Institute of Transport Studies, University of Leeds
Abstract
As part of a SERC funded project investigating outlier detection and replacement with transport data, univariate Box-Jenkins (1976) models have already been successfully applied to traffic count series (see Redfern et al, 1992). However, the underlying assumption of normality for ARIMA models implies they are not ideally suited for time series exhibiting certain behavioural characteristics. The limitations of ARIMA models are discussed in some detail by Tong (1983), including problems with time irreversibility, non-normality, cyclicity and asymmetry. Data with irregularly spaced extreme values are unlikely to be modelled well by ARIMA models, which are better suited to data where the probability of a very high value is small. Tong (1983) argues that one way of modelling such non-normal behaviour might be to retain the general ARIMA framework and allow the white noise element to be non-gaussian. As an alternative he proposes abandoning the linearity assumption and defines a group of non linear structures, one of which is the Self-Exciting Threshold Autoregressive (SETAR) model. The model form is described in more detail below but basically consists of two (or more) piecewise linear models, with the time series "tripping" between each model according to its value with respect to a threshold point. The model is called "Self-Exciting" because the indicator variable determining the appropriate linear model for each piece of data is itself a function of the data series. Intuitively this means the mechanism driving the alternation between each model form is not an external input such as a related time series (other models can be defined where this exists), but is actually contained within the series itself. The series is thus Self-Exciting.
The three concepts embedded within the SETAR model structure are those of the threshold, limit cycle and time delay, each of which can be illustrated by the diverse applications such models can take.
The threshold can be defined as some point beyond which, if the data falls, the series structure changes inherently and so an alternative linear model form would be appropriate. In hydrology this is seen as the non-linearity of soil infiltration, where at the soil saturation point (threshold) a new model for infiltration would become appropriate.
Limit cycles describe the stable cyclical phenomena which we sometimes observe within time series. The cyclical behaviour is stationary, ie consists of regular, sustained oscillations and is an intrinsic property of the data. The limit cycle phenomena is physically observable in the field of radio-engineering where a triode valve is used to generate oscillations (see Tong, 1983 for a full description). Essentially the triode value produces self-sustaining oscillations between emitting and collecting electrons, according to the voltage value of a grid placed between the anode and cathode (thereby acting as the threshold indicator).
The third essential concept within the SETAR structure is that of the time delay and is perhaps intuitively the easiest to grasp. It can be seen within the field of population biology where many types of non-linear model may apply. For example within the cyclical oscillations of blowfly population data there is an inbuilt "feedback" mechanism given by the hatching period for eggs, which would give rise to a time delay parameter within the model. For some processes this inherent delay may be so small as to be virtually instantaneous and so the delay parameter could be omitted.
In general time series Tong (1983) found the SETAR model well suited to the cyclical nature of the Canadian Lynx trapping series and for modelling riverflow systems (Tong, Thanoon & Gudmundsson, 1984). Here we investigate their applicability with time series traffic counts, some of which have exhibited the type of non-linear and cyclical characteristics which could undermine a straightforward linear modelling process
