NAME

SYNOPSIS

  # 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

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.

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

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

COPYRIGHT AND LICENSE

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.