library: libHist #include "TMultiDimFit.h"

# TMultiDimFit

class description - source file - inheritance tree (.pdf)

## class TMultiDimFit : public TNamed

| Inheritance Chart:
`TObject`
| <- | `TNamed`
| <- | `TMultiDimFit` |

        protected:

          virtual Double_t EvalControl(const Int_t* powers)
          virtual Double_t EvalFactor(Int_t p, Double_t x)
              virtual void MakeCandidates()
              virtual void MakeCoefficientErrors()
              virtual void MakeCoefficients()
              virtual void MakeCorrelation()
          virtual Double_t MakeGramSchmidt(Int_t function)
              virtual void MakeNormalized()
              virtual void MakeParameterization()
              virtual void MakeRealCode(const char* filename, const char* classname, Option_t* option = "")
            virtual Bool_t Select(const Int_t* iv)
            virtual Bool_t TestFunction(Double_t squareResidual, Double_t dResidur)

        public:

                               TMultiDimFit()
                               TMultiDimFit(Int_t dimension, TMultiDimFit::EMDFPolyType type = kMonomials, Option_t* option = "")
                               TMultiDimFit(const TMultiDimFit&)
                       virtual ~TMultiDimFit()
                  virtual void AddRow(const Double_t* x, Double_t D, Double_t E = 0)
                  virtual void AddTestRow(const Double_t* x, Double_t D, Double_t E = 0)
                  virtual void Browse(TBrowser* b)
                static TClass* Class()
                  virtual void Clear(Option_t* option = "")
                  virtual void Draw(Option_t* = "d")
              virtual Double_t Eval(const Double_t* x, const Double_t* coeff = 0)
                  virtual void FindParameterization(Option_t* option = "")
                  virtual void Fit(Option_t* option = "")
                      Double_t GetChi2() const
               const TVectorD* GetCoefficients() const
               const TMatrixD* GetCorrelationMatrix() const
                      Double_t GetError() const
                        Int_t* GetFunctionCodes() const
               const TMatrixD* GetFunctions() const
                virtual TList* GetHistograms() const
                      Double_t GetMaxAngle() const
                         Int_t GetMaxFunctions() const
                        Int_t* GetMaxPowers() const
                      Double_t GetMaxQuantity() const
                         Int_t GetMaxStudy() const
                         Int_t GetMaxTerms() const
               const TVectorD* GetMaxVariables() const
                      Double_t GetMeanQuantity() const
               const TVectorD* GetMeanVariables() const
                      Double_t GetMinAngle() const
                      Double_t GetMinQuantity() const
                      Double_t GetMinRelativeError() const
               const TVectorD* GetMinVariables() const
                         Int_t GetNCoefficients() const
                         Int_t GetNVariables() const
                         Int_t GetPolyType() const
                        Int_t* GetPowerIndex() const
                      Double_t GetPowerLimit() const
                  const Int_t* GetPowers() const
                      Double_t GetPrecision() const
               const TVectorD* GetQuantity() const
                      Double_t GetResidualMax() const
                         Int_t GetResidualMaxRow() const
                      Double_t GetResidualMin() const
                         Int_t GetResidualMinRow() const
                      Double_t GetResidualSumSq() const
                      Double_t GetRMS() const
                         Int_t GetSampleSize() const
               const TVectorD* GetSqError() const
                      Double_t GetSumSqAvgQuantity() const
                      Double_t GetSumSqQuantity() const
                      Double_t GetTestError() const
                      Double_t GetTestPrecision() const
               const TVectorD* GetTestQuantity() const
                         Int_t GetTestSampleSize() const
               const TVectorD* GetTestSqError() const
               const TVectorD* GetTestVariables() const
               const TVectorD* GetVariables() const
          static TMultiDimFit* Instance()
               virtual TClass* IsA() const
                virtual Bool_t IsFolder() const
              virtual Double_t MakeChi2(const Double_t* coeff = 0)
                  virtual void MakeCode(const char* functionName = "MDF", Option_t* option = "")
                  virtual void MakeHistograms(Option_t* option = "A")
                  virtual void MakeMethod(const Char_t* className = "MDF", Option_t* option = "")
                 TMultiDimFit& operator=(const TMultiDimFit&)
                  virtual void Print(Option_t* option = "ps") const
                          void SetMaxAngle(Double_t angle = 0)
                          void SetMaxFunctions(Int_t n)
                          void SetMaxPowers(const Int_t* powers)
                          void SetMaxStudy(Int_t n)
                          void SetMaxTerms(Int_t terms)
                          void SetMinAngle(Double_t angle = 1)
                          void SetMinRelativeError(Double_t error)
                          void SetPowerLimit(Double_t limit = 1e-3)
                  virtual void SetPowers(const Int_t* powers, Int_t terms)
                  virtual void ShowMembers(TMemberInspector& insp, char* parent)
                  virtual void Streamer(TBuffer& b)
                          void StreamerNVirtual(TBuffer& b)

