JMSLTM Numerical Library 6.1

com.imsl.stat
Class MultiCrossCorrelation

java.lang.Object
  extended by com.imsl.stat.MultiCrossCorrelation
All Implemented Interfaces:
Serializable, Cloneable

public class MultiCrossCorrelation
extends Object
implements Serializable, Cloneable

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

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.length and p = x[0].length. 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.length and q = y[0].length. 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 = (hatmu _{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}sumlimits_{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}sumlimits_{t}(X_{ti} - {hat mu _{X_i}})(Y_{t+k,j} - {hatmu _{Y_j}}) &{k = 0,1, dots,K} \
 frac{1}{N}sumlimits_{t}(X_{ti} - {hat mu _{X_i}})(Y_{t+k,j} - {hatmu _{Y_j}}) &{k = -1,-2, dots,-K} 
end {array} right.

where i = 1, ..., p, j = 1, ..., q, and K = maximum_lag. 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} 
 sumlimits_{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)}}{ {[ hatsigma _{X_i}(0)hatsigma _{X_j}(0)]}^{frac{1}{2}}} 
    ;;;;;k =0,pm1,dots, pm K

See Also:
Example, Serialized Form

Nested Class Summary
static class MultiCrossCorrelation.NonPosVariancesException
          The problem is ill-conditioned.
 
Constructor Summary
MultiCrossCorrelation(double[][] x, double[][] y, int maximum_lag)
          Constructor to compute the multichannel cross-correlation function of two mutually stationary multichannel time series.
 
Method Summary
 double[][][] getCrossCorrelation()
          Returns the cross-correlations between the channels of x and y.
 double[][][] getCrossCovariance()
          Returns the cross-covariances between the channels of x and y.
 double[] getMeanX()
          Returns the mean of each channel of x.
 double[] getMeanY()
          Returns the mean of each channel of y.
 double[] getVarianceX()
          Returns the variances of the channels of x.
 double[] getVarianceY()
          Returns the variances of the channels of y.
 void setMeanX(double[] mean)
          Estimate of the mean of each channel of x.
 void setMeanY(double[] mean)
          Estimate of the mean of each channel of y.
 
Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
 

Constructor Detail

MultiCrossCorrelation

public MultiCrossCorrelation(double[][] x,
                             double[][] y,
                             int maximum_lag)
Constructor to compute the multichannel cross-correlation function of two mutually stationary multichannel time series.

Parameters:
x - A two-dimensional double array containing the first multichannel stationary time series. Each row of x corresponds to an observation of a multivariate time series and each column of x corresponds to a univariate time series.
y - A two-dimensional double array containing the second multichannel stationary time series. Each row of y corresponds to an observation of a multivariate time series and each column of y corresponds to a univariate 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 number of observations of x and y.
Method Detail

getCrossCorrelation

public double[][][] getCrossCorrelation()
                                 throws MultiCrossCorrelation.NonPosVariancesException
Returns the cross-correlations between the channels of x and y.

Returns:
A double array of size 2 * maximum_lag +1 by x[0].length by y[0].length containing the cross-correlations between the time series x and y. The cross-correlation between channel i of the x series and channel j of the y series at lag k, where k = -maximum_lag, ..., 0, 1, ..., maximum_lag, corresponds to output array element with index [k][i][j] where k= 0,1,...,(2*maximum_lag), i = 1, ..., x[0].length, and j = 1, ..., y[0].length.
Throws:
MultiCrossCorrelation.NonPosVariancesException

getCrossCovariance

public double[][][] getCrossCovariance()
                                throws MultiCrossCorrelation.NonPosVariancesException
Returns the cross-covariances between the channels of x and y.

Returns:
A double array of size 2 * maximum_lag +1 by x[0].length by y[0].length containing the cross-covariances between the time series x and y. The cross-covariances between channel i of the x series and channel j of the y series at lag k where k = -maximum_lag, ..., 0, 1, ..., maximum_lag, corresponds to output array element with index [k][i][j] where k= 0,1,...,(2*maximum_lag), i = 1, ..., x[0].length, and j = 1, ..., y[0].length.
Throws:
MultiCrossCorrelation.NonPosVariancesException

getMeanX

public double[] getMeanX()
Returns the mean of each channel of x.

Returns:
A one-dimensional double containing the mean of each channel in the time series x.

getMeanY

public double[] getMeanY()
Returns the mean of each channel of y.

Returns:
A one-dimensional double containing the estimate mean of each channel in the time series y.

getVarianceX

public double[] getVarianceX()
                      throws MultiCrossCorrelation.NonPosVariancesException
Returns the variances of the channels of x.

Returns:
A one-dimensional double containing the variances of each channel in the time series x.
Throws:
MultiCrossCorrelation.NonPosVariancesException

getVarianceY

public double[] getVarianceY()
                      throws MultiCrossCorrelation.NonPosVariancesException
Returns the variances of the channels of y.

Returns:
A one-dimensional double containing the variances of each channel in the time series y.
Throws:
MultiCrossCorrelation.NonPosVariancesException

setMeanX

public void setMeanX(double[] mean)
Estimate of the mean of each channel of x.

Parameters:
mean - A one-dimensional double containing the estimate of the mean of each channel in time series x.

setMeanY

public void setMeanY(double[] mean)
Estimate of the mean of each channel of y.

Parameters:
mean - A one-dimensional double containing the estimate of the mean of each channel in the time series y.

JMSLTM Numerical Library 6.1

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Built July 30 2010.