Statistical modeling and inference for multivariate functional responses

In recent decades, with rapid development in data collection and storage technology, multivariate functional data are increasingly common in various fields of study. For example, movement of a subject in a series of activities is monitored with accelerometers and gyroscopes in gait analysis, daily temperature, precipitation and other meteorological variables at a weather station are recorded, etc. One goal of this research project is to make inference about the underlying population based on such multivariate functional data at hand. When there may be sub-populations, testing equality of the corresponding vectors of mean functions, i.e. analysis of variance of the multivariate functional data, is a fundamental problem. Several tests have been proposed and studied to achieve this goal in the univariate case, but only a few methods have been developed in the multivariate case and them have their own limitations. We propose and study a new test that takes into account dependence between different functional observations on the same subject and heteroscedasticity in the matrices of covariance functions. Our test is based on asymptotic null distribution of the proposed test statistic, thus is computationally fast. Further, since the asymptotic null distribution is generally skewed, we utilize chi-square approximation which is superior to normal approximation in this case. Theoretical results will be derived to demonstrate asymptotic properties including consistency and power of our test. Extensive simulation studies will be conducted to examine finite sample performance of the proposed method and compare it with existing methods. The proposed method will be applied to analyze the above-mentioned motivating datasets to demonstrate its utility. In some cases, in addition to the group effect there may be some covariate effects and trend that are important in their own rights. For example, there may be a trend component if the multivariate functional data are collected along time. Another goal of this research project is to extend our model and testing procedure to accommodate the trend and covariate effects based on multivariate functional linear models or semiparametric models for multiple functional responses. While the studied problems emerge as the need to understand the complex and dynamic structures behind the multivariate functional responses, the literature so far on semiparametric regression for multivariate functional responses is limited. We will conduct a simulation study and apply the proposed method to analyze some multivariate functional time series data sets concerning climate change and traffic flow