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<a name="Resampling-from-2D-histograms"></a>
<div class="header">
<p>
Next: <a href="Example-programs-for-2D-histograms.html#Example-programs-for-2D-histograms" accesskey="n" rel="next">Example programs for 2D histograms</a>, Previous: <a href="Reading-and-writing-2D-histograms.html#Reading-and-writing-2D-histograms" accesskey="p" rel="previous">Reading and writing 2D histograms</a>, Up: <a href="Histograms.html#Histograms" accesskey="u" rel="up">Histograms</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="Resampling-from-2D-histograms-1"></a>
<h3 class="section">23.21 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 (<em>x</em>,<em>y</em>) falls in
the range of that bin.  For a two-dimensional histogram the probability
distribution takes the form <em>p(x,y) dx dy</em> where,
</p>
<div class="example">
<pre class="example">p(x,y) = n_{ij}/ (N A_{ij})
</pre></div>

<p>In this equation 
<em>n_{ij}</em> is the number of events in the bin which
contains <em>(x,y)</em>, 
<em>A_{ij}</em> is the area of the bin and <em>N</em> is
the total number of events.  The distribution of events within each bin
is assumed to be uniform.
</p>
<dl>
<dt><a name="index-gsl_005fhistogram2d_005fpdf"></a>Data Type: <strong>gsl_histogram2d_pdf</strong></dt>
<dd><dl compact="compact">
<dt><code>size_t nx, ny</code></dt>
<dd><p>This is the number of histogram bins used to approximate the probability
distribution function in the x and y directions.
</p></dd>
<dt><code>double * xrange</code></dt>
<dd><p>The ranges of the bins in the x-direction are stored in an array of
<em><var>nx</var> + 1</em> elements pointed to by <var>xrange</var>.
</p></dd>
<dt><code>double * yrange</code></dt>
<dd><p>The ranges of the bins in the y-direction are stored in an array of
<em><var>ny</var> + 1</em> pointed to by <var>yrange</var>.
</p></dd>
<dt><code>double * sum</code></dt>
<dd><p>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>.
</p></dd>
</dl>
</dd></dl>

<p>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.
</p>
<dl>
<dt><a name="index-gsl_005fhistogram2d_005fpdf_005falloc"></a>Function: <em>gsl_histogram2d_pdf *</em> <strong>gsl_histogram2d_pdf_alloc</strong> <em>(size_t <var>nx</var>, size_t <var>ny</var>)</em></dt>
<dd><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></dd></dl>

<dl>
<dt><a name="index-gsl_005fhistogram2d_005fpdf_005finit"></a>Function: <em>int</em> <strong>gsl_histogram2d_pdf_init</strong> <em>(gsl_histogram2d_pdf * <var>p</var>, const gsl_histogram2d * <var>h</var>)</em></dt>
<dd><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></dd></dl>

<dl>
<dt><a name="index-gsl_005fhistogram2d_005fpdf_005ffree"></a>Function: <em>void</em> <strong>gsl_histogram2d_pdf_free</strong> <em>(gsl_histogram2d_pdf * <var>p</var>)</em></dt>
<dd><p>This function frees the two-dimensional probability distribution
function <var>p</var> and all of the memory associated with it.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fhistogram2d_005fpdf_005fsample"></a>Function: <em>int</em> <strong>gsl_histogram2d_pdf_sample</strong> <em>(const gsl_histogram2d_pdf * <var>p</var>, double <var>r1</var>, double <var>r2</var>, double * <var>x</var>, double * <var>y</var>)</em></dt>
<dd><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></dd></dl>

<hr>
<div class="header">
<p>
Next: <a href="Example-programs-for-2D-histograms.html#Example-programs-for-2D-histograms" accesskey="n" rel="next">Example programs for 2D histograms</a>, Previous: <a href="Reading-and-writing-2D-histograms.html#Reading-and-writing-2D-histograms" accesskey="p" rel="previous">Reading and writing 2D histograms</a>, Up: <a href="Histograms.html#Histograms" accesskey="u" rel="up">Histograms</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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