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<p>
Next: <a href="References-and-Further-Reading-for-Nonlinear-Least_002dSquares-Fitting.html#References-and-Further-Reading-for-Nonlinear-Least_002dSquares-Fitting" accesskey="n" rel="next">References and Further Reading for Nonlinear Least-Squares Fitting</a>, Previous: <a href="Computing-the-covariance-matrix-of-best-fit-parameters.html#Computing-the-covariance-matrix-of-best-fit-parameters" accesskey="p" rel="previous">Computing the covariance matrix of best fit parameters</a>, Up: <a href="Nonlinear-Least_002dSquares-Fitting.html#Nonlinear-Least_002dSquares-Fitting" accesskey="u" rel="up">Nonlinear Least-Squares Fitting</a> [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="Examples-29"></a>
<h3 class="section">38.11 Examples</h3>
<p>The following example program fits a weighted exponential model with
background to experimental data, <em>Y = A \exp(-\lambda t) + b</em>. The
first part of the program sets up the functions <code>expb_f</code> and
<code>expb_df</code> to calculate the model and its Jacobian. The appropriate
fitting function is given by,
where we have chosen <em>t_i = i</em>. The Jacobian matrix <em>J</em> is
the derivative of these functions with respect to the three parameters
(<em>A</em>, <em>\lambda</em>, <em>b</em>). It is given by,
where <em>x_0 = A</em>, <em>x_1 = \lambda</em> and <em>x_2 = b</em>.
</p>
<div class="example">
<pre class="verbatim">/* expfit.c -- model functions for exponential + background */
struct data {
size_t n;
double * y;
double * sigma;
};
int
expb_f (const gsl_vector * x, void *data,
gsl_vector * f)
{
size_t n = ((struct data *)data)->n;
double *y = ((struct data *)data)->y;
double *sigma = ((struct data *) data)->sigma;
double A = gsl_vector_get (x, 0);
double lambda = gsl_vector_get (x, 1);
double b = gsl_vector_get (x, 2);
size_t i;
for (i = 0; i < n; i++)
{
/* Model Yi = A * exp(-lambda * i) + b */
double t = i;
double Yi = A * exp (-lambda * t) + b;
gsl_vector_set (f, i, (Yi - y[i])/sigma[i]);
}
return GSL_SUCCESS;
}
int
expb_df (const gsl_vector * x, void *data,
gsl_matrix * J)
{
size_t n = ((struct data *)data)->n;
double *sigma = ((struct data *) data)->sigma;
double A = gsl_vector_get (x, 0);
double lambda = gsl_vector_get (x, 1);
size_t i;
for (i = 0; i < n; i++)
{
/* Jacobian matrix J(i,j) = dfi / dxj, */
/* where fi = (Yi - yi)/sigma[i], */
/* Yi = A * exp(-lambda * i) + b */
/* and the xj are the parameters (A,lambda,b) */
double t = i;
double s = sigma[i];
double e = exp(-lambda * t);
gsl_matrix_set (J, i, 0, e/s);
gsl_matrix_set (J, i, 1, -t * A * e/s);
gsl_matrix_set (J, i, 2, 1/s);
}
return GSL_SUCCESS;
}
int
expb_fdf (const gsl_vector * x, void *data,
gsl_vector * f, gsl_matrix * J)
{
expb_f (x, data, f);
expb_df (x, data, J);
return GSL_SUCCESS;
}
</pre></div>
<p>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).
