Fixed errors in multiple regression section. JIRA: MATH-407.
git-svn-id: https://svn.apache.org/repos/asf/commons/proper/math/branches/MATH_2_X@998761 13f79535-47bb-0310-9956-ffa450edef68
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Phil Steitz
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@ -473,37 +473,47 @@ System.out.println(regression.getSlopeStdErr());
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</subsection>
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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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MultipleLinearRegression</a> provides ordinary least squares regression
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with a generic multiple variable linear model, which in matrix notation
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can be expressed as:
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<a href="../apidocs/org/apache/commons/math/stat/regression/OLSMultipleLinearRegression.html">
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OLSMultipleLinearRegression</a> and
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<a href="../apidocs/org/apache/commons/math/stat/regression/GLSMultipleLinearRegression.html">
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GLSMultipleLinearRegression</a> provide least squares regression to fit the linear model:
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</p>
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<p>
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<code> y=X*b+u </code>
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<code> Y=X*b+u </code>
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</p>
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<p>
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where y is an <code>n-vector</code> <b>regressand</b>, X is a <code>[n,k]</code> matrix whose <code>k</code> columns are called
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<b>regressors</b>, b is <code>k-vector</code> of <b>regression parameters</b> and <code>u</code> is an <code>n-vector</code>
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of <b>error terms</b> or <b>residuals</b>. The notation is quite standard in literature,
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cf eg <a href="http://www.econ.queensu.ca/ETM">Davidson and MacKinnon, Econometrics Theory and Methods, 2004</a>.
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where Y is an n-vector <b>regressand</b>, X is a [n,k] matrix whose k columns are called
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<b>regressors</b>, b is k-vector of <b>regression parameters</b> and u is an n-vector
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of <b>error terms</b> or <b>residuals</b>.
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</p>
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<p>
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Two implementations are provided: <a href="../apidocs/org/apache/commons/math/stat/regression/OLSMultipleLinearRegression.html">
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OLSMultipleLinearRegression</a> and
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<a href="../apidocs/org/apache/commons/math/stat/regression/OLSMultipleLinearRegression.html">
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OLSMultipleLinearRegression</a> provides Ordinary Least Squares Regression, and
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<a href="../apidocs/org/apache/commons/math/stat/regression/GLSMultipleLinearRegression.html">
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GLSMultipleLinearRegression</a>
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GLSMultipleLinearRegression</a> implements Generalized Least Squares. See the javadoc for these
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classes for details on the algorithms and forumlas used.
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</p>
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<p>
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Observations (x,y and covariance data matrices) can be added to the model via the <code>addData(double[] y, double[][] x, double[][] covariance)</code> method.
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The observations are stored in memory until the next time the addData method is invoked.
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Data for OLS models can be loaded in a single double[] array, consisting of concatenated rows of data, each containing
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the regressand (Y) value, followed by regressor values; or using a double[][] array with rows corresponding to
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observations. GLS models also require a double[][] array representing the covariance matrix of the error terms. See
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<a href="../apidocs/org/apache/commons/math/stat/regression/AbstractMultipleLinearRegression.html#newSampleData(double[], int, int)">
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AbstractMultipleLinearRegression#newSampleData(double[],int,int)</a>,
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<a href="../apidocs/org/apache/commons/math/stat/regression/OLSMultipleLinearRegression.html#newSampleData(double[], double[][])">
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OLSMultipleLinearRegression#newSampleData(double[], double[][])</a> and
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<a href="../apidocs/org/apache/commons/math/stat/regression/GLSMultipleLinearRegression.html#newSampleData(double[], double[][], double[][])">
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GLSMultipleLinearRegression#newSampleData(double[],double[][],double[][])</a> for details.
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</p>
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<p>
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<strong>Usage Notes</strong>: <ul>
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<li> Data is validated when invoking the <code>addData(double[] y, double[][] x, double[][] covariance)</code> method and
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<code>IllegalArgumentException</code> is thrown when inappropriate.
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<li> Data are validated when invoking any of the newSample, newX, newY or newCovariance methods and
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<code>IllegalArgumentException</code> is thrown when input data arrays do not have matching dimensions
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or do not contain sufficient data to estimate the model.
