Exponential Smoothing for Forecasting and Bayesian Validation of Computer Models
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Despite their success and widespread usage in industry and business, ES methods have received little attention from the statistical community. We investigate three types of statistical models that have been found to underpin ES methods. They are ARIMA models, state space models with multiple sources of error (MSOE), and state space models with a single source of error (SSOE). We establish the relationship among the three classes of models and conclude that the class of SSOE state space models is broader than the other two and provides a formal statistical foundation for ES methods. To better understand ES methods, we investigate the behaviors of ES methods for time series generated from different processes. We mainly focus on time series of ARIMA type. ES methods forecast a time series using only the series own history. To include covariates into ES methods for better forecasting a time series, we propose a new forecasting method, Exponential Smoothing with Covariates (ESCov). ESCov uses an ES method to model what left unexplained in a time series by covariates. We establish the optimality of ESCov, identify SSOE state space models underlying ESCov, and derive analytically the variances of forecasts by ESCov. Empirical studies show that ESCov outperforms ES methods and regression with ARIMA errors. We suggest a model selection procedure for choosing appropriate covariates and ES methods in practice. Computer models have been commonly used to investigate complex systems for which physical experiments are highly expensive or very time-consuming. Before using a computer model, we need to address an important question ``How well does the computer model represent the real system?" The process of addressing this question is called computer model validation that generally involves the comparison of computer outputs and physical observations. In this thesis, we propose a Bayesian approach to computer model validation. This approach integrates together computer outputs and physical observation to give a better prediction of the real system output. This prediction is then used to validate the computer model. We investigate the impacts of several factors on the performance of the proposed approach and propose a generalization to the proposed approach.