class TPrincipal: public TNamed
>


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:

\begin{displaymath} \mathbf{x} = \left(x_0, x_1, \ldots, x_7\right) \end{displaymath}

in 8-dimensional pattern space.

One proceeds by generating a a representative tracks sample and building up the covariance matrix $\mathsf{C}$. 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,

\begin{displaymath} \mathbf{x}_m = \sum^7_{i=0} a_{m_i} \mathbf{e}_i \quad \mbox{where} \quad a_{m_i} = \mathbf{x}^T\bullet\mathbf{e}_i \end{displaymath}

allows the study of the behavior of the coefficients $a_{m_i}$ for all the tracks of the sample. The eigenvectors which are insignificant for the trajectory description in the expansion will have their corresponding coefficients $a_{m_i}$ 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 ($a_{m_i}$), 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 $M$ prototypes each being characterized by $P$ variables $x_0, x_1, \ldots, x_{P-1}$. Each prototype is a point, or a column vector, in a $P$-dimensional pattern space.

\begin{displaymath} \mathbf{x} = \left[\begin{array}{c} x_0\\ x_1\\ \vdots\\ x_{P-1}\end{array}\right]\,, \end{displaymath} (1)

where each $x_n$ represents the particular value associated with the $n$-dimension.

Those $P$ 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:

\begin{displaymath} \mathsf{C} = \left\langle\mathbf{y}\mathbf{y}^T\right\rangl... ...athbf{y} = \mathbf{x} - \left\langle\mathbf{x}\right\rangle\,, \end{displaymath} (2)

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

This matrix $\mathsf{C}$ 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 $N$ basis vectors $\mathbf{e}_n, n=0,\ldots,N,N+1, \ldots, P-1$,

\begin{displaymath} \mathbf{y}_i = \sum^N_{i=0} a_{i_n} \mathbf{e}_n, \quad i = 0, \ldots, M, \quad N < P-1 \end{displaymath} (3)

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

\begin{displaymath} \min\left(E_N\right) = \min\left[\left\langle\left(\mathb... ...\sum^N_{i=0} a_{i_n} \mathbf{e}_n\right)^2\right\rangle\right] \end{displaymath} (4)

with the conditions:
\begin{displaymath} \mathbf{e}_k\bullet\mathbf{e}_j = \delta_{jk} = \left\{\b... ...for} & k = j\\ 0 & \mbox{for} & k \neq j \end{array}\right. \end{displaymath} (5)

Multiplying (3) by $\mathbf{e}^T_n$ using (5), we get

\begin{displaymath} a_{i_n} = \mathbf{y}_i^T\bullet\mathbf{e}_n\,, \end{displaymath} (6)

so the error becomes
$\displaystyle E_N$ $\textstyle =$ $\displaystyle \left\langle\left[\sum_{n=N+1}^{P-1} a_{i_n}\mathbf{e}_n\right]^2\right\rangle$  
  $\textstyle =$ $\displaystyle \left\langle\left[\sum_{n=N+1}^{P-1} \mathbf{y}_i^T\bullet\mathbf{e}_n\mathbf{e}_n\right]^2\right\rangle$  
  $\textstyle =$ $\displaystyle \left\langle\sum_{n=N+1}^{P-1} \mathbf{e}_n^T\mathbf{y}_i\mathbf{y}_i^T\mathbf{e}_n\right\rangle$  
  $\textstyle =$ $\displaystyle \sum_{n=N+1}^{P-1} \mathbf{e}_n^T\mathsf{C}\mathbf{e}_n$ (7)

The minimization of the sum in (7) is obtained when each term $\mathbf{e}_n^\mathsf{C}\mathbf{e}_n$ is minimum, since $\mathsf{C}$ is positive definite. By the method of Lagrange multipliers, and the condition (5), we get


\begin{displaymath} E_N = \sum^{P-1}_{n=N+1} \left(\mathbf{e}_n^T\mathsf{C}\mathbf{e}_n - l_n\mathbf{e}_n^T\bullet\mathbf{e}_n + l_n\right) \end{displaymath} (8)

