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<h3 class="section" id="Correlation-and-Regression-Analysis-1"><span>26.4 Correlation and Regression Analysis<a class="copiable-link" href="#Correlation-and-Regression-Analysis-1"> ¶</a></span></h3>
<a class="anchor" id="XREFcov"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-cov"><span><code class="def-type"><var class="var">c</var> =</code> <strong class="def-name">cov</strong> <code class="def-code-arguments">(<var class="var">x</var>)</code><a class="copiable-link" href="#index-cov"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-cov-1"><span><code class="def-type"><var class="var">c</var> =</code> <strong class="def-name">cov</strong> <code class="def-code-arguments">(<var class="var">x</var>, <var class="var">y</var>)</code><a class="copiable-link" href="#index-cov-1"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-cov-2"><span><code class="def-type"><var class="var">c</var> =</code> <strong class="def-name">cov</strong> <code class="def-code-arguments">(…, <var class="var">opt</var>)</code><a class="copiable-link" href="#index-cov-2"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-cov-3"><span><code class="def-type"><var class="var">c</var> =</code> <strong class="def-name">cov</strong> <code class="def-code-arguments">(…, <var class="var">nanflag</var>)</code><a class="copiable-link" href="#index-cov-3"> ¶</a></span></dt>
<dd><p>Compute the covariance matrix.
</p>
<p>The covariance between two variable vectors <var class="var">A</var> and <var class="var">B</var> is
calculated as:
</p>
<div class="example">
<pre class="example-preformatted">cov (<var class="var">a</var>,<var class="var">b</var>) = 1/(N-1) * SUM_i (<var class="var">a</var>(i) - mean (<var class="var">a</var>)) * (<var class="var">b</var>(i) - mean (<var class="var">b</var>))
</pre></div>
<p>where <em class="math">N</em> is the length of the vectors <var class="var">a</var> and <var class="var">b</var>.
</p>
<p>If called with one argument, compute <code class="code">cov (<var class="var">x</var>, <var class="var">x</var>)</code>. If
<var class="var">x</var> is a vector, this is the scalar variance of <var class="var">x</var>. If <var class="var">x</var> is
a matrix, each row of <var class="var">x</var> is treated as an observation, and each column
as a variable, and the (<var class="var">i</var>, <var class="var">j</var>)-th<!-- /@w --> entry of
<code class="code">cov (<var class="var">x</var>)</code> is the covariance between the <var class="var">i</var>-th and
<var class="var">j</var>-th columns in <var class="var">x</var>. If <var class="var">x</var> has dimensions n x m, the output
<var class="var">c</var> will be a m x m square covariance matrix.
</p>
<p>If called with two arguments, compute <code class="code">cov (<var class="var">x</var>, <var class="var">y</var>)</code>, the
covariance between two random variables <var class="var">x</var> and <var class="var">y</var>. <var class="var">x</var> and
<var class="var">y</var> must have the same number of elements, and will be treated as
vectors with the covariance computed as
<code class="code">cov (<var class="var">x</var>(:), <var class="var">y</var>(:))</code>. The output will be a 2 x 2
covariance matrix.
</p>
<p>The optional argument <var class="var">opt</var> determines the type of normalization to
use. Valid values are
</p>
<dl class="table">
<dt>0 [default]:</dt>
<dd><p>Normalize with <em class="math">N-1</em>. This provides the best unbiased estimator of
the covariance.
</p>
</dd>
<dt>1:</dt>
<dd><p>Normalize with <em class="math">N</em>. This provides the second moment around the
mean. <var class="var">opt</var> is set to 1 for N = 1.
</p></dd>
</dl>
<p>The optional argument <var class="var">nanflag</var> must appear last in the argument list
and controls how NaN values are handled by <code class="code">cov</code>. The three valid
values are:
</p>
<dl class="table">
<dt>includenan [default]:</dt>
<dd><p>Leave NaN values in <var class="var">x</var> and <var class="var">y</var>. Output will follow the normal
rules for handling NaN values in arithmetic operations.
</p>
</dd>
<dt>omitrows:</dt>
<dd><p>Rows containing NaN values are trimmed from both <var class="var">x</var> and <var class="var">y</var>
prior to calculating the covariance. A NaN in one variable will remove
that row from both <var class="var">x</var> and <var class="var">y</var>.
</p>
</dd>
<dt>partialrows:</dt>
<dd><p>Rows containing NaN values are ignored from both <var class="var">x</var> and <var class="var">y</var>
independently for each <var class="var">i</var>-th and <var class="var">j</var>-th covariance
calculation. This may result in a different number of observations,
<em class="math">N</em>, being used to calculated each element of the covariance matrix.
