Package com.imsl.stat

Class ARMAMaxLikelihood

java.lang.Object
com.imsl.stat.ARMAMaxLikelihood
All Implemented Interfaces:
Serializable
public class ARMAMaxLikelihood extends Object implements Serializable
Computes maximum likelihood estimates of parameters for an ARMA model with p and q autoregressive and moving average terms respectively.

ARMAMaxLikelihood computes estimates of parameters for a nonseasonal ARMA model given a sample of observations, \(W_t\), for \(t=1, 2,\ldots, n, \) where \(n=\) z.length. The class is derived from the maximum likelihood estimation algorithm described by Akaike, Kitagawa, Arahata and Tada (1979), and the XSARMA routine published in the TIMSAC-78 Library.

The stationary time series \(W_t\) with mean \(\mu\) can be represented by the nonseasonal autoregressive moving average (ARMA) model by the following relationship: $$ {\phi}(B)(W_t-\mu)=\theta(B)a_t$$ where $$ t{\in}Z, ~~ Z=\{\ldots,-2,-1,0,1,2,\ldots\}\rm{,}$$ B is the backward shift operator defined by \(B^kW_t=W_{t-k}\rm{,}\) $$ \phi(B)=1-{\phi_1}B-{\phi_2}{B^2}-\ldots-{\phi_p}{B^p}, ~ p \ge 0\rm{.}$$ and $$ \theta(B)=1-{\theta_1}B-{\theta_2}{B^2}-\ldots-{\theta_q}{B^q}, ~ q \ge 0\rm{.}$$ The ARMAMaxLikelihood class estimates the AR coefficients \(\phi_1,\phi_2,\ldots,\phi_p\) and the MA coefficients \(\theta_1,\theta_2,\ldots,\theta_p\) using maximum likelihood estimation.

ARMAMaxLikelihood checks the initial estimates for both the autoregressive and moving average coefficients to ensure that they represent a stationary and invertible series respectively.

If $${\phi_1},{\phi_2},\ldots,{\phi_p}$$ are the initial estimates for a stationary series then all (complex) roots of the following polynomial will fall outside the unit circle: $$1-{\phi_1}z-{\phi_2}{z^2}-\ldots-{\phi_p}z^p\rm{.}$$

If $${\theta_1},{\theta_2},\ldots,{\theta_p}$$ are initial estimates for an invertible series then all (complex) roots of the polynomial $$1-{\theta_1}z-{\theta_2}{z^2}-\ldots-{\theta_q}z^q$$ will fall outside the unit circle.

By default, the order of the lags for the autoregressive terms is \(1, 2, \ldots, p\) and \(1, 2, \ldots, q\) for the moving average terms. However, this cannot be overridden.

Initial values for the AR and MA coefficients can be supplied via the setAR and setMA methods. Otherwise, initial estimates are computed internally by the method of moments. The class computes the roots of the associated polynomials. If the AR estimates represent a nonstationary series, ARMAMaxLikelihood issues a warning message and replaces the intial AR estimates with initial values that are stationary. If the MA estimates represent a noninvertible series, a terminal error is issued and new initial values must be sought.

ARMAMaxLikelihood also validates the final estimates of the AR coefficients to ensure that they too represent a stationary series. This is done to guard against the possibility that the internal log-likelihood optimizer converged to a nonstationary solution. If nonstationary estimates are encountered, a fatal error message is issued.

For model selection, the ARMA model with the minimum value for AIC might be preferred, \(\mbox{AIC} = -2\ln(L)+2(p+q)\), where L is the value of the maximum likelihood function evaluated at the parameter estimates.

ARMAMaxLikelihood can also handle white noise processes, i.e. \(ARMA(0,0)\) processes.

See Also:

  • Nested Class Summary

    Nested Classes
    Modifier and Type
    Class
    Description
    static class 
    ARMAMaxLikelihood.NonInvertibleException
    The solution is noninvertible.
    static class 
    ARMAMaxLikelihood.NonStationaryException
    The solution is nonstationary.

