MultiCrossCorrelation Class

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

For a list of all members of this type, see MultiCrossCorrelation Members.

System.Object
Imsl.Stat.MultiCrossCorrelation

public class MultiCrossCorrelation

Thread Safety

Public static (Shared in Visual Basic) members of this type are safe for multithreaded operations. Instance members are not guaranteed to be thread-safe.

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

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

Requirements

Namespace: Imsl.Stat

Assembly: ImslCS (in ImslCS.dll)

MultiCrossCorrelation Class

Thread Safety

Remarks

Requirements

See Also