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<title>MISER - GNU Scientific Library -- Reference Manual</title>
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<h3 class="section">24.3 MISER</h3>
<p><a name="index-MISER-monte-carlo-integration-2097"></a><a name="index-recursive-stratified-sampling_002c-MISER-2098"></a>
The <span class="sc">miser</span> algorithm of Press and Farrar is based on recursive
stratified sampling. This technique aims to reduce the overall
integration error by concentrating integration points in the regions of
highest variance.
<p>The idea of stratified sampling begins with the observation that for two
disjoint regions a and b with Monte Carlo estimates of the
integral E_a(f) and E_b(f) and variances
\sigma_a^2(f) and \sigma_b^2(f), the variance
\Var(f) of the combined estimate
<!-- {$E(f) = {1\over 2} (E_a(f) + E_b(f))$} -->
E(f) = (1/2) (E_a(f) + E_b(f))
is given by,
<pre class="example"> \Var(f) = (\sigma_a^2(f) / 4 N_a) + (\sigma_b^2(f) / 4 N_b).
</pre>
<p class="noindent">It can be shown that this variance is minimized by distributing the
points such that,
<pre class="example"> N_a / (N_a + N_b) = \sigma_a / (\sigma_a + \sigma_b).
</pre>
<p class="noindent">Hence the smallest error estimate is obtained by allocating sample
points in proportion to the standard deviation of the function in each
sub-region.
<p>The <span class="sc">miser</span> algorithm proceeds by bisecting the integration region
along one coordinate axis to give two sub-regions at each step. The
direction is chosen by examining all d possible bisections and
selecting the one which will minimize the combined variance of the two
sub-regions. The variance in the sub-regions is estimated by sampling
with a fraction of the total number of points available to the current
step. The same procedure is then repeated recursively for each of the
two half-spaces from the best bisection. The remaining sample points are
allocated to the sub-regions using the formula for N_a and
N_b. This recursive allocation of integration points continues
down to a user-specified depth where each sub-region is integrated using
a plain Monte Carlo estimate. These individual values and their error
estimates are then combined upwards to give an overall result and an
estimate of its error.
<p>The functions described in this section are declared in the header file
<samp><span class="file">gsl_monte_miser.h</span></samp>.
<div class="defun">
— Function: gsl_monte_miser_state * <b>gsl_monte_miser_alloc</b> (<var>size_t dim</var>)<var><a name="index-gsl_005fmonte_005fmiser_005falloc-2099"></a></var><br>
<blockquote><p><a name="index-gsl_005fmonte_005fmiser_005fstate-2100"></a>This function allocates and initializes a workspace for Monte Carlo
integration in <var>dim</var> dimensions. The workspace is used to maintain
the state of the integration.
</p></blockquote></div>
<div class="defun">
— Function: int <b>gsl_monte_miser_init</b> (<var>gsl_monte_miser_state* s</var>)<var><a name="index-gsl_005fmonte_005fmiser_005finit-2101"></a></var><br>
<blockquote><p>This function initializes a previously allocated integration state.
This allows an existing workspace to be reused for different
integrations.
</p></blockquote></div>
<div class="defun">
— Function: int <b>gsl_monte_miser_integrate</b> (<var>gsl_monte_function * f, const double xl</var>[]<var>, const double xu</var>[]<var>, size_t dim, size_t calls, gsl_rng * r, gsl_monte_miser_state * s, double * result, double * abserr</var>)<var><a name="index-gsl_005fmonte_005fmiser_005fintegrate-2102"></a></var><br>
<blockquote><p>This routines uses the <span class="sc">miser</span> Monte Carlo algorithm to integrate the
function <var>f</var> over the <var>dim</var>-dimensional hypercubic region
defined by the lower and upper limits in the arrays <var>xl</var> and
<var>xu</var>, each of size <var>dim</var>. The integration uses a fixed number
of function calls <var>calls</var>, and obtains random sampling points using
the random number generator <var>r</var>. A previously allocated workspace
<var>s</var> must be supplied. The result of the integration is returned in
<var>result</var>, with an estimated absolute error <var>abserr</var>.
</p></blockquote></div>
<div class="defun">
— Function: void <b>gsl_monte_miser_free</b> (<var>gsl_monte_miser_state * s</var>)<var><a name="index-gsl_005fmonte_005fmiser_005ffree-2103"></a></var><br>
<blockquote><p>This function frees the memory associated with the integrator state
<var>s</var>.
