Differences a seasonal or nonseasonal time series.

The type double function is imsls_d_difference.

Pointer to an array of length n_observations containing the differenced series.

Function imsls_f_difference performs m = n_differences successive backward differences of period si = periods [i - 1] and order di = orders [i - 1] for i = 1, ..., m on the n = n_observations observations {Zt} for t = 1, 2, ..., n.

Consider the backward shift operator B given by

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

where d ≥ 0 is the order of differencing. Note that

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

The general difference formula used in the function imsls_f_difference is given by

where nL represents the number of observations “lost” because of differencing and NaN represents the missing value code. See the functions imsls_f_machine and imsls_d_machineto retrieve missing values. Note that

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.

Consider the Airline Data (Box and Jenkins 1976, p. 531) consisting of the monthly total number of international airline passengers from January 1949 through December 1960. Function imsls_f_difference is used to compute

for t = 14, 15, ..., 24.

The data for this example is the same as that for the initial example. The first n_lost observations are excluded from W due to differencing, and n_lost is also output.