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ROOT 6.10/09 Reference Guide |
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ROOT 6.10/09 Reference Guide |
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.
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).
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:
\[ \mathbf{x} = \left(x_0, x_1, \ldots, x_7\right) \]
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,
\[ \mathbf{x}_m = \sum^7_{i=0} a_{m_i} \mathbf{e}_i \quad \mbox{where} \quad a_{m_i} = \mathbf{x}^T\bullet\mathbf{e}_i \]
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 their data.
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.
\[ \mathbf{x} = \left[\begin{array}{c} x_0\\x_1\\\vdots\\x_{P-1}\end{array}\right]\,, \]
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:
\[ \mathsf{C} = \left\langle\mathbf{y}\mathbf{y}^T\right\rangle \quad\mbox{where}\quad \mathbf{y} = \mathbf{x} - \left\langle\mathbf{x}\right\rangle\,, \]
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 \(\mathbf{e}_n, n=0,\ldots,N,N+1, \ldots, P-1\)
\[ \mathbf{y}_i = \sum^N_{i=0} a_{i_n} \mathbf{e}_n, \quad i = 1, \ldots, M, \quad N < P-1 \]
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.,
\[ \min\left(E_N\right) = \min\left[\left\langle\left(\mathbf{y}_i - \sum^N_{i=0} a_{i_n} \mathbf{e}_n\right)^2\right\rangle\right] \]
with the conditions:
\[ \mathbf{e}_k\bullet\mathbf{e}_j = \delta_{jk} = \left\{\begin{array}{rcl} 1 & \mbox{for} & k = j\\ 0 & \mbox{for} & k \neq j \end{array}\right. \]
Multiplying (3) by \(\mathbf{e}^T_n\) using (5), we get
\[ a_{i_n} = \mathbf{y}_i^T\bullet\mathbf{e}_n\,, \]
so the error becomes
\begin{eqnarray*} E_N &=& \left\langle\left[\sum_{n=N+1}^{P-1} a_{i_n}\mathbf{e}_n\right]^2\right\rangle\nonumber\\ &=& \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\\ &=& \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\\ &=& \sum_{n=N+1}^{P-1} \mathbf{e}_n^T\mathsf{C}\mathbf{e}_n \end{eqnarray*}
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
\[ E_N = \sum^{P-1}_{n=N+1} \left(\mathbf{e}_n^T\mathsf{C}\mathbf{e}_n - l_n\mathbf{e}_n^T\bullet\mathbf{e}_n + l_n\right) \]
The minimum condition \(\frac{dE_N}{d\mathbf{e}^T_n} = 0\) leads to the equation
\[ \mathsf{C}\mathbf{e}_n = l_n\mathbf{e}_n\,, \]
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
\[ E_N \sim \sum^{P-1}_{n=N+1} \mathbf{e}_n^T\bullet l_n\mathbf{e}_n = \sum^{P-1}_{n=N+1} l_n\,, \]
where \(l_n,\,n=N+1,\ldots,P\) \(l_n,\,n=N+1,\ldots,P-1\) 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 \(\mathbf{e}_1,\ldots,\mathbf{e}_P\) \(\mathbf{e}_0,\ldots,\mathbf{e}_{P-1}\) for its columns
\[ \mathsf{T} = \left[ \begin{array}{cccc} \mathbf{e}_0 & \mathbf{e}_1 & \vdots & \mathbf{e}_{P-1} \end{array}\right] = \left[ \begin{array}{cccc} \mathbf{e}_{0_0} & \mathbf{e}_{1_0} & \cdots & \mathbf{e}_{{P-1}_0}\\ \mathbf{e}_{0_1} & \mathbf{e}_{1_1} & \cdots & \mathbf{e}_{{P-1}_1}\\ \vdots & \vdots & \ddots & \vdots \\ \mathbf{e}_{0_{P-1}} & \mathbf{e}_{1_{P-1}} & \cdots & \mathbf{e}_{{P-1}_{P-1}}\\ \end{array}\right] \]
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
Definition at line 20 of file TPrincipal.h.
