AutoCorrelation Class |
Namespace: Imsl.Stat
The AutoCorrelation type exposes the following members.
Name | Description | |
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AutoCorrelation |
Constructor to compute the sample autocorrelation function of a
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.) | |
GetAutoCorrelations |
Returns the autocorrelations of the time series x.
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GetAutoCovariances |
Returns the variance and autocovariances of the time series x.
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GetHashCode | Serves as a hash function for a particular type. (Inherited from Object.) | |
GetPartialAutoCorrelations |
Returns the sample partial autocorrelation function of the stationary
time series x.
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GetStandardErrors |
Returns the standard errors of the autocorrelations of the
time series x.
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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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Mean |
The mean of the time series x.
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NumberOfProcessors |
Perform the parallel calculations with the maximum possible number of
processors set to NumberOfProcessors.
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Variance |
Returns the variance of the time series x.
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AutoCorrelation estimates the autocorrelation function of a stationary time series given a sample of n observations for .
Let
be the estimate of the mean of the time series where The autocovariance function is estimated bywhere K = maximumLag. Note that is an estimate of the sample variance. The autocorrelation function is estimated by
Note that by definition.
The standard errors of sample autocorrelations may be optionally computed according to the GetStandardErrors method argument stderrMethod. One method (Bartlett 1946) is based on a general asymptotic expression for the variance of the sample autocorrelation coefficient of a stationary time series with independent, identically distributed normal errors. The theoretical formula is
where assumes is unknown. For computational purposes, the autocorrelations are replaced by their estimates for , and the limits of summation are bounded because of the assumption that for all such that .
A second method (Moran 1947) utilizes an exact formula for the variance of the sample autocorrelation coefficient of a random process with independent, identically distributed normal errors. The theoretical formula is
where is assumed to be equal to zero. Note that this formula does not depend on the autocorrelation function.
The method GetPartialAutoCorrelations returns the estimated partial autocorrelations of the stationary time series given K = maximumLag sample autocorrelations for k=0,1,...,K. Consider the AR(k) process defined by
where denotes the j-th coefficient in the process. The set of estimates for k = 1, ..., K is the sample partial autocorrelation function. The autoregressive parameters for j = 1, ..., k are approximated by Yule-Walker estimates for successive AR(k) models where k = 1, ..., K. Based on the sample Yule-Walker equations a recursive relationship for k=1, ..., K was developed by Durbin (1960). The equations are given by andThis procedure is sensitive to rounding error and should not be used if the parameters are near the nonstationarity boundary. A possible alternative would be to estimate for successive AR(k) models using least or maximum likelihood. Based on the hypothesis that the true process is AR(p), Box and Jenkins (1976, page 65) note
See Box and Jenkins (1976, pages 82-84) for more information concerning the partial autocorrelation function.