    // @(#)root/hist:$Name:  $:$Id: TPrincipal.cxx,v 1.27 2004/06/09 13:40:02 brun Exp $
    // Author: Christian Holm Christensen    1/8/2000

    /*************************************************************************
     * Copyright (C) 1995-2004, Rene Brun and Fons Rademakers.               *
     * All rights reserved.                                                  *
     *                                                                       *
     * For the licensing terms see $ROOTSYS/LICENSE.                         *
     * For the list of contributors see $ROOTSYS/README/CREDITS.             *
     *************************************************************************/

    //____________________________________________________________________
    //

#
Principal Components Analysis (PCA)

The current implementation is based on the LINTRA package from CERNLIB by R. Brun, H. Hansroul, and J. Kubler. The class has been implemented by Christian Holm Christensen in August 2000.

##
Introduction

In many applications of various fields of research, the treatment of large amounts of data requires powerful techniques capable of rapid data reduction and analysis. Usually, the quantities most conveniently measured by the experimentalist, are not necessarily the most significant for classification and analysis of the data. It is then useful to have a way of selecting an optimal set of variables necessary for the recognition process and reducing the dimensionality of the problem, resulting in an easier classification procedure.

This paper describes the implementation of one such method of feature selection, namely the principal components analysis. This multidimensional technique is well known in the field of pattern recognition and and its use in Particle Physics has been documented elsewhere (cf. H. Wind, Function Parameterization, CERN 72-21).

##
Overview

Suppose we have prototypes which are trajectories of particles, passing through a spectrometer. If one measures the passage of the particle at say 8 fixed planes, the trajectory is described by an 8-component vector:

in 8-dimensional pattern space.

One proceeds by generating a a representative tracks sample and building up the covariance matrix . Its eigenvectors and eigenvalues are computed by standard methods, and thus a new basis is obtained for the original 8-dimensional space the expansion of the prototypes,

allows the study of the behavior of the coefficients for all the tracks of the sample. The eigenvectors which are insignificant for the trajectory description in the expansion will have their corresponding coefficients close to zero for all the prototypes.

On one hand, a reduction of the dimensionality is then obtained by omitting these least significant vectors in the subsequent analysis.

On the other hand, in the analysis of real data, these least significant variables(?) can be used for the pattern recognition problem of extracting the valid combinations of coordinates describing a true trajectory from the set of all possible wrong combinations.

The program described here performs this principal components analysis on a sample of data provided by the user. It computes the covariance matrix, its eigenvalues ands corresponding eigenvectors and exhibits the behavior of the principal components (), thus providing to the user all the means of understanding his data.

A short outline of the method of Principal Components is given in subsection 1.3.

##
Principal Components Method

Let's consider a sample of prototypes each being characterized by variables . Each prototype is a point, or a column vector, in a -dimensional pattern space.

| (1)

where each represents the particular value associated with the -dimension.

Those variables are the quantities accessible to the experimentalist, but are not necessarily the most significant for the classification purpose.

The Principal Components Method consists of applying a linear transformation to the original variables. This transformation is described by an orthogonal matrix and is equivalent to a rotation of the original pattern space into a new set of coordinate vectors, which hopefully provide easier feature identification and dimensionality reduction.

Let's define the covariance matrix:

| (2)

and the brackets indicate mean value over the sample of prototypes.

This matrix is real, positive definite, symmetric, and will have all its eigenvalues greater then zero. It will now be show that among the family of all the complete orthonormal bases of the pattern space, the base formed by the eigenvectors of the covariance matrix and belonging to the largest eigenvalues, corresponds to the most significant features of the description of the original prototypes.

let the prototypes be expanded on into a set of basis vectors ,

| (3)

The `best' feature coordinates , spanning a feature space, will be obtained by minimizing the error due to this truncated expansion, i.e.,

| (4)

with the conditions:

| (5)

Multiplying (3) by using (5), we get

| (6)

so the error becomes

|  |  | (7)

The minimization of the sum in (7) is obtained when each term is minimum, since is positive definite. By the method of Lagrange multipliers, and the condition (5), we get

| (8)

The minimum condition leads to the equation

| (9)

which shows that is an eigenvector of the covariance matrix with eigenvalue . The estimated minimum error is then given by

| (10)

where are the eigenvalues associated with the omitted eigenvectors in the expansion (3). Thus, by choosing the largest eigenvalues, and their associated eigenvectors, the error is minimized.

