CrossCorrelation Class

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

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

System.Object
Imsl.Stat.CrossCorrelation

public class CrossCorrelation

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

CrossCorrelation estimates the cross-correlation function of two jointly stationary time series given a sample of n = x.Length observations $\{X_t\}$ and $\{Y_t\}$ for t = 1,2, ..., n.

Let

$\hat \mu _x = \rm{xmean}$

be the estimate of the mean $\mu _X$ of the time series $\{X_t\}$ where

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

The autocovariance function of $\{X_t\}$ , $\sigma _X(k)$ , is estimated by

$\hat \sigma _X\left( k \right) = \frac{1}{n} \sum\limits_{t = 1}^{n - k} {\left( {X_t - \hat \mu _X} \right)} \left( {X_{t + k} - \hat \mu _X} \right), \mbox{\hspace{20pt}k=0,1,\dots,K}$

where K = maximumLag. Note that $\hat \sigma _X(0)$ is equivalent to the sample variance of x returned by property VarianceX. The autocorrelation function $\rho _X(k)$ is estimated by

$\hat\rho _X(k) = \frac{\hat \sigma _X(k)}{\hat \sigma _X(0)},\mbox{\hspace{20pt}} k=0,1,\dots,K$

Note that $\hat \rho _x(0) \equiv 1$ by definition. Let

$\hat \mu _Y = {\rm ymean}, \hat \sigma _Y(k), {\rm and} \hat \rho _Y(k)$

be similarly defined.

The cross-covariance function $\sigma _{XY}(k)$ is estimated by

$\hat \sigma _{XY}(k) = \left\{ \begin{array}{ll} \frac{1}{n}\sum\limits_{t=1}^{n-k}(X_t - {\hat \mu _X})(Y_{t+k} - {\hat\mu _Y}) & {k = 0,1, \dots,K} \\ \frac{1}{n}\sum\limits_{t=1-k}^{n}(X_t - {\hat \mu _X})(Y_{t+k} - {\hat\mu _Y}) &{k = -1,-2, \dots,-K} \end {array} \right.$

The cross-correlation function $\rho _{XY}(k)$ is estimated by

$\hat \rho _{XY}(k) = \frac{\hat \sigma _{XY}(k)} {[\hat\sigma _X(0) \hat\sigma _Y(0) ]^{\frac{1}{2}}} \;\;\; {k = 0,\pm1, \dots,\pm K}$

The standard errors of the sample cross-correlations may be optionally computed according to the GetStandardErrors method argument stderrMethod. One method is based on a general asymptotic expression for the variance of the sample cross-correlation coefficient of two jointly stationary time series with independent, identically distributed normal errors given by Bartlet (1978, page 352). The theoretical formula is

$\begin{array}{c} {\rm var} \left \{ \hat \rho _{XY}(k) \right \} = \frac{1}{n-k}\sum\limits_{i=-\infty}^{\infty} \left [\right. {\rho _X(i)}+\rho _{XY}(i-k)\rho _{XY}(i+k) \\ -2\rho _{XY}(k)\{\rho _X(i)\rho _{XY}(i+k)+\rho _{XY}(-i)\rho _Y(i+k)\} \\ +\rho^2_{XY}(k)\{\rho_X(i) + \frac{1}{2}\rho^2_X(i) + \frac{1}{2}\rho^2_Y(i)\} \left. \right ] \end{array}$

For computational purposes, the autocorrelations $\rho_X(k)$ and $\rho_Y(k)$ and the cross-correlations $\rho _{XY}(k)$ are replaced by their corresponding estimates for $\left|k\right|\le K$ , and the limits of summation are equal to zero for all k such that $\left|k\right| > K$ .

A second method evaluates Bartlett's formula under the additional assumption that the two series have no cross-correlation. The theoretical formula is

${\rm var}\{\hat \rho_{XY}(k)\} = \frac{1}{n-k}\sum\limits_{i=-\infty}^{\infty}{\rho_X(i)\rho_Y(i)} \;\;\;\;\; {k \ge 0}$

For additional special cases of Bartlett's formula, see Box and Jenkins (1976, page 377).

An important property of the cross-covariance coefficient is $\sigma _{XY}(k) = \sigma _{YX}(-k)$ for $k \ge 0$ . This result is used in the computation of the standard error of the sample cross-correlation for lag $k \lt 0$ . In general, the cross-covariance function is not symmetric about zero so both positive and negative lags are of interest.

Requirements

Namespace: Imsl.Stat

Assembly: ImslCS (in ImslCS.dll)

CrossCorrelation Class

Thread Safety

Remarks

Requirements

See Also