Pre-Treatment of Time Series and Assessment of Models
Hans-Peter Deutsch · Palgrave Macmillan UK eBooks · 2009
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 gt, a seasonal component,St, and a random component1 Z t : (34.1) $$\matrix{ {{X_t}\, = \,{g_t}\, + \,{s_t}\, + \,{Z_t}} \cr }$$ Z f then represents the stationary time series with E[Z t ] = 0. The trend g t is a deterministic function of the time variable f, which represents a long-term development, for example a polynomial or an exponential function.2 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: (34.2) $$\matrix{ {{s_t}\, = \,{s_{t\, + \,p}}} \cr }$$ It follows that the sum $$\sum olimits_{i\, = \,1}^p {{s_{t\, + \,1}}}$$ 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: $$\sum\limits_{i\, = \,1}^p {{s_{t\, + \,i}}} \, = \,0.$$