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<title>GNU Scientific Library – Reference Manual: The Multivariate Gaussian Distribution</title>
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<a name="The-Multivariate-Gaussian-Distribution"></a>
<div class="header">
<p>
Next: <a href="The-Exponential-Distribution.html#The-Exponential-Distribution" accesskey="n" rel="next">The Exponential Distribution</a>, Previous: <a href="The-Bivariate-Gaussian-Distribution.html#The-Bivariate-Gaussian-Distribution" accesskey="p" rel="previous">The Bivariate Gaussian Distribution</a>, Up: <a href="Random-Number-Distributions.html#Random-Number-Distributions" accesskey="u" rel="up">Random Number Distributions</a> [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="The-Multivariate-Gaussian-Distribution-1"></a>
<h3 class="section">20.5 The Multivariate Gaussian Distribution</h3>
<dl>
<dt><a name="index-gsl_005fran_005fmultivariate_005fgaussian"></a>Function: <em>int</em> <strong>gsl_ran_multivariate_gaussian</strong> <em>(const gsl_rng * <var>r</var>, const gsl_vector * <var>mu</var>, const gsl_matrix * <var>L</var>, gsl_vector * <var>result</var>)</em></dt>
<dd><a name="index-Bivariate-Gaussian-distribution-1"></a>
<a name="index-two-dimensional-Gaussian-distribution-1"></a>
<a name="index-Gaussian-distribution_002c-bivariate-1"></a>
<p>This function generates a random vector satisfying the <em>k</em>-dimensional multivariate Gaussian
distribution with mean <em>\mu</em> and variance-covariance matrix
<em>\Sigma</em>. On input, the <em>k</em>-vector <em>\mu</em> is given in <var>mu</var>, and
the Cholesky factor of the <em>k</em>-by-<em>k</em> matrix <em>\Sigma = L L^T</em> is
given in the lower triangle of <var>L</var>, as output from <code>gsl_linalg_cholesky_decomp</code>.
The random vector is stored in <var>result</var> on output. The probability distribution
for multivariate Gaussian random variates is
</p>
<div class="example">
<pre class="example">p(x_1,...,x_k) dx_1 ... dx_k = {1 \over \sqrt{(2 \pi)^k |\Sigma|} \exp \left(-{1 \over 2} (x - \mu)^T \Sigma^{-1} (x - \mu)\right) dx_1 \dots dx_k
</pre></div>
</dd></dl>
<dl>
<dt><a name="index-gsl_005fran_005fmultivariate_005fgaussian_005fpdf"></a>Function: <em>int</em> <strong>gsl_ran_multivariate_gaussian_pdf</strong> <em>(const gsl_vector * <var>x</var>, const gsl_vector * <var>mu</var>, const gsl_matrix * <var>L</var>, double * <var>result</var>, gsl_vector * <var>work</var>)</em></dt>
<dt><a name="index-gsl_005fran_005fmultivariate_005fgaussian_005flog_005fpdf"></a>Function: <em>int</em> <strong>gsl_ran_multivariate_gaussian_log_pdf</strong> <em>(const gsl_vector * <var>x</var>, const gsl_vector * <var>mu</var>, const gsl_matrix * <var>L</var>, double * <var>result</var>, gsl_vector * <var>work</var>)</em></dt>
<dd><p>These functions compute <em>p(x)</em> or <em>\log{p(x)}</em> at the point <var>x</var>, using mean vector
<var>mu</var> and variance-covariance matrix specified by its Cholesky factor <var>L</var> using the formula
above. Additional workspace of length <em>k</em> is required in <var>work</var>.
</p></dd></dl>
<dl>
<dt><a name="index-gsl_005fran_005fmultivariate_005fgaussian_005fmean"></a>Function: <em>int</em> <strong>gsl_ran_multivariate_gaussian_mean</strong> <em>(const gsl_matrix * <var>X</var>, gsl_vector * <var>mu_hat</var>)</em></dt>
<dd><p>Given a set of <em>n</em> samples <em>X_j</em> from a <em>k</em>-dimensional multivariate Gaussian distribution,
this function computes the maximum likelihood estimate of the mean of the distribution, given by
</p>
<div class="example">
<pre class="example">\Hat{\mu} = {1 \over n} \sum_{j=1}^n X_j
</pre></div>
<p>The samples <em>X_1,X_2,\dots,X_n</em> are given in the <em>n</em>-by-<em>k</em> matrix <var>X</var>, and the maximum
likelihood estimate of the mean is stored in <var>mu_hat</var> on output.
</p></dd></dl>
<dl>
<dt><a name="index-gsl_005fran_005fmultivariate_005fgaussian_005fvcov"></a>Function: <em>int</em> <strong>gsl_ran_multivariate_gaussian_vcov</strong> <em>(const gsl_matrix * <var>X</var>, gsl_matrix * <var>sigma_hat</var>)</em></dt>
<dd><p>Given a set of <em>n</em> samples <em>X_j</em> from a <em>k</em>-dimensional multivariate Gaussian distribution,
this function computes the maximum likelihood estimate of the variance-covariance matrix of the distribution,
given by
</p>
<div class="example">
<pre class="example">\Hat{\Sigma} = {1 \over n} \sum_{j=1}^n \left( X_j - \Hat{\mu} \right) \left( X_j - \Hat{\mu} \right)^T
</pre></div>
<p>The samples <em>X_1,X_2,\dots,X_n</em> are given in the <em>n</em>-by-<em>k</em> matrix <var>X</var> and the maximum
likelihood estimate of the variance-covariance matrix is stored in <var>sigma_hat</var> on output.
</p></dd></dl>
<hr>
<div class="header">
<p>
Next: <a href="The-Exponential-Distribution.html#The-Exponential-Distribution" accesskey="n" rel="next">The Exponential Distribution</a>, Previous: <a href="The-Bivariate-Gaussian-Distribution.html#The-Bivariate-Gaussian-Distribution" accesskey="p" rel="previous">The Bivariate Gaussian Distribution</a>, Up: <a href="Random-Number-Distributions.html#Random-Number-Distributions" accesskey="u" rel="up">Random Number Distributions</a> [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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