Fixed internal links and added covariance and correlation section.
git-svn-id: https://svn.apache.org/repos/asf/commons/proper/math/trunk@764314 13f79535-47bb-0310-9956-ffa450edef68
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Phil Steitz
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@ -25,20 +25,22 @@
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</properties>
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<body>
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<section name="1 Statistics">
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<subsection name="1.1 Overview" href="overview">
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<subsection name="1.1 Overview">
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<p>
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The statistics package provides frameworks and implementations for
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basic Descriptive statistics, frequency distributions, bivariate regression,
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and t-, chi-square and ANOVA test statistics.
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</p>
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<p>
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<a href="#1.2 Descriptive statistics">Descriptive statistics</a><br></br>
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<a href="#1.3 Frequency distributions">Frequency distributions</a><br></br>
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<a href="#1.4 Simple regression">Simple Regression</a><br></br>
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<a href="#1.5 Statistical tests">Statistical Tests</a><br></br>
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<a href="#a1.2_Descriptive_statistics">Descriptive statistics</a><br></br>
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<a href="#a1.3_Frequency_distributions">Frequency distributions</a><br></br>
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<a href="#a1.4_Simple_regression">Simple Regression</a><br></br>
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<a href="#a1.5_Multiple_linear_regression">Multiple Regression</a><br></br>
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<a href="#a1.6_Covariance_and_correlation">Covariance and correlation</a><br></br>
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<a href="#a1.7_Statistical_tests">Statistical Tests</a><br></br>
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</p>
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</subsection>
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<subsection name="1.2 Descriptive statistics" href="univariate">
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<subsection name="1.2 Descriptive statistics">
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<p>
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The stat package includes a framework and default implementations for
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the following Descriptive statistics:
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@ -217,7 +219,7 @@ DescriptiveStatistics stats = DescriptiveStatistics.newInstance(SynchronizedDesc
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</dl>
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</p>
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</subsection>
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<subsection name="1.3 Frequency distributions" href="frequency">
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<subsection name="1.3 Frequency distributions">
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<p>
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<a href="../apidocs/org/apache/commons/math/stat/Frequency.html">
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org.apache.commons.math.stat.descriptive.Frequency</a>
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@ -281,7 +283,7 @@ System.out.println(f.getCumPct("z")); // displays 1
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</dl>
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</p>
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</subsection>
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<subsection name="1.4 Simple regression" href="regression">
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<subsection name="1.4 Simple regression">
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<p>
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<a href="../apidocs/org/apache/commons/math/stat/regression/SimpleRegression.html">
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org.apache.commons.math.stat.regression.SimpleRegression</a>
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@ -398,7 +400,7 @@ System.out.println(regression.getSlopeStdErr());
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</dl>
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</p>
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</subsection>
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<subsection name="1.5 Multiple linear regression" href="regression">
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<subsection name="1.5 Multiple linear regression">
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<p>
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<a href="../apidocs/org/apache/commons/math/stat/regression/MultipleLinearRegression.html">
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org.apache.commons.math.stat.regression.MultipleLinearRegression</a>
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@ -492,7 +494,121 @@ regression.addData(y, x, omega); // we do need covariance
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</dl>
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</p>
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</subsection>
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<subsection name="1.6 Statistical tests" href="tests">
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<subsection name="1.6 Covariance and correlation">
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<p>
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The <a href="../apidocs/org/apache/commons/math/stat/correlation/package-summary.html">
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org.apache.commons.math.stat.correlation</a> package computes covariances
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and correlations for pairs of arrays or columns of a matrix.
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<a href="../apidocs/org/apache/commons/math/stat/correlation/Covariance.html">
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Covariance</a> computes covariances and
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<a href="../apidocs/org/apache/commons/math/stat/correlation/PearsonsCorrelation.html">
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PearsonsCorrelation</a> provides Pearson's Product-Moment correlation coefficients.