### Data Members

        private:

          static TMultiDimFit* fgInstance  Static instance

        protected:

                            TVectorD fQuantity              Training sample, dependent quantity
                            TVectorD fSqError               Training sample, error in quantity
                            Double_t fMeanQuantity          Mean of dependent quantity
                            Double_t fMaxQuantity           Max value of dependent quantity
                            Double_t fMinQuantity           Min value of dependent quantity
                            Double_t fSumSqQuantity         SumSquare of dependent quantity
                            Double_t fSumSqAvgQuantity      Sum of squares away from mean
                            TVectorD fVariables             Training sample, independent variables
                               Int_t fNVariables            Number of independent variables
                            TVectorD fMeanVariables         mean value of independent variables
                            TVectorD fMaxVariables          max value of independent variables
                            TVectorD fMinVariables          min value of independent variables
                               Int_t fSampleSize            Size of training sample
                            TVectorD fTestQuantity          Test sample, dependent quantity
                            TVectorD fTestSqError           Test sample, Error in quantity
                            TVectorD fTestVariables         Test sample, independent variables
                               Int_t fTestSampleSize        Size of test sample
                            Double_t fMinAngle              Min angle for acepting new function
                            Double_t fMaxAngle              Max angle for acepting new function
                               Int_t fMaxTerms              Max terms expected in final expr.
                            Double_t fMinRelativeError      Min relative error accepted
                              Int_t* fMaxPowers             [fNVariables] maximum powers
                            Double_t fPowerLimit            Control parameter
                            TMatrixD fFunctions             Functions evaluated over sample
                               Int_t fMaxFunctions          max number of functions
                              Int_t* fFunctionCodes         [fMaxFunctions] acceptance code
                               Int_t fMaxStudy              max functions to study
                            TMatrixD fOrthFunctions         As above, but orthogonalised
                            TVectorD fOrthFunctionNorms     Norm of the evaluated functions
                              Int_t* fMaxPowersFinal        [fNVariables] maximum powers from fit;
                              Int_t* fPowers                [fMaxFunctions*fNVariables]
                              Int_t* fPowerIndex            [fMaxTerms] Index of accepted powers
                            TVectorD fResiduals             Vector of the final residuals
                            Double_t fMaxResidual           Max redsidual value
                            Double_t fMinResidual           Min redsidual value
                               Int_t fMaxResidualRow        Row giving max residual
                               Int_t fMinResidualRow        Row giving min residual
                            Double_t fSumSqResidual         Sum of Square residuals
                               Int_t fNCoefficients         Dimension of model coefficients
                            TVectorD fOrthCoefficients      The model coefficients
                            TMatrixD fOrthCurvatureMatrix   Model matrix
                            TVectorD fCoefficients          Vector of the final coefficients
                            TVectorD fCoefficientsRMS       Vector of RMS of coefficients
                            Double_t fRMS                   Root mean square of fit
                            Double_t fChi2                  Chi square of fit
                               Int_t fParameterisationCode  Exit code of parameterisation
                            Double_t fError                 Error from parameterization
                            Double_t fTestError             Error from test
                            Double_t fPrecision             Relative precision of param
                            Double_t fTestPrecision         Relative precision of test
                            Double_t fCorrelationCoeff      Multi Correlation coefficient
                            TMatrixD fCorrelationMatrix     Correlation matrix
                            Double_t fTestCorrelationCoeff  Multi Correlation coefficient
                              TList* fHistograms            List of histograms
                       unsigned char fHistogramMask         Bit pattern of hisograms used
                     TVirtualFitter* fFitter                ! Fit object (MINUIT)
          TMultiDimFit::EMDFPolyType fPolyType              Type of polynomials to use
                              Bool_t fShowCorrelation       print correlation matrix
                              Bool_t fIsUserFunction        Flag for user defined function
                              Bool_t fIsVerbose