The minimum condition $\frac{dE_N}{d\mathbf{e}^T_n} = 0$ leads to the equation
\begin{displaymath} \mathsf{C}\mathbf{e}_n = l_n\mathbf{e}_n\,, \end{displaymath} (9)

which shows that $\mathbf{e}_n$ is an eigenvector of the covariance matrix $\mathsf{C}$ with eigenvalue $l_n$. The estimated minimum error is then given by
\begin{displaymath} E_N \sim \sum^{P-1}_{n=N+1} \mathbf{e}_n^T\bullet l_n\mathbf{e}_n = \sum^{P-1}_{n=N+1} l_n\,, \end{displaymath} (10)

where $l_n,\,n=N+1,\ldots,P-1$ are the eigenvalues associated with the omitted eigenvectors in the expansion (3). Thus, by choosing the $N$ largest eigenvalues, and their associated eigenvectors, the error $E_N$ is minimized.

The transformation matrix to go from the pattern space to the feature space consists of the ordered eigenvectors $\mathbf{e}_0,\ldots,\mathbf{e}_{P-1}$ for its columns

\begin{displaymath} \mathsf{T} = \left[ \begin{array}{cccc} \mathbf{e}_0 & \... ...bf{e}_{1_{P-1}} & \cdots & \mathbf{e}_{{P-1}_{P-1}}\\ \end{array}\right] \end{displaymath} (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 







Function Members (Methods)


public:



TPrincipal()


TPrincipal(Int_t nVariables, Option_t* opt = "ND")


 virtual~TPrincipal()


voidTObject::AbstractMethod(const char* method) const


virtual voidAddRow(const Double_t* x)


virtual voidTObject::AppendPad(Option_t* option = "")


virtual voidBrowse(TBrowser* b)


static TClass*Class()


virtual const char*TObject::ClassName() const


virtual voidClear(Option_t* option = "")


virtual TObject*TNamed::Clone(const char* newname = "") const


virtual Int_tTNamed::Compare(const TObject* obj) const


virtual voidTNamed::Copy(TObject& named) const


virtual voidTObject::Delete(Option_t* option = "")MENU 


virtual Int_tTObject::DistancetoPrimitive(Int_t px, Int_t py)


virtual voidTObject::Draw(Option_t* option = "")


virtual voidTObject::DrawClass() constMENU 


virtual TObject*TObject::DrawClone(Option_t* option = "") constMENU 


virtual voidTObject::Dump() constMENU 


virtual voidTObject::Error(const char* method, const char* msgfmt) const


virtual voidTObject::Execute(const char* method, const char* params, Int_t* error = 0)


virtual voidTObject::Execute(TMethod* method, TObjArray* params, Int_t* error = 0)


virtual voidTObject::ExecuteEvent(Int_t event, Int_t px, Int_t py)


virtual voidTObject::Fatal(const char* method, const char* msgfmt) const


virtual voidTNamed::FillBuffer(char*& buffer)


virtual TObject*TObject::FindObject(const char* name) const


virtual TObject*TObject::FindObject(const TObject* obj) const


const TMatrixD*GetCovarianceMatrix() const


virtual Option_t*TObject::GetDrawOption() const


static Long_tTObject::GetDtorOnly()


const TVectorD*GetEigenValues() const


const TMatrixD*GetEigenVectors() const


TList*GetHistograms() const


virtual const char*TObject::GetIconName() const


const TVectorD*GetMeanValues() const


virtual const char*TNamed::GetName() const


virtual char*TObject::GetObjectInfo(Int_t px, Int_t py) const


static Bool_tTObject::GetObjectStat()


virtual Option_t*TObject::GetOption() const


const Double_t*GetRow(Int_t row)


const TVectorD*GetSigmas() const


virtual const char*TNamed::GetTitle() const


virtual UInt_tTObject::GetUniqueID() const


const TVectorD*GetUserData() const


virtual Bool_tTObject::HandleTimer(TTimer* timer)


virtual ULong_tTNamed::Hash() const


virtual voidTObject::Info(const char* method, const char* msgfmt) const


virtual Bool_tTObject::InheritsFrom(const char* classname) const


virtual Bool_tTObject::InheritsFrom(const TClass* cl) const


virtual voidTObject::Inspect() constMENU 


voidTObject::InvertBit(UInt_t f)


