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<p>
<a name="Basic-Statistical-Functions"></a>
Next: <a rel="next" accesskey="n" href="Statistical-Plots.html#Statistical-Plots">Statistical Plots</a>,
Previous: <a rel="previous" accesskey="p" href="Descriptive-Statistics.html#Descriptive-Statistics">Descriptive Statistics</a>,
Up: <a rel="up" accesskey="u" href="Statistics.html#Statistics">Statistics</a>
<hr>
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<h3 class="section">25.2 Basic Statistical Functions</h3>
<p>Octave also supports various helpful statistical functions.
<!-- ./statistics/base/mahalanobis.m -->
<p><a name="doc_002dmahalanobis"></a>
<div class="defun">
— Function File: <b>mahalanobis</b> (<var>x, y</var>)<var><a name="index-mahalanobis-1840"></a></var><br>
<blockquote><p>Return the Mahalanobis' D-square distance between the multivariate
samples <var>x</var> and <var>y</var>, which must have the same number of
components (columns), but may have a different number of observations
(rows).
</p></blockquote></div>
<!-- ./statistics/base/center.m -->
<p><a name="doc_002dcenter"></a>
<div class="defun">
— Function File: <b>center</b> (<var>x</var>)<var><a name="index-center-1841"></a></var><br>
— Function File: <b>center</b> (<var>x, dim</var>)<var><a name="index-center-1842"></a></var><br>
<blockquote><p>If <var>x</var> is a vector, subtract its mean.
If <var>x</var> is a matrix, do the above for each column.
If the optional argument <var>dim</var> is given, perform the above
operation along this dimension
</p></blockquote></div>
<!-- ./statistics/base/studentize.m -->
<p><a name="doc_002dstudentize"></a>
<div class="defun">
— Function File: <b>studentize</b> (<var>x, dim</var>)<var><a name="index-studentize-1843"></a></var><br>
<blockquote><p>If <var>x</var> is a vector, subtract its mean and divide by its standard
deviation.
<p>If <var>x</var> is a matrix, do the above along the first non-singleton
dimension. If the optional argument <var>dim</var> is given then operate
along this dimension.
</p></blockquote></div>
<!-- ./specfun/nchoosek.m -->
<p><a name="doc_002dnchoosek"></a>
<div class="defun">
— Function File: <var>c</var> = <b>nchoosek</b> (<var>n, k</var>)<var><a name="index-nchoosek-1844"></a></var><br>
<blockquote>
<p>Compute the binomial coefficient or all combinations of <var>n</var>.
If <var>n</var> is a scalar then, calculate the binomial coefficient
of <var>n</var> and <var>k</var>, defined as
<pre class="example"> / \
| n | n (n-1) (n-2) ... (n-k+1) n!
| | = ------------------------- = ---------
| k | k! k! (n-k)!
\ /
</pre>
<p>If <var>n</var> is a vector generate all combinations of the elements
of <var>n</var>, taken <var>k</var> at a time, one row per combination. The
resulting <var>c</var> has size <code>[nchoosek (length (</code><var>n</var><code>),
</code><var>k</var><code>), </code><var>k</var><code>]</code>.
<p><code>nchoosek</code> works only for non-negative integer arguments; use
<code>bincoeff</code> for non-integer scalar arguments and for using vector
arguments to compute many coefficients at once.
<!-- Texinfo @sp should work but in practice produces ugly results for HTML. -->
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<!-- @sp 1 -->
<p class="noindent"><strong>See also:</strong> <a href="doc_002dbincoeff.html#doc_002dbincoeff">bincoeff</a>.
</p></blockquote></div>
<!-- ./statistics/base/histc.m -->
<p><a name="doc_002dhistc"></a>
<div class="defun">
— Function File: <var>n</var> = <b>histc</b> (<var>y, edges</var>)<var><a name="index-histc-1845"></a></var><br>
— Function File: <var>n</var> = <b>histc</b> (<var>y, edges, dim</var>)<var><a name="index-histc-1846"></a></var><br>
— Function File: [<var>n</var>, <var>idx</var>] = <b>histc</b> (<var><small class="dots">...</small></var>)<var><a name="index-histc-1847"></a></var><br>
<blockquote><p>Produce histogram counts.
<p>When <var>y</var> is a vector, the function counts the number of elements of
<var>y</var> that fall in the histogram bins defined by <var>edges</var>. This must be
a vector of monotonically non-decreasing values that define the edges of the
histogram bins. So, <var>n</var><code> (k)</code> contains the number of elements in
<var>y</var> for which <var>edges</var><code> (k) <= </code><var>y</var><code> < </code><var>edges</var><code> (k+1)</code>.
