Methods in Functional Data Analysis: Forecast Evaluation, Robust Serial Dependence Measures, and a Spatial Factor Copula Model

Abstract

With advancements in technology, new types of data have become available, including functional data, which observations in the form of functions or curves rather than scalar or vector-valued quantities. This emerging area presents unique challenges in handling intrinsically infinite-dimensional objects. In this thesis, we primarily focus on three problems, each of which has a distinct flavour of functional data analysis.

In Chapter 1, we provide an overview of the foundational concepts and methodologies that will serve as a basis for the subsequent chapters. This includes an exploration of topics such as functional data analysis, functional time series analysis, probabilistic forecasts, copula modelling, and robust methods that will be in later chapters. Additionally, we conclude this chapter by presenting a comprehensive list of the main contributions made by this thesis.

In Chapter 2, motivated by the goal of evaluating real-time forecasts of home team win probabilities in the National Basketball Association, we develop new tools for measuring the quality of continuously updated probabilistic forecasts. This includes introducing calibration surface plots, and simple graphical summaries of them, to evaluate at a glance whether a given continuously updated probability forecasting method is well-calibrated, as well as developing statistical tests and graphical tools to evaluate the skill, or relative performance, of two competing continuously updated forecasting methods. These tools are demonstrated in an application on evaluating the continuously updated forecasts published by United States-based multinational sports network ESPN on its principle webpage espn.com. This application provides statistical evidence that the forecasts published there are well-calibrated, and exhibit improved skill over several naïve models, but do not show significantly improved skill over simple logistic regression models based solely on a measurement of each teams’ relative strength, and the evolving score difference throughout the game.

In Chapter 3, we propose a new autocorrelation measure for functional time series that we term “spherical autocorrelation.” It is based on measuring the average angle between lagged pairs of series after having been projected onto a unit sphere. This new measure enjoys at least two complimentary advantages compared to existing autocorrelation measures for functional data, since it both 1) describes a notion of “sign” or “direction” of serial dependence in the series, and 2) is more robust to outliers. The asymptotic properties of estimators of the spherical autocorrelation are established, and used to construct confidence intervals and portmanteau white noise tests. These confidence intervals and tests are shown to be effective in simulation experiments, and in applications model selection for daily electricity price curves, and in measuring volatility in densely observed asset price data.

In Chapter 4, we propose a new model for spatial functional data that departs from the commonly adopted assumption of normality of the errors. Instead, we assume the existence of a common process that equally affects the measurements of the data at all locations at each time point. By using general copulas, our model can accommodate heavy tails and tail asymmetry, which the existing methods may suffer from. We then derive the closed-form expression of the likelihood function when the tail dependence is generated by an exponential distribution. The simulation studies show that the parameter estimates of the proposed method accurately capture the spatial and temporal dependence when the model is correctly specified. In the case where the model is misspecified, our method is still robust in capturing the spatial dependence and the general shape of the common mean function. We close the chapter by discussing some future works and potential extensions of the proposed model.

We conclude this thesis by presenting concise summaries of each chapter and engaging in further discussions in Chapter 5. Additionally, we also offer directions for future research in each chapter, highlighting potential applications of the proposed methods. Furthermore, we explore theoretical and computational avenues that may prove beneficial to practitioners and researchers, extending the scope of the proposed methods to encompass research, applications, and beyond.