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)
| library: libHist #include "TMultiDimFit.h" |
| Inheritance Chart: | |||||||||||||
|
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)
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
/*
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.
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)
are 
independent
variables, 
is the known, quantity dependent at
,
and 
is the square error in , the class
TMultiDimFit
will
try to find the parameterization
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.
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
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).
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
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 ![$ [-1,1]$](gif/multidimfit_img50.gif){kind=small}
for this to
succeed [5]
(TMultiDimFit::MakeNormalized). We then define a
matrix
of which the columns
are given by
is the component of
orthogonal
to
. Hence we obtain
[3],
We now take as a new model
. We thus want to
minimize
is a vector of the
dependent quantity in the sample. Differentiation with respect to
gives, using (6),
be the sum of squares of residuals when taking functions
into account. Then
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.
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
function to the reduction of S, is
given by
Two test are now applied to decide whether this
function is to be included in the final expression, or not.
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).
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))
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.
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
is an upper triangle matrix, which can be
readily inverted. So we now evaluate
can therefore be written as
is therefore identical with
this if
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).
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):
are the 5 dependent
quantities that define a track.
different values of
, the tracks
through the magnetic field, and determine the corresponding
.
. We call these values
.
a set of at least five relevant
coordinates
, using contrains, or
alternative:
, so that
are constrained and linear independent.
, to get linear
indenpendent (among themselves, but not independent of
) quantities
make a mutlidimensional fit,
using
as the variables, thus determing a set of
coefficents
.
To process data, using this parameterisation, do
within the domain of
the parameterization, using the result from the Principal Component
Analysis.
as before.
as before.
.
from
, using
the result from the Principal Component Analysis.
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
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.
*/
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 theclass 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 theclass 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 theclass 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 toclass description
void MakeCandidates()
PRIVATE METHOD: Create list of candidate functions for the parameterisation. See alsoclass 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 theclass descriptionFirst 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 theclass 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 toclass 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 theclass 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 alsoclass 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 alsoclass 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 theclass 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 theclass description
void SetMaxPowers(const Int_t* powers)
Set the maximum power to be considered in the fit for each variable. See alsoclass 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 theclass description
Bool_t TestFunction(Double_t squareResidual, Double_t dResidur)
PRIVATE METHOD: Test whether the currently considered function contributes to the fit. See alsoclass 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 1.22 2005/09/05 10:02:38 brun Exp $
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with
if
![$\displaystyle S_l = \left[\mathbf{D} - \sum^l_{k=1} a_k\mathbf{w}_k\right]^2 = ...
...2 - 2\mathbf{D} \sum^l_{k=1} a_k\mathbf{w}_k + \sum^l_{k=1} a_k^2\mathbf{w}_k^2$](gif/multidimfit_img70.gif)
and
![$\displaystyle \mathbf{c} = \left(\mathsf{B}^{-1}\mathbf{a}\right) = \left[\mathbf{a}^T\left(\mathsf{B}^{-1}\right)^T\right]^T .$](gif/multidimfit_img97.gif)