Statistical learning of temporally dependent high- &amp; multi-dimensional data
Creators: name: Chen, Yi
Publisher: name: No Publisher Supplied
Year: 2018

description: The growing capabilities in generating and collecting data has risen unique opportunities and challenges in Statistics and the emerging field of Data Science. The availability of data with complex structure, such as temporal dependence and multi-dimensional, provides scientists with more accurate ways to characterize intricate natural or social phenomenon. This thesis deals with statistical models, methods, theory, and algorithms for learning low-rank structures from temporal-dependent multi-dimensional data, including time series with matrix observations, dynamic networks, and multivariate spatial-temporal data. We established a unified framework of modeling such data as matrix-variate time series that faithfully preserves the structural properties and the temporal dependencies that are intrinsic to the data. The focus is to achieve dimension reduction and learn the underlying latent low-rank structure of the data. The models presented in this thesis extend the matrix factor model proposed by cite{wang2017factor} in three directions to fully exploit the structures and properties of the observed data.     Specifically, the constrained matrix factor models provide a general framework for incorporating domain or prior knowledge in the matrix factor model through linear constraints. The proposed framework is shown to be useful in achieving parsimonious parameterization, gaining efficiency in statistical inference, facilitating interpretation of the latent matrix factor, and identifying specific factors of interest. The factor models for dynamic networks target at a special kind of matrix time series where, at each time point, the observation is a square adjacency matrix whose rows and columns represent the same set of actors in the network. Most available probability and statistical models for dynamic network data are deduced from random graph theory where the networks are characterized on the level of node and edge. Our high-level modeling of the dynamic networks as a time series of relational matrices is less restrictive and more scaleable to high-dimensional dynamic network data which is very common nowadays.    The factor models for multivariate spatial-temporal data are designed to accommodate the smooth functional behavior of the underlying spatial process. The functional matrix factor model aims to explicitly express discrete observations from spatial continuum in the form of a function. It has the advantage of generating models that can describe continuous smooth spatial changes, which then allows for accurate estimates of parameters, effective data noise reduction through curve/surface smoothing, and applicability to data with irregular spatial sampling.    The estimating methods are generally based on moment matching and spectral decomposition of matrices constructed from the empirical auto-cross-covariance of the time series, thus capturing the temporal dynamics presented in the data. The latent low-rank structures are learned directly from the data with little subjective input or any restricted distributional assumptions. For the functional matrix factor model, the functional loadings are approximated non-parametrically. The estimated latent states or factors are of smaller dimensions and can be used as data in second-stage inference and prediction. Theoretical properties of the estimators are established. Simulation studies are carried out to demonstrate the finite-sample performance of the proposed methods and their associated asymptotic properties. The proposed methods are applied to a wide range of real datasets, such as multinational macroeconomic indices data, dynamic global trading networks, and the Comprehensive Climate Dataset among others.