The final element of <var>n</var> contains the number of elements of <var>y</var>
that was equal to the last element of <var>edges</var>.
<p>When <var>y</var> is a N-dimensional array, the same operation as above is
repeated along dimension <var>dim</var>. If this argument is given, the operation
is performed along the first non-singleton dimension.
<p>If a second output argument is requested an index matrix is also returned.
The <var>idx</var> matrix has same size as <var>y</var>. Each element of <var>idx</var>
contains the index of the histogram bin in which the corresponding element
of <var>y</var> was counted.
<!-- Texinfo @sp should work but in practice produces ugly results for HTML. -->
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<!-- @sp 1 -->
<p class="noindent"><strong>See also:</strong> <a href="doc_002dhist.html#doc_002dhist">hist</a>.
</p></blockquote></div>
<!-- ./specfun/perms.m -->
<p><a name="doc_002dperms"></a>
<div class="defun">
— Function File: <b>perms</b> (<var>v</var>)<var><a name="index-perms-1848"></a></var><br>
<blockquote>
<p>Generate all permutations of <var>v</var>, one row per permutation. The
result has size <code>factorial (</code><var>n</var><code>) * </code><var>n</var>, where <var>n</var>
is the length of <var>v</var>.
<p>As an example, <code>perms([1, 2, 3])</code> returns the matrix
<pre class="example"> 1 2 3
2 1 3
1 3 2
2 3 1
3 1 2
3 2 1
</pre>
</blockquote></div>
<!-- ./statistics/base/values.m -->
<p><a name="doc_002dvalues"></a>
<div class="defun">
— Function File: <b>values</b> (<var>x</var>)<var><a name="index-values-1849"></a></var><br>
<blockquote><p>Return the different values in a column vector, arranged in ascending
order.
<p>As an example, <code>values([1, 2, 3, 1])</code> returns the vector
<code>[1, 2, 3]</code>.
</p></blockquote></div>
<!-- ./statistics/base/table.m -->
<p><a name="doc_002dtable"></a>
<div class="defun">
— Function File: [<var>t</var>, <var>l_x</var>] = <b>table</b> (<var>x</var>)<var><a name="index-table-1850"></a></var><br>
— Function File: [<var>t</var>, <var>l_x</var>, <var>l_y</var>] = <b>table</b> (<var>x, y</var>)<var><a name="index-table-1851"></a></var><br>
<blockquote><p>Create a contingency table <var>t</var> from data vectors. The <var>l</var>
vectors are the corresponding levels.
<p>Currently, only 1- and 2-dimensional tables are supported.
</p></blockquote></div>
<!-- ./statistics/base/spearman.m -->
<p><a name="doc_002dspearman"></a>
<div class="defun">
— Function File: <b>spearman</b> (<var>x, y</var>)<var><a name="index-spearman-1852"></a></var><br>
<blockquote><p>Compute Spearman's rank correlation coefficient <var>rho</var> for each of
the variables specified by the input arguments.
<p>For matrices, each row is an observation and each column a variable;
vectors are always observations and may be row or column vectors.
<p><code>spearman (</code><var>x</var><code>)</code> is equivalent to <code>spearman (</code><var>x</var><code>,
</code><var>x</var><code>)</code>.
<p>For two data vectors <var>x</var> and <var>y</var>, Spearman's <var>rho</var> is the
correlation of the ranks of <var>x</var> and <var>y</var>.
<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></blockquote></div>
<!-- ./statistics/base/run_count.m -->
<p><a name="doc_002drun_005fcount"></a>
<div class="defun">
— Function File: <b>run_count</b> (<var>x, n</var>)<var><a name="index-run_005fcount-1853"></a></var><br>
<blockquote><p>Count the upward runs along the first non-singleton dimension of
<var>x</var> of length 1, 2, <small class="dots">...</small>, <var>n</var>-1 and greater than or equal
to <var>n</var>. If the optional argument <var>dim</var> is given operate
along this dimension
</p></blockquote></div>
<!-- ./statistics/base/ranks.m -->
<p><a name="doc_002dranks"></a>
<div class="defun">
— Function File: <b>ranks</b> (<var>x, dim</var>)<var><a name="index-ranks-1854"></a></var><br>
<blockquote><p>Return the ranks of <var>x</var> along the first non-singleton dimension
adjust for ties. If the optional argument <var>dim</var> is
given, operate along this dimension.
