@cindex nonlinear least squares fitting
@cindex least squares fitting, nonlinear

This chapter describes functions for multidimensional nonlinear
least-squares fitting.  The library provides low level components for a
variety of iterative solvers and convergence tests.  These can be
combined by the user to achieve the desired solution, with full access
to the intermediate steps of the iteration.  Each class of methods uses
the same framework, so that you can switch between solvers at runtime
without needing to recompile your program.  Each instance of a solver
keeps track of its own state, allowing the solvers to be used in
multi-threaded programs.

The header file @file{gsl_multifit_nlin.h} contains prototypes for the
multidimensional nonlinear fitting functions and related declarations.

@menu
* Overview of Nonlinear Least-Squares Fitting::  
* Initializing the Nonlinear Least-Squares Solver::  
* Providing the Function to be Minimized::  
* Iteration of the Minimization Algorithm::  
* Search Stopping Parameters for Minimization Algorithms::  
* Minimization Algorithms using Derivatives::  
* Minimization Algorithms without Derivatives::  
* Computing the covariance matrix of best fit parameters::  
* Example programs for Nonlinear Least-Squares Fitting::  
* References and Further Reading for Nonlinear Least-Squares Fitting::  
@end menu

@node Overview of Nonlinear Least-Squares Fitting
@section Overview
@cindex nonlinear least squares fitting, overview

The problem of multidimensional nonlinear least-squares fitting requires
the minimization of the squared residuals of @math{n} functions,
@math{f_i}, in @math{p} parameters, @math{x_i},
@tex
\beforedisplay
$$
\Phi(x)  = {1 \over 2} || F(x) ||^2
         = {1 \over 2} \sum_{i=1}^{n} f_i (x_1, \dots, x_p)^2
$$
\afterdisplay
@end tex
@ifinfo

@example
\Phi(x) = (1/2) || F(x) ||^2
        = (1/2) \sum_@{i=1@}^@{n@} f_i(x_1, ..., x_p)^2 
@end example

@end ifinfo
@noindent
All algorithms proceed from an initial guess using the linearization,
@tex
\beforedisplay
$$
\psi(p) = || F(x+p) || \approx || F(x) + J p\, ||
$$
\afterdisplay
@end tex
@ifinfo

@example
\psi(p) = || F(x+p) || ~=~ || F(x) + J p ||
@end example

@end ifinfo
@noindent
where @math{x} is the initial point, @math{p} is the proposed step
and @math{J} is the
Jacobian matrix @c{$J_{ij} = \partial f_i / \partial x_j$}
@math{J_@{ij@} = d f_i / d x_j}.  
Additional strategies are used to enlarge the region of convergence.
These include requiring a decrease in the norm @math{||F||} on each
step or using a trust region to avoid steps which fall outside the linear 
regime.

To perform a weighted least-squares fit of a nonlinear model
@math{Y(x,t)} to data (@math{t_i}, @math{y_i}) with independent gaussian
errors @math{\sigma_i}, use function components of the following form,
@tex
\beforedisplay
$$
f_i = {(Y(x, t_i) - y_i) \over \sigma_i}
$$
\afterdisplay
@end tex
@ifinfo

@example
f_i = (Y(x, t_i) - y_i) / \sigma_i
@end example

@end ifinfo
@noindent
Note that the model parameters are denoted by @math{x} in this chapter
since the non-linear least-squares algorithms are described
geometrically (i.e. finding the minimum of a surface).  The
independent variable of any data to be fitted is denoted by @math{t}.

@cindex Jacobian matrix, fitting
With the definition above the Jacobian is 
@c{$J_{ij} = (1 / \sigma_i) \partial Y_i / \partial x_j$}
@math{J_@{ij@} =(1 / \sigma_i)  d Y_i / d x_j}, where @math{Y_i = Y(x,t_i)}.  