  • Constructor Summary

    Constructors
    Constructor
    Description
    ARMAMaxLikelihood(int p, int q, double[] z)
    Constructor for ARMAMaxLikelihood.

  • Method Summary

    Modifier and Type
    Method
    Description
    void
    compute()
    Computes the exact maximum likelihood estimates for the autoregressive and moving average parameters of an ARMA time series
    double[][]
    forecast(int nForecast)
    Returns forecasts for lead times \(l=1,2,\ldots,\rm{nForecast}\) at origins z.length-backwardOrigin-1+j where \(j=1,\ldots,\rm{backwardOrigin}+1\).
    double[]
    getAR()
    Returns the final autoregressive parameter estimates.
    int
    getBackwardOrigin()
    Returns the current value for forecasting backward origin.
    double
    getConfidence()
    Returns the confidence level used for calculating deviations in getDeviations.
    double
    getConstant()
    Returns the estimate for the constant parameter in the ARMA series.
    double[]
    getDeviations()
    Returns the deviations from each forecast used for calculating the forecast confidence limits.
    double[]
    getForecast(int nForecast)
    Returns forecasts
    double[]
    getGradients()
    Returns the gradients for the final parameter estimates.
    double
    getGradientTolerance()
    Returns the gradient tolerance for the convergence algorithm.
    double
    getInnovationVariance()
    Returns the estimated innovation variance of this series.
    double
    getLikelihood()
    Returns the final estimate for \(-2\ln(L)\), where \(L\) is equal to the likelihood function evaluated using the final parameter estimates.
    double[]
    getMA()
    Returns the final moving average parameter estimates.
    int
    getMaxIterations()
    Returns the maximum number of iterations.
    double
    getMean()
    Returns the mean used to center the time series.
    int
    getP()
    Returns the number of autoregressive terms in the ARMA model
    double[]
    getPsiWeights()
    Returns the psi weights used for calculating forecasts from the infinite order moving average form of the ARMA model.
    int
    getQ()
    Returns the number of moving average terms in the ARMA model
    double[]
    getResiduals()
    Returns the current values of the vector of residuals.
    double[]
    getTimeSeries()
    Returns the time series used to construct ARMAMaxLikelihood.
    double
    getTolerance()
    Returns the tolerance for the convergence algorithm.
    boolean
    isInvertible(double[] ma)
    Tests whether the coefficients in ma are invertible
    boolean
    isStationary(double[] ar)
    Tests whether the coefficients in ar are stationary.
    void
    setAR(double[] ar)
    Sets the initial values for the autoregressive terms to the p values in ar.
    void
    setBackwardOrigin(int backwardOrigin)
    Sets the maximum backward origin.
    void
    setConfidence(double confidence)
    Sets the confidence level for calculating confidence limit deviations returned from getDeviations().
    void
    setConstant(double constant)
    Sets the initial value for the constant term in the ARMA model.
    void
    setGradientTolerance(double gradientTolerance)
    Sets the tolerance for the convergence algorithm.
    void
    setMA(double[] ma)
    Sets the initial values for the moving average terms to the q values in ma.
    void
    setMaxIterations(int maxIterations)
    Sets the maximum number of iterations.
    void
    setMean(double wMean)
    Sets the mean used for centering the series.
    void
    setTolerance(double tolerance)
    Sets the tolerance for the convergence algorithm.

    Methods inherited from class java.lang.Object

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

  • Constructor Details

  • Method Details

    • compute

      public void compute() throws ARMAMaxLikelihood.NonStationaryException, ARMAMaxLikelihood.NonInvertibleException
      Computes the exact maximum likelihood estimates for the autoregressive and moving average parameters of an ARMA time series
      Throws:
      ARMAMaxLikelihood.NonStationaryException - is thrown if the final maximum likelihood estimates for the time series are nonstationary.
      ARMAMaxLikelihood.NonInvertibleException - is thrown if the final maximum likelihood estimates for the time series are noninvertible.

