Quasi-Periodic Process Modeling

The early development of time series has been related to a quest for an understanding of cyclical phenomena. For example, Shuster’s periodogram (1898, 1906), and Yule’s (1927) introduction of autoregressive models, were devoted to the analysis of cyclical sunspot numbers and Whittle’s (1954) analysis of the water level in a rock channel on the Wellington coast of New Zealand is also related to a cyclical phenomenon. In fact many time series exhibit cyclical behavior in the sense there appears to be an approximate repetition of a pattern with a not very well defined period or amplitude. Rather both the period and amplitude appear to change gradually. Typically ecological data, air pollution data and several other physical phenomena exhibit such behavior. Two of the most familiar examples of time series which exhibit such behavior and which have been extensively analyzed are the Canadian lynx data, (for example see Campbell and Walker 1977, Tong 1977, Bhansali 1979 and Priestly 1981), and the Wolfer sunspot series, (Morris 1977, Tong 1983). Such series are frequently modeled by AR, ARMA or AR plus sinusoidal models. However, none of these modeling methods are very satisfactory for the prediction of more than one lead time, (Tong 1982). As pointed out in Akaike (1977b) in his discussion of several analyses of the Canadian lynx data, those analyses were unconvincing and that the critical issue in modeling time series is the selection of a proper model.