AutoCorrelation Class

Computes the sample autocorrelation function of a stationary time series.

Inheritance Hierarchy

SystemObject
Imsl.StatAutoCorrelation

Namespace: Imsl.Stat
Assembly: ImslCS (in ImslCS.dll) Version: 6.5.2.0

Syntax

C++

Copy

[SerializableAttribute]
public class AutoCorrelation

<SerializableAttribute>
Public Class AutoCorrelation

[SerializableAttribute]
public ref class AutoCorrelation

[<SerializableAttribute>]
type AutoCorrelation =  class end

The AutoCorrelation type exposes the following members.

Constructors

	Name	Description
	AutoCorrelation	Constructor to compute the sample autocorrelation function of a stationary time series.

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Methods

	Name	Description
	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.
	GetAutoCovariances	Returns the variance and autocovariances of the time series x.
	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.
	GetStandardErrors	Returns the standard errors of the autocorrelations of the time series x.
	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.)

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Properties

	Name	Description
	Mean	The mean of the time series x.
	NumberOfProcessors	Perform the parallel calculations with the maximum possible number of processors set to NumberOfProcessors.
	Variance	Returns the variance of the time series x.

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Remarks

AutoCorrelation estimates the autocorrelation function of a stationary time series given a sample of n observations $\{X_t\}$ for ${\rm t = 1, 2, \dots, n}$ .

Let

$\hat \mu = {\rm {xmean}}$

be the estimate of the mean $\mbox{\hspace{14pt}}\mu$ of the time series $\{X_t\}$ where

$\hat \mu = \left\{pa \begin{array}{ll} \mu & {\rm for}\;\mu\; {\rm known} \\ \frac{1}{n}\sum\limits_{t=1}^n {X_t } & {\rm for}\;\mu\; {\rm unknown} \end{array} \right.$

The autocovariance function $\sigma(k)$ is estimated by

$\hat \sigma \left( k \right) = \frac{1}{n} \sum\limits_{t = 1}^{n - k} {\left( {X_t - \hat \mu } \right)} \left( {X_{t + k} - \hat \mu } \right), \mbox{\hspace{20pt}k=0,1,\dots,K}$

where K = maximumLag. Note that $\hat \sigma(0)$ is an estimate of the sample variance. The autocorrelation function $\rho(k)$ is estimated by

$\hat\rho(k) = \frac{\hat \sigma(k)}{\hat \sigma(0)},\mbox{\hspace{20pt}} k=0,1,\dots,K$

Note that $\hat \rho(0) \equiv 1$ 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

$\mbox{var}\{\hat \rho(k)\} = \frac{1}{n}\sum\limits_{i=-\infty}^{\infty} \left[{\rho^2(i)}+\rho(i-k)\rho(i+k)-4\rho(i) \rho(k)\rho(i-k)+2\rho^2(i)\rho^2(k)\right]$

where $\hat \rho(k)$ assumes $\mu$ is unknown. For computational purposes, the autocorrelations $\rho(k)$ are replaced by their estimates $\hat \rho(k)$ for $\left|k\right|\leq K$ , and the limits of summation are bounded because of the assumption that $\rho(k) = 0$ for all such that $\left|k\right|> K$ .

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

$var\{\hat \rho(k)\} = \frac{n-k}{n(n+2)}$

where $\mu$ 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 $\hat \rho(k)$ for k=0,1,...,K. Consider the AR(k) process defined by

$X_t = {\phi_{k1}}X_{t-1}+{\phi_{k2}}X_{t-2}+ \dots+{\phi_{kk}}X_{t-k}+A_t$

where $\phi_{kj}$ denotes the j-th coefficient in the process. The set of estimates ${\{\hat \phi_{kk}\}}$ for k = 1, ..., K is the sample partial autocorrelation function. The autoregressive parameters $\{\hat \phi_{kj}\}$ 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

$\hat\rho(j) = {\hat\phi_{k1}}\hat\rho(j-1) + {\hat\phi_{k2}}\hat\rho(j-2) + \dots + {\hat\phi_{kk}}\hat\rho(j-k), \mbox{\hspace{20pt}j = 1,2,\dots,k}$

a recursive relationship for k=1, ..., K was developed by Durbin (1960). The equations are given by

$\hat \phi_{kk} = \left\{\begin{array}{ll} \hat\rho(1) & {\rm for}\;{\rm k}\; {\rm = 1} \\ \frac{\hat\rho(k)\; - \sum\limits_{j=1}^{k-1} {\hat\phi_{k-1,j}\hat\rho(k-j) }} {1\;-\; \sum\limits_{j=1}^{k-1}{\hat\phi_{k-1,j}\hat\rho(j)} } & {\rm for}\;{\rm k = 2,}\;\dots\; {\rm ,K} \end{array} \right.$

and

$\hat \phi_{kj} = \left\{\begin{array}{ll} \hat\phi_{k-1,j}-\hat\phi_{kk}\hat\phi_{k-1,k-j} & {\rm for}\; {\rm j}\; {\rm = 1,2,}\; \dots {\rm,k-1} \\ \hat \phi_{kk} & {\rm for}\;{\rm j = k} \end{array} \right.$

This 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 ${\{\phi_{kk}\}}$ 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

${\rm var}{\{ \hat\phi_{kk}\}} \simeq \frac {1}{n} \;\;\;\;\; {\rm k}\; \geq \; {\rm p + 1}$

See Box and Jenkins (1976, pages 82-84) for more information concerning the partial autocorrelation function.

Reference

Imsl.Stat Namespace

Other Resources

Example