5. Modelling Stochastic Processes with Time Series Analysis

This chapter introduces the reader to the concept of time series analysis using transfer functions, state-space models, and spectral decomposition. Time series analysis is used to develop stochastic, or probabilistic, models. First, the theoretical properties of different model types, including standard autoregressive moving-average models, integrating models, and seasonal models, are examined and compared in both the time and frequency domains. The results obtained here can then be used to determine the appropriate model structure for a given data set. Spectral methods are also introduced at this point to assist in explaining various seasonal or periodic components in the data set. Next, the topic of parameter estimation is considered, and results are obtained for different methods and approaches, including the Yule–Walker for autoregressive models, the log-likelihood method for generalised autoregressive moving-average models, and the Kalman filter for state-space models. Finally, appropriate model validation methods are presented for time series analysis. Throughout this chapter, the Edmonton temperature data series is used to illustrate the concepts involved in time series analysis. By the end of the chapter, the reader should have a thorough understanding of the principles of time series analysis, including model structure determination, parameter estimation, and model validation.