Public Member Functions | |
| TPrincipal () | |
| Empty constructor. Do not use. More... | |
| TPrincipal (Int_t nVariables, Option_t *opt="ND") | |
| Constructor. More... | |
| virtual | ~TPrincipal () |
| Destructor. More... | |
| virtual void | AddRow (const Double_t *x) |
| Add a data point and update the covariance matrix. More... | |
| virtual void | Browse (TBrowser *b) |
| Browse the TPrincipal object in the TBrowser. More... | |
| virtual void | Clear (Option_t *option="") |
| Clear the data in Object. More... | |
| const TMatrixD * | GetCovarianceMatrix () const |
| const TVectorD * | GetEigenValues () const |
| const TMatrixD * | GetEigenVectors () const |
| TList * | GetHistograms () const |
| const TVectorD * | GetMeanValues () const |
| const Double_t * | GetRow (Int_t row) |
| Return a row of the user supplied data. More... | |
| const TVectorD * | GetSigmas () const |
| const TVectorD * | GetUserData () const |
| Bool_t | IsFolder () const |
| Returns kTRUE in case object contains browsable objects (like containers or lists of other objects). More... | |
| 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 .cxx or .C. More... | |
| virtual void | MakeHistograms (const char *name="pca", Option_t *option="epsdx") |
| Make histograms of the result of the analysis. More... | |
| virtual void | MakeMethods (const char *classname="PCA", Option_t *option="") |
| Generate the file <classname>PCA.cxx which contains the implementation of two methods: which does the same as TPrincipal::X2P and TPrincipal::P2X respectively. More... | |
| virtual void | MakePrincipals () |
| Perform the principal components analysis. More... | |
| 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, and return it in x. More... | |
| virtual void | Print (Option_t *opt="MSE") const |
| Print the statistics Options are. More... | |
| virtual void | SumOfSquareResiduals (const Double_t *x, Double_t *s) |
| Calculates the sum of the square residuals, that is. More... | |
| void | Test (Option_t *option="") |
| Test the PCA, bye calculating the sum square of residuals (see method SumOfSquareResiduals), and display the histogram. More... | |
| 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. More... | |
 Public Member Functions inherited from TNamed | |
| TNamed () | |
| TNamed (const char *name, const char *title) | |
| TNamed (const TString &name, const TString &title) | |
| TNamed (const TNamed &named) | |
| TNamed copy ctor. More... | |
| virtual | ~TNamed () |
| virtual TObject * | Clone (const char *newname="") const |
| Make a clone of an object using the Streamer facility. More... | |
| virtual Int_t | Compare (const TObject *obj) const |
| Compare two TNamed objects. More... | |
| virtual void | Copy (TObject &named) const |
| Copy this to obj. More... | |
| virtual void | FillBuffer (char *&buffer) |
| Encode TNamed into output buffer. More... | |
| virtual const char * | GetName () const |
| Returns name of object. More... | |
| virtual const char * | GetTitle () const |
| Returns title of object. More... | |
| virtual ULong_t | Hash () const |
| Return hash value for this object. More... | |
| virtual Bool_t | IsSortable () const |
| virtual void | ls (Option_t *option="") const |
| List TNamed name and title. More... | |
| TNamed & | operator= (const TNamed &rhs) |
| TNamed assignment operator. More... | |
| virtual void | SetName (const char *name) |
| Set the name of the TNamed. More... | |
| virtual void | SetNameTitle (const char *name, const char *title) |
| Set all the TNamed parameters (name and title). More... | |
| virtual void | SetTitle (const char *title="") |
| Set the title of the TNamed. More... | |
| virtual Int_t | Sizeof () const |
| Return size of the TNamed part of the TObject. More... | |
| Public Member Functions inherited from TObject | |
| TObject () | |
| TObject constructor. More... | |
| TObject (const TObject &object) | |
| TObject copy ctor. More... | |
| virtual | ~TObject () |
| TObject destructor. More... | |
| void | AbstractMethod (const char *method) const |
| Use this method to implement an "abstract" method that you don't want to leave purely abstract. More... | |
| virtual void | AppendPad (Option_t *option="") |
| Append graphics object to current pad. More... | |
| virtual const char * | ClassName () const |
| Returns name of class to which the object belongs. More... | |
| virtual void | Delete (Option_t *option="") |
| Delete this object. More... | |
