library: libMinuit #include "TLinearFitter.h" |
TLinearFitter
class description - header file - source file - inheritance tree (.pdf)
private:
void AddToDesign(Double_t* x, Double_t y, Double_t e)
void ComputeTValues()
void CreateSubset(Int_t ntotal, Int_t h, Int_t* index)
Double_t CStep(Int_t step, Int_t h, Double_t* residuals, Int_t* index, Int_t* subdat, Int_t start, Int_t end)
void Graph2DLinearFitter(Double_t h)
void GraphLinearFitter(Double_t h)
void HistLinearFitter()
Bool_t Linf()
void MultiGraphLinearFitter(Double_t h)
Int_t Partition(Int_t nmini, Int_t* indsubdat)
void RDraw(Int_t* subdat, Int_t* indsubdat)
protected:
TLinearFitter& operator=(const TLinearFitter&)
public:
TLinearFitter()
TLinearFitter(Int_t ndim, const char* formula, Option_t* opt = "D")
TLinearFitter(Int_t ndim)
TLinearFitter(TFormula* function, Option_t* opt = "D")
TLinearFitter(const TLinearFitter&)
virtual ~TLinearFitter()
virtual void AddPoint(Double_t* x, Double_t y, Double_t e = 1)
virtual void AssignData(Int_t npoints, Int_t xncols, Double_t* x, Double_t* y, Double_t* e = 0)
virtual void Chisquare()
virtual Double_t Chisquare(Int_t, Double_t*) const
static TClass* Class()
virtual void Clear(Option_t* option = "")
virtual void ClearPoints()
virtual void Eval()
virtual void EvalRobust(Double_t h = -1)
virtual Int_t ExecuteCommand(const char* command, Double_t* args, Int_t nargs)
virtual void FixParameter(Int_t ipar)
virtual void FixParameter(Int_t ipar, Double_t parvalue)
virtual Double_t GetChisquare()
virtual void GetConfidenceIntervals(Int_t n, Int_t ndim, const Double_t* x, Double_t* ci, Double_t cl = 0.95)
virtual void GetConfidenceIntervals(TObject* obj, Double_t cl = 0.95)
virtual Double_t* GetCovarianceMatrix() const
virtual void GetCovarianceMatrix(TMatrixD& matr)
virtual Double_t GetCovarianceMatrixElement(Int_t i, Int_t j) const
virtual void GetErrors(TVectorD& vpar)
virtual Int_t GetErrors(Int_t, Double_t&, Double_t&, Double_t&, Double_t&) const
virtual void GetFitSample(TBits& bits)
virtual Int_t GetNumberFreeParameters() const
virtual Int_t GetNumberTotalParameters() const
virtual Double_t GetParameter(Int_t ipar) const
virtual Int_t GetParameter(Int_t ipar, char* name, Double_t& value, Double_t&, Double_t&, Double_t&) const
virtual void GetParameters(TVectorD& vpar)
virtual Double_t GetParError(Int_t ipar) const
virtual const char* GetParName(Int_t ipar) const
virtual Double_t GetParSignificance(Int_t ipar)
virtual Double_t GetParTValue(Int_t ipar)
virtual Int_t GetStats(Double_t&, Double_t&, Double_t&, Int_t&, Int_t&) const
virtual Double_t GetSumLog(Int_t)
virtual TClass* IsA() const
virtual Bool_t IsFixed(Int_t ipar) const
virtual void PrintResults(Int_t level, Double_t amin = 0) const
virtual void ReleaseParameter(Int_t ipar)
virtual void SetDim(Int_t n)
virtual void SetFitMethod(const char*)
virtual void SetFormula(const char* formula)
virtual void SetFormula(TFormula* function)
virtual Int_t SetParameter(Int_t, const char*, Double_t, Double_t, Double_t, Double_t)
virtual void ShowMembers(TMemberInspector& insp, char* parent)
virtual void StoreData(Bool_t store)
virtual void Streamer(TBuffer& b)
void StreamerNVirtual(TBuffer& b)
virtual Bool_t UpdateMatrix()
private:
TVectorD fParams vector of parameters
TMatrixDSym fParCovar matrix of parameters' covariances
TVectorD fTValues T-Values of parameters
TVectorD fParSign significance levels of parameters
TMatrixDSym fDesign matrix AtA
TMatrixDSym fDesignTemp ! temporary matrix, used for num.stability
TMatrixDSym fDesignTemp2 !
