On a General Class of Count Time Series Models

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This research studies data recorded in time that is count-valued, e.g., the annual number of Californian wildfires, yearly Alaskan polar bear sightings, monthly flu deaths, annual North Atlantic severe hurricanes, etc. The data may be correlated in time, implying that counts observed today may be influenced by (correlated with) counts occurring in the immediate past. This work develops time series methods that take into account the particular type of distribution appropriate for the counts, e.g., Poisson, geometric, binomial, etc.. Using the correct distribution facilitates accurate forecasts and inferences. The models developed here allow both positive and negative correlations, a feature absent from current statistical count time series models. For example, annual Pacific and Atlantic hurricane counts are well described by a Poisson-type distribution, but are negatively correlated: when Atlantic hurricane counts are high, Pacific counts tend to be low (and vice versa). It is important to account for such correlations to make accurate climate change conclusions.On technical levels, a discrete-time renewal process is used as a general model for a binary (zero-one) series. The renewal process is rendered stationary by selecting a special initial renewal life length. Copies of the stationary binary sequence are then superimposed in various ways to build the marginal distribution sought. It is easy to construct stationary series with Poisson, geometric, and binomial marginal distributions. The methods easily generate stationary count series with negative correlations and/or long-memory --- aspects not achievable from classical integer autoregressive moving-average count techniques. Inference issues, covariates, and multivariate series are also considered.