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</p>
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<p>
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<strong>Implementation Notes</strong>
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<ul>
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<li>
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Unbiased covariances are given by the formula <br></br>
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<code>cov(X, Y) = sum [(x<sub>i</sub> - E(X))(y<sub>i</sub> - E(Y))] / (n - 1)</code>
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where <code>E(X)</code> is the mean of <code>X</code> and <code>E(Y)</code>
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is the mean of the <code>Y</code> values. Non-bias-corrected estimates use
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<code>n</code> in place of <code>n - 1.</code> Whether or not covariances are
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bias-corrected is determined by the optional constructor parameter,
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"biasCorrected," which defaults to <code>true.</code>
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</li>
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<li>
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<a href="../apidocs/org/apache/commons/math/stat/correlation/PearsonsCorrelation.html">
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PearsonsCorrelation</a> computes corralations defined by the formula <br></br>
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<code>cor(X, Y) = sum[(x<sub>i</sub> - E(X))(y<sub>i</sub> - E(Y))] / [(n - 1)s(X)s(Y)]</code>
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where <code>E(X)</code> and <code>E(Y)</code> are means of <code>X</code> and <code>Y</code>
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and <code>s(X)</code>, <code>s(Y)</code> are standard deviations.
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</li>
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</ul>
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</p>
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<p>
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<strong>Examples:</strong>
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<dl>
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<dt><strong>Covariance of 2 arrays</strong></dt>
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<br></br>
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<dd>To compute the unbiased covariance between 2 double arrays,
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<code>x</code> and <code>y</code>, use:
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<source>
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new Covariance().covariance(x, y)
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</source>
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For non-bias-corrected covariances, use
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<source>
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covariance(x, y, false)
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</source>
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</dd>
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<br></br>
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<dt><strong>Covariance matrix</strong></dt>
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<br></br>
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<dd> A covariance matrix over the columns of a source matrix <code>data</code>
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can be computed using
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<source>
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new Covariance().computeCovarianceMatrix(data)
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</source>
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The i-jth entry of the returned matrix is the unbiased covariance of the ith and jth
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columns of <code>data.</code> As above, to get non-bias-corrected covariances,
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use
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<source>
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computeCovarianceMatrix(data, false)
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</source>
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</dd>
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<br></br>
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<dt><strong>Pearson's correlation of 2 arrays</strong></dt>
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<br></br>
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<dd>To compute the Pearson's product-moment correlation between two double arrays
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<code>x</code> and <code>y</code>, use:
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<source>
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new PearsonsCorrelation().correlation(x, y)
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</source>
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</dd>
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<br></br>
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<dt><strong>Pearson's correlation matrix</strong></dt>
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<br></br>
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<dd> A (Pearson's) correlation matrix over the columns of a source matrix <code>data</code>
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can be computed using
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<source>
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new PearsonsCorrelation().computeCorrelationMatrix(data)
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</source>
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The i-jth entry of the returned matrix is the Pearson's product-moment correlation between the
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ith and jth columns of <code>data.</code>
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</dd>
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<br></br>
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<dt><strong>Pearson's correlation significance and standard errors</strong></dt>
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<br></br>
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<dd> To compute standard errors and/or significances of correlation coefficients
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associated with Pearson's correlation coefficients, start by creating a PearsonsCorrelation
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instance from the data <code>data</code> using
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<source>
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PearsonsCorrelation correlation = new PearsonsCorrelation(data);
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</source>
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where <code>data</code> is either a rectangular array or a <code>RealMatrix.</code>
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Then the matrix of standard errors is
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<source>
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correlation.getCorrelationStandardErrors();
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</source>
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The formula used to compute the standard error is <br/>
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<code>SE<sub>r</sub> = ((1 - r<sup>2</sup>) / (n - 2))<sup>1/2</sup></code><br/>
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where <code>r</code> is the estimated correlation coefficient and
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<code>n</code> is the number of observations in the source dataset.<br/><br/>
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<strong>p-values</strong> for the null hypothesis that respective coefficients are zero (also known as
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<i>significances</i>) populate the <code>RealMatrix</code> returned by
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<source>
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correlation.getCorrelationPValues();
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</source>
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<code>getCorrelationPValues().getEntry(i,j)</code> is the probability
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that a random variable distributed as <code>t<sub>n-2</sub></code> takes
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a value with absolute value greater than or equal to <br></br>
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<code>|r|((n - 2) / (1 - r<sup>2</sup>))<sup>1/2</sup></code>, where <code>r</code>
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is the estimated correlation coefficient.
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</dd>
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<br></br>
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</dl>
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</p>
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</subsection>
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<subsection name="1.7 Statistical tests">
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<p>
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The interfaces and implementations in the
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<a href="../apidocs/org/apache/commons/math/stat/inference/">
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