</p>
<div class="example">
<pre class="verbatim">#include <stdlib.h>
#include <stdio.h>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_randist.h>
#include <gsl/gsl_vector.h>
#include <gsl/gsl_blas.h>
#include <gsl/gsl_multifit_nlin.h>
#include "expfit.c"
#define N 40
void print_state (size_t iter, gsl_multifit_fdfsolver * s);
int
main (void)
{
const gsl_multifit_fdfsolver_type *T;
gsl_multifit_fdfsolver *s;
int status;
unsigned int i, iter = 0;
const size_t n = N;
const size_t p = 3;
gsl_matrix *covar = gsl_matrix_alloc (p, p);
double y[N], sigma[N];
struct data d = { n, y, sigma};
gsl_multifit_function_fdf f;
double x_init[3] = { 1.0, 0.0, 0.0 };
gsl_vector_view x = gsl_vector_view_array (x_init, p);
const gsl_rng_type * type;
gsl_rng * r;
gsl_rng_env_setup();
type = gsl_rng_default;
r = gsl_rng_alloc (type);
f.f = &expb_f;
f.df = &expb_df;
f.fdf = &expb_fdf;
f.n = n;
f.p = p;
f.params = &d;
/* This is the data to be fitted */
for (i = 0; i < n; i++)
{
double t = i;
y[i] = 1.0 + 5 * exp (-0.1 * t)
+ gsl_ran_gaussian (r, 0.1);
sigma[i] = 0.1;
printf ("data: %u %g %g\n", i, y[i], sigma[i]);
};
T = gsl_multifit_fdfsolver_lmsder;
s = gsl_multifit_fdfsolver_alloc (T, n, p);
gsl_multifit_fdfsolver_set (s, &f, &x.vector);
print_state (iter, s);
do
{
iter++;
status = gsl_multifit_fdfsolver_iterate (s);
printf ("status = %s\n", gsl_strerror (status));
print_state (iter, s);
if (status)
break;
status = gsl_multifit_test_delta (s->dx, s->x,
1e-4, 1e-4);
}
while (status == GSL_CONTINUE && iter < 500);
gsl_multifit_covar (s->J, 0.0, covar);
#define FIT(i) gsl_vector_get(s->x, i)
#define ERR(i) sqrt(gsl_matrix_get(covar,i,i))
{
double chi = gsl_blas_dnrm2(s->f);
double dof = n - p;
double c = GSL_MAX_DBL(1, chi / sqrt(dof));
printf("chisq/dof = %g\n", pow(chi, 2.0) / dof);
printf ("A = %.5f +/- %.5f\n", FIT(0), c*ERR(0));
printf ("lambda = %.5f +/- %.5f\n", FIT(1), c*ERR(1));
printf ("b = %.5f +/- %.5f\n", FIT(2), c*ERR(2));
}
printf ("status = %s\n", gsl_strerror (status));
gsl_multifit_fdfsolver_free (s);
gsl_matrix_free (covar);
gsl_rng_free (r);
return 0;
}
void
print_state (size_t iter, gsl_multifit_fdfsolver * s)
{
printf ("iter: %3u x = % 15.8f % 15.8f % 15.8f "
"|f(x)| = %g\n",
iter,
gsl_vector_get (s->x, 0),
gsl_vector_get (s->x, 1),
gsl_vector_get (s->x, 2),
gsl_blas_dnrm2 (s->f));
}
</pre></div>
<p>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:
</p>
<div class="smallexample">
<pre class="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
</pre></div>
<p>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.
</p>
<p>If the chi-squared value shows a poor fit (i.e. <em>chi^2/dof >> 1</em>) then the error estimates obtained from the
covariance matrix will be too small. In the example program the error estimates
are multiplied by <em>\sqrt{\chi^2/dof}</em> 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.
</p>
<hr>
<div class="header">
<p>
Next: <a href="References-and-Further-Reading-for-Nonlinear-Least_002dSquares-Fitting.html#References-and-Further-Reading-for-Nonlinear-Least_002dSquares-Fitting" accesskey="n" rel="next">References and Further Reading for Nonlinear Least-Squares Fitting</a>, Previous: <a href="Computing-the-covariance-matrix-of-best-fit-parameters.html#Computing-the-covariance-matrix-of-best-fit-parameters" accesskey="p" rel="previous">Computing the covariance matrix of best fit parameters</a>, Up: <a href="Nonlinear-Least_002dSquares-Fitting.html#Nonlinear-Least_002dSquares-Fitting" accesskey="u" rel="up">Nonlinear Least-Squares Fitting</a> [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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