Difference Class

Differences a seasonal or nonseasonal time series.

For a list of all members of this type, see Difference Members.

System.Object
Imsl.Stat.Difference

public class Difference

Thread Safety

Public static (Shared in Visual Basic) members of this type are safe for multithreaded operations. Instance members are not guaranteed to be thread-safe.

Remarks

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 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 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.

Requirements

Namespace: Imsl.Stat

Assembly: ImslCS (in ImslCS.dll)

Difference Class

Thread Safety

Remarks

Requirements

See Also