Due essentially to the difficulties associated with obtaining explicit forms of stationary marginal distributions of nonlinear stationary processes, appropriate characterizations of such processes are worked upon little. After discussing an elaborate motivation behind this thesis and presenting preliminaries in Chapter 1, we characterize, in Chapter 2, the stationary marginal distributions of certain nonlinear multivariate stationary processes. To do so, we show that the stationary marginal distributions of these processes belong to specific skewdistribution families, and for a given skewdistribution from the corresponding family, a process, with stationary marginal distribution identical to that given skewdistribution, can be found. While conventional time series analysis greatly depends on the assumption of stationarity, measurements taken from many physical systems, which consist of both periodicity and randomness, often exhibit cyclostationarity (i.e., a periodic structure in their first and secondorder moments). Identifying the hourly global horizontal irradiances (GHIs), collected at a solar monitoring station of Saudi Arabia, as a cyclostationary process and considering the significant impact of that on the energy production in Saudi Arabia, Chapter 3 provides a temporal model of GHIs. Chapter 4 extends the analysis to a spatiotemporal cyclostationary modeling of 45 different solar monitoring stations of the Kingdom. Both the proposed models are shown to produce better forecasts, more realistic simulations, and reliable photovoltaic power estimates in comparison to a classical model that fails to recognize the GHI data as cyclostationary. Chapter 5 extends the notion of cyclostationarity to a novel and flexible class of processes, coined evolving period and amplitude cyclostationary (EPACS) processes, that allows periods and amplitudes of the mean and covariance functions to evolve and, therefore, accommodates a much larger class of processes than the cyclostationary processes. Thereafter, we investigate its properties, provide methodologies for statistical inference, and illustrate the presented methods using a simulation study and a real data example, from the heavens, of the magnitudes of the light emitted from the variable star R Hydrae. Finally, Chapter 6 summarizes the findings of the thesis and discusses its significance and possible future extensions.
Date of Award  Jul 10 2021 

Original language  English (US) 

Awarding Institution   Computer, Electrical and Mathematical Science and Engineering


Supervisor  Marc Genton (Supervisor) 

 cyclostationarity
 nonstationarity
 periodically correlated
 solar irradiance
 stationary distribution
 time series