AutoCorrelation (JMSL Numerical Library)

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com.imsl.stat
Class AutoCorrelation

java.lang.Object
  com.imsl.stat.AutoCorrelation

All Implemented Interfaces:: Cloneable, Serializable

public class AutoCorrelation
extends Object
implements Serializable, Cloneable

Computes the sample autocorrelation function of a stationary time series.

AutoCorrelation estimates the autocorrelation function of a stationary time series given a sample of n observations ${X_t}$ for .

Let

be the estimate of the mean

of the time series ${X_t}$ where

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

The autocovariance function

is estimated by

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

where K = maximum_lag. Note that is an estimate of the sample variance. The autocorrelation function is estimated by

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

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

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

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

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

where is assumed to be equal to zero. Note that this formula does not depend on the autocorrelation function.

The method getPartialAutoCorrelations estimates the partial autocorrelations of the stationary time series given K = maximum_lag sample autocorrelations 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

$hatrho(j) = {hatphi_{k1}}hatrho(j-1) + {hatphi_{k2}}hatrho(j-2) + dots + {hatphi_{kk}}hatrho(j-k), mbox{hspace{20pt}j = 1,2,dots,k}$

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

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

and

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

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

${rm var}{{ hatphi_{kk}}} simeq frac {1}{n} ;;;;; {rm k}; geq ; {rm p + 1}$

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

See Also:: Example, Serialized Form

Nested Class Summary
`static class`	`AutoCorrelation.NonPosVariancesException` The problem is ill-conditioned.

Field Summary
`static int`	`BARTLETTS_FORMULA` Indicates standard error computation using Bartlett's formula.
`static int`	`MORANS_FORMULA` Indicates standard error computation using Moran's formula.

Constructor Summary
`AutoCorrelation(double[] x, int maximum_lag)` Constructor to compute the sample autocorrelation function of a stationary time series.

Method Summary
`double[]`	`getAutoCorrelations()` Returns the autocorrelations of the time series `x`.
`double[]`	`getAutoCovariances()` Returns the variance and autocovariances of the time series `x`.
`double`	`getMean()` Returns the mean of the time series `x`.
`double[]`	`getPartialAutoCorrelations()` Returns the sample partial autocorrelation function of the stationary time series `x`.
`double[]`	`getStandardErrors(int stderrMethod)` Returns the standard errors of the autocorrelations of the time series `x`.
`double`	`getVariance()` Returns the variance of the time series `x`.
`void`	`setMean(double mean)` Estimate mean of the time series `x`.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Detail

BARTLETTS_FORMULA

public static final int BARTLETTS_FORMULA

Indicates standard error computation using Bartlett's formula.

See Also:: Constant Field Values

MORANS_FORMULA

public static final int MORANS_FORMULA

Indicates standard error computation using Moran's formula.

See Also:: Constant Field Values

Constructor Detail

AutoCorrelation

public AutoCorrelation(double[] x,
                       int maximum_lag)

Constructor to compute the sample autocorrelation function of a stationary time series.
Parameters:: x - a one-dimensional double array containing the stationary time series; maximum_lag - an int containing the maximum lag of autocovariance, autocorrelations, and standard errors of autocorrelations to be computed. maximum_lag must be greater than or equal to 1 and less than the number of observations in x

Method Detail

getAutoCorrelations

public double[] getAutoCorrelations()

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 where k = 1, ..., maximum_lag.

getAutoCovariances

public double[] getAutoCovariances()
                            throws AutoCorrelation.NonPosVariancesException

Returns the variance and 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:: AutoCorrelation.NonPosVariancesException - is thrown if the problem is ill-conditioned

getMean

public double getMean()

Returns the mean of the time series x.

Returns:: a double containing the mean

getPartialAutoCorrelations

public double[] getPartialAutoCorrelations()

Returns the sample partial autocorrelation function of the stationary time series x.

Returns:: a double array of length maximum_lag containing the partial autocorrelations of the time series x.

getStandardErrors

public double[] getStandardErrors(int stderrMethod)

Returns the standard errors of the autocorrelations of the time series x. Method of computation for standard errors of the autocorrelation is chosen by the stderrMethod parameter. If stderrMethod is set to BARTLETTS_FORMULA, Bartlett's formula is used to compute the standard errors of autocorrelations. If stderrMethod is set to MORANS_FORMULA, Moran's formula is used to compute the standard errors of autocorrelations.

Parameters:: stderrMethod - an int specifying the method to compute the standard errors of autocorrelations of the time series x
Returns:: a double array of length maximum_lag containing the standard errors of the autocorrelations of the time series x

getVariance

public double getVariance()

Returns the variance of the time series x.

Returns:: a double containing the variance of the time series x

setMean

public void setMean(double mean)

Estimate mean of the time series x.

Parameters:: mean - a double containing the estimate mean of the time series x.