    • setConfidence

      public void setConfidence(double confidence)
      Sets the confidence level for calculating confidence limit deviations returned from getDeviations().
      Parameters:
      confidence - a double scalar specifying the confidence level used in computing forecast confidence intervals. Typical choices for confidence are 0.90, 0.95, and 0.99. confidence must be greater than 0.0 and less than 1.0. Default: confidence = 0.95.

    • getBackwardOrigin

      public int getBackwardOrigin()
      Returns the current value for forecasting backward origin.
      Returns:
      A double scalar value representing the current value of backwardOrigin

    • getConfidence

      public double getConfidence()
      Returns the confidence level used for calculating deviations in getDeviations.
      Returns:
      A double scalar value of confidence used for computing the (1-confidence)*100% forecast confidence interval.

    • getP

      public int getP()
      Returns the number of autoregressive terms in the ARMA model
      Returns:
      An int scalar value of p, the number of autoregressive terms in the ARMA model.

    • getQ

      public int getQ()
      Returns the number of moving average terms in the ARMA model
      Returns:
      An int scalar value of q, the number of moving average terms in the ARMA model.

    • getGradients

      public double[] getGradients()
      Returns the gradients for the final parameter estimates.
      Returns:
      A double array of length p+q containing the gradients of the final parameter estimates.

    • getAR

      public double[] getAR()
      Returns the final autoregressive parameter estimates.
      Returns:
      A double array of length p containing the final autoregressive parameter estimates.

    • setAR

      public void setAR(double[] ar)
      Sets the initial values for the autoregressive terms to the p values in ar.
      Parameters:
      ar - An input double array of length p containing the initial values for the autoregressive terms. If this method is not called, initial values are computed by method of moments in the ARMA class.

    • setBackwardOrigin

      public void setBackwardOrigin(int backwardOrigin)
      Sets the maximum backward origin.
      Parameters:
      backwardOrigin - An int scalar specifying the maximum backward origin used in forecasting. backwardOrigin must be greater than or equal to 0 and less than or equal to z.length - StrictMath.max(p, q). Default: backwardOrigin = 0.

    • getMA

      public double[] getMA()
      Returns the final moving average parameter estimates.
      Returns:
      A double array of length q containing the final moving average parameter estimates.

    • setMA

      public void setMA(double[] ma)
      Sets the initial values for the moving average terms to the q values in ma.
      Parameters:
      ma - A double array of length q containing the initial values for the moving average terms. If this method is not called, initial values are computed by method of moments in the ARMA class.

    • getConstant

      public double getConstant()
      Returns the estimate for the constant parameter in the ARMA series.
      Returns:
      A double scalar equal to the estimate for the constant parameter in the ARMA series.

    • getForecast

      public double[] getForecast(int nForecast) throws ARMAMaxLikelihood.NonStationaryException, ARMAMaxLikelihood.NonInvertibleException, ARMA.MatrixSingularException, ARMA.TooManyCallsException, ARMA.IncreaseErrRelException, ARMA.NewInitialGuessException, ARMA.IllConditionedException, ARMA.TooManyITNException, ARMA.TooManyFcnEvalException, ARMA.TooManyJacobianEvalException
      Returns forecasts
      Parameters:
      nForecast - An input int representing the number of requested forecasts beyond the last value in the series.
      Returns:
      A double array containing the nForecast+backwardOrigin forecasts. The first backwardOrigin forecasts are one-step ahead forecasts for the last backwardOrigin values in the series. The next nForecast values in the returned series are forecasts for the next values beyond the series.
      Throws:
      ARMAMaxLikelihood.NonStationaryException - is thrown if the final maximum likelihood estimates for the time series are nonstationary.
      ARMAMaxLikelihood.NonInvertibleException - is thrown if the final maximum likelihood estimates for the time series are noninvertible.
      ARMAMaxLikelihood.InitialMAException - is thrown if the initial values provided for the moving average terms using setMA are noninvertible. In this case, ARMAMaxLikelihood terminates and does not compute the time series estimates.
      ARMA.MatrixSingularException - is thrown if the input matrix is singular.
      ARMA.TooManyCallsException - is thrown if the number of calls to the function has exceeded the maximum number of iterations times the number of moving average (MA) parameters + 1.
      ARMA.IncreaseErrRelException - is thrown if the bound for the relative error is too small.
      ARMA.NewInitialGuessException - is thrown if the iteration has not made good progress.
      ARMA.IllConditionedException - is thrown if the problem is ill-conditioned.
      ARMA.TooManyITNException - is thrown if the maximum number of iterations is exceeded.
      ARMA.TooManyFcnEvalException - is thrown if the maximum number of function evaluations exceeded.
      ARMA.TooManyJacobianEvalException - is thrown if the maximum number of Jacobian evaluations is exceeded.