        public:

          static const TMultiDimFit::EMDFPolyType kMonomials
          static const TMultiDimFit::EMDFPolyType kChebyshev
          static const TMultiDimFit::EMDFPolyType kLegendre

## Class Description

/*

# Multidimensional Fits in ROOT

#
Overview

A common problem encountered in different fields of applied science is to find an expression for one physical quantity in terms of several others, which are directly measurable.

An example in high energy physics is the evaluation of the momentum of a charged particle from the observation of its trajectory in a magnetic field. The problem is to relate the momentum of the particle to the observations, which may consists of of positional measurements at intervals along the particle trajectory.

The exact functional relationship between the measured quantities (e.g., the space-points) and the dependent quantity (e.g., the momentum) is in general not known, but one possible way of solving the problem, is to find an expression which reliably approximates the dependence of the momentum on the observations.

This explicit function of the observations can be obtained by a least squares fitting procedure applied to a representive sample of the data, for which the dependent quantity (e.g., momentum) and the independent observations are known. The function can then be used to compute the quantity of interest for new observations of the independent variables.

This class `TMultiDimFit` implements such a procedure in ROOT. It is largely based on the CERNLIB MUDIFI package [2]. Though the basic concepts are still sound, and therefore kept, a few implementation details have changed, and this class can take advantage of MINUIT [4] to improve the errors of the fitting, thanks to the class `TMinuit`.

In [5] and [6] H. Wind demonstrates the utility of this procedure in the context of tracking, magnetic field parameterisation, and so on. The outline of the method used in this class is based on Winds discussion, and I refer these two excellents text for more information.

And example of usage is given in `$ROOTSYS/tutorials/multidimfit.C`.

#
The Method

Let by the dependent quantity of interest, which depends smoothly on the observable quantities , which we'll denote by . Given a training sample of tuples of the form, (`TMultiDimFit::AddRow`)

where are independent variables, is the known, quantity dependent at , and is the square error in , the class `TMultiDimFit` will try to find the parameterization

| (1)

such that

| (2)

is minimal. Here are monomials, or Chebyshev or Legendre polynomials, labelled , in each variable , .

So what `TMultiDimFit` does, is to determine the number of terms , and then terms (or functions) , and the coefficients , so that is minimal (`TMultiDimFit::FindParameterization`).

Of course it's more than a little unlikely that will ever become exact zero as a result of the procedure outlined below. Therefore, the user is asked to provide a minimum relative error (`TMultiDimFit::SetMinRelativeError`), and will be considered minimized when

Optionally, the user may impose a functional expression by specifying the powers of each variable in specified functions (`TMultiDimFit::SetPowers`). In that case, only the coefficients is calculated by the class.

##
Limiting the Number of Terms

As always when dealing with fits, there's a real chance of over fitting. As is well-known, it's always possible to fit an polynomial in to points with , but the polynomial is not likely to fit new data at all [1]. Therefore, the user is asked to provide an upper limit, to the number of terms in (`TMultiDimFit::SetMaxTerms`).

However, since there's an infinite number of to choose from, the user is asked to give the maximum power. , of each variable to be considered in the minimization of (`TMultiDimFit::SetMaxPowers`).

