Further developments of two point process models for fine-scale time series : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Statistics at Massey University, Albany (Auckland), New Zealand

Abstract

Two point processes, the Autoregressive Conditional Duration (ACD) model and the Bartlett-Lewis Pulse (BLP) model, are further developed and used to model fine scale time series. Six different ACD models are specified and fitted to two sets of real stock transaction data. The Akaike Information Criterion (AIC), Takeuchi Information Criterion (TIC) and Generalized Information Criterion (GIC) are the information-theoretic criteria for model evaluation. This is the first time that ACD models have been evaluated and ranked based on the information-theoretic criteria, and makes the comparison of ACD models with different autoregressive and error structures straightforward. A newly proposed ACD model with a mixed lognormal-gamma error term distribution is identified as the best model with minimum predictive error. The original BLP model was developed and fitted to 60 years of 5-minute rainfall series from Kelburn, a place near Wellington, New Zealand, by Cowpertwait et al. (2007). The BLP model can be used to produce realistic simulation samples of fine scale rainfall series that are required in many applications, e.g. urban drainage system design. Following the continuous distributions of storm types approach as first proposed by Cowpertwait (2010), a more general BLP process characterization framework is formulated under which the original BLP model can be recovered as a special case. Statistical properties up to third order are derived for the BLP model characterized by continuous distributions of storm types. Without an increase in model parameters, a modified BLP model is specified, in which a conditional mean exponential distribution is used to represent the pulse depths distribution and a continuum of storm types within a process is assumed. Simulation studies show that the modified model improves the fit to the observed proportion of dry periods significantly, whilst retaining a good fit to moment properties. Also a better fit is obtained to the annual extreme rainfall values at the 5 minutes and 12 hours aggregation levels whilst retaining a good fit at other aggregation levels, and significant improvement is achieved in the goodness-of-fit to extremes for the individual months. The improvements of the original BLP model’s performance are mainly due to the successful implementation of the within cell pulse depths dependence structure using a conditional mean exponential distribution.