    • forecast

      public double[][] forecast(int nForecast) throws ARMAMaxLikelihood.NonStationaryException, ARMAMaxLikelihood.NonInvertibleException, ARMA.MatrixSingularException, ARMA.TooManyCallsException, ARMA.IncreaseErrRelException, ARMA.NewInitialGuessException, ARMA.IllConditionedException, ARMA.TooManyITNException, ARMA.TooManyFcnEvalException, ARMA.TooManyJacobianEvalException
      Returns forecasts for lead times \(l=1,2,\ldots,\rm{nForecast}\) at origins z.length-backwardOrigin-1+j where \(j=1,\ldots,\rm{backwardOrigin}+1\).
      Parameters:
      nForecast - An int scalar equal to the number of requested forecasts
      Returns:
      a double matrix of dimensions of nForecast by backwardOrigin + 1 containing the forecasts. The forecasts are for lead times \(l=1,2,\ldots,\rm{nForecast}\) at origins z.length-backwardOrigin-1+j where \(j=1,\ldots,\rm{backwardOrigin}+1\).
      Throws:
      ARMAMaxLikelihood.NonStationaryException - is thrown if the final maximum likelihood estimates for the time series are non-stationary.
      ARMAMaxLikelihood.NonInvertibleException - is thrown if the final maximum likelihood estimates for the time series are noninvertible.
      ARMAMaxLikelihood.InitialMAException - is thrown if the initial values provided for the moving average terms using setMA are noninvertible. In this case, ARMAMaxLikelihood terminates and does not compute the time series estimates.
      ARMA.MatrixSingularException - is thrown if the input matrix is singular.
      ARMA.TooManyCallsException - is thrown if the number of calls to the function has exceeded the maximum number of iterations times the number of moving average (MA) parameters + 1.
      ARMA.IncreaseErrRelException - is thrown if the bound for the relative error is too small.
      ARMA.NewInitialGuessException - is thrown if the iteration has not made good progress.
      ARMA.IllConditionedException - is thrown if the problem is ill-conditioned.
      ARMA.TooManyITNException - is thrown if the maximum number of iterations is exceeded.
      ARMA.TooManyFcnEvalException - is thrown if the maximum number of function evaluations exceeded.
      ARMA.TooManyJacobianEvalException - is thrown if the maximum number of Jacobian evaluations is exceeded.

    • isStationary

      public boolean isStationary(double[] ar)
      Tests whether the coefficients in ar are stationary.
      Parameters:
      ar - A double array containing the coefficients for the autoregressive terms in an ARMA model
      Returns:
      A boolean scalar equal to true if the coefficients in ar are stationary and false otherwise.

    • isInvertible

      public boolean isInvertible(double[] ma)
      Tests whether the coefficients in ma are invertible
      Parameters:
      ma - A double array containing the coefficients for the moving average terms in an ARMA model.
      Returns:
      A boolean scalar equal to true if the coefficients in ma are invertible and false otherwise.

