com.imsl.stat

Class PartialCovariances

java.lang.Object

com.imsl.stat.PartialCovariances

All Implemented Interfaces:

Serializable

public class PartialCovariances
extends Object
implements Serializable

Class PartialCovariances computes the partial covariances or partial correlations from an input covariance or correlation matrix. If the "independent" variables (the linear "effect" of the independent variables is removed in computing the partial covariances/correlations) are linearly related to one another, PartialCovariances detects the linearity and eliminates one or more of the independent variables from the list of independent variables. The number of variables eliminated, if any, can be determined from getPartialDegreesOfFreedom.

Given a covariance or correlation matrix \(\Sigma\) partitioned as $$ \left( \begin{array}{cc} \Sigma_{11} & \Sigma_{12} \\ \Sigma_{21} & \Sigma_{22} \end{array} \right) $$ class PartialCovariances computes the partial covariances (of the standardized variables if \(\Sigma\) is a correlation matrix) as $$ \Sigma_{22|1} = \Sigma_{22} - \Sigma_{21} \Sigma_{11}^{-1} \Sigma_{12} $$ A positive semidefinite solver is used to compute \(\Sigma_{11}^{-1} \Sigma_{12}\).

If partial correlations are desired, these are computed as $$ P_{22|1} = \left[ \textrm{diag}(\Sigma_{22|1}) \right]^{-1/2} \Sigma_{22|1} \left[ \textrm{diag}(\Sigma_{22|1}) \right]^{-1/2} $$ where \(\textrm{diag}(\Sigma)\) denotes the matrix containing the diagonal of its argument along its diagonal with zeros off the diagonal. If \(\Sigma_{11}\) is singular, then as many variables as required are deleted from \(\Sigma_{11}\) (and \(\Sigma_{12}\)) in order to eliminate the linear dependencies. The computations then proceed as above.

The p-value for a partial covariance tests the null hypothesis \(H_0: \sigma_{ij|1} = 0\), where \(\sigma_{ij|1}\) is the (i, j) element in matrix \(\Sigma_{22|1}\). The p-value for a partial correlation tests the null hypothesis \(H_0: \rho_{ij|1} = 0\), where \(\rho_{ij|1}\) is the (i, j) element in matrix \(P_{22|1}\). The p-values are returned by getPValues. If the degrees of freedom for sigma, df, is not known, the resulting p-values may be useful for comparison, but they should not by used as an approximation to the actual probabilities.

See Also:

Example1, Example2, Serialized Form

Nested Class Summary

Nested Classes

Modifier and Type	Class and Description
static class	PartialCovariances.InvalidMatrixException Exception thrown if a computed correlation is greater than one for some pair of variables.
static class	PartialCovariances.InvalidPartialCorrelationException Exception thrown if a computed partial correlation is greater than one for some pair of variables.

Constructor Summary

Constructors

Constructor and Description
PartialCovariances(int[] xIndices, double[][] sigma, int df) Creates a PartialCovariances object from a covariance or correleation matrix with a mix of dependent and independent variables.
PartialCovariances(int nIndependent, double[][] sigma, int df) Creates a PartialCovariances object from a covariance or correleation matrix with a the independent variables in the initial columns and the dependent variables in the final columns.

Method Summary

Modifier and Type	Method and Description
double[][]	getPartialCorrelationMatrix() Returns the partial correlation matrix.
double[][]	getPartialCovarianceMatrix() Returns the partial covariance matrix.
int	getPartialDegreesOfFreedom() Returns the degrees of freedom in the test that the partial correlation (covariance) is zero.
double[][]	getPValues() Calculates the p-values for testing the null hypothesis that the associated partial covariance/correlation is zero.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

PartialCovariances

public PartialCovariances(int nIndependent,
                          double[][] sigma,
                          int df)
                   throws PartialCovariances.InvalidMatrixException,
                          PartialCovariances.InvalidPartialCorrelationException

Creates a PartialCovariances object from a covariance or correleation matrix with a the independent variables in the initial columns and the dependent variables in the final columns.

Parameters:

nIndependent - is the number of "independent" variables to be used in the partial covariances/correlations. The partial covariances/correlations are the covariances/correlations between the dependent variables after removing the linear effect of the independent variables.

sigma - is a correlation or covariance matrix. The rows/columns must be ordered such that the first nIndependent rows/columns contain the independent variables, followed by the row/columns containing the dependent variables. The matrix must always be symmetric, positive semidefinite.

df - is an int indicating the number of degrees of freedom associated with the input matrix. If the number of degrees of freedom in the matrix varies from element to element, then a conservative choice for df is the minimum degrees of freedom for all elements in the matrix. The value of df must be at least one.

Throws:

PartialCovariances.InvalidMatrixException - is thrown if a computed correlation is greater than one for some pair of variables.

PartialCovariances.InvalidPartialCorrelationException - is thrown if a computed partial correlation is greater than one for some pair of variables. The input matrix to the constructor was not positive semidefinite.

PartialCovariances

public PartialCovariances(int[] xIndices,
                          double[][] sigma,
                          int df)
                   throws PartialCovariances.InvalidMatrixException,
                          PartialCovariances.InvalidPartialCorrelationException

Creates a PartialCovariances object from a covariance or correleation matrix with a mix of dependent and independent variables.

Parameters:

xIndices - is an array containing values indicating the status of the variable. If the i-th entry is 0 then the i-th column of the matrix contains a dependent variable. If the i-th entry is positive then the i-th column of the matrix contains an independent variable. If the i-th entry is negative then the i-th column of the matrix is not used in the analysis.

sigma - is a correlation or covariance matrix. The number of rows and columns in sigma must equal the length of the array xIndices. The matrix must always be symmetric, positive semidefinite.

df - is an int indicating the number of degrees of freedom associated with the input matrix. If the number of degrees of freedom in the matrix varies from element to element, then a conservative choice for df is the minimum degrees of freedom for all elements in the matrix. The value of df must be at least one.

Throws:

PartialCovariances.InvalidMatrixException - is thrown if a computed correlation is greater than one for some pair of variables.

PartialCovariances.InvalidPartialCorrelationException - is thrown if a computed partial correlation is greater than one for some pair of variables. The input matrix to the constructor was not positive semidefinite.

Method Detail

getPartialCovarianceMatrix

public double[][] getPartialCovarianceMatrix()

Returns the partial covariance matrix. This is valid only if the input to the constructor was a covariance matrix.

Returns:

The partial covariance matrix.

getPartialCorrelationMatrix

public double[][] getPartialCorrelationMatrix()

Returns the partial correlation matrix. This is valid only if the input to the constructor was a correlation matrix.

Returns:

The partial correlation matrix.

getPValues

public double[][] getPValues()

Calculates the p-values for testing the null hypothesis that the associated partial covariance/correlation is zero. It is assumed that the observations from which sigma was computed follows a multivariate normal distribution and that each element in sigma has df degrees of freedom.

Returns:

A square array of type double containing the p-values. The order of the matrix equals the number of dependent variables.

If the partial degrees of freedom is not greater than one then there are not enough degrees of freedom for hypothesis testing. The returned matrix will be set to all NaN values in this case. A warning is also issued in this case.

getPartialDegreesOfFreedom

public int getPartialDegreesOfFreedom()

Returns the degrees of freedom in the test that the partial correlation (covariance) is zero. This will usually be df minus the rank of the the independent variables. number of independent variables, but will be greater than this value if the independent variables are computationally linearly related.

Returns:

An int scalar value representing the degrees of freedom. If this value is not greater than one then there are not enough degrees of freedom for hypothesis testing. A warning is also issued in this case.