</p></blockquote></div>
<!-- ./statistics/base/range.m -->
<p><a name="doc_002drange"></a>
<div class="defun">
— Function File: <b>range</b> (<var>x</var>)<var><a name="index-range-1855"></a></var><br>
— Function File: <b>range</b> (<var>x, dim</var>)<var><a name="index-range-1856"></a></var><br>
<blockquote><p>If <var>x</var> is a vector, return the range, i.e., the difference
between the maximum and the minimum, of the input data.
<p>If <var>x</var> is a matrix, do the above for each column of <var>x</var>.
<p>If the optional argument <var>dim</var> is supplied, work along dimension
<var>dim</var>.
</p></blockquote></div>
<!-- ./statistics/base/probit.m -->
<p><a name="doc_002dprobit"></a>
<div class="defun">
— Function File: <b>probit</b> (<var>p</var>)<var><a name="index-probit-1857"></a></var><br>
<blockquote><p>For each component of <var>p</var>, return the probit (the quantile of the
standard normal distribution) of <var>p</var>.
</p></blockquote></div>
<!-- ./statistics/base/logit.m -->
<p><a name="doc_002dlogit"></a>
<div class="defun">
— Function File: <b>logit</b> (<var>p</var>)<var><a name="index-logit-1858"></a></var><br>
<blockquote><p>For each component of <var>p</var>, return the logit of <var>p</var> defined as
<pre class="example"> logit(<var>p</var>) = log (<var>p</var> / (1-<var>p</var>))
</pre>
</blockquote></div>
<!-- ./statistics/base/cloglog.m -->
<p><a name="doc_002dcloglog"></a>
<div class="defun">
— Function File: <b>cloglog</b> (<var>x</var>)<var><a name="index-cloglog-1859"></a></var><br>
<blockquote><p>Return the complementary log-log function of <var>x</var>, defined as
<pre class="example"> cloglog(x) = - log (- log (<var>x</var>))
</pre>
</blockquote></div>
<!-- ./statistics/base/kendall.m -->
<p><a name="doc_002dkendall"></a>
<div class="defun">
— Function File: <b>kendall</b> (<var>x, y</var>)<var><a name="index-kendall-1860"></a></var><br>
<blockquote><p>Compute Kendall's <var>tau</var> for each of the variables specified by
the input arguments.
<p>For matrices, each row is an observation and each column a variable;
vectors are always observations and may be row or column vectors.
<p><code>kendall (</code><var>x</var><code>)</code> is equivalent to <code>kendall (</code><var>x</var><code>,
</code><var>x</var><code>)</code>.
<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
<pre class="example"> 1
tau = ------- SUM sign (q(i) - q(j)) * sign (r(i) - r(j))
n (n-1) i,j
</pre>
<p class="noindent">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>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</code><var>n</var><code>+5)) / (9 * </code><var>n</var><code> * (</code><var>n</var><code>-1))</code>.
</p></blockquote></div>
<!-- ./statistics/base/iqr.m -->
<p><a name="doc_002diqr"></a>
<div class="defun">
— Function File: <b>iqr</b> (<var>x, dim</var>)<var><a name="index-iqr-1861"></a></var><br>
<blockquote><p>If <var>x</var> is a vector, return the interquartile range, i.e., the
difference between the upper and lower quartile, of the input data.
<p>If <var>x</var> is a matrix, do the above for first non-singleton
dimension of <var>x</var>. If the option <var>dim</var> argument is given,
then operate along this dimension.
</p></blockquote></div>
<!-- ./statistics/base/cut.m -->
<p><a name="doc_002dcut"></a>
<div class="defun">
— Function File: <b>cut</b> (<var>x, breaks</var>)<var><a name="index-cut-1862"></a></var><br>
<blockquote><p>Create categorical data out of numerical or continuous data by
cutting into intervals.
<p>If <var>breaks</var> is a scalar, the data is cut into that many
equal-width intervals. If <var>breaks</var> is a vector of break points,
the category has <code>length (</code><var>breaks</var><code>) - 1</code> groups.
<p>The returned value is a vector of the same size as <var>x</var> telling
which group each point in <var>x</var> belongs to. Groups are labelled
from 1 to the number of groups; points outside the range of
<var>breaks</var> are labelled by <code>NaN</code>.
</p></blockquote></div>
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