Package 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:

  • Nested Class Summary

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

  • Field Summary

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

  • Constructor Summary

    Constructors
    Constructor
    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
    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 Details

    • BARTLETTS_FORMULA

      public static final int BARTLETTS_FORMULA
      Indicates standard error computation using Bartlett's formula.
      See Also:

    • 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:

  • Constructor Details

    • 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 Details

    • 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