One way of obtaining values for the maximum power in variable , is to perform a regular fit to the dependent quantity , using a polynomial only in . The maximum power is is then the power that does not significantly improve the one-dimensional least-square fit over to [5].

There are still a huge amount of possible choices for ; in fact there are possible choices. Obviously we need to limit this. To this end, the user is asked to set a power control limit, (`TMultiDimFit::SetPowerLimit`), and a function is only accepted if

where is the leading power of variable in function . (`TMultiDimFit::MakeCandidates`). So the number of functions increase with (1, 2 is fine, 5 is way out).

## Gram-Schmidt Orthogonalisation

To further reduce the number of functions in the final expression, only those functions that significantly reduce is chosen. What `significant' means, is chosen by the user, and will be discussed below (see 2.3).

The functions are generally not orthogonal, which means one will have to evaluate all possible 's over all data-points before finding the most significant [1]. We can, however, do better then that. By applying the modified Gram-Schmidt orthogonalisation algorithm [5] [3] to the functions , we can evaluate the contribution to the reduction of from each function in turn, and we may delay the actual inversion of the curvature-matrix (`TMultiDimFit::MakeGramSchmidt`).

So we are let to consider an matrix , an element of which is given by

with | (3)

where labels the rows in the training sample and labels functions of variables, and . That is, is the term (or function) numbered evaluated at the data point . We have to normalise to for this to succeed [5] (`TMultiDimFit::MakeNormalized`). We then define a matrix of which the columns are given by

|  |  | (4)
|  |  | (5)

and is the component of orthogonal to . Hence we obtain [3],

if | (6)

We now take as a new model . We thus want to minimize

| (7)

where is a vector of the dependent quantity in the sample. Differentiation with respect to gives, using (6),

| (8)

or

| (9)

Let be the sum of squares of residuals when taking functions into account. Then

| (10)

Using (9), we see that

|  |  | (11)

So for each new function included in the model, we get a reduction of the sum of squares of residuals of , where is given by (4) and by (9). Thus, using the Gram-Schmidt orthogonalisation, we can decide if we want to include this function in the final model, before the matrix inversion.

##
Function Selection Based on Residual

Supposing that steps of the procedure have been performed, the problem now is to consider the function.

The sum of squares of residuals can be written as

| (12)

where the relation (9) have been taken into account. The contribution of the function to the reduction of S, is given by

| (13)

Two test are now applied to decide whether this function is to be included in the final expression, or not.

###
Test 1

Denoting by the subspace spanned by the function is by construction (see (4)) the projection of the function onto the direction perpendicular to . Now, if the length of (given by ) is very small compared to the length of this new function can not contribute much to the reduction of the sum of squares of residuals. The test consists then in calculating the angle between the two vectors and (see also figure 1) and requiring that it's greater then a threshold value which the user must set (`TMultiDimFit::SetMinAngle`).

Figure 1: (a) Angle between and , (b) angle between and

###
Test 2

Let be the data vector to be fitted. As illustrated in figure 1, the function will contribute significantly to the reduction of , if the angle between and is smaller than an upper limit , defined by the user (`TMultiDimFit::SetMaxAngle`)

However, the method automatically readjusts the value of this angle while fitting is in progress, in order to make the selection criteria less and less difficult to be fulfilled. The result is that the functions contributing most to the reduction of are chosen first (`TMultiDimFit::TestFunction`).

In case isn't defined, an alternative method of performing this second test is used: The function is accepted if (refer also to equation (13))

| (14)

where is the sum of the first residuals from the functions previously accepted; and is the total number of functions allowed in the final expression of the fit (defined by user).

>From this we see, that by restricting -- the number of terms in the final model -- the fit is more difficult to perform, since the above selection criteria is more limiting.

The more coefficients we evaluate, the more the sum of squares of residuals will be reduced. We can evaluate before inverting as shown below.