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</li>
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<li> Only the GLS regressions require the covariance matrix, so in the OLS regression it is ignored and can be safely
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inputted as <code>null</code>.</li>
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<li> By default, regression models are estimated with intercept terms. In the notation above, this implies that the
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X matrix contains an initial row identically equal to 1. X data supplied to the newX or newSample methods should not
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include this column - the data loading methods will create it automatically. To estimate a model without an intercept
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term, set the <code>noIntercept</code> property to <code>true.</code></li>
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</ul>
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</p>
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<p>
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@ -511,44 +521,48 @@ System.out.println(regression.getSlopeStdErr());
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<dl>
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<dt>OLS regression</dt>
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<br></br>
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<dd>Instantiate an OLS regression object and load dataset
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<dd>Instantiate an OLS regression object and load a dataset:
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<source>
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MultipleLinearRegression regression = new OLSMultipleLinearRegression();
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OLSMultipleLinearRegression regression = new OLSMultipleLinearRegression();
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double[] y = new double[]{11.0, 12.0, 13.0, 14.0, 15.0, 16.0};
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double[] x = new double[6][];
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x[0] = new double[]{1.0, 0, 0, 0, 0, 0};
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x[1] = new double[]{1.0, 2.0, 0, 0, 0, 0};
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x[2] = new double[]{1.0, 0, 3.0, 0, 0, 0};
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x[3] = new double[]{1.0, 0, 0, 4.0, 0, 0};
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x[4] = new double[]{1.0, 0, 0, 0, 5.0, 0};
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x[5] = new double[]{1.0, 0, 0, 0, 0, 6.0};
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regression.addData(y, x, null); // we don't need covariance
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x[0] = new double[]{0, 0, 0, 0, 0};
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x[1] = new double[]{2.0, 0, 0, 0, 0};
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x[2] = new double[]{0, 3.0, 0, 0, 0};
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x[3] = new double[]{0, 0, 4.0, 0, 0};
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x[4] = new double[]{0, 0, 0, 5.0, 0};
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x[5] = new double[]{0, 0, 0, 0, 6.0};
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regression.newSample(y, x);
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</source>
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</dd>
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<dd>Estimate of regression values honours the <code>MultipleLinearRegression</code> interface:
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<dd>Get regression parameters and diagnostics:
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<source>
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double[] beta = regression.estimateRegressionParameters();
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double[] beta = regression.estimateRegressionParameters();
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double[] residuals = regression.estimateResiduals();
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double[][] parametersVariance = regression.estimateRegressionParametersVariance();
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double regressandVariance = regression.estimateRegressandVariance();
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double rSquared = regression.caclulateRSquared();
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double sigma = regression.estimateRegressionStandardError();
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</source>
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</dd>
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<dt>GLS regression</dt>
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<br></br>
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<dd>Instantiate an GLS regression object and load dataset
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<dd>Instantiate a GLS regression object and load a dataset:
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<source>
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MultipleLinearRegression regression = new GLSMultipleLinearRegression();
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GLSMultipleLinearRegression regression = new GLSMultipleLinearRegression();
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double[] y = new double[]{11.0, 12.0, 13.0, 14.0, 15.0, 16.0};
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double[] x = new double[6][];
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x[0] = new double[]{1.0, 0, 0, 0, 0, 0};
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x[1] = new double[]{1.0, 2.0, 0, 0, 0, 0};
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x[2] = new double[]{1.0, 0, 3.0, 0, 0, 0};
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x[3] = new double[]{1.0, 0, 0, 4.0, 0, 0};
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x[4] = new double[]{1.0, 0, 0, 0, 5.0, 0};
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x[5] = new double[]{1.0, 0, 0, 0, 0, 6.0};
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x[0] = new double[]{0, 0, 0, 0, 0};
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x[1] = new double[]{2.0, 0, 0, 0, 0};
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x[2] = new double[]{0, 3.0, 0, 0, 0};
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x[3] = new double[]{0, 0, 4.0, 0, 0};
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x[4] = new double[]{0, 0, 0, 5.0, 0};
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x[5] = new double[]{0, 0, 0, 0, 6.0};
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double[][] omega = new double[6][];
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omega[0] = new double[]{1.1, 0, 0, 0, 0, 0};
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omega[1] = new double[]{0, 2.2, 0, 0, 0, 0};
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@ -556,12 +570,9 @@ omega[2] = new double[]{0, 0, 3.3, 0, 0, 0};
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omega[3] = new double[]{0, 0, 0, 4.4, 0, 0};
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omega[4] = new double[]{0, 0, 0, 0, 5.5, 0};
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omega[5] = new double[]{0, 0, 0, 0, 0, 6.6};
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regression.addData(y, x, omega); // we do need covariance
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regression.newSampleData(y, x, omega);
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</source>
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</dd>
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<dd>Estimate of regression values honours the same <code>MultipleLinearRegression</code> interface as
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the OLS regression.
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</dd>
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</dl>
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</p>
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</subsection>
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