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<a name="Correlation-and-Regression-Analysis"></a>
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<p>
Next: <a href="Distributions.html#Distributions" accesskey="n" rel="next">Distributions</a>, Previous: <a href="Statistical-Plots.html#Statistical-Plots" accesskey="p" rel="prev">Statistical Plots</a>, Up: <a href="Statistics.html#Statistics" 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#Concept-Index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<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>Function File: <em></em> <strong>cov</strong> <em>(<var>x</var>)</em></dt>
<dt><a name="index-cov-1"></a>Function File: <em></em> <strong>cov</strong> <em>(<var>x</var>, <var>opt</var>)</em></dt>
<dt><a name="index-cov-2"></a>Function File: <em></em> <strong>cov</strong> <em>(<var>x</var>, <var>y</var>)</em></dt>
<dt><a name="index-cov-3"></a>Function File: <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 (x) = 1/N-1 * SUM_i (x(i) - mean(x)) * (y(i) - mean(y))
</pre></div>
<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><small>MATLAB</small> compatibility: Octave always computes the covariance matrix.
For two inputs, however, <small>MATLAB</small> 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 scalar rather than
a matrix output. Code relying on this odd 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>Function File: <em></em> <strong>corr</strong> <em>(<var>x</var>)</em></dt>
<dt><a name="index-corr-1"></a>Function File: <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 (x,y) = cov (x,y) / (std (x) * std (y))
</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="XREFspearman"></a><dl>
<dt><a name="index-spearman"></a>Function File: <em></em> <strong>spearman</strong> <em>(<var>x</var>)</em></dt>
<dt><a name="index-spearman-1"></a>Function File: <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’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>, 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>Function File: <em></em> <strong>kendall</strong> <em>(<var>x</var>)</em></dt>
<dt><a name="index-kendall-1"></a>Function File: <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’s <var>tau</var>.
</p>
<p>For two data vectors <var>x</var>, <var>y</var> of common length <var>n</var>,
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
tau = ------- SUM sign (q(i) - q(j)) * sign (r(i) - r(j))
n (n-1) i,j
</pre></div>
<p>in which the
<var>q</var>(<var>i</var>) and <var>r</var>(<var>i</var>)
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 * (2<var>n</var>+5)) / (9 * <var>n</var> * (<var>n</var>-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>
<a name="XREFlogistic_005fregression"></a><dl>
<dt><a name="index-logistic_005fregression"></a>Function File: <em>[<var>theta</var>, <var>beta</var>, <var>dev</var>, <var>dl</var>, <var>d2l</var>, <var>p</var>] =</em> <strong>logistic_regression</strong> <em>(<var>y</var>, <var>x</var>, <var>print</var>, <var>theta</var>, <var>beta</var>)</em></dt>
<dd><p>Perform ordinal logistic regression.
</p>
<p>Suppose <var>y</var> takes values in <var>k</var> ordered categories, and let
<code>gamma_i (<var>x</var>)</code> be the cumulative probability that <var>y</var>
falls in one of the first <var>i</var> categories given the covariate
<var>x</var>. Then
</p>
<div class="example">
<pre class="example">[theta, beta] = logistic_regression (y, x)
</pre></div>
<p>fits the model
</p>
<div class="example">
<pre class="example">logit (gamma_i (x)) = theta_i - beta' * x, i = 1 … k-1
</pre></div>
<p>The number of ordinal categories, <var>k</var>, is taken to be the number
of distinct values of <code>round (<var>y</var>)</code>. If <var>k</var> equals 2,
<var>y</var> is binary and the model is ordinary logistic regression. The
matrix <var>x</var> is assumed to have full column rank.
</p>
<p>Given <var>y</var> only, <code>theta = logistic_regression (y)</code>
fits the model with baseline logit odds only.
</p>
<p>The full form is
</p>
<div class="example">
<pre class="example">[theta, beta, dev, dl, d2l, gamma]
= logistic_regression (y, x, print, theta, beta)
</pre></div>
<p>in which all output arguments and all input arguments except <var>y</var>
are optional.
</p>
<p>Setting <var>print</var> to 1 requests summary information about the fitted
model to be displayed. Setting <var>print</var> to 2 requests information
about convergence at each iteration. Other values request no
information to be displayed. The input arguments <var>theta</var> and
<var>beta</var> give initial estimates for <var>theta</var> and <var>beta</var>.
</p>
<p>The returned value <var>dev</var> holds minus twice the log-likelihood.
</p>
<p>The returned values <var>dl</var> and <var>d2l</var> are the vector of first
and the matrix of second derivatives of the log-likelihood with
respect to <var>theta</var> and <var>beta</var>.
</p>
<p><var>p</var> holds estimates for the conditional distribution of <var>y</var>
given <var>x</var>.
</p></dd></dl>
<hr>
<div class="header">
<p>
Next: <a href="Distributions.html#Distributions" accesskey="n" rel="next">Distributions</a>, Previous: <a href="Statistical-Plots.html#Statistical-Plots" accesskey="p" rel="prev">Statistical Plots</a>, Up: <a href="Statistics.html#Statistics" 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#Concept-Index" title="Index" rel="index">Index</a>]</p>
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