virtual void MakeHistograms(const char *name="pca", Option_t *option="epsdx")
Make histograms of the result of the analysis.
Definition: TPrincipal.cxx:569
TVectorD fUserData
Definition: TPrincipal.h:35
virtual void MakeCode(const char *filename="pca", Option_t *option="")
Generates the file <filename>, with .C appended if it does argument doesn't end in ....
Definition: TPrincipal.cxx:544
TMatrixD fCovarianceMatrix
Definition: TPrincipal.h:28
Int_t fNumberOfVariables
Definition: TPrincipal.h:24
TVectorD fSigmas
Definition: TPrincipal.h:27
TVectorD fOffDiagonal
Definition: TPrincipal.h:33
TVectorD fEigenValues
Definition: TPrincipal.h:31
virtual ~TPrincipal()
Destructor.
Definition: TPrincipal.cxx:357
TVectorD fMeanValues
Definition: TPrincipal.h:26
TList * fHistograms
Definition: TPrincipal.h:39
void MakeRealCode(const char *filename, const char *prefix, Option_t *option="")
This is the method that actually generates the code for the transformations to and from feature space...
Definition: TPrincipal.cxx:891
Bool_t fStoreData
Definition: TPrincipal.h:42
TMatrixD fEigenVectors
Definition: TPrincipal.h:30
virtual void Clear(Option_t *option="")
Clear the data in Object.
Definition: TPrincipal.cxx:480
virtual void P2X(const Double_t *p, Double_t *x, Int_t nTest)
Calculate x as a function of nTest of the most significant principal components p,...
Int_t fNumberOfDataPoints
Definition: TPrincipal.h:23
void MakeNormalised()
Normalize the covariance matrix.
Definition: TPrincipal.cxx:794
virtual void MakePrincipals()
Perform the principal components analysis.
Definition: TPrincipal.cxx:869
virtual void SumOfSquareResiduals(const Double_t *x, Double_t *s)
Calculates the sum of the square residuals, that is.
void Test(Option_t *option="")
Test the PCA, bye calculating the sum square of residuals (see method SumOfSquareResiduals),...
Bool_t fIsNormalised
Definition: TPrincipal.h:41
TPrincipal & operator=(const TPrincipal &)
Assignment operator.
Definition: TPrincipal.cxx:333
Basic string class.
Definition: TString.h:131
Bool_t EndsWith(const char *pat, ECaseCompare cmp=kExact) const
Return true if string ends with the specified string.
Definition: TString.cxx:2177
const char * Data() const
Definition: TString.h:364
TVectorT< Element > & Zero()
Set vector elements to zero.
Definition: TVectorT.cxx:451
TVectorT< Element > & ResizeTo(Int_t lwb, Int_t upb)
Resize the vector to [lwb:upb] .
Definition: TVectorT.cxx:292
Int_t GetNrows() const
Definition: TVectorT.h:75
Bool_t IsValid() const
Definition: TVectorT.h:83
Double_t x[n]
Definition: legend1.C:17
static constexpr double s
Double_t Sqrt(Double_t x)
Definition: TMath.h:681
Short_t Abs(Short_t d)
Definition: TMathBase.h:120
auto * t1
Definition: textangle.C:20
#define sym(otri1, otri2)
Definition: triangle.c:932