The transformation matrix to go from the pattern space to the feature space consists of the ordered eigenvectors for its columns

| (11)

This is an orthogonal transformation, or rotation, of the pattern space and feature selection results in ignoring certain coordinates in the transformed space.

Christian Holm
August 2000, CERN

    // $Id: TPrincipal.cxx,v 1.27 2004/06/09 13:40:02 brun Exp $
    // $Date: 2004/06/09 13:40:02 $
    // $Author: brun $

    #include "TPrincipal.h"

    #include "TVectorD.h"
    #include "TMatrixD.h"
    #include "TMatrixDSymEigen.h"
    #include "TList.h"
    #include "TH2.h"
    #include "TDatime.h"
    #include "TBrowser.h"
    #include "TROOT.h"
    #include "Riostream.h"

    ClassImp(TPrincipal);

    //____________________________________________________________________
     TPrincipal::TPrincipal()
    {
      // Empty CTOR, Do not use.

          fHistograms = 0;
    }

    //____________________________________________________________________
     TPrincipal::TPrincipal(Int_t nVariables, Option_t *opt)
      : fMeanValues(nVariables),
        fSigmas(nVariables),
        fCovarianceMatrix(nVariables,nVariables),
        fEigenVectors(nVariables,nVariables),
        fEigenValues(nVariables),
        fOffDiagonal(nVariables),
        fStoreData(kFALSE)
    {
      // Ctor. Argument is number of variables in the sample of data
      // Options are:
      //   N       Normalize the covariance matrix (default)
      //   D       Store input data (default)
      //
      // The created object is  named "principal" by default.
       if (nVariables <= 1) {
        Error("TPrincipal", "You can't be serious - nVariables == 1!!!");
        return;
      }

      SetName("principal");

      fTrace              = 0;
      fHistograms         = 0;
      fIsNormalised       = kFALSE;
      fNumberOfDataPoints = 0;
      fNumberOfVariables  = nVariables;
      while (strlen(opt) > 0) {
        switch(*opt++) {
        case 'N':
        case 'n':
          fIsNormalised = kTRUE;
          break;
        case 'D':
        case 'd':
          fStoreData    = kTRUE;
          break;
        default:
          break;
        }
      }

      if (!fMeanValues.IsValid())
        Error("TPrincipal","Couldn't create vector mean values");
      if (!fSigmas.IsValid())
        Error("TPrincipal","Couldn't create vector sigmas");
      if (!fCovarianceMatrix.IsValid())
        Error("TPrincipal","Couldn't create covariance matrix");
      if (!fEigenVectors.IsValid())
        Error("TPrincipal","Couldn't create eigenvector matrix");
      if (!fEigenValues.IsValid())
        Error("TPrincipal","Couldn't create eigenvalue vector");
      if (!fOffDiagonal.IsValid())
        Error("TPrincipal","Couldn't create offdiagonal vector");
      if (fStoreData) {
        fUserData.ResizeTo(nVariables*1000);
        fUserData.Zero();
        if (!fUserData.IsValid())
          Error("TPrincipal","Couldn't create user data vector");
      }
    }

    //____________________________________________________________________
     TPrincipal::~TPrincipal()
    {
      // destructor

       if (fHistograms) {
          fHistograms->Delete();
          delete fHistograms;
       }
    }

    //____________________________________________________________________
     void TPrincipal::AddRow(const Double_t *p)
    {
      //

/* Add a data point and update the covariance matrix. The input array must be `fNumberOfVariables` long.

