Package com.imsl.stat

Class AutoCorrelation

java.lang.Object
com.imsl.stat.AutoCorrelation
All Implemented Interfaces:
Serializable, Cloneable
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 \({\rm t = 1, 2, \dots, n}\).

Let $$\hat \mu = {\rm {xmean}}$$ be the estimate of the mean \(\mu\) of the time series \(\{X_t\} \) where

$$\hat \mu = \left\{ \begin{array}{ll} \mu & {\rm for}\;\mu\; {\rm known} \\ \frac{1}{n}\sum\limits_{t=1}^n {X_t } & {\rm for}\;\mu\; {\rm unknown} \end{array} \right.$$ The autocovariance function \(\sigma(k)\) is estimated by $$\hat \sigma ( k ) = \frac{1}{n} \sum\limits_{t = 1}^{n - k} {\left( {X_t - \hat \mu } \right)} \left({X_{t + k} - \hat \mu } \right), \mbox{ } k=0,1,\dots, K $$

where K = maximum_lag. Note that \(\hat \sigma(0)\) is an estimate of the sample variance. The autocorrelation function \(\rho(k)\) is estimated by $$\hat\rho(k) = \frac{\hat \sigma(k)}{\hat \sigma(0)},\mbox{ } k=0,1,\dots,K$$

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 \(k\) 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 estimates the partial autocorrelations of the stationary time series given K = maximum_lag 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 $$\hat\rho(j) = {\hat\phi_{k1}}\hat\rho(j-1) + {\hat\phi_{k2}}\hat\rho(j-2) + \dots + {\hat\phi_{kk}}\hat\rho(j-k), \mbox{ } j = 1,2,\dots,k $$ 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.

See Also:

  • Nested Class Summary

    Nested Classes
    Modifier and Type
    Class
    Description
    static class 
    AutoCorrelation.NonPosVariancesException
    The problem is ill-conditioned.

  • Field Summary

    Fields
    Modifier and Type
    Field
    Description
    static final int
    BARTLETTS_FORMULA
    Indicates standard error computation using Bartlett's formula.
    static final int
    MORANS_FORMULA
    Indicates standard error computation using Moran's formula.

  • Constructor Summary

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

  • Method Summary

    Modifier and Type
    Method
    Description
    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.
    int
    getNumberOfThreads()
    Returns the number of java.lang.Thread instances used for parallel processing.
    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.
    void
    setNumberOfThreads(int numberOfThreads)
    Sets the number of java.lang.Thread instances to be used for parallel processing.

    Methods inherited from class java.lang.Object

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

  • Field Details

    • BARTLETTS_FORMULA

      public static final int BARTLETTS_FORMULA
      Indicates standard error computation using Bartlett's formula.
      See Also:

    • MORANS_FORMULA

      public static final int MORANS_FORMULA
      Indicates standard error computation using Moran's formula.
      See Also:

  • Constructor Details

    • 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 Details

    • setNumberOfThreads

      public void setNumberOfThreads(int numberOfThreads)
      Sets the number of java.lang.Thread instances to be used for parallel processing.
      Parameters:
      numberOfThreads - an int specifying the number of java.lang.Thread instances to be used for parallel processing.

      Default: numberOfThreads = 1.

    • getNumberOfThreads

      public int getNumberOfThreads()
      Returns the number of java.lang.Thread instances used for parallel processing.
      Returns:
      an int containing the number of java.lang.Thread instances used for parallel processing.

    • 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.

    • getMean

      public double getMean()
      Returns the mean of the time series x.
      Returns:
      a double containing the mean

    • 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

    • 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 \({\rm k}\) 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

    • getVariance

      public double getVariance()
      Returns the variance of the time series x.
      Returns:
      a double containing the variance of the time series x

    • 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.