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

Namespace: Imsl.Stat
Assembly: ImslCS (in ImslCS.dll) Version: 6.5.0.0

Syntax

C#
[SerializableAttribute]
public class MultiCrossCorrelation
Visual Basic (Declaration)
<SerializableAttribute> _
Public Class MultiCrossCorrelation
Visual C++
[SerializableAttribute]
public ref class MultiCrossCorrelation

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

Inheritance Hierarchy

System..::.Object
Imsl.Stat..::.MultiCrossCorrelation

See Also