%perlcode %{
@EXPORT_OK = qw/
               gsl_fit_linear 
               gsl_fit_wlinear 
               gsl_fit_linear_est 
               gsl_fit_mul 
               gsl_fit_wmul 
               gsl_fit_mul_est 
             /;
%EXPORT_TAGS = ( all => [ @EXPORT_OK ] );

__END__

=head1 NAME

Math::GSL::Fit - Least-squares functions for a general linear model with one- or two-parameter regression

=head1 SYNOPSIS

    use Math::GSL::Fit qw/:all/;

=head1 DESCRIPTION

The functions in this module perform least-squares fits to a general linear
model, y = X c where y is a vector of n observations, X is an n by p matrix of
predictor variables, and the elements of the vector c are the p unknown
best-fit parameters which are to be estimated.

Here is a list of all the functions in this module :

=over

=item gsl_fit_linear($x, $xstride, $y, $ystride, $n)

This function computes the best-fit linear regression coefficients (c0,c1) of
the model Y = c_0 + c_1 X for the dataset ($x, $y), two vectors (in form of
arrays) of length $n with strides $xstride and $ystride. The errors on y are
assumed unknown so the variance-covariance matrix for the parameters (c0, c1)
is estimated from the scatter of the points around the best-fit line and
returned via the parameters (cov00, cov01, cov11). The sum of squares of the
residuals from the best-fit line is returned in sumsq. Note: the correlation
coefficient of the data can be computed using gsl_stats_correlation (see
Correlation), it does not depend on the fit. The function returns the following
values in this order : 0 if the operation succeeded, 1 otherwise, c0, c1,
cov00, cov01, cov11 and sumsq.

=item gsl_fit_wlinear($x, $xstride, $w, $wstride, $y, $ystride, $n)

This function computes the best-fit linear regression coefficients (c0,c1) of
the model Y = c_0 + c_1 X for the weighted dataset ($x, $y), two vectors (in
form of arrays) of length $n with strides $xstride and $ystride. The vector
(also in the form of an array) $w, of length $n and stride $wstride, specifies
the weight of each datapoint. The weight is the reciprocal of the variance for
each datapoint in y. The covariance matrix for the parameters (c0, c1) is
computed using the weights and returned via the parameters (cov00, cov01,
cov11). The weighted sum of squares of the residuals from the best-fit line,
\chi^2, is returned in chisq. The function returns the following values in this
order : 0 if the operation succeeded, 1 otherwise, c0, c1, cov00, cov01, cov11
and sumsq.

=item gsl_fit_linear_est($x, $c0, $c1, $cov00, $cov01, $cov11)

This function uses the best-fit linear regression coefficients $c0, $c1 and
their covariance $cov00, $cov01, $cov11 to compute the fitted function y and
its standard deviation y_err for the model Y = c_0 + c_1 X at the point $x. The
function returns the following values in this order : 0 if the operation
succeeded, 1 otherwise, y and y_err.

=item gsl_fit_mul($x, $xstride, $y, $ystride, $n)

This function computes the best-fit linear regression coefficient c1 of the
model Y = c_1 X for the datasets ($x, $y), two vectors (in form of arrays) of
length $n with strides $xstride and $ystride. The errors on y are assumed
unknown so the variance of the parameter c1 is estimated from the scatter of
the points around the best-fit line and returned via the parameter cov11. The
sum of squares of the residuals from the best-fit line is returned in sumsq.
The function returns the following values in this order : 0 if the operation
succeeded, 1 otherwise, c1, cov11 and sumsq.

=item gsl_fit_wmul($x, $xstride, $w, $wstride, $y, $ystride, $n)

This function computes the best-fit linear regression coefficient c1 of the
model Y = c_1 X for the weighted datasets ($x, $y), two vectors (in form of
arrays) of length $n with strides $xstride and $ystride. The vector (also in
the form of an array) $w, of length $n and stride $wstride, specifies the
weight of each datapoint. The weight is the reciprocal of the variance for each
datapoint in y. The variance of the parameter c1 is computed using the weights
and returned via the parameter cov11. The weighted sum of squares of the
residuals from the best-fit line, \chi^2, is returned in chisq. The function
returns the following values in this order : 0 if the operation succeeded, 1
otherwise, c1, cov11 and sumsq.

=item gsl_fit_mul_est($x, $c1, $cov11)

This function uses the best-fit linear regression coefficient $c1 and its
covariance $cov11 to compute the fitted function y and its standard deviation
y_err for the model Y = c_1 X at the point $x. The function returns the
following values in this order : 0 if the operation succeeded, 1 otherwise, y
and y_err.

=back

For more informations on the functions, we refer you to the GSL offcial
documentation: L<http://www.gnu.org/software/gsl/manual/html_node/>



=head1 EXAMPLES

This example shows how to use the function gsl_fit_linear. It's important to
see that the array passed to to function must be an array reference, not a
simple array. Also when you use strides, you need to initialize all the value
in the range used, otherwise you will get warnings.

    my @norris_x = (0.2, 337.4, 118.2, 884.6, 10.1, 226.5, 666.3, 996.3,
                        448.6, 777.0, 558.2, 0.4, 0.6, 775.5, 666.9, 338.0, 
                        447.5, 11.6, 556.0, 228.1, 995.8, 887.6, 120.2, 0.3, 
                        0.3, 556.8, 339.1, 887.2, 999.0, 779.0, 11.1, 118.3,
                        229.2, 669.1, 448.9, 0.5 ) ;
    my @norris_y = ( 0.1, 338.8, 118.1, 888.0, 9.2, 228.1, 668.5, 998.5,
                        449.1, 778.9, 559.2, 0.3, 0.1, 778.1, 668.8, 339.3, 
                        448.9, 10.8, 557.7, 228.3, 998.0, 888.8, 119.6, 0.3, 
                        0.6, 557.6, 339.3, 888.0, 998.5, 778.9, 10.2, 117.6,
                        228.9, 668.4, 449.2, 0.2);
    my $xstride = 2;
    my $wstride = 3;
    my $ystride = 5;
    my ($x, $w, $y);
    for my $i (0 .. 175)
    {
        $x->[$i] = 0;
        $w->[$i] = 0;
        $y->[$i] = 0;
    }

    for my $i (0 .. 35)
    {
        $x->[$i*$xstride] = $norris_x[$i];
        $w->[$i*$wstride] = 1.0;
        $y->[$i*$ystride] = $norris_y[$i];
    }
    my ($status, @results) = gsl_fit_linear($x, $xstride, $y, $ystride, 36); 

=head1 AUTHORS

Jonathan "Duke" Leto <jonathan@leto.net> and Thierry Moisan <thierry.moisan@gmail.com>

=head1 COPYRIGHT AND LICENSE

Copyright (C) 2008-2011 Jonathan "Duke" Leto and Thierry Moisan

This program is free software; you can redistribute it and/or modify it
under the same terms as Perl itself.

=cut

%}
