Pre-Treatment of Time Series and Assessment of Models

Abstract

The pre-treatment for the transformation of a given data set into a stationary time series has been mentioned several times in the preceding sections and will receive detailed treatment in this section. The basis for pre-treating a time series is its decomposition into a trend component g _t , a seasonal component s _t , and a random component¹ Z _t :

$${X_t} = {g_t} + {s_t} + {Z_t}$$

((36.1))

Z _t then represents the stationary time series with E[Z _t ] = 0. The trend g _t is a deterministic function of the time variable t, which represents a long-term development, for example a polynomial or an exponential function². A weaker trend can sometimes be more readily recognized after a compression of the time axis. The season s _t represents a periodic component with a period p:

$${s_t} = {s_{t + p}}$$

((36.2))

It follows that the sum $\sum\nolimits_{i = 1}^p {{s_{t + i}}}$ of p successive values is a constant. This constant can be incorporated into the trend g _t so that, without loss of generality, the sum can be assumed to be equal to zero:

$$\sum\limits_{i = 1}^p {{s_{t + i}}} = 0$$