@cindex fitting
@cindex least squares fit
@cindex regression, least squares
@cindex weighted linear fits
@cindex unweighted linear fits
This chapter describes routines for performing least squares fits to
experimental data using linear combinations of functions.  The data
may be weighted or unweighted, i.e. with known or unknown errors.  For
weighted data the functions compute the best fit parameters and their
associated covariance matrix.  For unweighted data the covariance
matrix is estimated from the scatter of the points, giving a
variance-covariance matrix.

The functions are divided into separate versions for simple one- or
two-parameter regression and multiple-parameter fits.  The functions
are declared in the header file @file{gsl_fit.h}.

@menu
* Fitting Overview::            
* Linear regression::           
* Linear fitting without a constant term::  
* Multi-parameter fitting::     
* Fitting Examples::            
* Fitting References and Further Reading::  
@end menu

@node Fitting Overview
@section Overview

Least-squares fits are found by minimizing @math{\chi^2}
(chi-squared), the weighted sum of squared residuals over @math{n}
experimental datapoints @math{(x_i, y_i)} for the model @math{Y(c,x)},
@tex
\beforedisplay
$$
\chi^2 = \sum_i w_i (y_i - Y(c, x_i))^2
$$
\afterdisplay
@end tex
@ifinfo

@example
\chi^2 = \sum_i w_i (y_i - Y(c, x_i))^2
@end example

@end ifinfo
@noindent
The @math{p} parameters of the model are @c{$c = \{c_0, c_1, \dots\}$}
@math{c = @{c_0, c_1, @dots{}@}}.  The
weight factors @math{w_i} are given by @math{w_i = 1/\sigma_i^2},
where @math{\sigma_i} is the experimental error on the data-point
@math{y_i}.  The errors are assumed to be
gaussian and uncorrelated. 
For unweighted data the chi-squared sum is computed without any weight factors. 

The fitting routines return the best-fit parameters @math{c} and their
@math{p \times p} covariance matrix.  The covariance matrix measures the
statistical errors on the best-fit parameters resulting from the 
errors on the data, @math{\sigma_i}, and is defined
@cindex covariance matrix, linear fits
as @c{$C_{ab} = \langle \delta c_a \delta c_b \rangle$}
@math{C_@{ab@} = <\delta c_a \delta c_b>} where @c{$\langle \, \rangle$}
@math{< >} denotes an average over the gaussian error distributions of the underlying datapoints.

The covariance matrix is calculated by error propagation from the data
errors @math{\sigma_i}.  The change in a fitted parameter @math{\delta
c_a} caused by a small change in the data @math{\delta y_i} is given
by
@tex
\beforedisplay
$$
\delta c_a = \sum_i {\partial c_a \over \partial y_i} \delta y_i
$$
\afterdisplay
@end tex
@ifinfo

@example
\delta c_a = \sum_i (dc_a/dy_i) \delta y_i
@end example

@end ifinfo
@noindent
allowing the covariance matrix to be written in terms of the errors on the data,
@tex
\beforedisplay
$$
C_{ab} =  \sum_{i,j} {\partial c_a \over \partial y_i}
                     {\partial c_b \over \partial y_j} 
                     \langle \delta y_i \delta y_j \rangle
$$
\afterdisplay
@end tex
@ifinfo

@example
C_@{ab@} = \sum_@{i,j@} (dc_a/dy_i) (dc_b/dy_j) <\delta y_i \delta y_j>
@end example

@end ifinfo
@noindent
For uncorrelated data the fluctuations of the underlying datapoints satisfy
@c{$\langle \delta y_i \delta y_j \rangle = \sigma_i^2 \delta_{ij}$}
@math{<\delta y_i \delta y_j> = \sigma_i^2 \delta_@{ij@}}, giving a 
corresponding parameter covariance matrix of
@tex
\beforedisplay
$$
C_{ab} = \sum_{i} {1 \over w_i} {\partial c_a \over \partial y_i} {\partial c_b \over \partial y_i} 
$$
\afterdisplay
@end tex
@ifinfo

@example
C_@{ab@} = \sum_i (1/w_i) (dc_a/dy_i) (dc_b/dy_i) 
@end example

@end ifinfo
@noindent
When computing the covariance matrix for unweighted data, i.e. data with unknown errors, 
the weight factors @math{w_i} in this sum are replaced by the single estimate @math{w =
1/\sigma^2}, where @math{\sigma^2} is the computed variance of the
residuals about the best-fit model, @math{\sigma^2 = \sum (y_i - Y(c,x_i))^2 / (n-p)}.  
This is referred to as the @dfn{variance-covariance matrix}.
@cindex variance-covariance matrix, linear fits

The standard deviations of the best-fit parameters are given by the
square root of the corresponding diagonal elements of
the covariance matrix, @c{$\sigma_{c_a} = \sqrt{C_{aa}}$}
@math{\sigma_@{c_a@} = \sqrt@{C_@{aa@}@}}.

