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<title>Resampling from 2D histograms - GNU Scientific Library -- Reference Manual</title>
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<h3 class="section">21.21 Resampling from 2D histograms</h3>
<p>As in the one-dimensional case, a two-dimensional histogram made by
counting events can be regarded as a measurement of a probability
distribution. Allowing for statistical error, the height of each bin
represents the probability of an event where (x,y) falls in
the range of that bin. For a two-dimensional histogram the probability
distribution takes the form p(x,y) dx dy where,
<pre class="example"> p(x,y) = n_{ij}/ (N A_{ij})
</pre>
<p class="noindent">In this equation
<!-- {$n_{ij}$} -->
n_{ij} is the number of events in the bin which
contains (x,y),
<!-- {$A_{ij}$} -->
A_{ij} is the area of the bin and N is
the total number of events. The distribution of events within each bin
is assumed to be uniform.
<div class="defun">
— Data Type: <b>gsl_histogram2d_pdf</b><var><a name="index-gsl_005fhistogram2d_005fpdf-1957"></a></var><br>
<blockquote><dl>
<dt><code>size_t nx, ny</code><dd>This is the number of histogram bins used to approximate the probability
distribution function in the x and y directions.
<br><dt><code>double * xrange</code><dd>The ranges of the bins in the x-direction are stored in an array of
<var>nx</var> + 1 elements pointed to by <var>xrange</var>.
<br><dt><code>double * yrange</code><dd>The ranges of the bins in the y-direction are stored in an array of
<var>ny</var> + 1 pointed to by <var>yrange</var>.
<br><dt><code>double * sum</code><dd>The cumulative probability for the bins is stored in an array of
<var>nx</var>*<var>ny</var> elements pointed to by <var>sum</var>.
</dl>
</p></blockquote></div>
<p class="noindent">The following functions allow you to create a <code>gsl_histogram2d_pdf</code>
struct which represents a two dimensional probability distribution and
generate random samples from it.
<div class="defun">
— Function: gsl_histogram2d_pdf * <b>gsl_histogram2d_pdf_alloc</b> (<var>size_t nx, size_t ny</var>)<var><a name="index-gsl_005fhistogram2d_005fpdf_005falloc-1958"></a></var><br>
<blockquote><p>This function allocates memory for a two-dimensional probability
distribution of size <var>nx</var>-by-<var>ny</var> and returns a pointer to a
newly initialized <code>gsl_histogram2d_pdf</code> struct. If insufficient
memory is available a null pointer is returned and the error handler is
invoked with an error code of <code>GSL_ENOMEM</code>.
</p></blockquote></div>
<div class="defun">
— Function: int <b>gsl_histogram2d_pdf_init</b> (<var>gsl_histogram2d_pdf * p, const gsl_histogram2d * h</var>)<var><a name="index-gsl_005fhistogram2d_005fpdf_005finit-1959"></a></var><br>
<blockquote><p>This function initializes the two-dimensional probability distribution
calculated <var>p</var> from the histogram <var>h</var>. If any of the bins of
<var>h</var> are negative then the error handler is invoked with an error
code of <code>GSL_EDOM</code> because a probability distribution cannot
contain negative values.
</p></blockquote></div>
<div class="defun">
— Function: void <b>gsl_histogram2d_pdf_free</b> (<var>gsl_histogram2d_pdf * p</var>)<var><a name="index-gsl_005fhistogram2d_005fpdf_005ffree-1960"></a></var><br>
<blockquote><p>This function frees the two-dimensional probability distribution
function <var>p</var> and all of the memory associated with it.
</p></blockquote></div>
<div class="defun">
— Function: int <b>gsl_histogram2d_pdf_sample</b> (<var>const gsl_histogram2d_pdf * p, double r1, double r2, double * x, double * y</var>)<var><a name="index-gsl_005fhistogram2d_005fpdf_005fsample-1961"></a></var><br>
<blockquote><p>This function uses two uniform random numbers between zero and one,
<var>r1</var> and <var>r2</var>, to compute a single random sample from the
two-dimensional probability distribution <var>p</var>.
</p></blockquote></div>
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