</p></dd>
</dl>
<p>Compatibility Note: Before Octave v9.1.0, <code class="code">cov</code> treated rows
<var class="var">x</var> and <var class="var">y</var> as multivariate random variables. Newer versions
attempt to maintain full compatibility with <small class="sc">MATLAB</small> by treating
<var class="var">x</var> and <var class="var">y</var> as two univariate distributions regardless of shape,
resulting in a 2x2 output matrix. Code relying on Octave’s previous
definition will need to be modified when running this newer version of
<code class="code">cov</code>. The previous behavior can be obtained by using the
NaN package’s <code class="code">covm</code> function as <code class="code">covm (<var class="var">x</var>, <var class="var">y</var>, "D")</code>.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFcorr">corr</a>.
</p></dd></dl>
<a class="anchor" id="XREFcorr"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-corr"><span><code class="def-type"><var class="var">r</var> =</code> <strong class="def-name">corr</strong> <code class="def-code-arguments">(<var class="var">x</var>)</code><a class="copiable-link" href="#index-corr"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-corr-1"><span><code class="def-type"><var class="var">r</var> =</code> <strong class="def-name">corr</strong> <code class="def-code-arguments">(<var class="var">x</var>, <var class="var">y</var>)</code><a class="copiable-link" href="#index-corr-1"> ¶</a></span></dt>
<dd><p>Compute matrix of correlation coefficients.
</p>
<p>If each row of <var class="var">x</var> and <var class="var">y</var> is an observation and each column is
a variable, then the (<var class="var">i</var>, <var class="var">j</var>)-th<!-- /@w --> entry of
<code class="code">corr (<var class="var">x</var>, <var class="var">y</var>)</code> is the correlation between the
<var class="var">i</var>-th variable in <var class="var">x</var> and the <var class="var">j</var>-th variable in <var class="var">y</var>.
<var class="var">x</var> and <var class="var">y</var> must have the same number of rows (observations). The
correlation coefficient is calculated for two variable vectors <var class="var">A</var> and
<var class="var">B</var> (columns of <var class="var">x</var> and <var class="var">y</var>) as:
</p>
<div class="example">
<pre class="example-preformatted">corr (<var class="var">A</var>,<var class="var">B</var>) = cov (<var class="var">A</var>,<var class="var">B</var>) / (std (<var class="var">A</var>) * std (<var class="var">B</var>))
</pre></div>
<p>The output variable <var class="var">r</var> will have size n x m, where n and m are the
number of variables (columns) in <var class="var">x</var> and <var class="var">y</var>, respectively. Note
that as the standard deviation of any scalar is zero, the correlation
coefficient will be returned as NaN for any scalar or single-row inputs.
</p>
<p>If called with one argument, compute <code class="code">corr (<var class="var">x</var>, <var class="var">x</var>)</code>,
the correlation between the each pair of columns of <var class="var">x</var>.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFcov">cov</a>, <a class="ref" href="#XREFcorrcoef">corrcoef</a>.
</p></dd></dl>
<a class="anchor" id="XREFcorrcoef"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-corrcoef"><span><code class="def-type"><var class="var">r</var> =</code> <strong class="def-name">corrcoef</strong> <code class="def-code-arguments">(<var class="var">x</var>)</code><a class="copiable-link" href="#index-corrcoef"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-corrcoef-1"><span><code class="def-type"><var class="var">r</var> =</code> <strong class="def-name">corrcoef</strong> <code class="def-code-arguments">(<var class="var">x</var>, <var class="var">y</var>)</code><a class="copiable-link" href="#index-corrcoef-1"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-corrcoef-2"><span><code class="def-type"><var class="var">r</var> =</code> <strong class="def-name">corrcoef</strong> <code class="def-code-arguments">(…, <var class="var">param</var>, <var class="var">value</var>, …)</code><a class="copiable-link" href="#index-corrcoef-2"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-corrcoef-3"><span><code class="def-type">[<var class="var">r</var>, <var class="var">p</var>] =</code> <strong class="def-name">corrcoef</strong> <code class="def-code-arguments">(…)</code><a class="copiable-link" href="#index-corrcoef-3"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-corrcoef-4"><span><code class="def-type">[<var class="var">r</var>, <var class="var">p</var>, <var class="var">lci</var>, <var class="var">hci</var>] =</code> <strong class="def-name">corrcoef</strong> <code class="def-code-arguments">(…)</code><a class="copiable-link" href="#index-corrcoef-4"> ¶</a></span></dt>
<dd><p>Compute a matrix of correlation coefficients.
</p>
<p><var class="var">x</var> is an array where each column contains a variable and each row is
an observation.
</p>
<p>If a second input <var class="var">y</var> (of the same size as <var class="var">x</var>) is given then
calculate the correlation coefficients between <var class="var">x</var> and <var class="var">y</var>.
</p>
<p><var class="var">param</var>, <var class="var">value</var> are optional pairs of parameters and values which
modify the calculation. Valid options are:
</p>
<dl class="table">
<dt><code class="code">"alpha"</code></dt>
<dd><p>Confidence level used for the bounds of the confidence interval, <var class="var">lci</var>
and <var class="var">hci</var>. Default is 0.05, i.e., 95% confidence interval.
