Recent technological advances have led to an exponential growth in the volume of data generated. The quest to make sense of these data, some of which are usually complex, has led to recent interest in development of statistical methods for analysing data with complex structures. One such field of interest is functional data analysis (FDA), which deals with the analysis of data that can be considered as functions, curves, or surfaces observed over a domain set. Outlier detection is a challenging but important part of the exploratory analysis process in FDA because functional observations can exhibit outlyingness in various ways compared to the bulk of the data. This thesis addresses the problem of detecting and classifying outliers in functional data with three main contributions.

First, the fdaoutlier R package is presented in Chapter 2. The package contains implementations of some of the state-of-the-art functional outlier detection methods in the literature. Some of the methods implemented include directional outlyingness, magnitude-shape plot, sequential transformations, total variation depth, and modified shape similarity index. Detailed illustrations of the functions of the package are provided, using various simulated and real functional datasets curated from the functional outlier detection literature. Overviews of the functional outlier detection methods implemented in the package are also presented in Chapter 2. This chapter therefore, serves as a review of some of the current literature in outlier detection for functional data.

Next, two new methods, named Semifast-MUOD and Fast-MUOD, are presented in Chapter 3. These methods work by computing for each curve three indices (magnitude, amplitude and shape index) that measure the outlyingness of that curve in terms of its magnitude, amplitude and shape. Semifast-MUOD computes these indices with respect to (w.r.t.) a random sample of the dataset, while Fast-MUOD computes these indices w.r.t. to the point-wise or L1 median. The classical boxplot is then used as a cutoff on the three indices to identify curves that are outliers of different types. A by-product of the methods is an unsupervised classification of the outliers into different types, without the need for visualisation. Performance evaluation of the methods, using various real and simulated datasets, shows that Fast-MUOD is the better of the two new proposed methods for outlier detection, in addition to being very scalable. Comparisons with latest functional outlier detection methods in the literature also show superior or comparable outlier detection performance.

In Chapter 4, some theoretical properties of the Fast-MUOD indices are presented. These include some definitions of the indices, as well as convergence proofs of the sample approximations. Some properties of the indices under simple transformations are also presented in this chapter. Finally, three techniques are presented in Chapter 5 for extending the Fast-MUOD indices to outlier detection in multivariate functional data observed on the same domain. These techniques include the use of random projections and identifying outliers on the marginal components of the multivariate functional data. The use of random projections showed the best result in performance evaluations with various real and simulated datasets.

Chapter 6 contains some concluding remarks and possible future research work.

About Oluwasegun Ojo

Oluwasegun Ojo is currently a Research Assistant in the Global Computing Group at IMDEA Networks Institute and a Mathematical Engineering PhD student at Carlos III University of Madrid. He completed a master’s degree in mathematical sciences at the African Institute for Mathematical Sciences (AIMS), Cameroon. Prior to that, he obtained his bachelor’s degree in statistics in 2014 from the Federal University of Technology, Akure, Nigeria. His current research interest is in the analysis of functional data.

PhD Thesis Advisor: Dr. Antonio Fernández Anta, IMDEA Networks Institute, and Dr. Rosa Elvira Lillo, UC3M

University: University Carlos III of Madrid, Spain

Doctoral Program: Mathematical Engineering

PhD Committee members:

• President: Francisco Javier Prieto Fernández, Professor, Universidad Carlos III de Madrid, Spain
• Secretary: Alba Franco Pereira, Professor, Universidad Complutense de Madrid, Spain
• Panel member: Fabian Scheipl, Lecturer, Ludwig Maximilian University of Munich, Germany