| virtual Int_t | DistancetoPrimitive (Int_t px, Int_t py) |
| Computes distance from point (px,py) to the object. More... | |
| virtual void | Draw (Option_t *option="") |
| Default Draw method for all objects. More... | |
| virtual void | DrawClass () const |
| Draw class inheritance tree of the class to which this object belongs. More... | |
| virtual TObject * | DrawClone (Option_t *option="") const |
Draw a clone of this object in the current selected pad for instance with: gROOT->SetSelectedPad(gPad). More... | |
| virtual void | Dump () const |
| Dump contents of object on stdout. More... | |
| virtual void | Error (const char *method, const char *msgfmt,...) const |
| Issue error message. More... | |
| virtual void | Execute (const char *method, const char *params, Int_t *error=0) |
| Execute method on this object with the given parameter string, e.g. More... | |
| virtual void | Execute (TMethod *method, TObjArray *params, Int_t *error=0) |
| Execute method on this object with parameters stored in the TObjArray. More... | |
| virtual void | ExecuteEvent (Int_t event, Int_t px, Int_t py) |
| Execute action corresponding to an event at (px,py). More... | |
| virtual void | Fatal (const char *method, const char *msgfmt,...) const |
| Issue fatal error message. More... | |
| virtual TObject * | FindObject (const char *name) const |
| Must be redefined in derived classes. More... | |
| virtual TObject * | FindObject (const TObject *obj) const |
| Must be redefined in derived classes. More... | |
| virtual Option_t * | GetDrawOption () const |
| Get option used by the graphics system to draw this object. More... | |
| virtual const char * | GetIconName () const |
| Returns mime type name of object. More... | |
| virtual char * | GetObjectInfo (Int_t px, Int_t py) const |
| Returns string containing info about the object at position (px,py). More... | |
| virtual Option_t * | GetOption () const |
| virtual UInt_t | GetUniqueID () const |
| Return the unique object id. More... | |
| virtual Bool_t | HandleTimer (TTimer *timer) |
| Execute action in response of a timer timing out. More... | |
| virtual void | Info (const char *method, const char *msgfmt,...) const |
| Issue info message. More... | |
| virtual Bool_t | InheritsFrom (const char *classname) const |
| Returns kTRUE if object inherits from class "classname". More... | |
| virtual Bool_t | InheritsFrom (const TClass *cl) const |
| Returns kTRUE if object inherits from TClass cl. More... | |
| virtual void | Inspect () const |
| Dump contents of this object in a graphics canvas. More... | |
| void | InvertBit (UInt_t f) |
| virtual Bool_t | IsEqual (const TObject *obj) const |
| Default equal comparison (objects are equal if they have the same address in memory). More... | |
| Bool_t | IsOnHeap () const |
| Bool_t | IsZombie () const |
| void | MayNotUse (const char *method) const |
| Use this method to signal that a method (defined in a base class) may not be called in a derived class (in principle against good design since a child class should not provide less functionality than its parent, however, sometimes it is necessary). More... | |
| virtual Bool_t | Notify () |
| This method must be overridden to handle object notification. More... | |
| void | Obsolete (const char *method, const char *asOfVers, const char *removedFromVers) const |
| Use this method to declare a method obsolete. More... | |
| void | operator delete (void *ptr) |
| Operator delete. More... | |
| void | operator delete[] (void *ptr) |
| Operator delete []. More... | |
| void * | operator new (size_t sz) |
| void * | operator new (size_t sz, void *vp) |
| void * | operator new[] (size_t sz) |
| void * | operator new[] (size_t sz, void *vp) |
| TObject & | operator= (const TObject &rhs) |
| TObject assignment operator. More... | |
| virtual void | Paint (Option_t *option="") |
| This method must be overridden if a class wants to paint itself. More... | |
| virtual void | Pop () |
| Pop on object drawn in a pad to the top of the display list. More... | |
| virtual Int_t | Read (const char *name) |
| Read contents of object with specified name from the current directory. More... | |
| virtual void | RecursiveRemove (TObject *obj) |
| Recursively remove this object from a list. More... | |
| void | ResetBit (UInt_t f) |
| virtual void | SaveAs (const char *filename="", Option_t *option="") const |
| Save this object in the file specified by filename. More... | |