The Covariance matrix and mean values of the input data is caculated on the fly by the following equations:

since this is a really fast method, with no rounding errors (please refer to CERN 72-21 pp. 54-106).

The data is stored internally in a `TVectorD`, in the following way:

With as defined in the class description. */ //

if (!p) return; // Increment the data point counter Int_t i,j; if (++fNumberOfDataPoints == 1) { for (Int_t i = 0; i < fNumberOfVariables; i++) fMeanValues(i) = p[i]; } else { Double_t cor = 1 - 1./Double_t(fNumberOfDataPoints); for (i = 0; i < fNumberOfVariables; i++) { fMeanValues(i) *= cor; fMeanValues(i) += p[i] / Double_t(fNumberOfDataPoints); Double_t t1 = (p[i] - fMeanValues(i)) / (fNumberOfDataPoints - 1); // Setting Matrix (lower triangle) elements for (j = 0; j < i + 1; j++) { fCovarianceMatrix(i,j) *= cor; fCovarianceMatrix(i,j) += t1 * (p[j] - fMeanValues(j)); } } } // Store data point in internal vector // If the vector isn't big enough to hold the new data, then // expand the vector by half it's size. if (!fStoreData) return; Int_t size = fUserData.GetNrows(); if (fNumberOfDataPoints * fNumberOfVariables > size) fUserData.ResizeTo(size + size/2); for (i = 0; i < fNumberOfVariables; i++) { j = (fNumberOfDataPoints-1) * fNumberOfVariables + i; fUserData(j) = p[i]; } } //____________________________________________________________________ void TPrincipal::Browse(TBrowser *b) { // Browse the TPrincipal object in the TBrowser. if (fHistograms) { TIter next(fHistograms); TH1* h = 0; while ((h = (TH1*)next())) b->Add(h,h->GetName()); } if (fStoreData) b->Add(&fUserData,"User Data"); b->Add(&fCovarianceMatrix,"Covariance Matrix"); b->Add(&fMeanValues,"Mean value vector"); b->Add(&fSigmas,"Sigma value vector"); b->Add(&fEigenValues,"Eigenvalue vector"); b->Add(&fEigenVectors,"Eigenvector Matrix"); } //____________________________________________________________________ void TPrincipal::Clear(Option_t *opt) { // Clear the data in Object. Notice, that's not possible to change // the dimension of the original data. if (fHistograms) { fHistograms->Delete(opt); } fNumberOfDataPoints = 0; fTrace = 0; fCovarianceMatrix.Zero(); fEigenVectors.Zero(); fEigenValues.Zero(); fMeanValues.Zero(); fSigmas.Zero(); fOffDiagonal.Zero(); if (fStoreData) { fUserData.ResizeTo(fNumberOfVariables * 1000); fUserData.Zero(); } } //____________________________________________________________________ const Double_t *TPrincipal::GetRow(Int_t row) { // Return a row of the user supplied data. // If row is out of bounds, 0 is returned. // It's up to the user to delete the returned array. // Row 0 is the first row; if (row >= fNumberOfDataPoints) return 0; if (!fStoreData) return 0; Int_t index = row * fNumberOfVariables; return &fUserData(index); } //____________________________________________________________________ void TPrincipal::MakeCode(const char *filename, Option_t *opt) { // Generates the file <filename>, with .C appended if it does // argument doesn't end in .cxx or .C. // // The file contains the implementation of two functions // // void X2P(Double_t *x, Double *p) // void P2X(Double_t *p, Double *x, Int_t nTest) // // which does the same as TPrincipal::X2P and TPrincipal::P2X // respectively. Please refer to these methods. // // Further, the static variables: // // Int_t gNVariables // Double_t gEigenValues[] // Double_t gEigenVectors[] // Double_t gMeanValues[] // Double_t gSigmaValues[] // // are initialized. The only ROOT header file needed is Rtypes.h // // See TPrincipal::MakeRealCode for a list of options TString outName(filename); if (!outName.EndsWith(".C") && !outName.EndsWith(".cxx")) outName += ".C"; MakeRealCode(outName.Data(),"",opt); } //____________________________________________________________________ void TPrincipal::MakeHistograms(const char *name, Option_t *opt) { // Make histograms of the result of the analysis. // The option string say which histograms to create // X