@node Initializing the Nonlinear Least-Squares Solver
@section Initializing the Solver

@deftypefun {gsl_multifit_fsolver *} gsl_multifit_fsolver_alloc (const gsl_multifit_fsolver_type * @var{T}, size_t @var{n}, size_t @var{p})
@tpindex gsl_multifit_fsolver
@tpindex gsl_multifit_fsolver_type
This function returns a pointer to a newly allocated instance of a
solver of type @var{T} for @var{n} observations and @var{p} parameters.
The number of observations @var{n} must be greater than or equal to
parameters @var{p}. 

If there is insufficient memory to create the solver then the function
returns a null pointer and the error handler is invoked with an error
code of @code{GSL_ENOMEM}.
@end deftypefun

@deftypefun {gsl_multifit_fdfsolver *} gsl_multifit_fdfsolver_alloc (const gsl_multifit_fdfsolver_type * @var{T}, size_t @var{n}, size_t @var{p})
@tpindex gsl_multifit_fdfsolver
@tpindex gsl_multifit_fdfsolver_type
This function returns a pointer to a newly allocated instance of a
derivative solver of type @var{T} for @var{n} observations and @var{p}
parameters.  For example, the following code creates an instance of a
Levenberg-Marquardt solver for 100 data points and 3 parameters,

@example
const gsl_multifit_fdfsolver_type * T 
    = gsl_multifit_fdfsolver_lmder;
gsl_multifit_fdfsolver * s 
    = gsl_multifit_fdfsolver_alloc (T, 100, 3);
@end example

@noindent
The number of observations @var{n} must be greater than or equal to
parameters @var{p}.

If there is insufficient memory to create the solver then the function
returns a null pointer and the error handler is invoked with an error
code of @code{GSL_ENOMEM}.
@end deftypefun

@deftypefun int gsl_multifit_fsolver_set (gsl_multifit_fsolver * @var{s}, gsl_multifit_function * @var{f}, const gsl_vector * @var{x})
This function initializes, or reinitializes, an existing solver @var{s}
to use the function @var{f} and the initial guess @var{x}.
@end deftypefun

@deftypefun int gsl_multifit_fdfsolver_set (gsl_multifit_fdfsolver * @var{s}, gsl_multifit_function_fdf * @var{fdf}, const gsl_vector * @var{x})
This function initializes, or reinitializes, an existing solver @var{s}
to use the function and derivative @var{fdf} and the initial guess
@var{x}.
@end deftypefun

@deftypefun void gsl_multifit_fsolver_free (gsl_multifit_fsolver * @var{s})
@deftypefunx void gsl_multifit_fdfsolver_free (gsl_multifit_fdfsolver * @var{s})
These functions free all the memory associated with the solver @var{s}.
@end deftypefun

@deftypefun {const char *} gsl_multifit_fsolver_name (const gsl_multifit_fsolver * @var{s})
@deftypefunx {const char *} gsl_multifit_fdfsolver_name (const gsl_multifit_fdfsolver * @var{s})
These functions return a pointer to the name of the solver.  For example,

@example
printf ("s is a '%s' solver\n", 
        gsl_multifit_fdfsolver_name (s));
@end example

@noindent
would print something like @code{s is a 'lmder' solver}.
@end deftypefun

@node Providing the Function to be Minimized
@section Providing the Function to be Minimized

You must provide @math{n} functions of @math{p} variables for the
minimization algorithms to operate on.  In order to allow for
arbitrary parameters the functions are defined by the following data
types:

@deftp {Data Type} gsl_multifit_function 
This data type defines a general system of functions with arbitrary parameters.  

@table @code
@item int (* f) (const gsl_vector * @var{x}, void * @var{params}, gsl_vector * @var{f})
this function should store the vector result
@c{$f(x,\hbox{\it params})$}
@math{f(x,params)} in @var{f} for argument @var{x} and arbitrary parameters @var{params},
returning an appropriate error code if the function cannot be computed.

@item size_t n
the number of functions, i.e. the number of components of the
vector @var{f}.

@item size_t p
the number of independent variables, i.e. the number of components of
the vector @var{x}.

