CrossCorrelation Class |
Namespace: Imsl.Stat
The CrossCorrelation type exposes the following members.
Name | Description | |
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CrossCorrelation |
Constructor to compute the sample cross-correlation function of two
stationary time series.
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Name | Description | |
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Equals | Determines whether the specified object is equal to the current object. (Inherited from Object.) | |
Finalize | Allows an object to try to free resources and perform other cleanup operations before it is reclaimed by garbage collection. (Inherited from Object.) | |
GetAutoCorrelationX |
Returns the autocorrelations of the time series x.
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GetAutoCorrelationY |
Returns the autocorrelations of the time series y.
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GetAutoCovarianceX |
Returns the autocovariances of the time series x.
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GetAutoCovarianceY |
Returns the autocovariances of the time series y.
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GetCrossCorrelations |
Returns the cross-correlations between the time series x and
y.
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GetCrossCovariances |
Returns the cross-covariances between the time series x and
y.
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GetHashCode | Serves as a hash function for a particular type. (Inherited from Object.) | |
GetStandardErrors |
Returns the standard errors of the cross-correlations between the
time series x and y.
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GetType | Gets the Type of the current instance. (Inherited from Object.) | |
MemberwiseClone | Creates a shallow copy of the current Object. (Inherited from Object.) | |
ToString | Returns a string that represents the current object. (Inherited from Object.) |
Name | Description | |
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MeanX |
Estimate of the mean of time series x.
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MeanY |
Estimate of the mean of time series y.
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NumberOfProcessors |
Perform the parallel calculations with the maximum possible number of
processors set to NumberOfProcessors.
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VarianceX |
Returns the variance of time series x.
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VarianceY |
Returns the variance of time series y.
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CrossCorrelation estimates the cross-correlation function of two jointly stationary time series given a sample of n = x.Length observations and for t = 1,2, ..., n.
Let
be the estimate of the mean of the time series whereThe autocovariance function of , , is estimated by
where K = maximumLag. Note that is equivalent to the sample variance of x returned by property VarianceX. The autocorrelation function is estimated byNote that by definition. Let
be similarly defined.The cross-covariance function is estimated by
The cross-correlation function is estimated byThe 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
For computational purposes, the autocorrelations and and the cross-correlations are replaced by their corresponding estimates for , and the limits of summation are equal to zero for all k such that .A second method evaluates Bartlett's formula under the additional assumption that the two series have no cross-correlation. The theoretical formula is
For additional special cases of Bartlett's formula, see Box and Jenkins (1976, page 377).An important property of the cross-covariance coefficient is for . This result is used in the computation of the standard error of the sample cross-correlation for lag . In general, the cross-covariance function is not symmetric about zero so both positive and negative lags are of interest.