@node   Linear regression
@section Linear regression
@cindex linear regression
The functions described in this section can be used to perform
least-squares fits to a straight line model, @math{Y(c,x) = c_0 + c_1 x}.

@cindex covariance matrix, from linear regression
@deftypefun int gsl_fit_linear (const double * @var{x}, const size_t @var{xstride}, const double * @var{y}, const size_t @var{ystride}, size_t @var{n}, double * @var{c0}, double * @var{c1}, double * @var{cov00}, double * @var{cov01}, double * @var{cov11}, double * @var{sumsq})
This function computes the best-fit linear regression coefficients
(@var{c0},@var{c1}) of the model @math{Y = c_0 + c_1 X} for the dataset
(@var{x}, @var{y}), two vectors of length @var{n} with strides
@var{xstride} and @var{ystride}.  The errors on @var{y} are assumed unknown so 
the variance-covariance matrix for the
parameters (@var{c0}, @var{c1}) is estimated from the scatter of the
points around the best-fit line and returned via the parameters
(@var{cov00}, @var{cov01}, @var{cov11}).   
The sum of squares of the residuals from the best-fit line is returned
in @var{sumsq}.
@end deftypefun

@deftypefun int gsl_fit_wlinear (const double * @var{x}, const size_t @var{xstride}, const double * @var{w}, const size_t @var{wstride}, const double * @var{y}, const size_t @var{ystride}, size_t @var{n}, double * @var{c0}, double * @var{c1}, double * @var{cov00}, double * @var{cov01}, double * @var{cov11}, double * @var{chisq})
This function computes the best-fit linear regression coefficients
(@var{c0},@var{c1}) of the model @math{Y = c_0 + c_1 X} for the weighted
dataset (@var{x}, @var{y}), two vectors of length @var{n} with strides
@var{xstride} and @var{ystride}.  The vector @var{w}, of length @var{n}
and stride @var{wstride}, specifies the weight of each datapoint. The
weight is the reciprocal of the variance for each datapoint in @var{y}.

The covariance matrix for the parameters (@var{c0}, @var{c1}) is
computed using the weights and returned via the parameters
(@var{cov00}, @var{cov01}, @var{cov11}).  The weighted sum of squares
of the residuals from the best-fit line, @math{\chi^2}, is returned in
@var{chisq}.
@end deftypefun

@deftypefun int gsl_fit_linear_est (double @var{x}, double @var{c0}, double @var{c1}, double @var{c00}, double @var{c01}, double @var{c11}, double * @var{y}, double * @var{y_err})
This function uses the best-fit linear regression coefficients
@var{c0},@var{c1} and their covariance
@var{cov00},@var{cov01},@var{cov11} to compute the fitted function
@var{y} and its standard deviation @var{y_err} for the model @math{Y =
c_0 + c_1 X} at the point @var{x}.
@end deftypefun

@node Linear fitting without a constant term
@section Linear fitting without a constant term

The functions described in this section can be used to perform
least-squares fits to a straight line model without a constant term,
@math{Y = c_1 X}.

@deftypefun int gsl_fit_mul (const double * @var{x}, const size_t @var{xstride}, const double * @var{y}, const size_t @var{ystride}, size_t @var{n}, double * @var{c1}, double * @var{cov11}, double * @var{sumsq})
This function computes the best-fit linear regression coefficient
@var{c1} of the model @math{Y = c_1 X} for the datasets (@var{x},
@var{y}), two vectors of length @var{n} with strides @var{xstride} and
@var{ystride}.  The errors on @var{y} are assumed unknown so the 
variance of the parameter @var{c1} is estimated from
the scatter of the points around the best-fit line and returned via the
parameter @var{cov11}.  The sum of squares of the residuals from the
best-fit line is returned in @var{sumsq}.
@end deftypefun

@deftypefun int gsl_fit_wmul (const double * @var{x}, const size_t @var{xstride}, const double * @var{w}, const size_t @var{wstride}, const double * @var{y}, const size_t @var{ystride}, size_t @var{n}, double * @var{c1}, double * @var{cov11}, double * @var{sumsq})
This function computes the best-fit linear regression coefficient
@var{c1} of the model @math{Y = c_1 X} for the weighted datasets
(@var{x}, @var{y}), two vectors of length @var{n} with strides
@var{xstride} and @var{ystride}.  The vector @var{w}, of length @var{n}
and stride @var{wstride}, specifies the weight of each datapoint. The
weight is the reciprocal of the variance for each datapoint in @var{y}.