Histogram original data // P Histogram principal components corresponding to // original data // D Histogram the difference between the original data // and the projection of principal unto a lower // dimensional subspace (2D histograms) // E Histogram the eigenvalues // S Histogram the square of the residues // (see TPrincipal::SumOfSquareResidues) // The histograms will be named <name>_<type><number>, where <name> // is the first argument, <type> is one of X,P,D,E,S, and <number> // is the variable. Bool_t makeX = kFALSE; Bool_t makeD = kFALSE; Bool_t makeP = kFALSE; Bool_t makeE = kFALSE; Bool_t makeS = kFALSE; Int_t len = strlen(opt); Int_t i,j,k; for (i = 0; i < len; i++) { switch (opt[i]) { case 'X': case 'x': if (fStoreData) makeX = kTRUE; break; case 'd': case 'D': if (fStoreData) makeD = kTRUE; break; case 'P': case 'p': if (fStoreData) makeP = kTRUE; break; case 'E': case 'e': makeE = kTRUE; break; case 's': case 'S': if (fStoreData) makeS = kTRUE; break; default: Warning("MakeHistograms","Unknown option: %c",opt[i]); } } // If no option was given, then exit gracefully if (!makeX && !makeD && !makeP && !makeE && !makeS) return; // If the list of histograms doesn't exist, create it. if (!fHistograms) fHistograms = new TList; // Don't create the histograms if they are already in the TList. if (makeX && fHistograms->FindObject(Form("%s_x000",name))) makeX = kFALSE; if (makeD && fHistograms->FindObject(Form("%s_d000",name))) makeD = kFALSE; if (makeP && fHistograms->FindObject(Form("%s_p000",name))) makeP = kFALSE; if (makeE && fHistograms->FindObject(Form("%s_e",name))) makeE = kFALSE; if (makeS && fHistograms->FindObject(Form("%s_s",name))) makeS = kFALSE; TH1F **hX = 0; TH2F **hD = 0; TH1F **hP = 0; TH1F *hE = 0; TH1F *hS = 0; // Initialize the arrays of histograms needed if (makeX) hX = new TH1F * [fNumberOfVariables]; if (makeD) hD = new TH2F * [fNumberOfVariables]; if (makeP) hP = new TH1F * [fNumberOfVariables]; if (makeE){ hE = new TH1F(Form("%s_e",name), "Eigenvalues of Covariance matrix", fNumberOfVariables,0,fNumberOfVariables); hE->SetXTitle("Eigenvalue"); fHistograms->Add(hE); } if (makeS) { hS = new TH1F(Form("%s_s",name),"E_{N}", fNumberOfVariables-1,1,fNumberOfVariables); hS->SetXTitle("N"); hS->SetYTitle("#sum_{i=1}^{M} (x_{i} - x'_{N,i})^{2}"); fHistograms->Add(hS); } // Initialize sub elements of the histogram arrays for (i = 0; i < fNumberOfVariables; i++) { if (makeX) { // We allow 4 sigma spread in the original data in our // histogram. Double_t xlowb = fMeanValues(i) - 4 * fSigmas(i); Double_t xhighb = fMeanValues(i) + 4 * fSigmas(i); Int_t xbins = fNumberOfDataPoints/100; hX[i] = new TH1F(Form("%s_x%03d", name, i), Form("Pattern space, variable %d", i), xbins,xlowb,xhighb); hX[i]->SetXTitle(Form("x_{%d}",i)); fHistograms->Add(hX[i]); } if(makeD) { // The upper limit below is arbitrary!!! Double_t dlowb = 0; Double_t dhighb = 20; Int_t dbins = fNumberOfDataPoints/100; hD[i] = new TH2F(Form("%s_d%03d", name, i), Form("Distance from pattern to " "feature space, variable %d", i), dbins,dlowb,dhighb, fNumberOfVariables-1, 1, fNumberOfVariables); hD[i]->SetXTitle(Form("|x_{%d} - x'_{%d,N}|/#sigma_{%d}",i,i,i)); hD[i]->SetYTitle("N"); fHistograms->Add(hD[i]); } if(makeP) { // For some reason, the trace of the none-scaled matrix // (see TPrincipal::MakeNormalised) should enter here. Taken // from LINTRA code. Double_t plowb = -10 * TMath::Sqrt(fEigenValues(i) * fTrace); Double_t phighb = -plowb; Int_t pbins = 100; hP[i] = new TH1F(Form("%s_p%03d", name, i), Form("Feature space, variable %d", i), pbins,plowb,phighb); hX[i]->SetXTitle(Form("p_{%d}",i)); fHistograms->Add(hP[i]); } if (makeE) // The Eigenvector histogram is easy hE->Fill(i,fEigenValues(i)); } if (!makeX && !makeP && !makeD && !makeS) return; Double_t *x = 0; Double_t *p = new Double_t[fNumberOfVariables]; Double_t *d = new Double_t[fNumberOfVariables]; for (i = 0; i < fNumberOfDataPoints; i++) { // Zero arrays for (j = 