    • getLikelihood

      public double getLikelihood() throws ARMAMaxLikelihood.NonStationaryException, ARMAMaxLikelihood.NonInvertibleException
      Returns the final estimate for \(-2\ln(L)\), where \(L\) is equal to the likelihood function evaluated using the final parameter estimates.
      Returns:
      A double scalar equal to the log likelihood function, \(-2\ln(L)\).
      Throws:
      ARMAMaxLikelihood.NonStationaryException - is thrown if the final maximum likelihood estimates for the time series are nonstationary.
      ARMAMaxLikelihood.NonInvertibleException - is thrown if the final maximum likelihood estimates for the time series are noninvertible.
      InitialMAException - is thrown if the initial values provided for the moving average terms using setMA are noninvertible. In this case, ARMAMaxLikelihood terminates and does not compute the time series estimates.

    • getMaxIterations

      public int getMaxIterations()
      Returns the maximum number of iterations.
      Returns:
      An int scalar containing the maximum number of iterations allowed in the nonlinear equation solver used in both the method of moments and least-squares algorithms. Default: maxIterations = 300.

    • getResiduals

      public double[] getResiduals()
      Returns the current values of the vector of residuals.
      Returns:
      A double array of length backwardOrigin containing the residuals for the last backwardOrigin values in the time series. The compute and either the forecast or getForecast methods must be called before calling this method.

    • setMaxIterations

      public void setMaxIterations(int maxIterations) throws IllegalArgumentException
      Sets the maximum number of iterations.
      Parameters:
      maxIterations - An int scalar specifying the maximum number of iterations allowed in the nonlinear equation solver used in both the method of moments and least-squares algorithms. Default: maxIterations = 300.
      Throws:
      IllegalArgumentException

    • setConstant

      public void setConstant(double constant)
      Sets the initial value for the constant term in the ARMA model.
      Parameters:
      constant - A double scalar specifying the initial value for the constant term in the ARMA model. By default, the constant term is initially estimated using ARMA method of moments estimation.

    • setMean

      public void setMean(double wMean)
      Sets the mean used for centering the series.
      Parameters:
      wMean - A double scalar specifying the value to use for centering the time series. By default the series is centered using the mean of the series.

    • setTolerance

      public void setTolerance(double tolerance)
      Sets the tolerance for the convergence algorithm.
      Parameters:
      tolerance - A double scalar specifying the value to use for the convergence tolerance. Default: tolerance = 2.220446049e-016.

    • getGradientTolerance

      public double getGradientTolerance()
      Returns the gradient tolerance for the convergence algorithm.
      Returns:
      A double scalar specifying the value used for the gradient tolerance.

    • setGradientTolerance

      public void setGradientTolerance(double gradientTolerance)
      Sets the tolerance for the convergence algorithm.
      Parameters:
      gradientTolerance - A double scalar specifying the the tolerance used for numerically estimating the gradient by differences. Default: gradientTolerance = 1e-04.

    • getTolerance

      public double getTolerance()
      Returns the tolerance for the convergence algorithm.
      Returns:
      A double scalar containing the value of the tolerance used during maximum likelihood estimation.

    • getMean

      public double getMean()
      Returns the mean used to center the time series.
      Returns:
      A double scalar specifying the value to used for centering the time series.

    • getTimeSeries

      public double[] getTimeSeries()
      Returns the time series used to construct ARMAMaxLikelihood.
      Returns:
      A double containing the values of the time series passed to the class constructor.

    • getInnovationVariance

      public double getInnovationVariance()
      Returns the estimated innovation variance of this series.
      Returns:
      A double scalar equal to the estimated innovation variance for the time series.

    • getDeviations

      public double[] getDeviations()
      Returns the deviations from each forecast used for calculating the forecast confidence limits.
      Returns:
      A double array of length backwardOrigin+nForecast containing the deviations for calculating forecast confidence intervals. The confidence level is specified in confidence. By default, confidence= 0.95.

    • getPsiWeights

      public double[] getPsiWeights()
      Returns the psi weights used for calculating forecasts from the infinite order moving average form of the ARMA model.
      Returns:
      A double array of length nForecast containing the psi weights of the infinite order moving average form of the model.