@item void * params
a pointer to the arbitrary parameters of the function.
@end table
@end deftp

@deftp {Data Type} gsl_multifit_function_fdf
This data type defines a general system of functions with arbitrary parameters and
the corresponding Jacobian matrix of derivatives,

@table @code
@item int (* f) (const gsl_vector * @var{x}, void * @var{params}, gsl_vector * @var{f})
this function should store the vector result
@c{$f(x,\hbox{\it params})$}
@math{f(x,params)} in @var{f} for argument @var{x} and arbitrary parameters @var{params},
returning an appropriate error code if the function cannot be computed.

@item int (* df) (const gsl_vector * @var{x}, void * @var{params}, gsl_matrix * @var{J})
this function should store the @var{n}-by-@var{p} matrix result
@c{$J_{ij} = \partial f_i(x,\hbox{\it params}) / \partial x_j$}
@math{J_ij = d f_i(x,params) / d x_j} in @var{J} for argument @var{x} 
and arbitrary parameters @var{params}, returning an appropriate error code if the
function cannot be computed.

@item int (* fdf) (const gsl_vector * @var{x}, void * @var{params}, gsl_vector * @var{f}, gsl_matrix * @var{J})
This function should set the values of the @var{f} and @var{J} as above,
for arguments @var{x} and arbitrary parameters @var{params}.  This function
provides an optimization of the separate functions for @math{f(x)} and
@math{J(x)}---it is always faster to compute the function and its
derivative at the same time.

@item size_t n
the number of functions, i.e. the number of components of the
vector @var{f}.

@item size_t p
the number of independent variables, i.e. the number of components of
the vector @var{x}.

@item void * params
a pointer to the arbitrary parameters of the function.
@end table
@end deftp

Note that when fitting a non-linear model against experimental data,
the data is passed to the functions above using the
@var{params} argument and the trial best-fit parameters through the
@var{x} argument.

@node Iteration of the Minimization Algorithm
@section Iteration

The following functions drive the iteration of each algorithm.  Each
function performs one iteration to update the state of any solver of the
corresponding type.  The same functions work for all solvers so that
different methods can be substituted at runtime without modifications to
the code.

@deftypefun int gsl_multifit_fsolver_iterate (gsl_multifit_fsolver * @var{s})
@deftypefunx int gsl_multifit_fdfsolver_iterate (gsl_multifit_fdfsolver * @var{s})
These functions perform a single iteration of the solver @var{s}.  If
the iteration encounters an unexpected problem then an error code will
be returned.  The solver maintains a current estimate of the best-fit
parameters at all times. 
@end deftypefun

The solver struct @var{s} contains the following entries, which can
be used to track the progress of the solution:

@table @code
@item gsl_vector * x
The current position.

@item gsl_vector * f
The function value at the current position.

@item gsl_vector * dx
The difference between the current position and the previous position,
i.e. the last step, taken as a vector.

@item gsl_matrix * J
The Jacobian matrix at the current position (for the
@code{gsl_multifit_fdfsolver} struct only)
@end table

The best-fit information also can be accessed with the following
auxiliary functions,

@deftypefun {gsl_vector *} gsl_multifit_fsolver_position (const gsl_multifit_fsolver * @var{s})
@deftypefunx {gsl_vector *} gsl_multifit_fdfsolver_position (const gsl_multifit_fdfsolver * @var{s})
These functions return the current position (i.e. best-fit parameters)
@code{s->x} of the solver @var{s}.
@end deftypefun

@node Search Stopping Parameters for Minimization Algorithms
@section Search Stopping Parameters
@cindex nonlinear fitting, stopping parameters

A minimization procedure should stop when one of the following conditions is
true:

@itemize @bullet
@item
A minimum has been found to within the user-specified precision.

@item
A user-specified maximum number of iterations has been reached.

@item
An error has occurred.
@end itemize

@noindent
The handling of these conditions is under user control.  The functions
below allow the user to test the current estimate of the best-fit
parameters in several standard ways.