## Coefficients and Coefficient Errors

Having found a parameterization, that is the 's and , that minimizes , we still need to determine the coefficients . However, it's a feature of how we choose the significant functions, that the evaluation of the 's becomes trivial [5]. To derive , we first note that equation (4) can be written as

| (15)

where

| (16)

Consequently, is an upper triangle matrix, which can be readily inverted. So we now evaluate

| (17)

The model can therefore be written as

The original model is therefore identical with this if

| (18)

The reason we use rather then is to save storage, since can be stored in the same matrix as (`TMultiDimFit::MakeCoefficients`). The errors in the coefficients is calculated by inverting the curvature matrix of the non-orthogonal functions [1] (`TMultiDimFit::MakeCoefficientErrors`).

##
Considerations

It's important to realize that the training sample should be representive of the problem at hand, in particular along the borders of the region of interest. This is because the algorithm presented here, is a interpolation, rahter then a extrapolation [5].

Also, the independent variables need to be linear independent, since the procedure will perform poorly if they are not [5]. One can find an linear transformation from ones original variables to a set of linear independent variables , using a Principal Components Analysis (see `TPrincipal`), and then use the transformed variable as input to this class [5] [6].

H. Wind also outlines a method for parameterising a multidimensional dependence over a multidimensional set of variables. An example of the method from [5], is a follows (please refer to [5] for a full discussion):

1. Define are the 5 dependent quantities that define a track.
2. Compute, for different values of , the tracks through the magnetic field, and determine the corresponding .
3. Use the simulated observations to determine, with a simple approximation, the values of . We call these values .
4. Determine from a set of at least five relevant coordinates , using contrains, or alternative:
5. Perform a Principal Component Analysis (using `TPrincipal`), and use to get a linear transformation , so that are constrained and linear independent.
6. Perform a Principal Component Analysis on , to get linear indenpendent (among themselves, but not independent of ) quantities
7. For each component make a mutlidimensional fit, using as the variables, thus determing a set of coefficents .

To process data, using this parameterisation, do

1. Test wether the observation within the domain of the parameterization, using the result from the Principal Component Analysis.
2. Determine as before.
3. Detetmine as before.
4. Use the result of the fit to determind .
5. Transform back to from , using the result from the Principal Component Analysis.

##
Testing the parameterization

The class also provides functionality for testing the, over the training sample, found parameterization (`TMultiDimFit::Fit`). This is done by passing the class a test sample of tuples of the form , where are the independent variables, the known, dependent quantity, and is the square error in (`TMultiDimFit::AddTestRow`).

The parameterization is then evaluated at every in the test sample, and

is evaluated. The relative error over the test sample

should not be to low or high compared to from the training sample. Also, multiple correlation coefficient from both samples should be fairly close, otherwise one of the samples is not representive of the problem. A large difference in the reduced over the two samples indicate an over fit, and the maximum number of terms in the parameterisation should be reduced.

It's possible to use Minuit [4] to further improve the fit, using the test sample.

Christian Holm
November 2000, NBI

## Bibliography

1
  Philip R. Bevington and D. Keith Robinson.
  Data Reduction and Error Analysis for the Physical Sciences.
  McGraw-Hill, 2 edition, 1992.

2
  René Brun et al.
  Mudifi.
  Long writeup DD/75-23, CERN, 1980.

3
  Gene H. Golub and Charles F. van Loan.
  Matrix Computations.
  John Hopkins Univeristy Press, Baltimore, 3 edition, 1996.

4
  F. James.
  Minuit.
  Long writeup D506, CERN, 1998.

5
  H. Wind.
  Function parameterization.
  In Proceedings of the 1972 CERN Computing and Data Processing School, volume 72-21 of Yellow report. CERN, 1972.

6
  H. Wind.
  1. principal component analysis, 2. pattern recognition for track finding, 3. interpolation and functional representation.
  Yellow report EP/81-12, CERN, 1981.

