MultiCrossCorrelation Class

Computes the multichannel cross-correlation function of two mutually stationary multichannel time series.

Inheritance Hierarchy

System.Object
Imsl.Stat.MultiCrossCorrelation

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

Syntax

C++

Copy

[SerializableAttribute]
public class MultiCrossCorrelation

<SerializableAttribute>
Public Class MultiCrossCorrelation

[SerializableAttribute]
public ref class MultiCrossCorrelation

[<SerializableAttribute>]
type MultiCrossCorrelation =  class end

The MultiCrossCorrelation type exposes the following members.

Constructors

	Name	Description
	MultiCrossCorrelation	Constructor to compute the multichannel cross-correlation function of two mutually stationary mulitchannel time series.

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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.)
	GetCrossCorrelation	Returns the cross-correlations between the channels of x and y.
	GetCrossCovariance	Returns the cross-covariances between the channels of x and y.
	GetHashCode	Serves as a hash function for a particular type. (Inherited from Object.)
	GetMeanX	Returns an estimate of the mean of each channel of x.
	GetMeanY	Returns an estimate of the mean of each channel of y.
	GetType	Gets the Type of the current instance. (Inherited from Object.)
	GetVarianceX	Returns the variances of the channels of x.
	GetVarianceY	Returns the variances of the channels of y.
	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.

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Remarks

MultiCrossCorrelation estimates the multichannel cross-correlation function of two mutually stationary multichannel time series. Define the multichannel time series X by

$X = (X_1, X_2, \dots, X_p)$

where

$X_j = {(X_{1j}, X_{2j}, \dots, X_{nj})}^T, \;\;\;\;\; j = 1,2, \dots, p$

with n = x.GetLength(0) and p = x.GetLength(1). Similarly, define the multichannel time series Y by

$Y = (Y_1, Y_2, \dots, Y_q)$

where

$Y_j = {(Y_{1j}, Y_{2j}, \dots, Y_{mj})}^T, \;\;\;\;\; j = 1,2, \dots, q$

with m = y.GetLength(0) and q = y.GetLength(1). The columns of X and Y correspond to individual channels of multichannel time series and may be examined from a univariate perspective. The rows of X and Y correspond to observations of p-variate and q-variate time series, respectively, and may be examined from a multivariate perspective. Note that an alternative characterization of a multivariate time series X considers the columns to be observations of the multivariate time series while the rows contain univariate time series. For example, see Priestley (1981, page 692) and Fuller (1976, page 14).

Let $\hat \mu _X$ = xmean be the row vector containing the means of the channels of X. In particular,

$\hat \mu _X = (\hat\mu _{X_1}, \hat \mu _{X_2}, \dots, \hat \mu _{X_p})$

where for j = 1, 2, ..., p

$\hat \mu _{X_j} = \left\{ \begin{array}{ll} \mu _{X_j} & {\rm for}\;\mu _{X_j}\; {\rm known} \\ \frac{1}{n}\sum\limits_{t=1}^n {X_{tj}} & {\rm for}\;\mu _{X_j}\; {\rm unknown} \end{array} \right.$

Let $\hat \mu _Y$ = ymean be similarly defined. The cross-covariance of lag k between channel i of X and channel j of Y is estimated by

$\hat \sigma _{X_iY_j}(k) = \left\{ \begin{array}{ll} \frac{1}{N}\sum\limits_{t}(X_{ti} - {\hat \mu _{X_i}}) (Y_{t+k,j} - {\hat\mu _{Y_j}}) &{k = 0,1, \dots,K} \\ \frac{1}{N} \sum\limits_{t}(X_{ti} - {\hat \mu _{X_i}})(Y_{t+k,j} - {\hat\mu _{Y_j}}) &{k = -1,-2, \dots,-K} \end {array} \right.$

where i = 1, ..., p, j = 1, ..., q, and K = maximumLag. The summation on t extends over all possible cross-products with N equal to the number of cross-products in the sum.

Let $\hat \sigma _X(0)$ = xvar, where xvar is the variance of X, be the row vector consisting of estimated variances of the channels of X. In particular,

$\hat \sigma _X(0) = (\hat \sigma _{X_1}(0), \hat \sigma _{X_2}(0), \dots, \hat \sigma _{X_p}(0))$

where

$\hat \sigma _{X_j}(0) = \frac{1}{n} \sum\limits_{t = 1}^{n} {\left( {X_{tj} - \hat \mu _{X_j}} \right)}^2 {, \mbox{\hspace{20pt}j=0,1,\dots,p}}$

Let $\hat \sigma _Y(0)$ = yvar, where yvar is the variance of Y, be similarly defined. The cross-correlation of lag k between channel i of X and channel j of Y is estimated by

$\hat \rho _{X_jY_j}(k) = \frac{\hat \sigma _{{X_j}{Y_j}(k)}}{ {[ \hat\sigma _{X_i}(0)\hat\sigma _{X_j}(0)]}^{\frac{1}{2}}} \;\;\;\;\;k = 0,\pm1,\dots, \pm K$

Reference

Imsl.Stat Namespace

Other Resources

Example