0; j < fNumberOfVariables; j++) p[j] = d[j] = 0; // update the original data histogram x = (Double_t*)(GetRow(i)); if (makeP||makeD||makeS) // calculate the corresponding principal component X2P(x,p); if (makeD || makeS) { // Calculate the difference between the original data, and the // same project onto principal components, and then to a lower // dimensional sub-space for (j = fNumberOfVariables; j > 0; j--) { P2X(p,d,j); for (k = 0; k < fNumberOfVariables; k++) { // We use the absolute value of the difference! d[k] = x[k] - d[k]; if (makeS) hS->Fill(j,d[k]*d[k]); if (makeD) { d[k] = TMath::Abs(d[k]) / (fIsNormalised ? fSigmas(k) : 1); (hD[k])->Fill(d[k],j); } } } } if (makeX||makeP) { // If we are asked to make any of these histograms, we have to // go here for (j = 0; j < fNumberOfVariables; j++) { if (makeX) (hX[j])->Fill(x[j]); if (makeP) (hP[j])->Fill(p[j]); } } } // Clean up if (d) delete [] d; if (p) delete [] p; // Normalize the residues if (makeS) hS->Scale(Double_t(1.)/fNumberOfDataPoints); } //____________________________________________________________________ void TPrincipal::MakeNormalised() { // PRIVATE METHOD: Normalize the covariance matrix Int_t i,j; for (i = 0; i < fNumberOfVariables; i++) { fSigmas(i) = TMath::Sqrt(fCovarianceMatrix(i,i)); if (fIsNormalised) for (j = 0; j <= i; j++) fCovarianceMatrix(i,j) /= (fSigmas(i) * fSigmas(j)); fTrace += fCovarianceMatrix(i,i); } // Fill remaining parts of matrix, and scale. for (i = 0; i < fNumberOfVariables; i++) for (j = 0; j <= i; j++) { fCovarianceMatrix(i,j) /= fTrace; fCovarianceMatrix(j,i) = fCovarianceMatrix(i,j); } } //____________________________________________________________________ void TPrincipal::MakeMethods(const char *classname, Option_t *opt) { // Generate the file <classname>PCA.cxx which contains the // implementation of two methods: // // void <classname>::X2P(Double_t *x, Double *p) // void <classname>::P2X(Double_t *p, Double *x, Int_t nTest) // // which does the same as TPrincipal::X2P and TPrincipal::P2X // respectivly. Please refer to these methods. // // Further, the public static members: // // Int_t <classname>::fgNVariables // Double_t <classname>::fgEigenValues[] // Double_t <classname>::fgEigenVectors[] // Double_t <classname>::fgMeanValues[] // Double_t <classname>::fgSigmaValues[] // // are initialized, and assumed to exist. The class declaration is // assumed to be in <classname>.h and assumed to be provided by the // user. // // See TPrincipal::MakeRealCode for a list of options // // The minimal class definition is: // // class <classname> { // public: // static Int_t fgNVariables; // static Double_t fgEigenVectors[]; // static Double_t fgEigenValues[]; // static Double_t fgMeanValues[]; // static Double_t fgSigmaValues[]; // // void X2P(Double_t *x, Double_t *p); // void P2X(Double_t *p, Double_t *x, Int_t nTest); // }; // // Whether the methods <classname>::X2P and <classname>::P2X should // be static or not, is up to the user. MakeRealCode(Form("%sPCA.cxx", classname), classname, opt); } //____________________________________________________________________ void TPrincipal::MakePrincipals() { // Perform the principal components analysis. // This is done in several stages in the TMatrix::EigenVectors method: // * Transform the covariance matrix into a tridiagonal matrix. // * Find the eigenvalues and vectors of the tridiagonal matrix. // Normalize matrix covariance matrix MakeNormalised(); TMatrixDSym sym; sym.Use(fCovarianceMatrix.GetNrows(),fCovarianceMatrix.GetMatrixArray()); TMatrixDSymEigen eigen(sym); fEigenVectors = eigen.GetEigenVectors(); fEigenValues = eigen.GetEigenValues(); } //____________________________________________________________________ void TPrincipal::MakeRealCode(const char *filename, const char *classname, Option_t *) { // PRIVATE METHOD: // This is the method that actually generates the code for the // transformations to and from feature space and pattern space // It's called