###

     */

     TMultiDimFit()

     Empty CTOR. Do not use

     TMultiDimFit(Int_t dimension,
    			   EMDFPolyType type,
    			   Option_t *option)
      : TNamed("multidimfit","Multi-dimensional fit object"),
        fQuantity(dimension),
        fSqError(dimension),
        fVariables(dimension*100),
        fMeanVariables(dimension),
        fMaxVariables(dimension),
        fMinVariables(dimension)

     Constructor
     Second argument is the type of polynomials to use in
     parameterisation, one of:
          TMultiDimFit::kMonomials
          TMultiDimFit::kChebyshev
          TMultiDimFit::kLegendre

     Options:
       K      Compute (k)correlation matrix
       V      Be verbose

     Default is no options.

     ~TMultiDimFit()

     DTOR

    void AddRow(const Double_t *x, Double_t D, Double_t E)

     Add a row consisting of fNVariables independent variables, the
     known, dependent quantity, and optionally, the square error in
     the dependent quantity, to the training sample to be used for the
     parameterization.
     The mean of the variables and quantity is calculated on the fly,
     as outlined in TPrincipal::AddRow.
     This sample should be representive of the problem at hand.
     Please note, that if no error is given Poisson statistics is
     assumed and the square error is set to the value of dependent
     quantity.  See also the

    class description

    void AddTestRow(const Double_t *x, Double_t D, Double_t E)

     Add a row consisting of fNVariables independent variables, the
     known, dependent quantity, and optionally, the square error in
     the dependent quantity, to the test sample to be used for the
     test of the parameterization.
     This sample needn't be representive of the problem at hand.
     Please note, that if no error is given Poisson statistics is
     assumed and the square error is set to the value of dependent
     quantity.  See also the

    class description

    void Browse(TBrowser* b)

     Browse the TMultiDimFit object in the TBrowser.

    void Clear(Option_t *option)

     Clear internal structures and variables

    Double_t Eval(const Double_t *x, const Double_t* coeff)

     Evaluate parameterization at point x. Optional argument coeff is
     a vector of coefficients for the parameterisation, fNCoefficients
     elements long.

    Double_t EvalControl(const Int_t *iv)

     PRIVATE METHOD:
     Calculate the control parameter from the passed powers

    Double_t EvalFactor(Int_t p, Double_t x)

     PRIVATE METHOD:
     Evaluate function with power p at variable value x

    void FindParameterization(Option_t *)

     Find the parameterization

     Options:
         None so far

     For detailed description of what this entails, please refer to the

    class description

    void Fit(Option_t *option)

     Try to fit the found parameterisation to the test sample.

     Options
         M     use Minuit to improve coefficients

     Also, refer to

    class description

    void MakeCandidates()

     PRIVATE METHOD:
     Create list of candidate functions for the parameterisation. See
     also

    class description

    Double_t MakeChi2(const Double_t* coeff)

     Calculate Chi square over either the test sample. The optional
     argument coeff is a vector of coefficients to use in the
     evaluation of the parameterisation. If coeff == 0, then the found
     coefficients is used.
     Used my MINUIT for fit (see TMultDimFit::Fit)

    void MakeCode(const char* filename, Option_t *option)

     Generate the file <filename> with .C appended if argument doesn't
     end in .cxx or .C. The contains the implementation of the
     function:

       Double_t <funcname>(Double_t *x)

     which does the same as TMultiDimFit::Eval. Please refer to this
     method.

     Further, the static variables:

         Int_t    gNVariables
         Int_t    gNCoefficients
         Double_t gDMean
         Double_t gXMean[]
         Double_t gXMin[]
         Double_t gXMax[]
         Double_t gCoefficient[]
         Int_t    gPower[]

     are initialized. The only ROOT header file needed is Rtypes.h

     See TMultiDimFit::MakeRealCode for a list of options

    void MakeCoefficientErrors()

     PRIVATE METHOD:
     Compute the errors on the coefficients. For this to be done, the
     curvature matrix of the non-orthogonal functions, is computed.

    void MakeCoefficients()

     PRIVATE METHOD:
     Invert the model matrix B, and compute final coefficients. For a
     more thorough discussion of what this means, please refer to the

    class description

     First we invert the lower triangle matrix fOrthCurvatureMatrix
     and store the inverted matrix in the upper triangle.