virtual TClass*IsA() const


virtual Bool_tTObject::IsEqual(const TObject* obj) const


virtual Bool_tIsFolder() const


Bool_tTObject::IsOnHeap() const


virtual Bool_tTNamed::IsSortable() const


Bool_tTObject::IsZombie() const


virtual voidTNamed::ls(Option_t* option = "") const


virtual voidMakeCode(const char* filename = "pca", Option_t* option = "")MENU 


virtual voidMakeHistograms(const char* name = "pca", Option_t* option = "epsdx")MENU 


virtual voidMakeMethods(const char* classname = "PCA", Option_t* option = "")MENU 


virtual voidMakePrincipals()MENU 


voidTObject::MayNotUse(const char* method) const


virtual Bool_tTObject::Notify()


static voidTObject::operator delete(void* ptr)


static voidTObject::operator delete(void* ptr, void* vp)


static voidTObject::operator delete[](void* ptr)


static voidTObject::operator delete[](void* ptr, void* vp)


void*TObject::operator new(size_t sz)


void*TObject::operator new(size_t sz, void* vp)


void*TObject::operator new[](size_t sz)


void*TObject::operator new[](size_t sz, void* vp)


virtual voidP2X(const Double_t* p, Double_t* x, Int_t nTest)


virtual voidTObject::Paint(Option_t* option = "")


virtual voidTObject::Pop()


virtual voidPrint(Option_t* opt = "MSE") constMENU 


virtual Int_tTObject::Read(const char* name)


virtual voidTObject::RecursiveRemove(TObject* obj)


voidTObject::ResetBit(UInt_t f)


virtual voidTObject::SaveAs(const char* filename = "", Option_t* option = "") constMENU 


virtual voidTObject::SavePrimitive(basic_ostream<char,char_traits<char> >& out, Option_t* option = "")


voidTObject::SetBit(UInt_t f)


voidTObject::SetBit(UInt_t f, Bool_t set)


virtual voidTObject::SetDrawOption(Option_t* option = "")MENU 


static voidTObject::SetDtorOnly(void* obj)


virtual voidTNamed::SetName(const char* name)MENU 


virtual voidTNamed::SetNameTitle(const char* name, const char* title)


static voidTObject::SetObjectStat(Bool_t stat)


virtual voidTNamed::SetTitle(const char* title = "")MENU 


virtual voidTObject::SetUniqueID(UInt_t uid)


virtual voidShowMembers(TMemberInspector& insp, char* parent)


virtual Int_tTNamed::Sizeof() const


virtual voidStreamer(TBuffer& b)


voidStreamerNVirtual(TBuffer& b)


virtual voidSumOfSquareResiduals(const Double_t* x, Double_t* s)


virtual voidTObject::SysError(const char* method, const char* msgfmt) const


voidTest(Option_t* option = "")MENU 


Bool_tTObject::TestBit(UInt_t f) const


Int_tTObject::TestBits(UInt_t f) const


virtual voidTObject::UseCurrentStyle()


virtual voidTObject::Warning(const char* method, const char* msgfmt) const


virtual Int_tTObject::Write(const char* name = 0, Int_t option = 0, Int_t bufsize = 0)


virtual Int_tTObject::Write(const char* name = 0, Int_t option = 0, Int_t bufsize = 0) const


virtual voidX2P(const Double_t* x, Double_t* p)





protected:



TPrincipal(const TPrincipal&)


virtual voidTObject::DoError(int level, const char* location, const char* fmt, va_list va) const


voidMakeNormalised()


voidMakeRealCode(const char* filename, const char* prefix, Option_t* option = "")


voidTObject::MakeZombie()


TPrincipal&operator=(const TPrincipal&)










Data Members


public:



enum TObject::EStatusBits { kCanDelete


kMustCleanup


kObjInCanvas


kIsReferenced


kHasUUID


kCannotPick


kNoContextMenu


kInvalidObject


};


enum TObject::[unnamed] { kIsOnHeap


kNotDeleted


kZombie


kBitMask


kSingleKey


kOverwrite


kWriteDelete


};





protected:



TMatrixDfCovarianceMatrixCovariance matrix


TVectorDfEigenValuesEigenvalue vector of trans


TMatrixDfEigenVectorsEigenvector matrix of trans


TList*fHistogramsList of histograms


Bool_tfIsNormalisedNormalize matrix?


