Definitions

The stationarity of time series is assumed in ARMA, ANN, and BL models. That is a simplification of reality. A well known source of non-stationarity is a linear component, the trend. One can eliminate the trend by differencing, since derivatives of linear functions are constant. The elegant extension of this idea is the Auto-Regression Fractionally-Integrated Moving-Average Model (ARFIMA).

We define an ARFIMA process as the folowing time series $z_t$

$$A(L)(1-L)^d z_t= B(L) \epsilon_t.$$ (35)

Here

$$A(L) w_t = w_t -\sum_{i=1}^p a_i w_{t-i}$$ (36)

and

$$B(L) \epsilon_t = \epsilon_t -\sum_{i=1}^q b_i \epsilon_{t-i},$$ (37)

where $\epsilon_t= Gaussian\ \{0, \sigma^2\}$ .

We define the transformation $(1-L)^d$ as follows:

$$w_t=(1-L)^d z_t = z_t-\sum_{i=1}^{\infty} d_i z_{t-i}.$$ (38)

Here

$$d_i= {\Gamma (i-d) \over \Gamma(i+1) \Gamma(-d)},$$ (39)

where $d$ is a fractional integration parameter, and $\Gamma(.)$ is a gamma function.

We assume that

$$z_{t-i}=0,\ w_{t-i}=0,\ \epsilon_{t-i}=0,\ \ if\ \ t \le i.$$ (40)

We truncate sequence (1.39)

$$d_i=0,\ \ if\ \ i > R.$$ (41)

Here $R$ is the truncation parameter, the number of non-zero components.