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<span id="Correlation-and-Regression-Analysis"></span><div class="header">
<p>
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<hr>
<span id="Correlation-and-Regression-Analysis-1"></span><h3 class="section">26.4 Correlation and Regression Analysis</h3>
<span id="XREFcov"></span><dl>
<dt id="index-cov">: <em></em> <strong>cov</strong> <em>(<var>x</var>)</em></dt>
<dt id="index-cov-1">: <em></em> <strong>cov</strong> <em>(<var>x</var>, <var>opt</var>)</em></dt>
<dt id="index-cov-2">: <em></em> <strong>cov</strong> <em>(<var>x</var>, <var>y</var>)</em></dt>
<dt id="index-cov-3">: <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>, <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>’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>
<span id="XREFcorr"></span><dl>
<dt id="index-corr">: <em></em> <strong>corr</strong> <em>(<var>x</var>)</em></dt>
<dt id="index-corr-1">: <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>, <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>
<span id="XREFcorrcoef"></span><dl>
<dt id="index-corrcoef">: <em><var>r</var> =</em> <strong>corrcoef</strong> <em>(<var>x</var>)</em></dt>
<dt id="index-corrcoef-1">: <em><var>r</var> =</em> <strong>corrcoef</strong> <em>(<var>x</var>, <var>y</var>)</em></dt>
<dt id="index-corrcoef-2">: <em><var>r</var> =</em> <strong>corrcoef</strong> <em>(…, <var>param</var>, <var>value</var>, …)</em></dt>
<dt id="index-corrcoef-3">: <em>[<var>r</var>, <var>p</var>] =</em> <strong>corrcoef</strong> <em>(…)</em></dt>
<dt id="index-corrcoef-4">: <em>[<var>r</var>, <var>p</var>, <var>lci</var>, <var>hci</var>] =</em> <strong>corrcoef</strong> <em>(…)</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>"alpha"</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>"rows"</code></dt>
<dd><p>Determine processing of NaN values. Acceptable values are <code>"all"</code>,
<code>"complete"</code>, and <code>"pairwise"</code>. Default is <code>"all"</code>.
With <code>"complete"</code>, only the rows without NaN values are considered.
With <code>"pairwise"</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’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>
<span id="XREFspearman"></span><dl>
<dt id="index-spearman">: <em></em> <strong>spearman</strong> <em>(<var>x</var>)</em></dt>
<dt id="index-spearman-1">: <em></em> <strong>spearman</strong> <em>(<var>x</var>, <var>y</var>)</em></dt>
<dd><span id="index-Spearman_0027s-Rho"></span>
<p>Compute Spearman’s rank correlation coefficient
<var>rho</var>.
</p>
<p>For two data vectors <var>x</var> and <var>y</var>, Spearman’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>
<span id="XREFkendall"></span><dl>
<dt id="index-kendall">: <em></em> <strong>kendall</strong> <em>(<var>x</var>)</em></dt>
<dt id="index-kendall-1">: <em></em> <strong>kendall</strong> <em>(<var>x</var>, <var>y</var>)</em></dt>
<dd><span id="index-Kendall_0027s-Tau"></span>
<p>Compute Kendall’s
<var>tau</var>.
</p>
<p>For two data vectors <var>x</var>, <var>y</var> of common length <em>N</em>, Kendall’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’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>
<hr>
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Next: <a href="Distributions.html" accesskey="n" rel="next">Distributions</a>, Previous: <a href="Basic-Statistical-Functions.html" accesskey="p" rel="prev">Basic Statistical Functions</a>, Up: <a href="Statistics.html" accesskey="u" rel="up">Statistics</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html" title="Index" rel="index">Index</a>]</p>
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