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<section id="numpy-random-randomstate-multivariate-normal">
<h1>numpy.random.RandomState.multivariate_normal<a class="headerlink" href="#numpy-random-randomstate-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.RandomState.multivariate_normal">
<span class="sig-prename descclassname"><span class="pre">random.RandomState.</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><span class="sig-paren">)</span><a class="headerlink" href="#numpy.random.RandomState.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 (standard deviation, or “width,”
squared) of the one-dimensional normal distribution.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>New code should use the
<a class="reference internal" href="numpy.random.Generator.multivariate_normal.html#numpy.random.Generator.multivariate_normal" title="numpy.random.Generator.multivariate_normal"><code class="xref py py-obj docutils literal notranslate"><span class="pre">multivariate_normal</span></code></a>
method of a <a class="reference internal" href="../generator.html#numpy.random.Generator" title="numpy.random.Generator"><code class="xref py py-obj docutils literal notranslate"><span class="pre">Generator</span></code></a> instance instead;
please see the <a class="reference internal" href="../index.html#random-quick-start"><span class="std std-ref">Quick start</span></a>.</p>
</div>
<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>
</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>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<dl class="simple">
<dt><a class="reference internal" href="numpy.random.Generator.multivariate_normal.html#numpy.random.Generator.multivariate_normal" title="numpy.random.Generator.multivariate_normal"><code class="xref py py-obj docutils literal notranslate"><span class="pre">random.Generator.multivariate_normal</span></code></a></dt><dd><p>which should be used for new code.</p>
</dd>
</dl>
</div>
<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">>>> </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">>>> </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">>>> </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">>>> </span><span class="n">x</span><span class="p">,</span> <span class="n">y</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">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">>>> </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">'x'</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s1">'equal'</span><span class="p">)</span>
<span class="gp">>>> </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 class="rubric">References</p>
<div role="list" class="citation-list">
<div class="citation" id="r4da0e147c95f-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="r4da0e147c95f-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="6091ce11-382b-42ea-89b6-8edc14f50b64">
<div class="try_examples_button_container"><button class="try_examples_button" onclick="window.tryExamplesShowIframe('6091ce11-382b-42ea-89b6-8edc14f50b64','2a1e5a23-dcca-4dee-867f-8ca29fd28c94','a3079111-21a1-44c7-aba2-c5f7c1d6063f','../../../lite/tree/../notebooks/index.html?path=19a181ca_8296_4606_870c_76d72b2e4fec.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">>>> </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">>>> </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">>>> </span><span class="n">x</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">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">>>> </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>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">>>> </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">>>> </span><span class="n">pts</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">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">>>> </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">>>> </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">>>> </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">>>> </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">>>> </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">'.'</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">>>> </span><span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s1">'equal'</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
<span class="gp">>>> </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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