| virtual void | SavePrimitive (std::ostream &out, Option_t *option="") |
| Save a primitive as a C++ statement(s) on output stream "out". More... | |
| void | SetBit (UInt_t f, Bool_t set) |
| Set or unset the user status bits as specified in f. More... | |
| void | SetBit (UInt_t f) |
| virtual void | SetDrawOption (Option_t *option="") |
| Set drawing option for object. More... | |
| virtual void | SetUniqueID (UInt_t uid) |
| Set the unique object id. More... | |
| virtual void | SysError (const char *method, const char *msgfmt,...) const |
| Issue system error message. More... | |
| Bool_t | TestBit (UInt_t f) const |
| Int_t | TestBits (UInt_t f) const |
| virtual void | UseCurrentStyle () |
| Set current style settings in this object This function is called when either TCanvas::UseCurrentStyle or TROOT::ForceStyle have been invoked. More... | |
| virtual void | Warning (const char *method, const char *msgfmt,...) const |
| Issue warning message. More... | |
| virtual Int_t | Write (const char *name=0, Int_t option=0, Int_t bufsize=0) |
| Write this object to the current directory. More... | |
| virtual Int_t | Write (const char *name=0, Int_t option=0, Int_t bufsize=0) const |
| Write this object to the current directory. More... | |
Protected Member Functions | |
| TPrincipal (const TPrincipal &) | |
| Copy constructor. More... | |
| void | MakeNormalised () |
| Normalize the covariance matrix. More... | |
| 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 and pattern space It's called by TPrincipal::MakeCode and TPrincipal::MakeMethods. More... | |
| TPrincipal & | operator= (const TPrincipal &) |
| Assignment operator. More... | |
| Protected Member Functions inherited from TObject | |
| virtual void | DoError (int level, const char *location, const char *fmt, va_list va) const |
| Interface to ErrorHandler (protected). More... | |
| void | MakeZombie () |
Protected Attributes | |
| TMatrixD | fCovarianceMatrix |
| TVectorD | fEigenValues |
| TMatrixD | fEigenVectors |
| TList * | fHistograms |
| Bool_t | fIsNormalised |
| TVectorD | fMeanValues |
| Int_t | fNumberOfDataPoints |
| Int_t | fNumberOfVariables |
| TVectorD | fOffDiagonal |
| TVectorD | fSigmas |
| Bool_t | fStoreData |
| Double_t | fTrace |
| TVectorD | fUserData |
| Protected Attributes inherited from TNamed | |
| TString | fName |
| TString | fTitle |
Additional Inherited Members | |
| Public Types inherited from TObject | |
| enum | { kIsOnHeap = 0x01000000, kNotDeleted = 0x02000000, kZombie = 0x04000000, kBitMask = 0x00ffffff } |
| enum | { kSingleKey = BIT(0), kOverwrite = BIT(1), kWriteDelete = BIT(2) } |
| enum | EStatusBits { kCanDelete = BIT(0), kMustCleanup = BIT(3), kObjInCanvas = BIT(3), kIsReferenced = BIT(4), kHasUUID = BIT(5), kCannotPick = BIT(6), kNoContextMenu = BIT(8), kInvalidObject = BIT(13) } |
| Static Public Member Functions inherited from TObject | |
| static Long_t | GetDtorOnly () |
| Return destructor only flag. More... | |
| static Bool_t | GetObjectStat () |
| Get status of object stat flag. More... | |
| static void | SetDtorOnly (void *obj) |
| Set destructor only flag. More... | |
| static void | SetObjectStat (Bool_t stat) |
| Turn on/off tracking of objects in the TObjectTable. More... | |
#include <TPrincipal.h>
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protected |
Copy constructor.
Definition at line 312 of file TPrincipal.cxx.
| TPrincipal::TPrincipal | ( | ) |
Empty constructor. Do not use.
Definition at line 229 of file TPrincipal.cxx.
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virtual |
Destructor.
Definition at line 357 of file TPrincipal.cxx.
Constructor.
Argument is number of variables in the sample of data Options are:
The created object is named "principal" by default.
Definition at line 253 of file TPrincipal.cxx.
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 calculated on the fly by the following equations:
\[ \left<x_i\right>^{(0)} = x_{i0} \]
\[ \left<x_i\right>^{(n)} = \left<x_i\right>^{(n-1)} + \frac1n \left(x_{in} - \left<x_i\right>^{(n-1)}\right) \]
\[ C_{ij}^{(0)} = 0 \]
\[ C_{ij}^{(n)} = C_{ij}^{(n-1)} + \frac1{n-1}\left[\left(x_{in} - \left<x_i\right>^{(n)}\right) \left(x_{jn} - \left<x_j\right>^{(n)}\right)\right] - \frac1n C_{ij}^{(n-1)} \]
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:
\[ \mathbf{x} = \left[\left(x_{0_0},\ldots,x_{{P-1}_0}\right),\ldots, \left(x_{0_i},\ldots,x_{{P-1}_i}\right), \ldots\right] \]
With \(P\) as defined in the class description.
