com.imsl.stat

Class CrossCorrelation

java.lang.Object

com.imsl.stat.CrossCorrelation

All Implemented Interfaces:

Serializable, Cloneable

public class CrossCorrelation
extends Object
implements Serializable, Cloneable

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

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 = maximum_lag. Note that $\hat \sigma _X(0)$ is equivalent to the sample variance of x returned by method getVarianceX. 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.

See Also:

Example, Serialized Form

Nested Class Summary

Nested Classes

Modifier and Type	Class and Description
static class	CrossCorrelation.NonPosVariancesException The problem is ill-conditioned.

Field Summary

Fields

Modifier and Type	Field and Description
static int	BARTLETTS_FORMULA Indicates standard error computation using Bartlett's formula.
static int	BARTLETTS_FORMULA_NOCC Indicates standard error computation using Bartlett's formula with the assumption of no cross-correlation.

Constructor Summary

Constructors

Constructor and Description
CrossCorrelation(double[] x, double[] y, int maximum_lag) Constructor to compute the sample cross-correlation function of two stationary time series.

Method Summary

Modifier and Type	Method and Description
double[]	getAutoCorrelationX() Returns the autocorrelations of the time series x.
double[]	getAutoCorrelationY() Returns the autocorrelations of the time series y.
double[]	getAutoCovarianceX() Returns the autocovariances of the time series x.
double[]	getAutoCovarianceY() Returns the autocovariances of the time series y.
double[]	getCrossCorrelation() Returns the cross-correlations between the time series x and y.
double[]	getCrossCovariance() Returns the cross-covariances between the time series x and y.
double	getMeanX() Returns the mean of the time series x.
double	getMeanY() Returns the mean of the time series y.
double[]	getStandardErrors(int stderrMethod) Returns the standard errors of the cross-correlations between the time series x and y.
double	getVarianceX() Returns the variance of time series x.
double	getVarianceY() Returns the variance of time series y.
void	setMeanX(double mean) Estimate of the mean of time series x.
void	setMeanY(double mean) Estimate of the mean of time series y.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Detail

BARTLETTS_FORMULA

public static final int BARTLETTS_FORMULA

Indicates standard error computation using Bartlett's formula.

See Also:

Constant Field Values

BARTLETTS_FORMULA_NOCC

public static final int BARTLETTS_FORMULA_NOCC

Indicates standard error computation using Bartlett's formula with the assumption of no cross-correlation.

See Also:

Constant Field Values

Constructor Detail

CrossCorrelation

public CrossCorrelation(double[] x,
                        double[] y,
                        int maximum_lag)

Constructor to compute the sample cross-correlation function of two stationary time series.

Parameters:

x - A one-dimensional double array containing the first stationary time series.

y - A one-dimensional double array containing the second stationary time series.

maximum_lag - An int containing the maximum lag of the cross-covariance and cross-correlations to be computed. maximum_lag must be greater than or equal to 1 and less than the minimum of the number of observations of x and y.

Method Detail

setMeanX

public void setMeanX(double mean)

Estimate of the mean of time series x.

Parameters:

mean - A double containing the estimate mean of the time series x.

getMeanX

public double getMeanX()

Returns the mean of the time series x.

Returns:

A double containing the mean of the time series x.

setMeanY

public void setMeanY(double mean)

Estimate of the mean of time series y.

Parameters:

mean - A double containing the estimate mean of the time series y.

getMeanY

public double getMeanY()

Returns the mean of the time series y.

Returns:

A double containing the mean of the time series y.

getVarianceX

public double getVarianceX()
                    throws CrossCorrelation.NonPosVariancesException

Returns the variance of time series x.

Returns:

A double containing the variance of the time series x.

Throws:

CrossCorrelation.NonPosVariancesException

getVarianceY

public double getVarianceY()
                    throws CrossCorrelation.NonPosVariancesException

Returns the variance of time series y.

