Package smile.timeseries

Class AR

java.lang.Object
smile.timeseries.AR
All Implemented Interfaces:
Serializable
public class AR extends Object implements Serializable
Autoregressive model. The autoregressive model specifies that the output variable depends linearly on its own previous values and on a stochastic term; thus the model is in the form of a stochastic difference equation. Together with the moving-average (MA) model, it is a special case and key component of the more general autoregressive–moving-average (ARMA) and autoregressive integrated moving average (ARIMA) models of time series, which have a more complicated stochastic structure; it is also a special case of the vector autoregressive model (VAR), which consists of a system of more than one interlocking stochastic difference equation in more than one evolving random variable.

Contrary to the moving-average (MA) model, the autoregressive model is not always stationary as it may contain a unit root.

The notation AR(p) indicates an autoregressive model of order p. For an AR(p) model to be weakly stationary, the inverses of the roots of the characteristic polynomial must be less than 1 in modulus.

Two general approaches are available for determining the order p. The first approach is to use the partial autocorrelation function, and the second approach is to use some information criteria.

Autoregression is a good start point for more complicated models. They often fit quite well (don't need the MA terms). And the fitting process is fast (MLEs require some iterations). In applications, easily fitting autoregressions is important for obtaining initial values of parameters and in getting estimates of the error process. Least squares is a popular choice, as is the Yule-Walker procedure. Unlike the Yule-Walker procedure, least squares can produce a non-stationary fitted model. Both the Yule-Walker and least squares estimators are non-iterative and consistent, so they can be used as starting values for iterative methods like MLE.

See Also:

  • Nested Class Summary

    Nested Classes
    Modifier and Type
    Class
    Description
    static enum 
    AR.Method
    The fitting method.

  • Constructor Summary

    Constructors
    Constructor
    Description
    AR(double[] x, double[] ar, double b, AR.Method method)
    Constructor.

  • Method Summary

    Modifier and Type
    Method
    Description
    double
    adjustedR2()
    Returns adjusted R2 statistic.
    double[]
    ar()
    Returns the linear coefficients of AR (without intercept).
    int
    df()
    Returns the degree-of-freedom of residual standard error.
    static AR
    fit(double[] x, int p)
    Fits an autoregressive model with Yule-Walker procedure.
    double[]
    fittedValues()
    Returns the fitted values.
    double
    forecast()
    Returns 1-step ahead forecast.
    double[]
    forecast(int l)
    Returns l-step ahead forecast.
    double
    intercept()
    Returns the intercept.
    double
    mean()
    Returns the mean of time series.
    static AR
    ols(double[] x, int p)
    Fits an autoregressive model with least squares method.
    static AR
    ols(double[] x, int p, boolean stderr)
    Fits an autoregressive model with least squares method.
    int
    p()
    Returns the order of AR.
    double
    R2()
    Returns R2 statistic.
    double[]
    residuals()
    Returns the residuals, that is response minus fitted values.
    double
    RSS()
    Returns the residual sum of squares.
    String
    toString()
     
    double[][]
    ttest()
    Returns the t-test of the coefficients (including intercept).
    double
    variance()
    Returns the residual variance.
    double[]
    x()
    Returns the time series.

    Methods inherited from class java.lang.Object

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

  • Constructor Details

    • AR

      public AR(double[] x, double[] ar, double b, AR.Method method)
      Constructor.
      Parameters:
      x - the time series
      ar - the estimated weight parameters of AR(p).
      b - the intercept.
      method - the fitting method.

  • Method Details

    • x

      public double[] x()
      Returns the time series.
      Returns:
      the time series.

    • mean

      public double mean()
      Returns the mean of time series.
      Returns:
      the mean of time series.

    • p

      public int p()
      Returns the order of AR.
      Returns:
      the order of AR.

    • ttest

      public double[][] ttest()
      Returns the t-test of the coefficients (including intercept). The first column is the coefficients, the second column is the standard error of coefficients, the third column is the t-score of the hypothesis test if the coefficient is zero, the fourth column is the p-values of test. The last row is of intercept.
      Returns:
      the t-test of the coefficients.

    • ar

      public double[] ar()
      Returns the linear coefficients of AR (without intercept).
      Returns:
      the linear coefficients.

    • intercept

      public double intercept()
      Returns the intercept.
      Returns:
      the intercept.

    • residuals

      public double[] residuals()
      Returns the residuals, that is response minus fitted values.
      Returns:
      the residuals.

    • fittedValues

      public double[] fittedValues()
      Returns the fitted values.
      Returns:
      the fitted values.

    • RSS

      public double RSS()
      Returns the residual sum of squares.
      Returns:
      the residual sum of squares.

    • variance

      public double variance()
      Returns the residual variance.
      Returns:
      the residual variance.

    • df

      public int df()
      Returns the degree-of-freedom of residual standard error.
      Returns:
      the degree-of-freedom of residual standard error.

    • R2

      public double R2()
      Returns R2 statistic. In regression, the R2 coefficient of determination is a statistical measure of how well the regression line approximates the real data points. An R2 of 1.0 indicates that the regression line perfectly fits the data.

      In the case of ordinary least-squares regression, R2 increases as we increase the number of variables in the model (R2 will not decrease). This illustrates a drawback to one possible use of R2, where one might try to include more variables in the model until "there is no more improvement". This leads to the alternative approach of looking at the adjusted R2.

      Returns:
      R2 statistic.

    • adjustedR2

      public double adjustedR2()
      Returns adjusted R2 statistic. The adjusted R2 has almost same explanation as R2 but it penalizes the statistic as extra variables are included in the model.
      Returns:
      Adjusted R2 statistic.

    • fit

      public static AR fit(double[] x, int p)
      Fits an autoregressive model with Yule-Walker procedure.
      Parameters:
      x - the time series.
      p - the order.
      Returns:
      the model.

    • ols

      public static AR ols(double[] x, int p)
      Fits an autoregressive model with least squares method.
      Parameters:
      x - the time series.
      p - the order.
      Returns:
      the model.

    • ols

      public static AR ols(double[] x, int p, boolean stderr)
      Fits an autoregressive model with least squares method.
      Parameters:
      x - the time series.
      p - the order.
      stderr - the flag if estimate the standard errors of parameters.
      Returns:
      the model.

    • forecast

      public double forecast()
      Returns 1-step ahead forecast.
      Returns:
      1-step ahead forecast.

    • forecast

      public double[] forecast(int l)
      Returns l-step ahead forecast.
      Parameters:
      l - the number of steps.
      Returns:
      l-step ahead forecast.

    • toString

      public String toString()
      Overrides:
      toString in class Object