</p>
</dd>
<dt><code class="code">"rows"</code></dt>
<dd><p>Determine processing of NaN values. Acceptable values are <code class="code">"all"</code>,
<code class="code">"complete"</code>, and <code class="code">"pairwise"</code>. Default is <code class="code">"all"</code>.
With <code class="code">"complete"</code>, only the rows without NaN values are considered.
With <code class="code">"pairwise"</code>, the selection of NaN-free rows is made for each
pair of variables.
</p></dd>
</dl>
<p>Output <var class="var">r</var> is a matrix of Pearson’s product moment correlation
coefficients for each pair of variables.
</p>
<p>Output <var class="var">p</var> is a matrix of pair-wise p-values testing for the null
hypothesis of a correlation coefficient of zero.
</p>
<p>Outputs <var class="var">lci</var> and <var class="var">hci</var> are matrices containing, respectively, the
lower and higher bounds of the 95% confidence interval of each correlation
coefficient.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFcorr">corr</a>, <a class="ref" href="#XREFcov">cov</a>, <a class="ref" href="Descriptive-Statistics.html#XREFstd">std</a>.
</p></dd></dl>
<a class="anchor" id="XREFspearman"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-spearman"><span><code class="def-type"><var class="var">rho</var> =</code> <strong class="def-name">spearman</strong> <code class="def-code-arguments">(<var class="var">x</var>)</code><a class="copiable-link" href="#index-spearman"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-spearman-1"><span><code class="def-type"><var class="var">rho</var> =</code> <strong class="def-name">spearman</strong> <code class="def-code-arguments">(<var class="var">x</var>, <var class="var">y</var>)</code><a class="copiable-link" href="#index-spearman-1"> ¶</a></span></dt>
<dd><a class="index-entry-id" id="index-Spearman_0027s-Rho"></a>
<p>Compute Spearman’s rank correlation coefficient
<var class="var">rho</var>.
</p>
<p>For two data vectors <var class="var">x</var> and <var class="var">y</var>, Spearman’s
<var class="var">rho</var>
is the correlation coefficient of the ranks of <var class="var">x</var> and <var class="var">y</var>.
</p>
<p>If <var class="var">x</var> and <var class="var">y</var> are drawn from independent distributions,
<var class="var">rho</var>
has zero mean and variance
<code class="code">1 / (N - 1)</code>,
where <em class="math">N</em> is the length of the <var class="var">x</var> and <var class="var">y</var> vectors, and is
asymptotically normally distributed.
</p>
<p><code class="code">spearman (<var class="var">x</var>)</code> is equivalent to
<code class="code">spearman (<var class="var">x</var>, <var class="var">x</var>)</code>.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="Basic-Statistical-Functions.html#XREFranks">ranks</a>, <a class="ref" href="#XREFkendall">kendall</a>.
</p></dd></dl>
<a class="anchor" id="XREFkendall"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-kendall"><span><code class="def-type"><var class="var">tau</var> =</code> <strong class="def-name">kendall</strong> <code class="def-code-arguments">(<var class="var">x</var>)</code><a class="copiable-link" href="#index-kendall"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-kendall-1"><span><code class="def-type"><var class="var">tau</var> =</code> <strong class="def-name">kendall</strong> <code class="def-code-arguments">(<var class="var">x</var>, <var class="var">y</var>)</code><a class="copiable-link" href="#index-kendall-1"> ¶</a></span></dt>
<dd><a class="index-entry-id" id="index-Kendall_0027s-Tau"></a>
<p>Compute Kendall’s
<var class="var">tau</var>.
</p>
<p>For two data vectors <var class="var">x</var>, <var class="var">y</var> of common length <em class="math">N</em>, Kendall’s
<var class="var">tau</var>
is the correlation of the signs of all rank differences of
<var class="var">x</var> and <var class="var">y</var>; i.e., if both <var class="var">x</var> and <var class="var">y</var> have distinct
entries, then
</p>
<div class="example">
<div class="group"><pre class="example-preformatted"> 1
<var class="var">tau</var> = ------- SUM sign (<var class="var">q</var>(i) - <var class="var">q</var>(j)) * sign (<var class="var">r</var>(i) - <var class="var">r</var>(j))
N (N-1) i,j
</pre></div></div>
<p>in which the
<var class="var">q</var>(i) and <var class="var">r</var>(i)
are the ranks of <var class="var">x</var> and <var class="var">y</var>, respectively.
</p>
<p>If <var class="var">x</var> and <var class="var">y</var> are drawn from independent distributions,
Kendall’s
<var class="var">tau</var>
is asymptotically normal with mean 0 and variance
<code class="code">(2 * (2N+5)) / (9 * N * (N-1))</code>.
</p>
<p><code class="code">kendall (<var class="var">x</var>)</code> is equivalent to <code class="code">kendall (<var class="var">x</var>,
<var class="var">x</var>)</code>.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="Basic-Statistical-Functions.html#XREFranks">ranks</a>, <a class="ref" href="#XREFspearman">spearman</a>.
</p></dd></dl>
</div>
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