Functional data analysis, which deals with a sample of functions or curves, plays an important role in modern data analysis. Nowadays in the era of "Big Data", multidimensional and multivariate functional data are becoming increasingly common, especially in biological, medical, and engineering applications, but the large dimension and complex structure of these data pose significant challenges. 

This project has substantially narrowed the gap between the increasing demand for handling multidimensional and multivariate functional data in practice, and the insufficient development of statistical methods and computational tools. This research has advanced mathematical and computational statistics, and the project outcomes have been applied to neuroimaging, oceanographic, and traffic data. This project has focused on three specific sub-projects:  

1. Covariance function estimation for multidimensional functional data: Covariance function estimation is regarded as a crucial step in functional data analysis. However, for multidimensional functional data, there are very few existing estimators due to the fundamental difficulty of multi-dimensionality. We proposed an efficient yet widely applicable modeling which addresses this. Not only do the resulting techniques provide powerful covariance estimators, but they also serve as a building block for subsequent statistical techniques for multidimensional functional data.

2. Independence tests for multivariate functional data: Independence tests are useful statistical tools for detecting relationships among different attributes. When these attributes are functions, we obtain a multivariate functional dataset, e.g., multiple signals measured over time (regarded as functions) from different human brain regions when the corresponding subject is performing a particular task. The corresponding relationship between these brain regions is known as functional connectivity. We proposed a novel and powerful test that can identify the dependency for multivariate functional data. Our work is empirically shown to be useful in detecting functional connectivity patterns.

3. Autoregressive models for multivariate functional time series: Historically functional data are collected from independent sources. Nowadays many functional data are recorded in a sequential manner, called "multivariate functional time series" of which measurements of a few variables are naturally separated by consecutive time intervals, e.g., hourly maximum temperature and hourly maximum wind speed in many days. Since functional time series often arise from a single source, the dependency between curves is usually expected and it is of interest to model this dependency by predicting future measurements using the history. Most existing techniques are based on a truncated procedure whose discrete nature often adversely affects the statistical performance. We developed an estimation that avoids this discrete procedure to improve estimation accuracy, computational stability and predictability.

Graduate students, including women, have participated in this project. They have learned advanced computational and mathematical skills. Some of the above sub-projects have naturally become part of their dissertations. The project outcomes have been disseminated in seminars and conferences. This project has also produced peer-reviewed publications and publicly accessible computer code and packages.

Last Modified: 10/12/2020
Modified by: Xiaoke Zhang