The variance of the parameter @var{c1} is computed using the weights
and returned via the parameter @var{cov11}.  The weighted sum of
squares of the residuals from the best-fit line, @math{\chi^2}, is
returned in @var{chisq}.
@end deftypefun

@deftypefun int gsl_fit_mul_est (double @var{x}, double @var{c1}, double @var{c11}, double * @var{y}, double * @var{y_err})
This function uses the best-fit linear regression coefficient @var{c1}
and its covariance @var{cov11} to compute the fitted function
@var{y} and its standard deviation @var{y_err} for the model @math{Y =
c_1 X} at the point @var{x}.
@end deftypefun

@node Multi-parameter fitting
@section Multi-parameter fitting
@cindex multi-parameter regression
@cindex fits, multi-parameter linear
The functions described in this section perform least-squares fits to a
general linear model, @math{y = X c} where @math{y} is a vector of
@math{n} observations, @math{X} is an @math{n} by @math{p} matrix of
predictor variables, and the elements of the vector @math{c} are the @math{p} unknown best-fit parameters which are to be estimated.  The chi-squared value is given by @c{$\chi^2 = \sum_i w_i (y_i - \sum_j X_{ij} c_j)^2$}
@math{\chi^2 = \sum_i w_i (y_i - \sum_j X_@{ij@} c_j)^2}.

This formulation can be used for fits to any number of functions and/or
variables by preparing the @math{n}-by-@math{p} matrix @math{X}
appropriately.  For example, to fit to a @math{p}-th order polynomial in
@var{x}, use the following matrix,
@tex
\beforedisplay
$$
X_{ij} = x_i^j
$$
\afterdisplay
@end tex
@ifinfo

@example
X_@{ij@} = x_i^j
@end example

@end ifinfo
@noindent
where the index @math{i} runs over the observations and the index
@math{j} runs from 0 to @math{p-1}.

To fit to a set of @math{p} sinusoidal functions with fixed frequencies
@math{\omega_1}, @math{\omega_2}, @dots{}, @math{\omega_p}, use,
@tex
\beforedisplay
$$
X_{ij} = \sin(\omega_j x_i)
$$
\afterdisplay
@end tex
@ifinfo

@example
X_@{ij@} = sin(\omega_j x_i)
@end example

@end ifinfo
@noindent
To fit to @math{p} independent variables @math{x_1}, @math{x_2}, @dots{},
@math{x_p}, use,
@tex
\beforedisplay
$$
X_{ij} = x_j(i)
$$
\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})
@deftypefunx int gsl_multifit_linear_svd (const gsl_matrix * @var{X}, const gsl_vector * @var{y}, double @var{tol}, size_t * @var{rank}, gsl_vector * @var{c}, gsl_matrix * @var{cov}, double * @var{chisq}, gsl_multifit_linear_workspace * @var{work})
These functions compute 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.
In the second form of the function the components are discarded if the
ratio of singular values @math{s_i/s_0} falls below the user-specified
tolerance @var{tol}, and the effective rank is returned in @var{rank}.
@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})
@deftypefunx int gsl_multifit_wlinear_svd (const gsl_matrix * @var{X}, const gsl_vector * @var{w}, const gsl_vector * @var{y}, double @var{tol}, size_t * @var{rank}, 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 weighted
model @math{y = X c} for the observations @var{y} with weights @var{w}
and the matrix of predictor variables @var{X}.  The covariance matrix of
the model parameters @var{cov} is computed with the given weights.  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.  In the second form of the function the
components are discarded if the ratio of singular values @math{s_i/s_0}
falls below the user-specified tolerance @var{tol}, and the effective
rank is returned in @var{rank}.
@end deftypefun

@deftypefun int gsl_multifit_linear_est (const gsl_vector * @var{x}, const gsl_vector * @var{c}, const gsl_matrix * @var{cov}, double * @var{y}, double * @var{y_err})
This function uses the best-fit multilinear regression coefficients
@var{c} and their covariance matrix
@var{cov} to compute the fitted function value
@var{y} and its standard deviation @var{y_err} for the model @math{y = x.c} 
at the point @var{x}.
@end deftypefun

@node Fitting Examples
@section Examples

The following program computes a least squares straight-line fit to a
simple 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,3.0in}
@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-wlinear2,3.0in}
@end iftex

@node Fitting References and Further Reading
@section References and Further Reading

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)
@uref{http://pdg.lbl.gov/}
@end itemize

@noindent
The Review of Particle Physics is available online at the website given
above.

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