Wei Wu, Ph.D. - US grants

The proposal aims to advance statistical theory for random
processes that exhibit features like long-range dependence and
nonlinearities, and to educate both statisticians and others
scientists in this new exciting area. Compared to the
well-developed theory under the independence assumption, it is
considerably more challenging to establish a limit theory for
processes with such features. The Principal Investigator proposes
a powerful martingale based method and studies spectral
estimation, empirical processes, nonparametric estimation and
other related asymptotic problems for such processes.

Processes with long-range dependence and nonlinearities occur in
various fields, including computer networks, communication,
finance, geology, hydrology, econometrics and atmospheric science
among others. Applications of the research results developed in
the proposal would help test and justify claims made by scientists
in such fields. In particular, the PI develops statistical
methodology to identify trends in temperature and ozone sequences
and provides statistical reasoning for meteorologists' claims on
climate change patterns.

The investigator studies estimation, testing and construction of simultaneous confidence regions for time-varying parameters in models with non-stationary and dependent errors. The simultaneous confidence regions can be used to test patterns of associations between covariates and responses. The investigator also develops new tools for asymptotic analysis of non-stationary time series.

During the last decade, models with time-varying data generating mechanisms have gained substantial attention in various fields including economics, finance, engineering, sociology, medical science, environmental science among others. Results from the proposal are useful for understanding the dynamic association between explanatory variables and responses for temporally observed data in which the underlying physical mechanism change with respect to time.

The goal of this project is to establish a systematic asymptotic theory for estimates of large dimensional covariance matrices in time series; a fundamental problem in high-dimensional inference. In particular, the investigator plans to study properties of sample covariances and sample covariance matrices for stationary processes; deal with consistent estimation of covariance matrices of stationary processes and its applications in prediction and other problems; and explore non-Gaussian features of random processes by estimating higher order cumulant tensors.

Covariance matrices play a fundamental role in various fields including environmental science, engineering, economics and finance. Estimation of covariance matrices is needed in analyzing, testing, monitoring and predicting of seismic, economic and financial and other time series. Results developed from this project can provide a theoretical foundation for estimation of covariance matrices and can potentially improve time series processing algorithms that are used in various applications.

During the last decade, analysis of high-dimensional independent data has gained substantial attention. The project involves a variety of estimation and statistical inference problems for high-dimensional multiple time series, a universal type of data seen in a broad spectrum of real applications in spatio-temporal statistics, biomedical engineering, environmental science, finance, and signal processing. As an important research problem, one should extract information from a large number of time series, where the second order structure plays a fundamental role in those applications.

Results developed from this project will provide the theoretical foundation for estimating and inference of the space-time covariance and precision matrix, their related functionals, and time-varying graphs of high-dimensional time series. All of the problems are linked together to characterize the second order properties of the high-dimensional time series with the non-linear and non-stationary time dependent features. We will also study enhanced methods that account for the temporal and spatial dependence structures. Results from this research are useful for understanding the dynamic features of high-dimensional dependent data. In particular, the techniques are applicable to biomedical engineering problems such as modeling brain connectivity networks by using fMRI data.

The information era has witnessed an explosion in the collection of high dimensional time series data across a wide range of areas, including finance, signal processing, neuroscience, meteorology, seismology, among others. For low dimensional time series, there is a well-developed estimation and inference theory. Inference theory in the high dimensional setting is of fundamental importance and has wide applications, but has been rarely studied. Researchers face a number of challenges in solving real-world problems: (i) complex dynamics of data generating systems, (ii) temporal and cross-sectional dependencies, (iii) high dimensionality and (iv) non-Gaussian distributions. The goal of this project is to develop and advance inference theory for high dimensional time series data by concerning all the above characteristics. The project will provide training to graduate students and publicly avaialble statistical packages.

This project involves developing a systematic asymptotic theory for estimation and inference for high dimensional time series, including parameter estimation, construction of simultaneous confidence intervals, prediction, model selection, Granger causality test, hypothesis testing, and spectral domain estimation. To this end, a new methodology for the estimation of parameters and second-order characteristics for high dimensional time series will be proposed. New tools and concentration inequalities for the asymptotic analysis of high-dimensional time series will be developed. To perform simultaneous inference and significance testing, the PIs will investigate the very deep Gaussian approximation problem and the high dimensional central limit theorems by taking both high dimensionality and temporal and cross-sectional dependencies into account.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

In the mobile and big data era, data on human mobility and interaction in both physical space and virtual space are pervasively available. The study of human dynamics with the assistance of big data analytics becomes a timely effort. The outcome from this study helps understand how human activities change over time and how they may change the environment, economy, and politics. At the micro scale, research on communities, influence propagation, anomaly detection, and mobility prediction can benefit marketing research, mitigate crimes, as well as mitigate and contain epidemics. Therefore, this project will advance not only mathematics and statistics, but also many other fields including human geography, business, and public health.

The project aims to analyze multi-relational data in large spatiotemporal datasets, and covers a broad range of topics pertaining to the study of human dynamics, including anomaly detection, trend discovery, hidden community detection, pattern mining, and role prediction, etc. The types of data analysis covers statistical inference on both unstructured data and structured data that are supported on a graph. The work includes four major thrusts: 1) latent network estimation from non-stationary time series, 2) online change-point detection and synchronization testing for high-dimensional time series, 3) multi-relational data analysis based on tensor factorization and validity testing, and 4) spatial and spectral analysis of graph signals. These research projects will contribute to not only time series analysis, tensor analysis, and graph signal processing, but also machine learning from large spatiotemporal datasets. The synergy between the three areas and machine learning enables powerful methodologies for modeling multi-relational data and mining data defined on both regular and irregular structures. This research will result in theoretical foundations underpinning time series and dynamic complex networks as well as practical software tools for a broad range of applications.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.