@deftypefun int gsl_multifit_test_delta (const gsl_vector * @var{dx}, const gsl_vector * @var{x}, double @var{epsabs}, double @var{epsrel})

This function tests for the convergence of the sequence by comparing the
last step @var{dx} with the absolute error @var{epsabs} and relative
error @var{epsrel} to the current position @var{x}.  The test returns
@code{GSL_SUCCESS} if the following condition is achieved,
@tex
\beforedisplay
$$
|dx_i| < \hbox{\it epsabs} + \hbox{\it epsrel\/}\, |x_i|
$$
\afterdisplay
@end tex
@ifinfo

@example
|dx_i| < epsabs + epsrel |x_i|
@end example

@end ifinfo
@noindent
for each component of @var{x} and returns @code{GSL_CONTINUE} otherwise.
@end deftypefun

@cindex residual, in nonlinear systems of equations
@deftypefun int gsl_multifit_test_gradient (const gsl_vector * @var{g}, double @var{epsabs})
This function tests the residual gradient @var{g} against the absolute
error bound @var{epsabs}.  Mathematically, the gradient should be
exactly zero at the minimum. The test returns @code{GSL_SUCCESS} if the
following condition is achieved,
@tex
\beforedisplay
$$
\sum_i |g_i| < \hbox{\it epsabs}
$$
\afterdisplay
@end tex
@ifinfo

@example
\sum_i |g_i| < epsabs
@end example

@end ifinfo
@noindent
and returns @code{GSL_CONTINUE} otherwise.  This criterion is suitable
for situations where the precise location of the minimum, @math{x},
is unimportant provided a value can be found where the gradient is small
enough.
@end deftypefun


@deftypefun int gsl_multifit_gradient (const gsl_matrix * @var{J}, const gsl_vector * @var{f}, gsl_vector * @var{g})
This function computes the gradient @var{g} of @math{\Phi(x) = (1/2)
||F(x)||^2} from the Jacobian matrix @math{J} and the function values
@var{f}, using the formula @math{g = J^T f}.
@end deftypefun

@node Minimization Algorithms using Derivatives
@section Minimization Algorithms using Derivatives

The minimization algorithms described in this section make use of both
the function and its derivative.  They require an initial guess for the
location of the minimum. There is no absolute guarantee of
convergence---the function must be suitable for this technique and the
initial guess must be sufficiently close to the minimum for it to work.

@comment ============================================================
@cindex Levenberg-Marquardt algorithms
@deffn {Derivative Solver} gsl_multifit_fdfsolver_lmsder
@cindex LMDER algorithm
@cindex MINPACK, minimization algorithms
This is a robust and efficient version of the Levenberg-Marquardt
algorithm as implemented in the scaled @sc{lmder} routine in
@sc{minpack}.  Minpack was written by Jorge J. Mor@'e, Burton S. Garbow
and Kenneth E. Hillstrom.

The algorithm uses a generalized trust region to keep each step under
control.  In order to be accepted a proposed new position @math{x'} must
satisfy the condition @math{|D (x' - x)| < \delta}, where @math{D} is a
diagonal scaling matrix and @math{\delta} is the size of the trust
region.  The components of @math{D} are computed internally, using the
column norms of the Jacobian to estimate the sensitivity of the residual
to each component of @math{x}.  This improves the behavior of the
algorithm for badly scaled functions.

On each iteration the algorithm attempts to minimize the linear system
@math{|F + J p|} subject to the constraint @math{|D p| < \Delta}.  The
solution to this constrained linear system is found using the
Levenberg-Marquardt method.

The proposed step is now tested by evaluating the function at the
resulting point, @math{x'}.  If the step reduces the norm of the
function sufficiently, and follows the predicted behavior of the
function within the trust region, then it is accepted and the size of the
trust region is increased.  If the proposed step fails to improve the
solution, or differs significantly from the expected behavior within
the trust region, then the size of the trust region is decreased and
another trial step is computed.

