fitting.texi

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X_{ij} = x_j(i)
$$
\afterdisplay
@end tex
@ifinfo
@example
X_@{ij@} = x_j(i)
@end example
@end ifinfo
@noindent
where @math{x_j(i)} is the @math{i}-th value of the predictor variable
@math{x_j}.

The functions described in this section are declared in the header file
@file{gsl_multifit.h}.

The solution of the general linear least-squares system requires an
additional working space for intermediate results, such as the singular
value decomposition of the matrix @math{X}.

@deftypefun {gsl_multifit_linear_workspace *} gsl_multifit_linear_alloc (size_t @var{n}, size_t @var{p})
This function allocates a workspace for fitting a model to @var{n}
observations using @var{p} parameters.
@end deftypefun

@deftypefun void gsl_multifit_linear_free (gsl_multifit_linear_workspace * @var{work})
This function frees the memory associated with the workspace @var{w}.
@end deftypefun

@deftypefun int gsl_multifit_linear (const gsl_matrix * @var{X}, const gsl_vector * @var{y}, gsl_vector * @var{c}, gsl_matrix * @var{cov}, double * @var{chisq}, gsl_multifit_linear_workspace * @var{work})
This function computes the best-fit parameters @var{c} of the model
@math{y = X c} for the observations @var{y} and the matrix of predictor
variables @var{X}.  The variance-covariance matrix of the model
parameters @var{cov} is estimated from the scatter of the observations
about the best-fit.  The sum of squares of the residuals from the
best-fit, @math{\chi^2}, is returned in @var{chisq}. 

The best-fit is found by singular value decomposition of the matrix
@var{X} using the preallocated workspace provided in @var{work}. The
modified Golub-Reinsch SVD algorithm is used, with column scaling to
improve the accuracy of the singular values. Any components which have
zero singular value (to machine precision) are discarded from the fit.
@end deftypefun

@deftypefun int gsl_multifit_wlinear (const gsl_matrix * @var{X}, const gsl_vector * @var{w}, const gsl_vector * @var{y}, gsl_vector * @var{c}, gsl_matrix * @var{cov}, double * @var{chisq}, gsl_multifit_linear_workspace * @var{work})

This function computes the best-fit parameters @var{c} of the model
@math{y = X c} for the observations @var{y} and the matrix of predictor
variables @var{X}.  The covariance matrix of the model parameters
@var{cov} is estimated from the weighted data.  The weighted sum of
squares of the residuals from the best-fit, @math{\chi^2}, is returned
in @var{chisq}.

The best-fit is found by singular value decomposition of the matrix
@var{X} using the preallocated workspace provided in @var{work}. Any
components which have zero singular value (to machine precision) are
discarded from the fit.
@end deftypefun

@node Fitting Examples
@section Examples

The following program computes a least squares straight-line fit to a
simple (fictitious) dataset, and outputs the best-fit line and its
associated one standard-deviation error bars.

@example
@verbatiminclude examples/fitting.c
@end example
@noindent
The following commands extract the data from the output of the program
and display it using the @sc{gnu} plotutils @code{graph} utility, 

@example
$ ./demo > tmp
$ more tmp
# best fit: Y = -106.6 + 0.06 X
# covariance matrix:
# [ 39602, -19.9
#   -19.9, 0.01]
# chisq = 0.8

$ for n in data fit hi lo ; 
   do 
     grep "^$n" tmp | cut -d: -f2 > $n ; 
   done
$ graph -T X -X x -Y y -y 0 20 -m 0 -S 2 -Ie data 
     -S 0 -I a -m 1 fit -m 2 hi -m 2 lo
@end example

@iftex
@sp 1
@center @image{fit-wlinear,3.4in}
@end iftex

The next program performs a quadratic fit @math{y = c_0 + c_1 x + c_2
x^2} to a weighted dataset using the generalised linear fitting function
@code{gsl_multifit_wlinear}.  The model matrix @math{X} for a quadratic
fit is given by,

@tex
\beforedisplay
$$
X=\pmatrix{1&x_0&x_0^2\cr
1&x_1&x_1^2\cr
1&x_2&x_2^2\cr
\dots&\dots&\dots\cr}
$$
\afterdisplay
@end tex
@ifinfo
@example
X = [ 1   , x_0  , x_0^2 ;
      1   , x_1  , x_1^2 ;
      1   , x_2  , x_2^2 ;
      ... , ...  , ...   ]
@end example
@end ifinfo
@noindent
where the column of ones corresponds to the constant term @math{c_0}.
The two remaining columns corresponds to the terms @math{c_1 x} and
@math{c_2 x^2}.

The program reads @var{n} lines of data in the format (@var{x}, @var{y},
@var{err}) where @var{err} is the error (standard deviation) in the
value @var{y}.

@example
@verbatiminclude examples/fitting2.c
@end example
@noindent
A suitable set of data for fitting can be generated using the following
program.  It outputs a set of points with gaussian errors from the curve
@math{y = e^x} in the region @math{0 < x < 2}.

@example
@verbatiminclude examples/fitting3.c
@end example
@noindent
The data can be prepared by running the resulting executable program,

@example
$ ./generate > exp.dat
$ more exp.dat
0.1 0.97935 0.110517
0.2 1.3359 0.12214
0.3 1.52573 0.134986
0.4 1.60318 0.149182
0.5 1.81731 0.164872
0.6 1.92475 0.182212
....
@end example
@noindent
To fit the data use the previous program, with the number of data points
given as the first argument.  In this case there are 19 data points.

@example
$ ./fit 19 < exp.dat
0.1 0.97935 +/- 0.110517
0.2 1.3359 +/- 0.12214
...
# best fit: Y = 1.02318 + 0.956201 X + 0.876796 X^2
# covariance matrix:
[ +1.25612e-02, -3.64387e-02, +1.94389e-02  
  -3.64387e-02, +1.42339e-01, -8.48761e-02  
  +1.94389e-02, -8.48761e-02, +5.60243e-02 ]
# chisq = 23.0987
@end example
@noindent
The parameters of the quadratic fit match the coefficients of the
expansion of @math{e^x}, taking into account the errors on the
parameters and the @math{O(x^3)} difference between the exponential and
quadratic functions for the larger values of @math{x}.  The errors on
the parameters are given by the square-root of the corresponding
diagonal elements of the covariance matrix.  The chi-squared per degree
of freedom is 1.4, indicating a reasonable fit to the data.

@iftex
@sp 1
@center @image{fit-wlinear2,3.4in}
@end iftex

@node Fitting References and Further Reading
@section References and Further Reading
@noindent
A summary of formulas and techniques for least squares fitting can be
found in the "Statistics" chapter of the Annual Review of Particle
Physics prepared by the Particle Data Group.

@itemize @asis
@item
@cite{Review of Particle Properties}
R.M. Barnett et al., Physical Review D54, 1 (1996)
@url{http://pdg.lbl.gov/}
@end itemize
@noindent
The Review of Particle Physics is available online at the website given
above.

@noindent
@cindex NIST Statistical Reference Datasets
@cindex Statistical Reference Datasets (StRD)
The tests used to prepare these routines are based on the NIST
Statistical Reference Datasets. The datasets and their documentation are
available from NIST at the following website,

@center @url{http://www.nist.gov/itl/div898/strd/index.html}.