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<a name="Correlation-and-Regression-Analysis"></a>
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<a name="Correlation-and-Regression-Analysis-1"></a>
<h3 class="section">26.4 Correlation and Regression Analysis</h3>


<a name="XREFcov"></a><dl>
<dt><a name="index-cov"></a><em></em> <strong>cov</strong> <em>(<var>x</var>)</em></dt>
<dt><a name="index-cov-1"></a><em></em> <strong>cov</strong> <em>(<var>x</var>, <var>opt</var>)</em></dt>
<dt><a name="index-cov-2"></a><em></em> <strong>cov</strong> <em>(<var>x</var>, <var>y</var>)</em></dt>
<dt><a name="index-cov-3"></a><em></em> <strong>cov</strong> <em>(<var>x</var>, <var>y</var>, <var>opt</var>)</em></dt>
<dd><p>Compute the covariance matrix.
</p>
<p>If each row of <var>x</var> and <var>y</var> is an observation, and each column is
a variable, then the (<var>i</var>,&nbsp;<var>j</var><span class="nolinebreak">)-th</span><!-- /@w --> entry of
<code>cov (<var>x</var>, <var>y</var>)</code> is the covariance between the <var>i</var>-th
variable in <var>x</var> and the <var>j</var>-th variable in <var>y</var>.
</p>
<div class="example">
<pre class="example">cov (<var>x</var>) = 1/(N-1) * SUM_i (<var>x</var>(i) - mean(<var>x</var>)) * (<var>y</var>(i) - mean(<var>y</var>))
</pre></div>

<p>where <em>N</em> is the length of the <var>x</var> and <var>y</var> vectors.
</p>

<p>If called with one argument, compute <code>cov (<var>x</var>, <var>x</var>)</code>, the
covariance between the columns of <var>x</var>.
</p>
<p>The argument <var>opt</var> determines the type of normalization to use.
Valid values are
</p>
<dl compact="compact">
<dt>0:</dt>
<dd><p>normalize with <em>N-1</em>, provides the best unbiased estimator of the
covariance [default]
</p>
</dd>
<dt>1:</dt>
<dd><p>normalize with <em>N</em>, this provides the second moment around the mean
</p></dd>
</dl>

<p>Compatibility Note:: Octave always treats rows of <var>x</var> and <var>y</var>
as multivariate random variables.
For two inputs, however, <small>MATLAB</small> treats <var>x</var> and <var>y</var> as two
univariate distributions regardless of their shapes, and will calculate
<code>cov ([<var>x</var>(:), <var>y</var>(:)])</code> whenever the number of elements in
<var>x</var> and <var>y</var> are equal.  This will result in a 2x2 matrix.
Code relying on <small>MATLAB</small>&rsquo;s definition will need to be changed when
running in Octave.
</p>
<p><strong>See also:</strong> <a href="#XREFcorr">corr</a>.
</p></dd></dl>


<a name="XREFcorr"></a><dl>
<dt><a name="index-corr"></a><em></em> <strong>corr</strong> <em>(<var>x</var>)</em></dt>
<dt><a name="index-corr-1"></a><em></em> <strong>corr</strong> <em>(<var>x</var>, <var>y</var>)</em></dt>
<dd><p>Compute matrix of correlation coefficients.
</p>
<p>If each row of <var>x</var> and <var>y</var> is an observation and each column is
a variable, then the (<var>i</var>,&nbsp;<var>j</var><span class="nolinebreak">)-th</span><!-- /@w --> entry of
<code>corr (<var>x</var>, <var>y</var>)</code> is the correlation between the
<var>i</var>-th variable in <var>x</var> and the <var>j</var>-th variable in <var>y</var>.
</p>
<div class="example">
<pre class="example">corr (<var>x</var>,<var>y</var>) = cov (<var>x</var>,<var>y</var>) / (std (<var>x</var>) * std (<var>y</var>))
</pre></div>

<p>If called with one argument, compute <code>corr (<var>x</var>, <var>x</var>)</code>,
the correlation between the columns of <var>x</var>.
</p>
<p><strong>See also:</strong> <a href="#XREFcov">cov</a>.
</p></dd></dl>