    void MakeCorrelation()

     PRIVATE METHOD:
     Compute the correlation matrix

    Double_t MakeGramSchmidt(Int_t function)

     PRIVATE METHOD:
     Make Gram-Schmidt orthogonalisation. The class description gives
     a thorough account of this algorithm, as well as
     references. Please refer to the

    class description

    void MakeHistograms(Option_t *option)

     Make histograms of the result of the analysis. This message
     should be sent after having read all data points, but before
     finding the parameterization

     Options:
         A         All the below
         X         Original independent variables
         D         Original dependent variables
         N         Normalised independent variables
         S         Shifted dependent variables
         R1        Residuals versus normalised independent variables
         R2        Residuals versus dependent variable
         R3        Residuals computed on training sample
         R4        Residuals computed on test sample

     For a description of these quantities, refer to

    class description

    void MakeMethod(const Char_t* classname, Option_t* option)

     Generate the file <classname>MDF.cxx which contains the
     implementation of the method:

       Double_t <classname>::MDF(Double_t *x)

     which does the same as  TMultiDimFit::Eval. Please refer to this
     method.

     Further, the public static members:

       Int_t    <classname>::fgNVariables
       Int_t    <classname>::fgNCoefficients
       Double_t <classname>::fgDMean
       Double_t <classname>::fgXMean[]       //[fgNVariables]
       Double_t <classname>::fgXMin[]        //[fgNVariables]
       Double_t <classname>::fgXMax[]        //[fgNVariables]
       Double_t <classname>::fgCoefficient[] //[fgNCoeffficents]
       Int_t    <classname>::fgPower[]       //[fgNCoeffficents*fgNVariables]

     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 TMultiDimFit::MakeRealCode for a list of options

     The minimal class definition is:

       class <classname> {
       public:
         Int_t    <classname>::fgNVariables;     // Number of variables
         Int_t    <classname>::fgNCoefficients;  // Number of terms
         Double_t <classname>::fgDMean;          // Mean from training sample
         Double_t <classname>::fgXMean[];        // Mean from training sample
         Double_t <classname>::fgXMin[];         // Min from training sample
         Double_t <classname>::fgXMax[];         // Max from training sample
         Double_t <classname>::fgCoefficient[];  // Coefficients
         Int_t    <classname>::fgPower[];        // Function powers

         Double_t Eval(Double_t *x);
       };

     Whether the method <classname>::Eval should be static or not, is
     up to the user.

    void MakeNormalized()

     PRIVATE METHOD:
     Normalize data to the interval [-1;1]. This is needed for the
     classes method to work.

    void MakeParameterization()

     PRIVATE METHOD:
     Find the parameterization over the training sample. A full account
     of the algorithm is given in the

    class description

    void MakeRealCode(const char *filename,
    				const char *classname,
    				Option_t *)

     PRIVATE METHOD:
     This is the method that actually generates the code for the
     evaluation the parameterization on some point.
     It's called by TMultiDimFit::MakeCode and TMultiDimFit::MakeMethod.

     The options are: NONE so far

    void Print(Option_t *option) const

     Print statistics etc.
     Options are
       P        Parameters
       S        Statistics
       C        Coefficients
       R        Result of parameterisation
       F        Result of fit
       K        Correlation Matrix
       M        Pretty print formula

    Bool_t Select(const Int_t *)

     Selection method. User can override this method for specialized
     selection of acceptable functions in fit. Default is to select
     all. This message is sent during the build-up of the function
     candidates table once for each set of powers in
     variables. Notice, that the argument array contains the powers
     PLUS ONE. For example, to De select the function
         f = x1^2 * x2^4 * x3^5,
     this method should return kFALSE if given the argument
         { 3, 4, 6 }

    void SetMaxAngle(Double_t ang)

     Set the max angle (in degrees) between the initial data vector to
     be fitted, and the new candidate function to be included in the
     fit.  By default it is 0, which automatically chooses another
     selection criteria. See also

    class description

    void SetMinAngle(Double_t ang)

