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  <section id="numpy-random-generator-multivariate-normal">
<h1>numpy.random.Generator.multivariate_normal<a class="headerlink" href="#numpy-random-generator-multivariate-normal" title="Link to this heading">#</a></h1>
<p>method</p>
<dl class="py method">
<dt class="sig sig-object py" id="numpy.random.Generator.multivariate_normal">
<span class="sig-prename descclassname"><span class="pre">random.Generator.</span></span><span class="sig-name descname"><span class="pre">multivariate_normal</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">mean</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">cov</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">size</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">check_valid</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'warn'</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">tol</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1e-8</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">method</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">'svd'</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#numpy.random.Generator.multivariate_normal" title="Link to this definition">#</a></dt>
<dd><p>Draw random samples from a multivariate normal distribution.</p>
<p>The multivariate normal, multinormal or Gaussian distribution is a
generalization of the one-dimensional normal distribution to higher
dimensions.  Such a distribution is specified by its mean and
covariance matrix.  These parameters are analogous to the mean
(average or “center”) and variance (the squared standard deviation,
or “width”) of the one-dimensional normal distribution.</p>
<dl class="field-list">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>mean</strong><span class="classifier">1-D array_like, of length N</span></dt><dd><p>Mean of the N-dimensional distribution.</p>
</dd>
<dt><strong>cov</strong><span class="classifier">2-D array_like, of shape (N, N)</span></dt><dd><p>Covariance matrix of the distribution. It must be symmetric and
positive-semidefinite for proper sampling.</p>
</dd>
<dt><strong>size</strong><span class="classifier">int or tuple of ints, optional</span></dt><dd><p>Given a shape of, for example, <code class="docutils literal notranslate"><span class="pre">(m,n,k)</span></code>, <code class="docutils literal notranslate"><span class="pre">m*n*k</span></code> samples are
generated, and packed in an <em class="xref py py-obj">m</em>-by-<em class="xref py py-obj">n</em>-by-<em class="xref py py-obj">k</em> arrangement.  Because
each sample is <em class="xref py py-obj">N</em>-dimensional, the output shape is <code class="docutils literal notranslate"><span class="pre">(m,n,k,N)</span></code>.
If no shape is specified, a single (<em class="xref py py-obj">N</em>-D) sample is returned.</p>
</dd>
<dt><strong>check_valid</strong><span class="classifier">{ ‘warn’, ‘raise’, ‘ignore’ }, optional</span></dt><dd><p>Behavior when the covariance matrix is not positive semidefinite.</p>
</dd>
<dt><strong>tol</strong><span class="classifier">float, optional</span></dt><dd><p>Tolerance when checking the singular values in covariance matrix.
cov is cast to double before the check.</p>
</dd>
<dt><strong>method</strong><span class="classifier">{ ‘svd’, ‘eigh’, ‘cholesky’}, optional</span></dt><dd><p>The cov input is used to compute a factor matrix A such that
<code class="docutils literal notranslate"><span class="pre">A</span> <span class="pre">&#64;</span> <span class="pre">A.T</span> <span class="pre">=</span> <span class="pre">cov</span></code>. This argument is used to select the method
used to compute the factor matrix A. The default method ‘svd’ is
the slowest, while ‘cholesky’ is the fastest but less robust than
the slowest method. The method <em class="xref py py-obj">eigh</em> uses eigen decomposition to
compute A and is faster than svd but slower than cholesky.</p>
</dd>
</dl>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><dl>
<dt><strong>out</strong><span class="classifier">ndarray</span></dt><dd><p>The drawn samples, of shape <em>size</em>, if that was provided.  If not,
the shape is <code class="docutils literal notranslate"><span class="pre">(N,)</span></code>.</p>
<p>In other words, each entry <code class="docutils literal notranslate"><span class="pre">out[i,j,...,:]</span></code> is an N-dimensional
value drawn from the distribution.</p>
</dd>
</dl>
</dd>
</dl>
<p class="rubric">Notes</p>
<p>The mean is a coordinate in N-dimensional space, which represents the
location where samples are most likely to be generated.  This is
analogous to the peak of the bell curve for the one-dimensional or
univariate normal distribution.</p>
<p>Covariance indicates the level to which two variables vary together.
From the multivariate normal distribution, we draw N-dimensional
samples, <span class="math notranslate nohighlight">\(X = [x_1, x_2, ... x_N]\)</span>.  The covariance matrix
element <span class="math notranslate nohighlight">\(C_{ij}\)</span> is the covariance of <span class="math notranslate nohighlight">\(x_i\)</span> and <span class="math notranslate nohighlight">\(x_j\)</span>.