by TPrincipal::MakeCode and TPrincipal::MakeMethods. // // The options are: NONE so far Bool_t isMethod = (classname[0] == '\0' ? kFALSE : kTRUE); const char *prefix = (isMethod ? Form("%s::", classname) : ""); const char *cv_qual = (isMethod ? "" : "static "); ofstream outFile(filename,ios::out if (!outFile) { Error("MakeRealCode","couldn't open output file '%s'",filename); return; } cout << "Writing on file \"" << filename << "\" ... " << flush; // // Write header of file // // Emacs mode line ;-) outFile << "// -*- mode: c++ -*-" << endl; // Info about creator outFile << "// " << endl << "// File " << filename << " generated by TPrincipal::MakeCode" << endl; // Time stamp TDatime date; outFile << "// on " << date.AsString() << endl; // ROOT version info outFile << "// ROOT version " << gROOT->GetVersion() << endl << "//" << endl; // General information on the code outFile << "// This file contains the functions " << endl << "//" << endl << "// void " << prefix << "X2P(Double_t *x, Double_t *p); " << endl << "// void " << prefix << "P2X(Double_t *p, Double_t *x, Int_t nTest);" << endl << "//" << endl << "// The first for transforming original data x in " << endl << "// pattern space, to principal components p in " << endl << "// feature space. The second function is for the" << endl << "// inverse transformation, but using only nTest" << endl << "// of the principal components in the expansion" << endl << "// " << endl << "// See TPrincipal class documentation for more " << "information " << endl << "// " << endl; // Header files outFile << "#ifndef __CINT__" << endl; if (isMethod) // If these are methods, we need the class header outFile << "#include \"" << classname << ".h\"" << endl; else // otherwise, we need the typedefs of Int_t and Double_t outFile << "#include <Rtypes.h> // needed for Double_t etc" << endl; // Finish the preprocessor block outFile << "#endif" << endl << endl; // // Now for the data // // We make the Eigenvector matrix, Eigenvalue vector, Sigma vector, // and Mean value vector static, since all are needed in both // functions. Also ,the number of variables are stored in a static // variable. outFile << "//" << endl << "// Static data variables" << endl << "//" << endl; outFile << cv_qual << "Int_t " << prefix << "gNVariables = " << fNumberOfVariables << ";" << endl; // Assign the values to the Eigenvector matrix. The elements are // stored row-wise, that is element // M[i][j] = e[i * nVariables + j] // where i and j are zero-based. outFile << endl << "// Assignment of eigenvector matrix." << endl << "// Elements are stored row-wise, that is" << endl << "// M[i][j] = e[i * nVariables + j] " << endl << "// where i and j are zero-based" << endl; outFile << cv_qual << "Double_t " << prefix << "gEigenVectors[] = {" << flush; Int_t i,j; for (i = 0; i < fNumberOfVariables; i++) { for (j = 0; j < fNumberOfVariables; j++) { Int_t index = i * fNumberOfVariables + j; outFile << (index != 0 ? "," : "" ) << endl << " " << fEigenVectors(i,j) << flush; } } outFile << "};" << endl << endl; // Assignment to eigenvalue vector. Zero-based. outFile << "// Assignment to eigen value vector. Zero-based." << endl; outFile << cv_qual << "Double_t " << prefix << "gEigenValues[] = {" << flush; for (i = 0; i < fNumberOfVariables; i++) outFile << (i != 0 ? "," : "") << endl << " " << fEigenValues(i) << flush; outFile << endl << "};" << endl << endl; // Assignment to mean Values vector. Zero-based. outFile << "// Assignment to mean value vector. Zero-based." << endl; outFile << cv_qual << "Double_t " << prefix << "gMeanValues[] = {" << flush; for (i = 0; i < fNumberOfVariables; i++) outFile << (i != 0 ? "," : "") << endl << " " << fMeanValues(i) << flush; outFile << endl << "};" << endl << endl; // Assignment to mean Values vector. Zero-based. outFile << "// Assignment to sigma value vector. Zero-based." << endl; outFile << cv_qual << "Double_t " << prefix << "gSigmaValues[] = {" << flush; for (i = 0; i < fNumberOfVariables; i++) outFile << (i != 0 ? "," : "") << endl << " " << (fIsNormalised ? fSigmas(i) : 1) << flush; // << " " << fSigmas(i) << flush; outFile << endl << "};" << endl << endl; // // Finally we reach the functions themselves // // First: void x2p(Double_t *x, Double_t *p); // outFile << "// " << endl << "// The " << (isMethod ? "method " : "function ") << " void " << prefix << "X2P(Double_t *x, Double_t *p)" << endl << "// " << endl; outFile << "void " << prefix << "X2P(Double_t *x, Double_t *p) {" << endl << " for (Int_t i = 0; i < gNVariables; i++) {" << endl << " p[i] = 0;" << endl << " for (Int_t j = 0; j < gNVariables; j++)" << endl << " p[i] += (x[j] - gMeanValues[j]) " << endl << " * gEigenVectors[j * gNVariables + i] " << "/ gSigmaValues[j];" << endl << endl << " }" << endl << "}" << endl << endl; // // Now: void p2x(Double_t *p, Double_t *x, Int_t nTest); // outFile << "// " << endl << "// The " << (isMethod ? "method " : "function ") << " void " << prefix << "P2X(Double_t *p, Double_t *x, Int_t nTest)" << endl << "// " << endl; outFile << "void " << prefix << "P2X(Double_t *p, Double_t *x, Int_t nTest) {" << endl << " for (Int_t i = 0; i < gNVariables; i++) {" << endl << " x[i] = gMeanValues[i];" << endl << " for (Int_t j = 0; j < nTest; j++)" << endl << " x[i] += p[j] * gSigmaValues[i] " << endl << " * gEigenVectors[i * gNVariables + j];" << endl << " }" << endl << "}" << endl << endl; // EOF outFile << "// EOF for " << filename << endl; // Close the file outFile.close(); cout << "done" << endl; } //____________________________________________________________________ void TPrincipal::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, and return it in x. // It's the users responsibility to make sure that both x and p are // of the right size (i.e., memory must be allocated for x). for (Int_t i = 0; i < fNumberOfVariables; i++){ x[i] = fMeanValues(i); for (Int_t j = 0; j < nTest; j++) x[i] += p[j] * (fIsNormalised ? fSigmas(i) : 1) * fEigenVectors(i,j); } } //____________________________________________________________________ void TPrincipal::Print(Option_t *opt) const { // Print the statistics // Options are // M Print mean values of original data // S Print sigma values of original data // E Print eigenvalues of covarinace matrix // V Print eigenvectors of covarinace matrix // Default is MSE Bool_t printV = kFALSE; Bool_t printM = kFALSE; Bool_t printS = kFALSE; Bool_t printE = kFALSE; Int_t len = strlen(opt); for (Int_t i = 0; i < len; i++) { switch (opt[i]) { case 'V': case 'v': printV = kTRUE; break; case 'M': case 'm': printM = kTRUE; break; case 'S': case 's': printS = kTRUE; break; case 'E': case 'e': printE = kTRUE; break; default: Warning("Print", "Unknown option '%c'",opt[i]); break; } } if (printM||printS||printE) { cout << " Variable # " << flush; if (printM) cout << "| Mean Value " << flush; if (printS) cout << "| Sigma " << flush; if (printE) cout << "| Eigenvalue" << flush; cout << endl; cout << "-------------" << flush; if (printM) cout << "+------------" << flush; if (printS) cout << "+------------" << flush; if (printE) cout << "+------------" << flush; cout << endl; for (Int_t i = 0; i < fNumberOfVariables; i++) { cout << setw(12) << i << " " << flush; if (printM) cout << "| " << setw(10) << setprecision(4) << fMeanValues(i) << " " << flush; if (printS) cout << "| " << setw(10) << setprecision(4) << fSigmas(i) << " " << flush; if (printE) cout << "| " << setw(10) << setprecision(4) << fEigenValues(i) << " " << flush; cout << endl; } cout << endl; } if(printV) { for (Int_t i = 0; i < fNumberOfVariables; i++) { cout << "Eigenvector # " << i << flush; TVectorD v(fNumberOfVariables); v = TMatrixDColumn_const(fEigenVectors,i); v.Print(); } } } //____________________________________________________________________ void TPrincipal::SumOfSquareResiduals(const Double_t *x, Double_t *s) { // PRIVATE METHOD: // /* Calculates the sum of the square residuals, that is