     Set the min angle (in degrees) between a new candidate function
     and the subspace spanned by the previously accepted
     functions. See also

    class description

    void SetPowers(const Int_t* powers, Int_t terms)

     Define a user function. The input array must be of the form
     (p11, ..., p1N, ... ,pL1, ..., pLN)
     Where N is the dimension of the data sample, L is the number of
     terms (given in terms) and the first number, labels the term, the
     second the variable.  More information is given in the

    class description

    void SetPowerLimit(Double_t limit)

     Set the user parameter for the function selection. The bigger the
     limit, the more functions are used. The meaning of this variable
     is defined in the

    class description

    void SetMaxPowers(const Int_t* powers)

     Set the maximum power to be considered in the fit for each
     variable. See also

    class description

    void SetMinRelativeError(Double_t error)

     Set the acceptable relative error for when sum of square
     residuals is considered minimized. For a full account, refer to
     the

    class description

    Bool_t TestFunction(Double_t squareResidual,
    				  Double_t dResidur)

     PRIVATE METHOD:
     Test whether the currently considered function contributes to the
     fit. See also

    class description

    Inline Functions

                       void Draw(Option_t* = "d")
                   Double_t GetChi2() const
            const TMatrixD* GetCorrelationMatrix() const
            const TVectorD* GetCoefficients() const
                   Double_t GetError() const
                     Int_t* GetFunctionCodes() const
            const TMatrixD* GetFunctions() const
                     TList* GetHistograms() const
                   Double_t GetMaxAngle() const
                      Int_t GetMaxFunctions() const
                     Int_t* GetMaxPowers() const
                   Double_t GetMaxQuantity() const
                      Int_t GetMaxStudy() const
                      Int_t GetMaxTerms() const
            const TVectorD* GetMaxVariables() const
                   Double_t GetMeanQuantity() const
            const TVectorD* GetMeanVariables() const
                   Double_t GetMinAngle() const
                   Double_t GetMinQuantity() const
                   Double_t GetMinRelativeError() const
            const TVectorD* GetMinVariables() const
                      Int_t GetNVariables() const
                      Int_t GetNCoefficients() const
                      Int_t GetPolyType() const
                     Int_t* GetPowerIndex() const
                   Double_t GetPowerLimit() const
               const Int_t* GetPowers() const
                   Double_t GetPrecision() const
            const TVectorD* GetQuantity() const
                   Double_t GetResidualMax() const
                   Double_t GetResidualMin() const
                      Int_t GetResidualMaxRow() const
                      Int_t GetResidualMinRow() const
                   Double_t GetResidualSumSq() const
                   Double_t GetRMS() const
                      Int_t GetSampleSize() const
            const TVectorD* GetSqError() const
                   Double_t GetSumSqAvgQuantity() const
                   Double_t GetSumSqQuantity() const
                   Double_t GetTestError() const
                   Double_t GetTestPrecision() const
            const TVectorD* GetTestQuantity() const
                      Int_t GetTestSampleSize() const
            const TVectorD* GetTestSqError() const
            const TVectorD* GetTestVariables() const
            const TVectorD* GetVariables() const
              TMultiDimFit* Instance()
                     Bool_t IsFolder() const
                       void SetMaxFunctions(Int_t n)
                       void SetMaxStudy(Int_t n)
                       void SetMaxTerms(Int_t terms)
                    TClass* Class()
                    TClass* IsA() const
                       void ShowMembers(TMemberInspector& insp, char* parent)
                       void Streamer(TBuffer& b)
                       void StreamerNVirtual(TBuffer& b)
               TMultiDimFit TMultiDimFit(const TMultiDimFit&)
              TMultiDimFit& operator=(const TMultiDimFit&)

    Author: Christian Holm Christensen 07/11/2000

    Last update: root/hist:$Name:  $:$Id: TMultiDimFit.cxx,v 1.18 2005/04/12 16:26:01 brun Exp $