PartialCovariances Class

Class PartialCovariances computes the partial covariances or partial correlations from an input covariance or correlation matrix.

Inheritance Hierarchy

SystemObject
Imsl.StatPartialCovariances

Namespace: Imsl.Stat
Assembly: ImslCS (in ImslCS.dll) Version: 6.5.2.0

Syntax

C++

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[SerializableAttribute]
public class PartialCovariances

<SerializableAttribute>
Public Class PartialCovariances

[SerializableAttribute]
public ref class PartialCovariances

[<SerializableAttribute>]
type PartialCovariances =  class end

The PartialCovariances type exposes the following members.

Constructors

	Name	Description
	PartialCovariances(Int32, Double, Int32)	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.
	PartialCovariances(Int32, Double, Int32)	Creates a PartialCovariances object from a covariance or correleation matrix with a mix of dependent and independent variables.

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Methods

	Name	Description
	Equals	Determines whether the specified object is equal to the current object. (Inherited from Object.)
	Finalize	Allows an object to try to free resources and perform other cleanup operations before it is reclaimed by garbage collection. (Inherited from Object.)
	GetHashCode	Serves as a hash function for a particular type. (Inherited from Object.)
	GetPartialCorrelationMatrix	Returns the partial correlation matrix.
	GetPartialCovarianceMatrix	Returns the partial covariance matrix.
	GetPValues	Calculates the p-values for testing the null hypothesis that the associated partial covariance/correlation is zero.
	GetType	Gets the Type of the current instance. (Inherited from Object.)
	MemberwiseClone	Creates a shallow copy of the current Object. (Inherited from Object.)
	ToString	Returns a string that represents the current object. (Inherited from Object.)

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Properties

	Name	Description
	NumberOfProcessors	Perform the parallel calculations with the maximum possible number of processors set to NumberOfProcessors.
	PartialDegreesOfFreedom	The degrees of freedom in the test that the partial correlation (covariance) is zero.

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Remarks

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 the property PartialDegreesOfFreedom.

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.

Reference

Imsl.Stat Namespace

Other Resources

Example 1

Example 2