where , is the component of the principal vector, corresponding to , the original data; I.e., the square distance to the space spanned by eigenvectors.
*/ //

if (!x) return; Double_t p[100]; Double_t xp[100]; X2P(x,p); for (Int_t i = fNumberOfVariables-1; i >= 0; i--) { P2X(p,xp,i); for (Int_t j = 0; j < fNumberOfVariables; j++) { s[i] += (x[j] - xp[j])*(x[j] - xp[j]); } } } //____________________________________________________________________ void TPrincipal::Test(Option_t *) { // Test the PCA, bye calculating the sum square of residuals // (see method SumOfSquareResiduals), and display the histogram MakeHistograms("pca","S"); if (!fStoreData) return; TH1 *pca_s = 0; if (fHistograms) pca_s = (TH1*)fHistograms->FindObject("pca_s"); if (!pca_s) { Warning("Test", "Couldn't get histogram of square residuals"); return; } pca_s->Draw(); } //____________________________________________________________________ void TPrincipal::X2P(const Double_t *x, Double_t *p) { // Calculate the principal components from the original data vector // x, and return it in p. // It's the users responsibility to make sure that both x and p are // of the right size (i.e., memory must be allocated for p). for (Int_t i = 0; i < fNumberOfVariables; i++){ p[i] = 0; for (Int_t j = 0; j < fNumberOfVariables; j++) p[i] += (x[j] - fMeanValues(j)) * fEigenVectors(j,i) / (fIsNormalised ? fSigmas(j) : 1); } }