<a name="XREFcorrcoef"></a><dl>
<dt><a name="index-corrcoef"></a><em><var>r</var> =</em> <strong>corrcoef</strong> <em>(<var>x</var>)</em></dt>
<dt><a name="index-corrcoef-1"></a><em><var>r</var> =</em> <strong>corrcoef</strong> <em>(<var>x</var>, <var>y</var>)</em></dt>
<dt><a name="index-corrcoef-2"></a><em><var>r</var> =</em> <strong>corrcoef</strong> <em>(&hellip;, <var>param</var>, <var>value</var>, &hellip;)</em></dt>
<dt><a name="index-corrcoef-3"></a><em>[<var>r</var>, <var>p</var>] =</em> <strong>corrcoef</strong> <em>(&hellip;)</em></dt>
<dt><a name="index-corrcoef-4"></a><em>[<var>r</var>, <var>p</var>, <var>lci</var>, <var>hci</var>] =</em> <strong>corrcoef</strong> <em>(&hellip;)</em></dt>
<dd><p>Compute a matrix of correlation coefficients.
</p>
<p><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>y</var> (of the same size as <var>x</var>) is given then
calculate the correlation coefficients between <var>x</var> and <var>y</var>.
</p>
<p><var>param</var>, <var>value</var> are optional pairs of parameters and values which
modify the calculation.  Valid options are:
</p>
<dl compact="compact">
<dt><code>&quot;alpha&quot;</code></dt>
<dd><p>Confidence level used for the bounds of the confidence interval, <var>lci</var>
and <var>hci</var>.  Default is 0.05, i.e., 95% confidence interval.
</p>
</dd>
<dt><code>&quot;rows&quot;</code></dt>
<dd><p>Determine processing of NaN values.  Acceptable values are <code>&quot;all&quot;</code>,
<code>&quot;complete&quot;</code>, and <code>&quot;pairwise&quot;</code>.  Default is <code>&quot;all&quot;</code>.
With <code>&quot;complete&quot;</code>, only the rows without NaN values are considered.
With <code>&quot;pairwise&quot;</code>, the selection of NaN-free rows is made for each
pair of variables.
</p></dd>
</dl>

<p>Output <var>r</var> is a matrix of Pearson&rsquo;s product moment correlation
coefficients for each pair of variables.
</p>
<p>Output <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>lci</var> and <var>hci</var> are matrices containing, respectively, the
lower and higher bounds of the 95% confidence interval of each correlation
coefficient.
</p>
<p><strong>See also:</strong> <a href="#XREFcorr">corr</a>, <a href="#XREFcov">cov</a>.
</p></dd></dl>


<a name="XREFspearman"></a><dl>
<dt><a name="index-spearman"></a><em></em> <strong>spearman</strong> <em>(<var>x</var>)</em></dt>
<dt><a name="index-spearman-1"></a><em></em> <strong>spearman</strong> <em>(<var>x</var>, <var>y</var>)</em></dt>
<dd><a name="index-Spearman_0027s-Rho"></a>
<p>Compute Spearman&rsquo;s rank correlation coefficient
<var>rho</var>.
</p>
<p>For two data vectors <var>x</var> and <var>y</var>, Spearman&rsquo;s
<var>rho</var>
is the correlation coefficient of the ranks of <var>x</var> and <var>y</var>.
</p>
<p>If <var>x</var> and <var>y</var> are drawn from independent distributions,
<var>rho</var>
has zero mean and variance
<code>1 / (N - 1)</code>,
where <em>N</em> is the length of the <var>x</var> and <var>y</var> vectors, and is
asymptotically normally distributed.
</p>
<p><code>spearman (<var>x</var>)</code> is equivalent to
<code>spearman (<var>x</var>, <var>x</var>)</code>.
</p>
<p><strong>See also:</strong> <a href="Basic-Statistical-Functions.html#XREFranks">ranks</a>, <a href="#XREFkendall">kendall</a>.
</p></dd></dl>


<a name="XREFkendall"></a><dl>
<dt><a name="index-kendall"></a><em></em> <strong>kendall</strong> <em>(<var>x</var>)</em></dt>
<dt><a name="index-kendall-1"></a><em></em> <strong>kendall</strong> <em>(<var>x</var>, <var>y</var>)</em></dt>
<dd><a name="index-Kendall_0027s-Tau"></a>
<p>Compute Kendall&rsquo;s
<var>tau</var>.
</p>
<p>For two data vectors <var>x</var>, <var>y</var> of common length <em>N</em>, Kendall&rsquo;s
<var>tau</var>
is the correlation of the signs of all rank differences of
<var>x</var> and <var>y</var>; i.e., if both <var>x</var> and <var>y</var> have distinct
entries, then
</p>

<div class="example">
<pre class="example">         1
<var>tau</var> = -------   SUM sign (<var>q</var>(i) - <var>q</var>(j)) * sign (<var>r</var>(i) - <var>r</var>(j))
      N (N-1)   i,j
</pre></div>

<p>in which the
<var>q</var>(i) and <var>r</var>(i)
are the ranks of <var>x</var> and <var>y</var>, respectively.
</p>
<p>If <var>x</var> and <var>y</var> are drawn from independent distributions,
Kendall&rsquo;s
<var>tau</var>
is asymptotically normal with mean 0 and variance
<code>(2 * (2N+5)) / (9 * N * (N-1))</code>.
</p>
<p><code>kendall (<var>x</var>)</code> is equivalent to <code>kendall (<var>x</var>,
<var>x</var>)</code>.
</p>
<p><strong>See also:</strong> <a href="Basic-Statistical-Functions.html#XREFranks">ranks</a>, <a href="#XREFspearman">spearman</a>.
</p></dd></dl>


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