Multivariate Functional Data Analysis and Visualization

  • Zhuo Qu

Student thesis: Doctoral Thesis


As a branch of statistics, functional data analysis (FDA) studies observations regarded as curves, surfaces, or other objects evolving over a continuum. Although one has seen a flourishing of methods and theories on FDA, two issues are observed. Firstly, the functional data are sampled from common time grids; secondly, methods developed only for univariate functional data are challenging to be applied to multivariate functional data. After exploring model-based fitting for regularly observed multivariate functional data, we explore new visualization tools, clustering, and multivariate functional depths for irregularly observed (sparse) multivariate functional data. The four main chapters that comprise the dissertation are organized as follows. First, median polish for functional multivariate analysis of variance (FMANOVA) is proposed with the implementation of multivariate functional depths in Chapter 2. Numerical studies and environmental datasets are considered to illustrate the robustness of median polish. Second, the sparse functional boxplot and the intensity sparse functional boxplot, as practical exploratory tools that make visualization possible for both complete and sparse functional data, are introduced in Chapter 3. These visualization tools depict sparseness characteristics in the proportion of sparseness and relative intensity of fitted sparse points inside the central region, respectively. Third, a robust distance-based robust two-layer partition (RTLP) clustering of sparse multivariate functional data is introduced in Chapter 4. The RTLP clustering is based on our proposed elastic time distance (ETD) specifically for sparse multivariate functional data. Lastly, the multivariate functional integrated depth and the multivariate functional extremal depth based on multivariate depths are proposed in Chapter 5. Global and local formulas for each depth are explored, with theoretical properties being proved and the finite sample depth estimation for irregularly observed multivariate functional data being investigated. In addition, the simplified sparse functional boxplot and simplified intensity sparse functional boxplot for visualization without data reconstruction are introduced. Together, these four extensions to multivariate functional data make them more general and of applicational interest in exploratory multivariate functional data analysis.
Date of AwardNov 2022
Original languageEnglish (US)
Awarding Institution
  • Computer, Electrical and Mathematical Sciences and Engineering
SupervisorMarc Genton (Supervisor)


  • clustering
  • median polish
  • multivariate functional data
  • outlier detection
  • robustness
  • visualization

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