The element <span class="math notranslate nohighlight">\(C_{ii}\)</span> is the variance of <span class="math notranslate nohighlight">\(x_i\)</span> (i.e. its
“spread”).</p>
<p>Instead of specifying the full covariance matrix, popular
approximations include:</p>
<ul class="simple">
<li><p>Spherical covariance (<a class="reference internal" href="../../generated/numpy.cov.html#numpy.cov" title="numpy.cov"><code class="xref py py-obj docutils literal notranslate"><span class="pre">cov</span></code></a> is a multiple of the identity matrix)</p></li>
<li><p>Diagonal covariance (<a class="reference internal" href="../../generated/numpy.cov.html#numpy.cov" title="numpy.cov"><code class="xref py py-obj docutils literal notranslate"><span class="pre">cov</span></code></a> has non-negative elements, and only on
the diagonal)</p></li>
</ul>
<p>This geometrical property can be seen in two dimensions by plotting
generated data-points:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">mean</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">cov</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">100</span><span class="p">]]</span>  <span class="c1"># diagonal covariance</span>
</pre></div>
</div>
<p>Diagonal covariance means that points are oriented along x or y-axis:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span><span class="w"> </span><span class="nn">matplotlib.pyplot</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">plt</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">rng</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">default_rng</span><span class="p">()</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">rng</span><span class="o">.</span><span class="n">multivariate_normal</span><span class="p">(</span><span class="n">mean</span><span class="p">,</span> <span class="n">cov</span><span class="p">,</span> <span class="mi">5000</span><span class="p">)</span><span class="o">.</span><span class="n">T</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="s1">&#39;x&#39;</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s1">&#39;equal&#39;</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
<p>Note that the covariance matrix must be positive semidefinite (a.k.a.
nonnegative-definite). Otherwise, the behavior of this method is
undefined and backwards compatibility is not guaranteed.</p>
<p>This function internally uses linear algebra routines, and thus results
may not be identical (even up to precision) across architectures, OSes,
or even builds. For example, this is likely if <code class="docutils literal notranslate"><span class="pre">cov</span></code> has multiple equal
singular values and <code class="docutils literal notranslate"><span class="pre">method</span></code> is <code class="docutils literal notranslate"><span class="pre">'svd'</span></code> (default). In this case,
<code class="docutils literal notranslate"><span class="pre">method='cholesky'</span></code> may be more robust.</p>
<p class="rubric">References</p>
<div role="list" class="citation-list">
<div class="citation" id="r73135c1078a9-1" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span>1<span class="fn-bracket">]</span></span>
<p>Papoulis, A., “Probability, Random Variables, and Stochastic
Processes,” 3rd ed., New York: McGraw-Hill, 1991.</p>
</div>
<div class="citation" id="r73135c1078a9-2" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span>2<span class="fn-bracket">]</span></span>
<p>Duda, R. O., Hart, P. E., and Stork, D. G., “Pattern
Classification,” 2nd ed., New York: Wiley, 2001.</p>
</div>
</div>
<p class="rubric">Examples</p>
<div class="try_examples_outer_container docutils container" id="6aa20f02-33ae-4fe6-aeec-be9af98a9901">
<div class="try_examples_button_container"><button class="try_examples_button" onclick="window.tryExamplesShowIframe('6aa20f02-33ae-4fe6-aeec-be9af98a9901','b2129613-257d-4163-8161-022ddc96d00b','83e3d3f8-6560-4f59-89d3-48c02275b590','../../../lite/tree/../notebooks/index.html?path=22712ad2_1b94_4367_89cd_814f208df19b.ipynb','None')">Try it in your browser!</button></div><div class="try_examples_content docutils container">
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">mean</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">cov</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]]</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">rng</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">default_rng</span><span class="p">()</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span> <span class="o">=</span> <span class="n">rng</span><span class="o">.</span><span class="n">multivariate_normal</span><span class="p">(</span><span class="n">mean</span><span class="p">,</span> <span class="n">cov</span><span class="p">,</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">x</span><span class="o">.</span><span class="n">shape</span>
<span class="go">(3, 3, 2)</span>
</pre></div>
</div>
<p>We can use a different method other than the default to factorize cov:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">y</span> <span class="o">=</span> <span class="n">rng</span><span class="o">.</span><span class="n">multivariate_normal</span><span class="p">(</span><span class="n">mean</span><span class="p">,</span> <span class="n">cov</span><span class="p">,</span> <span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">),</span> <span class="n">method</span><span class="o">=</span><span class="s1">&#39;cholesky&#39;</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">y</span><span class="o">.</span><span class="n">shape</span>
<span class="go">(3, 3, 2)</span>
</pre></div>
</div>
<p>Here we generate 800 samples from the bivariate normal distribution
with mean [0, 0] and covariance matrix [[6, -3], [-3, 3.5]].  The
expected variances of the first and second components of the sample
are 6 and 3.5, respectively, and the expected correlation
coefficient is -3/sqrt(6*3.5) ≈ -0.65465.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">cov</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">6</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mf">3.5</span><span class="p">]])</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">pts</span> <span class="o">=</span> <span class="n">rng</span><span class="o">.</span><span class="n">multivariate_normal</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">cov</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="mi">800</span><span class="p">)</span>
</pre></div>
</div>
<p>Check that the mean, covariance, and correlation coefficient of the
sample are close to the expected values:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">pts</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="go">array([ 0.0326911 , -0.01280782])  # may vary</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">np</span><span class="o">.</span><span class="n">cov</span><span class="p">(</span><span class="n">pts</span><span class="o">.</span><span class="n">T</span><span class="p">)</span>
<span class="go">array([[ 5.96202397, -2.85602287],</span>
<span class="go">       [-2.85602287,  3.47613949]])  # may vary</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">np</span><span class="o">.</span><span class="n">corrcoef</span><span class="p">(</span><span class="n">pts</span><span class="o">.</span><span class="n">T</span><span class="p">)[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
<span class="go">-0.6273591314603949  # may vary</span>
</pre></div>
</div>
<p>We can visualize this data with a scatter plot.  The orientation
of the point cloud illustrates the negative correlation of the
components of this sample.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">import</span><span class="w"> </span><span class="nn">matplotlib.pyplot</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">plt</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">pts</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">pts</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;.&#39;</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.5</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s1">&#39;equal&#39;</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
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