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Algorithm::CurveFit(3)User Contributed Perl DocumentatioAlgorithm::CurveFit(3)

NAME
       Algorithm::CurveFit - Nonlinear Least Squares Fitting

SYNOPSIS
       use Algorithm::CurveFit;

	 # Known form of the formula
	 my $formula = 'c + a * x^2';
	 my $variable = 'x';
	 my @xdata = read_file('xdata'); # The data corresponsing to $variable
	 my @ydata = read_file('ydata'); # The data on the other axis
	 my @parameters = (
	     # Name    Guess   Accuracy
	     ['a',     0.9,    0.00001],  # If an iteration introduces smaller
	     ['c',     20,     0.00005],  # changes that the accuracy, end.
	 );
	 my $max_iter = 100; # maximum iterations

	 my $square_residual = Algorithm::CurveFit->curve_fit(
	     formula		=> $formula, # may be a Math::Symbolic tree instead
	     params		=> \@parameters,
	     variable		=> $variable,
	     xdata		=> \@xdata,
	     ydata		=> \@ydata,
	     maximum_iterations => $max_iter,
	 );

	 use Data::Dumper;
	 print Dumper \@parameters;
	 # Prints
	 # $VAR1 = [
	 #	    [
	 #	      'a',
	 #	      '0.201366784209602',
	 #	      '1e-05'
	 #	    ],
	 #	    [
	 #	      'c',
	 #	      '1.94690440147554',
	 #	      '5e-05'
	 #	    ]
	 #	  ];
	 #
	 # Real values of the parameters (as demonstrated by noisy input data):
	 # a = 0.2
	 # c = 2

DESCRIPTION
       "Algorithm::CurveFit" implements a nonlinear least squares curve
       fitting algorithm. That means, it fits a curve of known form (sine-
       like, exponential, polynomial of degree n, etc.) to a given set of data
       points.

       For details about the algorithm and its capabilities and flaws, you're
       encouraged to read the MathWorld page referenced below. Note, however,
       that it is an iterative algorithm that improves the fit with each
       iteration until it converges. The following rule of thumb usually holds
       true:

       · A good guess improves the probability of convergence and the quality
	 of the fit.

       · Increasing the number of free parameters decreases the quality and
	 convergence speed.

       · Make sure that there are no correlated parameters such as in 'a + b *
	 e^(c+x)'.  (The example can be rewritten as 'a + b * e^c * e^x' in
	 which 'c' and 'b' are basically equivalent parameters.

       The curve fitting algorithm is accessed via the 'curve_fit' subroutine.
       It requires the following parameters as 'key => value' pairs:

       formula
	 The formula should be a string that can be parsed by Math::Symbolic.
	 Alternatively, it can be an existing Math::Symbolic tree.  Please
	 refer to the documentation of that module for the syntax.

	 Evaluation of the formula for a specific value of the variable
	 (X-Data) and the parameters (see below) should yield the associated
	 Y-Data value in case of perfect fit.

       variable
	 The 'variable' is the variable in the formula that will be replaced
	 with the X-Data points for evaluation. If omitted in the call to
	 "curve_fit", the name 'x' is default. (Hence 'xdata'.)

       params
	 The parameters are the symbols in the formula whose value is varied
	 by the algorithm to find the best fit of the curve to the data. There
	 may be one or more parameters, but please keep in mind that the
	 number of parameters not only increases processing time, but also
	 decreases the quality of the fit.

	 The value of this options should be an anonymous array. This array
	 should hold one anonymous array for each parameter. That array should
	 hold (in order) a parameter name, an initial guess, and optionally an
	 accuracy measure.

	 Example:

	   $params = [
	     ['parameter1', 5,	0.00001],
	     ['parameter2', 12, 0.0001 ],
	     ...
	   ];

	   Then later:
	   curve_fit(
	   ...
	     params => $params,
	   ...
	   );

	 The accuracy measure means that if the change of parameters from one
	 iteration to the next is below each accuracy measure for each
	 parameter, convergence is assumed and the algorithm stops iterating.

	 In order to prevent looping forever, you are strongly encouraged to
	 make use of the accuracy measure (see also: maximum_iterations).

	 The final set of parameters is not returned from the subroutine but
	 the parameters are modified in-place. That means the original data
	 structure will hold the best estimate of the parameters.

       xdata
	 This should be an array reference to an array holding the data for
	 the variable of the function. (Which defaults to 'x'.)

       ydata
	 This should be an array reference to an array holding the function
	 values corresponding to the x-values in 'xdata'.

       maximum_iterations
	 Optional parameter to make the process stop after a given number of
	 iterations.  Using the accuracy measure and this option together is
	 encouraged to prevent the algorithm from going into an endless loop
	 in some cases.

       The subroutine returns the sum of square residuals after the final
       iteration as a measure for the quality of the fit.

   EXPORT
       None by default, but you may choose to export "curve_fit" using the
       standard Exporter semantics.

   SUBROUTINES
       This is a list of public subroutines

       curve_fit
	 This subroutine implements the curve fitting as explained in
	 DESCRIPTION above.

NOTES AND CAVEATS
       · When computing the derivative symbolically using "Math::Symbolic",
	 the formula simplification algorithm can sometimes fail to find the
	 equivalent of "(x-x_0)/(x-x_0)". Typically, these would be hidden in
	 a more complex product. The effect is that for "x -> x_0", the
	 evaluation of the derivative becomes undefined.

	 Since version 1.05, we fall back to numeric differentiation using
	 five-point stencil in such cases. This should help with one of the
	 primary complaints about the reliability of the module.

       · This module is NOT fast.  For slightly better performance, the
	 formulas are compiled to Perl code if possible.

SEE ALSO
       The algorithm implemented in this module was taken from:

       Eric W. Weisstein. "Nonlinear Least Squares Fitting." From MathWorld--A
       Wolfram Web Resource.
       http://mathworld.wolfram.com/NonlinearLeastSquaresFitting.html

       New versions of this module can be found on http://steffen-mueller.net
       or CPAN.

       This module uses the following modules. It might be a good idea to be
       familiar with them. Math::Symbolic, Math::MatrixReal, Test::More

AUTHOR
       Steffen Mueller, <smueller@cpan.org<gt>

COPYRIGHT AND LICENSE
       Copyright (C) 2005-2010 by Steffen Mueller

       This library is free software; you can redistribute it and/or modify it
       under the same terms as Perl itself, either Perl version 5.6 or, at
       your option, any later version of Perl 5 you may have available.

perl v5.14.1			  2011-07-19		Algorithm::CurveFit(3)
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