TMatrixDSym fDesignTemp3 !
TVectorD fAtb vector Atb
TVectorD fAtbTemp ! temporary vector, used for num.stability
TVectorD fAtbTemp2 !
TVectorD fAtbTemp3 !
Bool_t* fFixedParams array of fixed/released params
TObjArray fFunctions array of basis functions
TVectorD fY the values being fit
Double_t fY2 sum of square of y, used for chisquare
Double_t fY2Temp ! temporary variable used for num.stability
TMatrixD fX values of x
TVectorD fE the errors if they are known
TFormula* fInputFunction the function being fit
Int_t fNpoints number of points
Int_t fNfunctions number of basis functions
Int_t fFormulaSize length of the formula
Int_t fNdim number of dimensions in the formula
Int_t fNfixed number of fixed parameters
Int_t fSpecial =100+n if fitting a polynomial of deg.n
char* fFormula the formula
Bool_t fIsSet Has the formula been set?
Bool_t fStoreData Is the data stored?
Double_t fChisquare Chisquare of the fit
Int_t fH number of good points in robust fit
Bool_t fRobust true when performing a robust fit
TBits fFitsample indices of points, used in the robust fit
The Linear Fitter - fitting functions that are LINEAR IN PARAMETERS
Linear fitter is used to fit a set of data points with a linear
combination of specified functions. Note, that "linear" in the name
stands only for the model dependency on parameters, the specified
functions can be nonlinear.
The general form of this kind of model is
y(x) = a[0] + a[1]*f[1](x)+...a[n]*f[n](x)
Functions f are fixed functions of x. For example, fitting with a
polynomial is linear fitting in this sense.
The fitting method
The fit is performed using the Normal Equations method with Cholesky
decomposition.
Why should it be used?
The linear fitter is considerably faster than general non-linear
fitters and doesn't require to set the initial values of parameters.
Using the fitter:
1.Adding the data points:
1.1 To store or not to store the input data?
- There are 2 options in the constructor - to store or not
store the input data. The advantages of storing the data
are that you'll be able to reset the fitting model without
adding all the points again, and that for very large sets
of points the chisquare is calculated more precisely.
The obvious disadvantage is the amount of memory used to
keep all the points.
- Before you start adding the points, you can change the
store/not store option by StoreData() method.
1.2 The data can be added:
- simply point by point - AddPoint() method
- an array of points at once:
If the data is already stored in some arrays, this data
can be assigned to the linear fitter without physically
coping bytes, thanks to the Use() method of
TVector and TMatrix classes - AssignData() method
2.Setting the formula
2.1 The linear formula syntax:
-Additive parts are separated by 2 plus signes "++"
--for example "1 ++ x" - for fitting a straight line
-All standard functions, undrestood by TFormula, can be used
as additive parts
--TMath functions can be used too
-Functions, used as additive parts, shouldn't have any parameters,
even if those parameters are set.
--for example, if normalizing a sum of a gaus(0, 1) and a
gaus(0, 2), don't use the built-in "gaus" of TFormula,
because it has parameters, take TMath::Gaus(x, 0, 1) instead.
-Polynomials can be used like "pol3", .."polN"
-If fitting a more than 3-dimensional formula, variables should
be numbered as follows:
-- x[0], x[1], x[2]... For example, to fit "1 ++ x[0] ++ x[1] ++ x[2] ++ x[3]*x[3]"
2.2 Setting the formula:
2.2.1 If fitting a 1-2-3-dimensional formula, one can create a
TF123 based on a linear expression and pass this function
to the fitter:
--Example:
TLinearFitter *lf = new TLinearFitter();
TF2 *f2 = new TF2("f2", "x ++ y ++ x*x*y*y", -2, 2, -2, 2);
lf->SetFormula(f2);
--The results of the fit are then stored in the function,
just like when the TH1::Fit or TGraph::Fit is used
--A linear function of this kind is by no means different
from any other function, it can be drawn, evaluated, etc.