Definition at line 410 of file TPrincipal.cxx.
Browse the TPrincipal object in the TBrowser.
Reimplemented from TObject.
Definition at line 457 of file TPrincipal.cxx.
Clear the data in Object.
Notice, that's not possible to change the dimension of the original data.
Reimplemented from TNamed.
Definition at line 480 of file TPrincipal.cxx.
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inline |
Definition at line 58 of file TPrincipal.h.
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inline |
Definition at line 59 of file TPrincipal.h.
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inline |
Definition at line 60 of file TPrincipal.h.
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inline |
Definition at line 61 of file TPrincipal.h.
|
inline |
Definition at line 62 of file TPrincipal.h.
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;
Definition at line 507 of file TPrincipal.cxx.
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inline |
Definition at line 64 of file TPrincipal.h.
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inline |
Definition at line 65 of file TPrincipal.h.
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inlinevirtual |
Returns kTRUE in case object contains browsable objects (like containers or lists of other objects).
Reimplemented from TObject.
Definition at line 66 of file TPrincipal.h.
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
which does the same as TPrincipal::X2P and TPrincipal::P2X respectively. Please refer to these methods.
Further, the static variables:
are initialized. The only ROOT header file needed is Rtypes.h
See TPrincipal::MakeRealCode for a list of options
Definition at line 544 of file TPrincipal.cxx.
Make histograms of the result of the analysis.
The option string say which histograms to create
TPrincipal::SumOfSquareResiduals) 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. Definition at line 569 of file TPrincipal.cxx.
Generate the file <classname>PCA.cxx which contains the implementation of two methods:
which does the same as TPrincipal::X2P and TPrincipal::P2X respectively.
Please refer to these methods.
Further, the public static members:
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:
Whether the methods <classname>X2P and <classname>P2X should be static or not, is up to the user.
Definition at line 849 of file TPrincipal.cxx.
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protected |
Normalize the covariance matrix.
Definition at line 787 of file TPrincipal.cxx.
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virtual |
Perform the principal components analysis.
This is done in several stages in the TMatrix::EigenVectors method:
Definition at line 862 of file TPrincipal.cxx.
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protected |
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
Definition at line 884 of file TPrincipal.cxx.
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protected |
Assignment operator.
Definition at line 333 of file TPrincipal.cxx.
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).
Definition at line 1053 of file TPrincipal.cxx.
Print the statistics Options are.
Reimplemented from TNamed.
Definition at line 1073 of file TPrincipal.cxx.
Calculates the sum of the square residuals, that is.
\[ E_N = \sum_{i=0}^{P-1} \left(x_i - x^\prime_i\right)^2 \]
where \(x^\prime_i = \sum_{j=i}^N p_i e_{n_j}\) 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.
Definition at line 1162 of file TPrincipal.cxx.
Test the PCA, bye calculating the sum square of residuals (see method SumOfSquareResiduals), and display the histogram.
Definition at line 1184 of file TPrincipal.cxx.
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).
Definition at line 1208 of file TPrincipal.cxx.
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protected |
Definition at line 28 of file TPrincipal.h.
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protected |
Definition at line 31 of file TPrincipal.h.
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protected |
Definition at line 30 of file TPrincipal.h.
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protected |
Definition at line 39 of file TPrincipal.h.
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protected |
Definition at line 41 of file TPrincipal.h.
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protected |
Definition at line 26 of file TPrincipal.h.
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protected |
Definition at line 23 of file TPrincipal.h.
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protected |
Definition at line 24 of file TPrincipal.h.
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protected |
Definition at line 33 of file TPrincipal.h.
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Definition at line 27 of file TPrincipal.h.
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Definition at line 42 of file TPrincipal.h.
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Definition at line 37 of file TPrincipal.h.
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Definition at line 35 of file TPrincipal.h.
ROOT 6.10/09 - Reference Guide Generated on Thu May 31 2018 12:15:05 using Doxygen 1.8.13.