UC Santa Cruz

Count time series are now widely encountered in practice. This dissertation contains three projects on count time series.

Our first project uses a recent advance in stationary count time series to develop a general seasonal count time series modeling paradigm. The model constructed here permits any marginal distribution for the series and the most flexible autocorrelations possible, including those with negative dependence. Likelihood methods of inference are explored. The project first develops the modeling methods, which entail a discrete transformation of a Gaussian process having seasonal dynamics. Properties of this model class are then established and particle filtering likelihood methods of parameter estimation are developed. A simulation study demonstrating the efficacy of the methods is presented and an application to the number of rainy days in successive weeks in Seattle, Washington is given.

Our second project reviews and compares popular methods that produce count time series having Poisson marginal distributions. The project begins by reviewing common ways that count series with Poisson marginal distributions can be produced. Statistical estimation methods are next discussed for some of the more worthy methods. Modeling nonstationary series with covariates motivates consideration of methods where the Poisson parameter depends on time. The methods are illustrated in the analysis of two series: 1) a count sequence of major hurricanes occurring in the North Atlantic Basin since 1970, and 2) the number of no-hitter games pitched in major league baseball since 1893.

Our third project develops a mathematical model and statistical methods to quantify trends in presence/absence observations of snow cover (not depths) and applies these in an analysis of Northern Hemispheric observations extracted from satellite flyovers during 1967-2021. A two-state Markov chain model with periodic dynamics is introduced to analyze changes in the data in a grid by grid fashion. Trends, converted to the number of weeks of snow cover lost/gained per century, are estimated for each study grid. Uncertainty margins for these trends are developed from the model and used to assess the significance of the trend estimates. Grids with questionable data quality are explicitly identified. Among trustworthy grids, snow presence is seen to be declining in almost twice as many cells as it is advancing. While Arctic and southern latitude snow presence is found to be rapidly receding, other locations, such as Eastern Canada, are experiencing advancing snow cover.