--For multidimensional fitting, TFormulas of the form:
x[0]++...++x[n] can be used
2.2.2 There is no need to create the function if you don't want to,
the formula can be set by expression:
--Example:
// 2 is the number of dimensions
TLinearFitter *lf = new TLinearFitter(2);
lf->SetFormula("x ++ y ++ x*x*y*y");
2.2.3 The fastest functions to compute are polynomials and hyperplanes.
--Polynomials are set the usual way: "pol1", "pol2",...
--Hyperplanes are set by expression "hyp3", "hyp4", ...
---The "hypN" expressions only work when the linear fitter
is used directly, not through TH1::Fit or TGraph::Fit.
To fit a graph or a histogram with a hyperplane, define
the function as "1++x++y".
---A constant term is assumed for a hyperplane, when using
the "hypN" expression, so "hyp3" is in fact fitting with
"1++x++y++z" function.
--Fitting hyperplanes is much faster than fitting other
expressions so if performance is vital, calculate the
function values beforehand and give them to the fitter
as variables
--Example:
You want to fit "sin(x)|cos(2*x)" very fast. Calculate
sin(x) and cos(2*x) beforehand and store them in array *data.
Then:
TLinearFitter *lf=new TLinearFitter(2, "hyp2");
lf->AssignData(npoint, 2, data, y);
2.3 Resetting the formula
2.3.1 If the input data is stored (or added via AssignData() function),
the fitting formula can be reset without re-adding all the points.
--Example:
TLinearFitter *lf=new TLinearFitter("1++x++x*x");
lf->AssignData(n, 1, x, y, e);
lf->Eval()
//looking at the parameter significance, you see,
// that maybe the fit will improve, if you take out
// the constant term
lf->SetFormula("x++x*x");
lf->Eval();
...
2.3.2 If the input data is not stored, the fitter will have to be
cleared and the data will have to be added again to try a
different formula.
3.Accessing the fit results
3.1 There are methods in the fitter to access all relevant information:
--GetParameters, GetCovarianceMatrix, etc
--the t-values of parameters and their significance can be reached by
GetParTValue() and GetParSignificance() methods
3.2 If fitting with a pre-defined TF123, the fit results are also
written into this function.
4.Robust fitting - Least Trimmed Squares regression (LTS)
Outliers are atypical(by definition), infrequant observations; data points
which do not appear to follow the characteristic distribution of the rest
of the data. These may reflect genuine properties of the underlying
phenomenon(variable), or be due to measurement errors or anomalies which
shouldn't be modelled. (StatSoft electronic textbook)
Even a single gross outlier can greatly influence the results of least-
squares fitting procedure, and in this case use of robust(resistant) methods
is recommended.
The method implemented here is based on the article and algorithm:
"Computing LTS Regression for Large Data Sets" by
P.J.Rousseeuw and Katrien Van Driessen
The idea of the method is to find the fitting coefficients for a subset
of h observations (out of n) with the smallest sum of squared residuals.
The size of the subset h should lie between (npoints + nparameters +1)/2
and n, and represents the minimal number of good points in the dataset.
The default value is set to (npoints + nparameters +1)/2, but of course
if you are sure that the data contains less outliers it's better to change
h according to your data.
To perform a robust fit, call EvalRobust() function instead of Eval() after
adding the points and setting the fitting function.
Note, that standard errors on parameters are not computed!
TLinearFitter()
default c-tor, input data is stored
If you don't want to store the input data,
run the function StoreData(kFALSE) after constructor
TLinearFitter(Int_t ndim)
The parameter stands for number of dimensions in the fitting formula
The input data is stored. If you don't want to store the input data,
run the function StoreData(kFALSE) after constructor
TLinearFitter(Int_t ndim, const char *formula, Option_t *opt)
First parameter stands for number of dimensions in the fitting formula
Second parameter is the fitting formula: see class description for formula syntax
Options:
The option is to store or not to store the data
If you don't want to store the data, choose "" for the option, or run
StoreData(kFalse) member function after the constructor
TLinearFitter(TFormula *function, Option_t *opt)
This constructor uses a linear function. How to create it?
TFormula now accepts formulas of the following kind:
TFormula("f", "x++y++z++x*x") or
TFormula("f", "x[0]++x[1]++x[2]*x[2]");
Other than the look, it's in no
way different from the regular formula, it can be evaluated,
drawn, etc.
