Multivariate spatio-temporal modeling and simulation

Abstract

In view of multivariate nature of general spatio-temporal data sets observed in various disciplines such as meteorology, engineering and medical science, we are interested in gaining some scientific insight into the underlying stochastic processes, which incorporate the complex dependence structure among different variables at different locations over time. To this end, this dissertation studies the methods and applications of the multivariate modelling, simulation and missing value imputation of data sets in space and time under different settings. Chapter I, Space-time data sets are often multivariate and collected at monitored discrete time lags, which are usually viewed as a component of time series in environmental science and related areas. Valid and practical covariance models are needed to characterize geostatistical formulations of these types of data sets in a wide range of applications. We propose several classes of multivariate spatio-temporal functions to characterize underlying random fields whose discrete temporal margins are some celebrated autoregressive and moving average (ARMA) models, and obtain sufficient and/or necessary conditions for them to be valid covariance matrix functions. The possibility of taking advantage of well-established time series and spatial statistics tools makes it relatively straightforward to identify and fit the proposed models in practice. Finally, applications of the proposed multivariate covariance matrix functions are illustrated on Kansas weather data in terms of co-kriging, compared with some traditional space-time models for prediction. Chapter II, We propose an efficient method for simulating multivariate spatio-temporal data on a compact two-point homogeneous space with sphere as a special case. These large scale global data sets are obtained based on truncating the series expansion of multivariate spatio-temporal random fields on this space. The algorithm can be boiled down to simply simulate a uniformly distributed random vector on a sphere, on which the great circle distance is defined. Multiple covariance models are compared to fit the simulated multivariate space-time data including the model proposed in the Chapter I. The simulation studies suggest some guideline for choosing appropriate models and parameterizations for different multivariate data in space and time. Chapter III, Motivated by dealing with incompleteness of energy data to study network situational awareness, we proposed a spatial Gaussian process variational autoencoders (GP-VAE) to impute the corrupted or missing information based on multivariate Gaussian processes and Bayesian deep learning. The missing data is learned by projecting the multivariate data space including both space and time into a lower dimensional latent space without missingness, where the low dimensional dynamics is modeled with vector Gaussian time series. Model comparison has been made with traditional data imputation method in literature on a simulated energy data set from smart meters, solar inverters, grid automation/SCADA sensors and micro PMU on a geographical lattice over time.