Class for non-linear least-squares fit. More...

#include <NonLinearFit.h>

Inheritance diagram for casa::NonLinearFit< T >:

Public Member Functions
	NonLinearFit (Bool svd=False)
	Create a fitter: the normal way to generate a fitter object.
	NonLinearFit (const NonLinearFit &other)
	Copy constructor (deep copy)
NonLinearFit &	operator= (const NonLinearFit &other)
	Assignment (deep copy)
virtual	~NonLinearFit ()
	Destructor.
void	setMaxIter (uInt maxIter=MAXITER)
	setMaxIter() sets the maximum number of iterations to do before stopping.
uInt	getMaxIter () const
	getMaxIter() queries what the maximum number of iterations currently is
uInt	currentIteration () const
	currentIteration() queries what the current iteration is
void	setCriteria (const Double criteria=CRITERIUM)
	setCriteria() sets the convergence criteria.
Double	getCriteria () const
	getCriteria() queries the current criteria
Bool	converged () const
	Check to see if the fit has converged.
Static Public Attributes
static const uInt	MAXITER
	Default maximum number of iterations (30)
static const Double	CRITERIUM
	Default convergence criterium (0.001)
Protected Member Functions
virtual Bool	fitIt (Vector< typename FunctionTraits< T >::BaseType > &sol, const Array< typename FunctionTraits< T >::BaseType > &x, const Vector< typename FunctionTraits< T >::BaseType > &y, const Vector< typename FunctionTraits< T >::BaseType > const sigma, const Vector< Bool > const mask=0)=0
	Generalised fitter.
Protected Attributes
uInt	maxiter_p
	Maximum number of iterations.
uInt	curiter_p
	Current iteration number.
Double	criterium_p
	Convergence criteria.
Bool	converge_p
	Has fit converged.

Detailed Description

template<class T>
class casa::NonLinearFit< T >

Class for non-linear least-squares fit.

Review Status

Reviewed By:: wbrouw
Date Reviewed:: 2006/06/15
Test programs:: tNonLinearFitLM

Prerequisite

Etymology

A nonlinear function is used to fit a set of data points.

Synopsis

NOTE: Constraints added. Documentation out of date at moment, check the tLinearFitSVD and tNonLinearFirLM programs for examples.

The following is a brief summary of the non-linear least-squares fit problem. See module header, Fitting , for a more complete description.

Given a set of N data points (measurements), (x(i), y(i)) i = 0,...,N-1, along with a set of standard deviations, sigma(i), for the data points, and a specified non-linear function, f(x;a) where a = a(j) j = 0,...,M-1 are a set of parameters to be determined, the non-linear least-squares fit tries to minimize

    chi-square = [(y(0)-f(x(0);a)/sigma(0)]^2 + [(y(1)-f(x(1);a)/sigma(1)]^2 + 
                 ... + [(y(N-1)-f(x(N-1);a))/sigma(N-1)]^2.

by adjusting {a(j)} in the equation.

For multidimensional functions, x(i) is a vector, and

    f(x(i);a) = f(x(i,0), x(i,1), x(i,2), ...;a)

If the measurement errors (standard deviation sigma) are not known at all, they can all be set to one initially. In this case, we assume all measurements have the same standard deviation, after minimizing chi-square, we recompute

    sigma^2 = {(y(0)-z(0))^2 + (y(1)-z(1))^2 + ..\.
              + (y(N-1)-z(N-1))^2}/(N-M) = chi-square/(N-M).

A statistic weight can be also be assigned to each measurement if the standard deviation is not available. sigma can be calculated from

    sigma = 1/ sqrt(weight)

Alternatively a 'weight' switch can be set with asWeight(). For best arithmetic performance, weight should be normalized to a maximum value of one. Having a large weight value can sometimes lead to overflow problems.

The function to be fitted to the data can be given as an instance of the Function class. One can also form a sum of functions using the CompoundFunction .

For small datasets the usage of the calls is:

Create a functional description of the parameters
Create a fitter: NonLinearFit<T> fitter();
Set the functional representation: fitter.setFunction()
Do the fit to the data: fitter.fit(x, data, sigma) (or do a number of calls to buildNormalMatrix(x, data, sigma) and finish of with fitter.fit() or fitter.sol())
if needed the covariance; residuals; chiSquared, parameter errors can all be obtained

Note that the fitter is reusable. An example is given in the following.

The solution of a fit always produces the total number of parameters given to the fitter. I.e. including any parameters that were fixed. In the latter case the solution returned will be the fixed value.

Template Type Argument Requirements (T)

Float
Double
Complex
DComplex

If there are a large number of unknowns or a large number of data points machine memory limits (or timing reasons) may not allow a complete in-core fitting to be performed. In this case one can incrementally build the normal equation (see buildNormalMatrix()).

The normal operation of the class tests for real inversion problems only. If tests are needed for almost collinear columns in the solution matrix, the collinearity can be set as the square of the sine of the minimum angle allowed.

Singular Value Decomposition is supported by setting the 'svd' switch, which has a behaviour completely identical to, apart from a default collinearity check of 1e-8.

Other information (see a.o. LSQaips ) can be set and obtained as well.

Motivation

The creation of the class module was driven by the need to write code to fit Gaussian functions to data points.

Example

Definition at line 153 of file NonLinearFit.h.

Constructor & Destructor Documentation

template<class T>

casa::NonLinearFit< T >::NonLinearFit ( Bool svd = False ) [explicit]

Create a fitter: the normal way to generate a fitter object.

Necessary data will be deduced from the Functional provided with setFunction(). Create optionally a fitter with SVD behaviour specified.

template<class T>

casa::NonLinearFit< T >::NonLinearFit ( const NonLinearFit< T > & other )

Copy constructor (deep copy)

template<class T>

virtual casa::NonLinearFit< T >::~NonLinearFit ( ) [virtual]

Destructor.

Member Function Documentation

template<class T>

Bool casa::NonLinearFit< T >::converged ( ) const [inline]

Check to see if the fit has converged.

Definition at line 195 of file NonLinearFit.h.

template<class T>

uInt casa::NonLinearFit< T >::currentIteration ( ) const [inline]

currentIteration() queries what the current iteration is

Definition at line 184 of file NonLinearFit.h.

template<class T>

virtual Bool casa::NonLinearFit< T >::fitIt	(	Vector< typename FunctionTraits< T >::BaseType > &	sol,
		const Array< typename FunctionTraits< T >::BaseType > &	x,
		const Vector< typename FunctionTraits< T >::BaseType > &	y,
		const Vector< typename FunctionTraits< T >::BaseType > *const	sigma,
		const Vector< Bool > *const	mask = `0`
	)		`[protected, pure virtual]`