The algorithm also monitors the progress of the solution and returns an
error if the changes in the solution are smaller than the machine
precision.  The possible error codes are,

@table @code
@item GSL_ETOLF
the decrease in the function falls below machine precision

@item GSL_ETOLX
the change in the position vector falls below machine precision

@item GSL_ETOLG
the norm of the gradient, relative to the norm of the function, falls
below machine precision

@item GSL_ENOPROG
the routine has made 10 or more attempts to find a suitable trial step
without success (but subsequent calls can be made to continue the
search).@footnote{The return code @code{GSL_CONTINUE} was used for
this case in versions prior to 1.14.}
@end table

@noindent
These error codes indicate that further iterations will be unlikely to
change the solution from its current value.  
@end deffn

@deffn {Derivative Solver} gsl_multifit_fdfsolver_lmder
This is an unscaled version of the @sc{lmder} algorithm.  The elements of the
diagonal scaling matrix @math{D} are set to 1.  This algorithm may be
useful in circumstances where the scaled version of @sc{lmder} converges too
slowly, or the function is already scaled appropriately.
@end deffn

@node Minimization Algorithms without Derivatives
@section Minimization Algorithms without Derivatives

There are no algorithms implemented in this section at the moment.

@node Computing the covariance matrix of best fit parameters
@section Computing the covariance matrix of best fit parameters
@cindex best-fit parameters, covariance
@cindex least squares, covariance of best-fit parameters
@cindex covariance matrix, nonlinear fits

@deftypefun int gsl_multifit_covar (const gsl_matrix * @var{J}, double @var{epsrel}, gsl_matrix * @var{covar})
This function uses the Jacobian matrix @var{J} to compute the covariance
matrix of the best-fit parameters, @var{covar}.  The parameter
@var{epsrel} is used to remove linear-dependent columns when @var{J} is
rank deficient.

The covariance matrix is given by,
@tex
\beforedisplay
$$
C = (J^T J)^{-1}
$$
\afterdisplay
@end tex
@ifinfo

@example
covar = (J^T J)^@{-1@}
@end example

@end ifinfo
@noindent
and is computed by QR decomposition of J with column-pivoting.  Any
columns of @math{R} which satisfy 
@tex
\beforedisplay
$$
|R_{kk}| \leq epsrel |R_{11}|
$$
\afterdisplay
@end tex
@ifinfo

@example
|R_@{kk@}| <= epsrel |R_@{11@}|
@end example

@end ifinfo
@noindent
are considered linearly-dependent and are excluded from the covariance
matrix (the corresponding rows and columns of the covariance matrix are
set to zero).

If the minimisation uses the weighted least-squares function
@math{f_i = (Y(x, t_i) - y_i) / \sigma_i} then the covariance
matrix above gives the statistical error on the best-fit parameters
resulting from the gaussian errors @math{\sigma_i} on 
the underlying data @math{y_i}.  This can be verified from the relation 
@math{\delta f = J \delta c} and the fact that the fluctuations in @math{f}
from the data @math{y_i} are normalised by @math{\sigma_i} and 
so satisfy @c{$\langle \delta f \delta f^T \rangle = I$}
@math{<\delta f \delta f^T> = I}.

For an unweighted least-squares function @math{f_i = (Y(x, t_i) -
y_i)} the covariance matrix above should be multiplied by the variance
of the residuals about the best-fit @math{\sigma^2 = \sum (y_i - Y(x,t_i))^2 / (n-p)}
to give the variance-covariance
matrix @math{\sigma^2 C}.  This estimates the statistical error on the
best-fit parameters from the scatter of the underlying data.

