Syntax

C#
[SerializableAttribute] public class Difference

Visual Basic (Declaration)
<SerializableAttribute> _ Public Class Difference

Visual C++
[SerializableAttribute] public ref class Difference

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.

Inheritance Hierarchy

System..::.Object
Imsl.Stat..::.Difference

Syntax

Remarks

Inheritance Hierarchy

See Also