The option is to store or not to store the data
If you don't want to store the data, choose "" for the option, or run
StoreData(kFalse) member function after the constructor
void AddPoint(Double_t *x, Double_t y, Double_t e)
Adds 1 point to the fitter.
First parameter stands for the coordinates of the point, where the function is measured
Second parameter - the value being fitted
Third parameter - weight(measurement error) of this point (=1 by default)
void AssignData(Int_t npoints, Int_t xncols, Double_t *x, Double_t *y, Double_t *e)
This function is to use when you already have all the data in arrays
and don't want to copy them into the fitter. In this function, the Use() method
of TVectorD and TMatrixD is used, so no bytes are physically moved around.
First parameter - number of points to fit
Second parameter - number of variables in the model
Third parameter - the variables of the model, stored in the following way:
(x0(0), x1(0), x2(0), x3(0), x0(1), x1(1), x2(1), x3(1),...
void ClearPoints()
To be used when different sets of points are fitted with the same formula.
void Eval()
Perform the fit and evaluate the parameters
void GetConfidenceIntervals(Int_t n, Int_t ndim, const Double_t *x, Double_t *ci, Double_t cl)
Computes point-by-point confidence intervals for the fitted function
Parameters:
n - number of points
ndim - dimensions of points
x - points, at which to compute the intervals, for ndim > 1
should be in order: (x0,y0, x1, y1, ... xn, yn)
ci - computed intervals are returned in this array
cl - confidence level, default=0.95
void GetConfidenceIntervals(TObject *obj, Double_t cl)
Computes confidence intervals at level cl. Default is 0.95
The TObject parameter can be a TGraphErrors, a TGraph2DErrors or a TH123.
For Graphs, confidence intervals are computed for each point,
the value of the graph at that point is set to the function value at that
point, and the graph y-errors (or z-errors) are set to the value of
the confidence interval at that point
For Histograms, confidence intervals are computed for each bin center
The bin content of this bin is then set to the function value at the bin
center, and the bin error is set to the confidence interval value.
Allowed combinations:
Fitted object Passed object
TGraph TGraphErrors, TH1
TGraphErrors, AsymmErrors TGraphErrors, TH1
TH1 TGraphErrors, TH1
TGraph2D TGraph2DErrors, TH2
TGraph2DErrors TGraph2DErrors, TH2
TH2 TGraph2DErrors, TH2
TH3 TH3
void GetFitSample(TBits &bits)
For robust lts fitting, returns the sample, on which the best fit was based
void SetFormula(const char *formula)
Additive parts should be separated by "++".
Examples (ai are parameters to fit):
1.fitting function: a0*x0 + a1*x1 + a2*x2
input formula "x[0]++x[1]++x[2]"
2.TMath functions can be used:
fitting function: a0*TMath::Gaus(x, 0, 1) + a1*y
input formula: "TMath::Gaus(x, 0, 1)++y"
fills the array of functions
void EvalRobust(Double_t h)
Finds the parameters of the fitted function in case data contains
outliers.
Parameter h stands for the minimal fraction of good points in the
dataset (h < 1, i.e. for 70% of good points take h=0.7).
The default value of h*Npoints is (Npoints + Nparameters+1)/2
If the user provides a value of h smaller than above, default is taken
See class description for the algorithm details
void CreateSubset(Int_t ntotal, Int_t h, Int_t *index)
Creates a p-subset to start
ntotal - total number of points from which the subset is chosen
Int_t Partition(Int_t nmini, Int_t *indsubdat)
divides the elements into approximately equal subgroups
number of elements in each subgroup is stored in indsubdat
number of subgroups is returned
void RDraw(Int_t *subdat, Int_t *indsubdat)
Draws ngroup nonoverlapping subdatasets out of a dataset of size n
such that the selected case numbers are uniformly distributed from 1 to n
Author: Anna Kreshuk 04/03/2005
Last update: root/minuit:$Name: $:$Id: TLinearFitter.cxx,v 1.28 2006/06/20 16:01:16 brun Exp $
Copyright (C) 1995-2005, Rene Brun and Fons Rademakers. *
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