For more information about covariance matrices see @ref{Fitting Overview}.
@end deftypefun

@comment ============================================================

@node Example programs for Nonlinear Least-Squares Fitting
@section Examples

The following example program fits a weighted exponential model with
background to experimental data, @math{Y = A \exp(-\lambda t) + b}. The
first part of the program sets up the functions @code{expb_f} and
@code{expb_df} to calculate the model and its Jacobian.  The appropriate
fitting function is given by,
@tex
\beforedisplay
$$
f_i = ((A \exp(-\lambda t_i) + b) - y_i)/\sigma_i
$$
\afterdisplay
@end tex
@ifinfo

@example
f_i = ((A \exp(-\lambda t_i) + b) - y_i)/\sigma_i
@end example

@end ifinfo
@noindent
where we have chosen @math{t_i = i}.  The Jacobian matrix @math{J} is
the derivative of these functions with respect to the three parameters
(@math{A}, @math{\lambda}, @math{b}).  It is given by,
@tex
\beforedisplay
$$
J_{ij} = {\partial f_i \over \partial x_j}
$$
\afterdisplay
@end tex
@ifinfo

@example
J_@{ij@} = d f_i / d x_j
@end example

@end ifinfo
@noindent
where @math{x_0 = A}, @math{x_1 = \lambda} and @math{x_2 = b}.

@example
@verbatiminclude examples/expfit.c
@end example

@noindent
The main part of the program sets up a Levenberg-Marquardt solver and
some simulated random data. The data uses the known parameters
(1.0,5.0,0.1) combined with gaussian noise (standard deviation = 0.1)
over a range of 40 timesteps. The initial guess for the parameters is
chosen as (0.0, 1.0, 0.0).

@example
@verbatiminclude examples/nlfit.c
@end example

@noindent
The iteration terminates when the change in x is smaller than 0.0001, as
both an absolute and relative change.  Here are the results of running
the program:

@smallexample
iter: 0 x=1.00000000 0.00000000 0.00000000 |f(x)|=117.349
status=success
iter: 1 x=1.64659312 0.01814772 0.64659312 |f(x)|=76.4578
status=success
iter: 2 x=2.85876037 0.08092095 1.44796363 |f(x)|=37.6838
status=success
iter: 3 x=4.94899512 0.11942928 1.09457665 |f(x)|=9.58079
status=success
iter: 4 x=5.02175572 0.10287787 1.03388354 |f(x)|=5.63049
status=success
iter: 5 x=5.04520433 0.10405523 1.01941607 |f(x)|=5.44398
status=success
iter: 6 x=5.04535782 0.10404906 1.01924871 |f(x)|=5.44397
chisq/dof = 0.800996
A      = 5.04536 +/- 0.06028
lambda = 0.10405 +/- 0.00316
b      = 1.01925 +/- 0.03782
status = success
@end smallexample

@noindent
The approximate values of the parameters are found correctly, and the
chi-squared value indicates a good fit (the chi-squared per degree of
freedom is approximately 1).  In this case the errors on the parameters
can be estimated from the square roots of the diagonal elements of the
covariance matrix.  

If the chi-squared value shows a poor fit (i.e. @c{$\chi^2/(n-p) \gg 1$}
@math{chi^2/dof >> 1}) then the error estimates obtained from the
covariance matrix will be too small.  In the example program the error estimates
are multiplied by @c{$\sqrt{\chi^2/(n-p)}$}
@math{\sqrt@{\chi^2/dof@}} in this case, a common way of increasing the
errors for a poor fit.  Note that a poor fit will result from the use
an inappropriate model, and the scaled error estimates may then
be outside the range of validity for gaussian errors.

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

@node References and Further Reading for Nonlinear Least-Squares Fitting
@section References and Further Reading

The @sc{minpack} algorithm is described in the following article,

@itemize @w{}
@item
J.J. Mor@'e, @cite{The Levenberg-Marquardt Algorithm: Implementation and
Theory}, Lecture Notes in Mathematics, v630 (1978), ed G. Watson.
@end itemize

@noindent
The following paper is also relevant to the algorithms described in this
section,

@itemize @w{}
@item 
J.J. Mor@'e, B.S. Garbow, K.E. Hillstrom, ``Testing Unconstrained
Optimization Software'', ACM Transactions on Mathematical Software, Vol
7, No 1 (1981), p 17--41.
@end itemize

