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Posted to commits@commons.apache.org by ps...@apache.org on 2009/04/12 20:52:37 UTC
svn commit: r764314 -
/commons/proper/math/trunk/src/site/xdoc/userguide/stat.xml
Author: psteitz
Date: Sun Apr 12 18:52:37 2009
New Revision: 764314
URL: http://svn.apache.org/viewvc?rev=764314&view=rev
Log:
Fixed internal links and added covariance and correlation section.
Modified:
commons/proper/math/trunk/src/site/xdoc/userguide/stat.xml
Modified: commons/proper/math/trunk/src/site/xdoc/userguide/stat.xml
URL: http://svn.apache.org/viewvc/commons/proper/math/trunk/src/site/xdoc/userguide/stat.xml?rev=764314&r1=764313&r2=764314&view=diff
==============================================================================
--- commons/proper/math/trunk/src/site/xdoc/userguide/stat.xml (original)
+++ commons/proper/math/trunk/src/site/xdoc/userguide/stat.xml Sun Apr 12 18:52:37 2009
@@ -25,20 +25,22 @@
</properties>
<body>
<section name="1 Statistics">
- <subsection name="1.1 Overview" href="overview">
+ <subsection name="1.1 Overview">
<p>
The statistics package provides frameworks and implementations for
basic Descriptive statistics, frequency distributions, bivariate regression,
and t-, chi-square and ANOVA test statistics.
</p>
<p>
- <a href="#1.2 Descriptive statistics">Descriptive statistics</a><br></br>
- <a href="#1.3 Frequency distributions">Frequency distributions</a><br></br>
- <a href="#1.4 Simple regression">Simple Regression</a><br></br>
- <a href="#1.5 Statistical tests">Statistical Tests</a><br></br>
+ <a href="#a1.2_Descriptive_statistics">Descriptive statistics</a><br></br>
+ <a href="#a1.3_Frequency_distributions">Frequency distributions</a><br></br>
+ <a href="#a1.4_Simple_regression">Simple Regression</a><br></br>
+ <a href="#a1.5_Multiple_linear_regression">Multiple Regression</a><br></br>
+ <a href="#a1.6_Covariance_and_correlation">Covariance and correlation</a><br></br>
+ <a href="#a1.7_Statistical_tests">Statistical Tests</a><br></br>
</p>
</subsection>
- <subsection name="1.2 Descriptive statistics" href="univariate">
+ <subsection name="1.2 Descriptive statistics">
<p>
The stat package includes a framework and default implementations for
the following Descriptive statistics:
@@ -217,7 +219,7 @@
</dl>
</p>
</subsection>
- <subsection name="1.3 Frequency distributions" href="frequency">
+ <subsection name="1.3 Frequency distributions">
<p>
<a href="../apidocs/org/apache/commons/math/stat/Frequency.html">
org.apache.commons.math.stat.descriptive.Frequency</a>
@@ -281,7 +283,7 @@
</dl>
</p>
</subsection>
- <subsection name="1.4 Simple regression" href="regression">
+ <subsection name="1.4 Simple regression">
<p>
<a href="../apidocs/org/apache/commons/math/stat/regression/SimpleRegression.html">
org.apache.commons.math.stat.regression.SimpleRegression</a>
@@ -398,7 +400,7 @@
</dl>
</p>
</subsection>
- <subsection name="1.5 Multiple linear regression" href="regression">
+ <subsection name="1.5 Multiple linear regression">
<p>
<a href="../apidocs/org/apache/commons/math/stat/regression/MultipleLinearRegression.html">
org.apache.commons.math.stat.regression.MultipleLinearRegression</a>
@@ -492,7 +494,121 @@
</dl>
</p>
</subsection>
- <subsection name="1.6 Statistical tests" href="tests">
+ <subsection name="1.6 Covariance and correlation">
+ <p>
+ The <a href="../apidocs/org/apache/commons/math/stat/correlation/package-summary.html">
+ org.apache.commons.math.stat.correlation</a> package computes covariances
+ and correlations for pairs of arrays or columns of a matrix.
+ <a href="../apidocs/org/apache/commons/math/stat/correlation/Covariance.html">
+ Covariance</a> computes covariances and
+ <a href="../apidocs/org/apache/commons/math/stat/correlation/PearsonsCorrelation.html">
+ PearsonsCorrelation</a> provides Pearson's Product-Moment correlation coefficients.
