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<!DOCTYPE html>
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<meta name="description" content="StatsLib is a templated C++ library for fast computation of statistical distribution functions.">
<meta name="author" content="Keith O'Hara">
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<title>StatsLib: Multivariate Normal Distribution</title>
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<a data-toggle="collapse" href="#collapse1"><h4><strong style="font-size: 120%;">StatsLib: Multivariate Normal Distribution</strong></h4></a>
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<a href="#intro">Introduction</a> <br>
<a href="#density">Density</a> <br>
<a href="#random">Random Sampling</a>
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<h3 style="text-align: left;" id="intro"><strong style="font-size: 120%;">Introduction</strong></h3>
<p>The density function of the Multivariate Normal distribution:</p>
$$
f(\mathbf{x}; \boldsymbol{\mu}, \boldsymbol{\Sigma}) = \dfrac{1}{\sqrt{(2\pi)^k |\boldsymbol{\Sigma}|}} \exp \left( - \frac{1}{2} (\mathbf{x} - \boldsymbol{\mu})^\top \boldsymbol{\Sigma}^{-1} (\mathbf{x} - \boldsymbol{\mu}) \right)
$$
<p>where $k$ is the dimension of the real-valued vector $\mathbf{x}$.</p>
<p>Parameters:</p>
<ul>
<li><code>mu_par</code> is $\boldsymbol{\mu}$</li>
<li><code>sigma_par</code> is $\boldsymbol{\Sigma}$</li>
</ul>
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<h3 style="text-align: left;" id="density"><strong style="font-size: 120%;">Density</strong></h3>
<br>
<p><strong>Definition:</strong></p>
<pre class="brush: cpp;">
template<typename mT, typename eT = double>
statslib_inline
eT dmvnorm(const mT& X, const mT& mu_par, const mT& Sigma_par, bool log_form = false);
</pre>
<p>Computes the density function.</p>
<hr>
<p><strong>Examples:</strong></p>
<pre class="brush: cpp;">
// parameters
arma::mat mu = arma::zeros(5,1);
arma::mat Sigma = arma::eye(5,5);
// Armadillo input
arma::mat X(5,1);
X.fill(1.5);
arma::mat dens_vals_mat = stats::dmvnorm(X,mu,Sigma);
arma::mat log_dens_vals_mat = stats::dmvnorm(X,mu,Sigma,true);
</pre>
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<h3 style="text-align: left;" id="random"><strong style="font-size: 120%;">Random Sampling</strong></h3>
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<p><strong>Definition:</strong></p>
<pre class="brush: cpp;">
// vector draw
template<typename T>
statslib_inline
T rmvnorm(const T& mu_par, const T& Sigma_par, const bool pre_chol = false);
// n samples, outputs n x k matrix
template<typename T>
statslib_inline
T rmvnorm(const uint_t n, const T& mu_par, const T& Sigma_par, const bool pre_chol = false);
</pre>
<p>Generates pseudo-random draws.</p>
<hr>
<p><strong>Examples:</strong></p>
<pre class="brush: cpp;">
// parameters
arma::mat mu = arma::zeros(5,1);
arma::mat Sigma = arma::eye(5,5);
// Armadillo output
arma::mat rand_mat = stats::rmvnorm<arma::mat>(mu,Sigma);
</pre>
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