Package com.imsl.stat

Class Difference

java.lang.Object
com.imsl.stat.Difference
All Implemented Interfaces:
Serializable, Cloneable
public class Difference extends Object implements Serializable, Cloneable
Differences a seasonal or nonseasonal time series.

Class Difference performs m = periods.length successive backward differences of period \(s_i = {\rm {periods}} [i - 1]\) and order \(d_i = {\rm {orders}} [i - 1] \,\, {\rm{for}} \,\, i = 1, \dots, m\) on the n = z.length observations \(\left\{ Z_t \right\} \,\, {\rm{for}} \,\, t = 1, 2, \dots, n\).

Consider the backward shift operator B given by

$$B^kZ_t = Z_{t-k}$$

for all k. Then, the backward difference operator with period s is defined by the following:

$$\Delta _s Z_t = \left( {1 - B^s } \right)Z_t = Z_t - Z_{t - s} \,\,\,\,\,\,{\rm{for}}\,s \ge 0$$

Note that \(B_sZ_t\) and \(\Delta_sZ_t\) are defined only for \(t = (s + 1), \dots, n\). Repeated differencing with period s is simply

$$\Delta _s^d Z_t = \left( {1 - B^s } \right)^d Z_t = \sum\limits_{j = 0}^d {\frac{{d!}}{{j!\left( {d - j} \right)!}}} \left( { - 1} \right)^j B^{sj} Z_t$$

where \(d \ge 0\) is the order of differencing. Note that

$$ \Delta _s^d Z_t$$

is defined only for \(t = (sd + 1), \dots, n\).

The general difference formula used in the class Difference is given by

$$W_T = \left\{ \begin{array}{ll} \rm{NaN} & {\rm for}\,\, t = 1, \ldots, n_L \\ \Delta _{s_1 }^{d_1 } \Delta _{s_2 }^{d2} \ldots \Delta _{s_m }^{d_m } Z_t & {\rm for}\,\, t = n_L + 1,\ldots, n \end{array} \right.$$

where \(n_L\) represents the number of observations "lost" because of differencing and NaN represents the missing value code. Note that

$$n_L = \sum\limits_j {s_j d_j }$$

A homogeneous, stationary time series can be arrived at by appropriately differencing a homogeneous, nonstationary time series (Box and Jenkins 1976, p. 85). Preliminary application of an appropriate transformation followed by differencing of a series can enable model identification and parameter estimation in the class of homogeneous stationary autoregressive moving average models.

See Also:

  • Constructor Summary

    Constructors
    Constructor
    Description
    Difference()
    Constructor for Difference.

  • Method Summary

    Modifier and Type
    Method
    Description
    final double[]
    compute(double[] z, int[] periods)
    Computes a Difference series.
    void
    excludeFirst(boolean exclude)
    If set to true, the observations lost due to differencing will be excluded.
    int
    getObservationsLost()
    Returns the number of observations lost because of differencing the time series.
    void
    setOrders(int[] orders)
    Sets the orders for the Difference object

    Methods inherited from class java.lang.Object

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

  • Constructor Details

    • Difference

      public Difference()
      Constructor for Difference.

  • Method Details

    • setOrders

      public void setOrders(int[] orders)
      Sets the orders for the Difference object
      Parameters:
      orders - an int array of length equal to length of periods, containing the order of each difference given in periods. The elements of orders must be greater than or equal to 0.

    • getObservationsLost

      public int getObservationsLost()
      Returns the number of observations lost because of differencing the time series. Note that the compute method must be invoked first before invoking this method. Otherwise, the return value is 0.
      Returns:
      an int containing the number of observations lost because of differencing the time series z.

    • excludeFirst

      public void excludeFirst(boolean exclude)
      If set to true, the observations lost due to differencing will be excluded. The differenced series will be the length of the number of observations minus the number of observations lost. If set to false, the observations lost due to differencing will be set to NaN (Not a number) and included in the differenced series. The default is to set the lost observations to NaN.
      Parameters:
      exclude - a boolean specifying whether or not to exclude lost observations due to differencing.

    • compute

      public final double[] compute(double[] z, int[] periods) throws IllegalArgumentException
      Computes a Difference series.
      Parameters:
      z - a double array containing the time series.
      periods - an int array containing the periods at which z is to be differenced.
      Returns:
      a double array containing the differenced series.
      Throws:
      IllegalArgumentException