+ </p>
+ <p>
+ <strong>Implementation Notes</strong>
+ <ul>
+ <li>
+ Unbiased covariances are given by the formula <br></br>
+ <code>cov(X, Y) = sum [(x<sub>i</sub> - E(X))(y<sub>i</sub> - E(Y))] / (n - 1)</code>
+ where <code>E(X)</code> is the mean of <code>X</code> and <code>E(Y)</code>
+ is the mean of the <code>Y</code> values. Non-bias-corrected estimates use
+ <code>n</code> in place of <code>n - 1.</code> Whether or not covariances are
+ bias-corrected is determined by the optional constructor parameter,
+ "biasCorrected," which defaults to <code>true.</code>
+ </li>
+ <li>
+ <a href="../apidocs/org/apache/commons/math/stat/correlation/PearsonsCorrelation.html">
+ PearsonsCorrelation</a> computes corralations defined by the formula <br></br>
+ <code>cor(X, Y) = sum[(x<sub>i</sub> - E(X))(y<sub>i</sub> - E(Y))] / [(n - 1)s(X)s(Y)]</code>
+ where <code>E(X)</code> and <code>E(Y)</code> are means of <code>X</code> and <code>Y</code>
+ and <code>s(X)</code>, <code>s(Y)</code> are standard deviations.
+ </li>
+ </ul>
+ </p>
+ <p>
+ <strong>Examples:</strong>
+ <dl>
+ <dt><strong>Covariance of 2 arrays</strong></dt>
+ <br></br>
+ <dd>To compute the unbiased covariance between 2 double arrays,
+ <code>x</code> and <code>y</code>, use:
+ <source>
+new Covariance().covariance(x, y)
+ </source>
+ For non-bias-corrected covariances, use
+ <source>
+covariance(x, y, false)
+ </source>
+ </dd>
+ <br></br>
+ <dt><strong>Covariance matrix</strong></dt>
+ <br></br>
+ <dd> A covariance matrix over the columns of a source matrix <code>data</code>
+ can be computed using
+ <source>
+new Covariance().computeCovarianceMatrix(data)
+ </source>
+ The i-jth entry of the returned matrix is the unbiased covariance of the ith and jth
+ columns of <code>data.</code> As above, to get non-bias-corrected covariances,
+ use
+ <source>
+computeCovarianceMatrix(data, false)
+ </source>
+ </dd>
+ <br></br>
+ <dt><strong>Pearson's correlation of 2 arrays</strong></dt>
+ <br></br>
+ <dd>To compute the Pearson's product-moment correlation between two double arrays
+ <code>x</code> and <code>y</code>, use:
+ <source>
+new PearsonsCorrelation().correlation(x, y)
+ </source>
+ </dd>
+ <br></br>
+ <dt><strong>Pearson's correlation matrix</strong></dt>
+ <br></br>
+ <dd> A (Pearson's) correlation matrix over the columns of a source matrix <code>data</code>
+ can be computed using
+ <source>
+new PearsonsCorrelation().computeCorrelationMatrix(data)
+ </source>
+ The i-jth entry of the returned matrix is the Pearson's product-moment correlation between the
+ ith and jth columns of <code>data.</code>
+ </dd>
+ <br></br>
+ <dt><strong>Pearson's correlation significance and standard errors</strong></dt>
+ <br></br>
+ <dd> To compute standard errors and/or significances of correlation coefficients
+ associated with Pearson's correlation coefficients, start by creating a PearsonsCorrelation
+ instance from the data <code>data</code> using
+ <source>
+PearsonsCorrelation correlation = new PearsonsCorrelation(data);
+ </source>
+ where <code>data</code> is either a rectangular array or a <code>RealMatrix.</code>
+ Then the matrix of standard errors is
+ <source>
+correlation.getCorrelationStandardErrors();
+ </source>
+ The formula used to compute the standard error is <br/>
+ <code>SE<sub>r</sub> = ((1 - r<sup>2</sup>) / (n - 2))<sup>1/2</sup></code><br/>
+ where <code>r</code> is the estimated correlation coefficient and
+ <code>n</code> is the number of observations in the source dataset.<br/><br/>
+ <strong>p-values</strong> for the null hypothesis that respective coefficients are zero (also known as
+ <i>significances</i>) populate the <code>RealMatrix</code> returned by
+ <source>
+correlation.getCorrelationPValues();
+ </source>
+ <code>getCorrelationPValues().getEntry(i,j)</code> is the probability
+ that a random variable distributed as <code>t<sub>n-2</sub></code> takes
+ a value with absolute value greater than or equal to <br></br>
+ <code>|r|((n - 2) / (1 - r<sup>2</sup>))<sup>1/2</sup></code>, where <code>r</code>
+ is the estimated correlation coefficient.
+ </dd>
+ <br></br>
+ </dl>
+ </p>
+ </subsection>
+ <subsection name="1.7 Statistical tests">
<p>
The interfaces and implementations in the
<a href="../apidocs/org/apache/commons/math/stat/inference/">