</p></blockquote></div>
<p>The <span class="sc">miser</span> algorithm has several configurable parameters which can
be changed using the following two functions.<a rel="footnote" href="#fn-1" name="fnd-1"><sup>1</sup></a>
<div class="defun">
— Function: void <b>gsl_monte_miser_params_get</b> (<var>const gsl_monte_miser_state * s, gsl_monte_miser_params * params</var>)<var><a name="index-gsl_005fmonte_005fmiser_005fparams_005fget-2104"></a></var><br>
<blockquote><p>This function copies the parameters of the integrator state into the
user-supplied <var>params</var> structure.
</p></blockquote></div>
<div class="defun">
— Function: void <b>gsl_monte_miser_params_set</b> (<var>gsl_monte_miser_state * s, const gsl_monte_miser_params * params</var>)<var><a name="index-gsl_005fmonte_005fmiser_005fparams_005fset-2105"></a></var><br>
<blockquote><p>This function sets the integrator parameters based on values provided
in the <var>params</var> structure.
</p></blockquote></div>
<p>Typically the values of the parameters are first read using
<code>gsl_monte_miser_params_get</code>, the necessary changes are made to
the fields of the <var>params</var> structure, and the values are copied
back into the integrator state using
<code>gsl_monte_miser_params_set</code>. The functions use the
<code>gsl_monte_miser_params</code> structure which contains the following
fields:
<div class="defun">
— Variable: double <b>estimate_frac</b><var><a name="index-estimate_005ffrac-2106"></a></var><br>
<blockquote><p>This parameter specifies the fraction of the currently available number of
function calls which are allocated to estimating the variance at each
recursive step. The default value is 0.1.
</p></blockquote></div>
<div class="defun">
— Variable: size_t <b>min_calls</b><var><a name="index-min_005fcalls-2107"></a></var><br>
<blockquote><p>This parameter specifies the minimum number of function calls required
for each estimate of the variance. If the number of function calls
allocated to the estimate using <var>estimate_frac</var> falls below
<var>min_calls</var> then <var>min_calls</var> are used instead. This ensures
that each estimate maintains a reasonable level of accuracy. The
default value of <var>min_calls</var> is <code>16 * dim</code>.
</p></blockquote></div>
<div class="defun">
— Variable: size_t <b>min_calls_per_bisection</b><var><a name="index-min_005fcalls_005fper_005fbisection-2108"></a></var><br>
<blockquote><p>This parameter specifies the minimum number of function calls required
to proceed with a bisection step. When a recursive step has fewer calls
available than <var>min_calls_per_bisection</var> it performs a plain Monte
Carlo estimate of the current sub-region and terminates its branch of
the recursion. The default value of this parameter is <code>32 *
min_calls</code>.
</p></blockquote></div>
<div class="defun">
— Variable: double <b>alpha</b><var><a name="index-alpha-2109"></a></var><br>
<blockquote><p>This parameter controls how the estimated variances for the two
sub-regions of a bisection are combined when allocating points. With
recursive sampling the overall variance should scale better than
1/N, since the values from the sub-regions will be obtained using
a procedure which explicitly minimizes their variance. To accommodate
this behavior the <span class="sc">miser</span> algorithm allows the total variance to
depend on a scaling parameter \alpha,
<pre class="example"> \Var(f) = {\sigma_a \over N_a^\alpha} + {\sigma_b \over N_b^\alpha}.
</pre>
<p class="noindent">The authors of the original paper describing <span class="sc">miser</span> recommend the
value \alpha = 2 as a good choice, obtained from numerical
experiments, and this is used as the default value in this
implementation.
</p></blockquote></div>
<div class="defun">
— Variable: double <b>dither</b><var><a name="index-dither-2110"></a></var><br>
<blockquote><p>This parameter introduces a random fractional variation of size
<var>dither</var> into each bisection, which can be used to break the
symmetry of integrands which are concentrated near the exact center of
the hypercubic integration region. The default value of dither is zero,
so no variation is introduced. If needed, a typical value of
<var>dither</var> is 0.1.
</p></blockquote></div>
<div class="footnote">
<hr>
<h4>Footnotes</h4><p class="footnote"><small>[<a name="fn-1" href="#fnd-1">1</a>]</small> The previous
method of accessing these fields directly through the
<code>gsl_monte_miser_state</code> struct is now deprecated.</p>
<hr></div>
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