TVectorDfMeanValuesMean value over all data points


TStringTNamed::fNameobject identifier


Int_tfNumberOfDataPointsNumber of data points


Int_tfNumberOfVariablesNumber of variables


TVectorDfOffDiagonalelements of the tridiagonal


TVectorDfSigmasvector of sigmas


Bool_tfStoreDataShould we store input data?


TStringTNamed::fTitleobject title


Double_tfTraceTrace of covarience matrix


TVectorDfUserDataVector of original data points







Class Charts

























Inheritance
Inherited Members
Includes
Libraries



Class Charts


Function documentation


 TPrincipal(const TPrincipal& )


 Empty CTOR, Do not use.





 TPrincipal(Int_t nVariables, Option_t* opt = "ND")


 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.





 TPrincipal(const TPrincipal& )


copy constructor





TPrincipal& operator=(const TPrincipal& )


assignement operator





 ~TPrincipal()


 destructor





void AddRow(const Double_t* x)


  /*
     >
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:





\begin{displaymath}
\left<x_i\right>^{(0)} = x_{i0}
\end{displaymath}











\begin{displaymath}
\left<x_i\right>^{(n)} = \left<x_i\right>^{(n-1)}
+ \frac1n \left(x_{in} - \left<x_i\right>^{(n-1)}\right)
\end{displaymath}











\begin{displaymath}
C_{ij}^{(0)} = 0
\end{displaymath}











\begin{displaymath}
C_{ij}^{(n)} = C_{ij}^{(n-1)}
+ \frac1{n-1}\left[\left(x_{i...
...\left<x_j\right>^{(n)}\right)\right]
- \frac1n C_{ij}^{(n-1)}
\end{displaymath}






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:





\begin{displaymath}
\mathbf{x} = \left[\left(x_{0_0},\ldots,x_{{P-1}_0}\right),\ldots,
\left(x_{0_i},\ldots,x_{{P-1}_i}\right), \ldots\right]
\end{displaymath}






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





void Browse(TBrowser* b)


 Browse the TPrincipal object in the TBrowser.





void Clear(Option_t* option = "")


 Clear the data in Object. Notice, that's not possible to change
 the dimension of the original data.





const Double_t * 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;





void MakeCode(const char* filename = "pca", Option_t* option = "")


 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





void MakeHistograms(const char* name = "pca", Option_t* option = "epsdx")


 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.





void MakeNormalised()


 PRIVATE METHOD: Normalize the covariance matrix





void MakeMethods(const char* classname = "PCA", Option_t* option = "")


 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.





void 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.





void MakeRealCode(const char* filename, const char* prefix, Option_t* option = "")


 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





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, 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).





void Print(Option_t* opt = "MSE") 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





void SumOfSquareResiduals(const Double_t* x, Double_t* s)


 PRIVATE METHOD:


   /*
    >
    Calculates the sum of the square residuals, that is
    

    

    
    \begin{displaymath}
    E_N = \sum_{i=0}^{P-1} \left(x_i - x^\prime_i\right)^2
    \end{displaymath}
    

    

    

    where
    
    $x^\prime_i = \sum_{j=i}^N p_i e_{n_j}$, $p_i$ is the
    $i^{\mbox{th}}$ component of the principal vector, corresponding to
    $x_i$, the original data; I.e., the square distance to the space
    spanned by $N$ eigenvectors.
    

    
   */
 





void Test(Option_t* option = "")


 Test the PCA, bye calculating the sum square of residuals
 (see method SumOfSquareResiduals), and display the histogram





void 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).





const TMatrixD    * GetCovarianceMatrix()



{return &fCovarianceMatrix;}






const TVectorD    * GetEigenValues()



{return &fEigenValues;}






const TMatrixD    * GetEigenVectors()



{return &fEigenVectors;}






TList             * GetHistograms()



{return fHistograms;}






const TVectorD    * GetMeanValues()



{return &fMeanValues;}






const TVectorD    * GetSigmas()



{return &fSigmas;}






const TVectorD    * GetUserData()



{return &fUserData;}






Bool_t IsFolder()



{ return kTRUE;}









Author:  Christian Holm Christensen    1/8/2000

Last change: root/hist:$Id: TPrincipal.h 20882 2007-11-19 11:31:26Z rdm $

Last generated: 2008-06-25 08:50

Copyright   (C) 1995-2004, Rene Brun and Fons Rademakers.               *




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