Returns:

A double containing the variance of the time series y.

Throws:

CrossCorrelation.NonPosVariancesException

getCrossCorrelation

public double[] getCrossCorrelation()
                             throws CrossCorrelation.NonPosVariancesException

Returns the cross-correlations between the time series x and y.

Returns:

A double array of length 2 * maximum_lag +1 containing the cross-correlations between the time series x and y. The cross-correlation between x and y at lag k, where k = -maximum_lag ,..., 0, 1,...,maximum_lag, corresponds to output array indices 0, 1,..., (2*maximum_lag).

Throws:

CrossCorrelation.NonPosVariancesException

getCrossCovariance

public double[] getCrossCovariance()

Returns the cross-covariances between the time series x and y.

Returns:

A double array of length 2 * maximum_lag +1 containing the cross-covariances between the time series x and y. The cross-covariance between x and y at lag k, where k = -maximum_lag ,..., 0, 1,...,maximum_lag, corresponds to output array indices 0, 1,..., (2*maximum_lag).

getAutoCorrelationX

public double[] getAutoCorrelationX()
                             throws CrossCorrelation.NonPosVariancesException

Returns the autocorrelations of the time series x.

Returns:

A double array of length maximum_lag +1 containing the autocorrelations of the time series x. The 0-th element of this array is 1. The k-th element of this array contains the autocorrelation of lag k where k = 1, ..., maximum_lag.

Throws:

CrossCorrelation.NonPosVariancesException

getAutoCorrelationY

public double[] getAutoCorrelationY()
                             throws CrossCorrelation.NonPosVariancesException

Returns the autocorrelations of the time series y.

Returns:

A double array of length maximum_lag +1 containing the autocorrelations of the time series y. The 0-th element of this array is 1. The k-th element of this array contains the autocorrelation of lag k where k = 1, ..., maximum_lag.

Throws:

CrossCorrelation.NonPosVariancesException

getAutoCovarianceX

public double[] getAutoCovarianceX()
                            throws CrossCorrelation.NonPosVariancesException

Returns the autocovariances of the time series x.

Returns:

A double array of length maximum_lag +1 containing the variances and autocovariances of the time series x. The 0-th element of the array contains the variance of the time series x. The k-th element contains the autocovariance of lag k where k = 1, ..., maximum_lag.

Throws:

CrossCorrelation.NonPosVariancesException

getAutoCovarianceY

public double[] getAutoCovarianceY()
                            throws CrossCorrelation.NonPosVariancesException

Returns the autocovariances of the time series y.

Returns:

A double array of length maximum_lag +1 containing the variances and autocovariances of the time series y. The 0-th element of the array contains the variance of the time series x. The k-th element contains the autocovariance of lag k where k = 1, ..., maximum_lag.

Throws:

CrossCorrelation.NonPosVariancesException

getStandardErrors

public double[] getStandardErrors(int stderrMethod)
                           throws CrossCorrelation.NonPosVariancesException

Returns the standard errors of the cross-correlations between the time series x and y. Method of computation for standard errors of the cross-correlation is determined by the stderrMethod parameter. If stderrMethod is set to BARTLETTS_FORMULA, Bartlett's formula is used to compute the standard errors of cross-correlations. If stderrMethod is set to BARTLETTS_FORMULA_NOCC, Bartlett's formula is used to compute the standard errors of cross-correlations, with the assumption of no cross-correlation.

Parameters:

stderrMethod - An int specifying the method to compute the standard errors of cross-correlations between the time series x and y.

Returns:

A double array of length 2 * maximum_lag + 1 containing the standard errors of the cross-correlations between the time series x and y. The standard error of cross-correlations between x and y at lag k, where k = -maximum_lag,..., 0, 1,..., maximum_lag, corresponds to output array indices 0, 1,..., (2*maximum_lag).

Throws:

CrossCorrelation.NonPosVariancesException