A kernel-enriched order-dependent nonparametric spatio-temporal process

Spatio-temporal processes are necessary modeling tools for various environmental, biological, and geographical problems. The underlying model is commonly considered to be parametric and to be a Gaussian process. Additionally, the covariance function is expected to be stationary and separable. This structure need not always be realistic. Moreover, attempts have been made to construct nonparametric processes of neither stationary nor separable covariance functions. Nevertheless, as we elucidate, some desirable and necessary spatio-temporal properties are not guaranteed by the existing approaches, thus, calling for further innovative ideas. In this article, using kernel convolution of order-based dependent Dirichlet process, we construct a novel spatio-temporal model. We show that this satisfies desirable properties and includes the stationary, separable, parametric processes as special cases. Our resultant posterior distribution is variable dimensional, which we attack using Transdimensional Transformation based Markov Chain Monte Carlo, which can update all the variables and change dimensions using deterministic transformations of a random variable drawn from some arbitrary density defined on relevant support. We demonstrate our model’s performance on simulated and real data sets. In all situations, the findings are highly encouraging

Statistical and stochastic post-processing of regional climate model data: copula-based downscaling, disaggregation and multivariate bias correction

In order to delineate management or climate change adaptation strategies for natural or technical water bodies, impact studies are necessary. To this end, impact models are set up for a given region which requires time series of meteorological data as driving data. Regional climate models (RCMs) are capable of simulating gridded data sets of several meteorological variables. The advantages over observed data are that the time series are complete and that meteorological information is also provided for ungauged locations. Furthermore, climate change impact studies can be conducted by driving the simulations with different forcing variables for future periods. While the performance of RCMs generally increases with a higher spatio-temporal resolution, the computational and storage demand increases non-linearly which can impede such highly resolved simulations in practice. Furthermore, systematic biases of the univariate distributions and multivariate dependence structures are a common problem of RCM simulations on all spatio-temporal scales. Depending on the case study, meteorological data must fulfill different criteria. For instance, the spatio-temporal resolution of precipitation time series should be as fine as 1 km and 5 minutes in order to be used for urban hydrological impact models. To bridge the gap between the demands of impact modelers and available meteorological RCM data, different computationally efficient statistical and stochastic post-processing techniques have been developed to correct the bias and to increase the spatio-temporal resolution. The main meteorological variable treated in this thesis is precipitation due to its importance for hydrological impact studies. The models presented in this thesis belong to the classes of bias correction, downscaling and temporal disaggregation techniques. The focus of the developed methods lies on multivariate copulas. Copulas constitute a promising modeling approach for highly-skewed and mixed discrete-continuous variables like precipitation since the marginal distribution is treated separately from the dependence structure. This feature makes them useful for the modeling of different meteorological variables as well. While copulas have been utilized in the past to model precipitation and other meteorological variables that are relevant in hydrology, applications to RCM simulations are not very common. The first method is a geostatistical estimation technique for distribution parameters of daily precipitation for ungauged locations, so that a bias correction with Quantile Mapping can be performed. The second method is a spatial downscaling of coarse scale RCM precipitation fields to a finer resolved domain. The model is based on the Gaussian Copula and generates ensembles of daily precipitation fields that resemble the precipitation fields of fine scale RCM simulations. The third method disaggregates hourly precipitation time series simulated by an RCM to a resolution of 5 minutes. The Gaussian Copula was utilized to condition the simulation on both spatial and temporal precipitation amounts to respect the spatio-temporal dependence structure. The fourth method is an approach to simulate a meteorological variable conditional on other variables at the same location and time step. The method was developed to improve the inter-variable dependence structure of univariately bias corrected RCM simulations in an hourly resolution

Local-Aggregate Modeling for Big-Data via Distributed Optimization: Applications to Neuroimaging

Technological advances have led to a proliferation of structured big data that have matrix-valued covariates. We are specifically motivated to build predictive models for multi-subject neuroimaging data based on each subject's brain imaging scans. This is an ultra-high-dimensional problem that consists of a matrix of covariates (brain locations by time points) for each subject; few methods currently exist to fit supervised models directly to this tensor data. We propose a novel modeling and algorithmic strategy to apply generalized linear models (GLMs) to this massive tensor data in which one set of variables is associated with locations. Our method begins by fitting GLMs to each location separately, and then builds an ensemble by blending information across locations through regularization with what we term an aggregating penalty. Our so called, Local-Aggregate Model, can be fit in a completely distributed manner over the locations using an Alternating Direction Method of Multipliers (ADMM) strategy, and thus greatly reduces the computational burden. Furthermore, we propose to select the appropriate model through a novel sequence of faster algorithmic solutions that is similar to regularization paths. We will demonstrate both the computational and predictive modeling advantages of our methods via simulations and an EEG classification problem.Comment: 41 pages, 5 figures and 3 table

A new class of multiscale lattice cell (MLC) models for spatio-temporal evolutionary image representation

Spatio-temporal evolutionary (STE) images are a class of complex dynamical systems that evolve over both space and time. With increased interest in the investigation of nonlinear complex phenomena, especially spatio-temporal behaviour governed by evolutionary laws that are dependent on both spatial and temporal dimensions, there has been an increased need to investigate model identification methods for this class of complex systems. Compared with pure temporal processes, the identification of spatio-temporal models from observed images is much more difficult and quite challenging. Starting with an assumption that there is no apriori information about the true model but only observed data are available, this study introduces a new class of multiscale lattice cell (MLC) models to represent the rules of the associated spatio-temporal evolutionary system. An application to a chemical reaction exhibiting a spatio-temporal evolutionary behaviour, is investigated to demonstrate the new modelling framework

Bayesian Recurrent Neural Network Models for Forecasting and Quantifying Uncertainty in Spatial-Temporal Data

Recurrent neural networks (RNNs) are nonlinear dynamical models commonly used in the machine learning and dynamical systems literature to represent complex dynamical or sequential relationships between variables. More recently, as deep learning models have become more common, RNNs have been used to forecast increasingly complicated systems. Dynamical spatio-temporal processes represent a class of complex systems that can potentially benefit from these types of models. Although the RNN literature is expansive and highly developed, uncertainty quantification is often ignored. Even when considered, the uncertainty is generally quantified without the use of a rigorous framework, such as a fully Bayesian setting. Here we attempt to quantify uncertainty in a more formal framework while maintaining the forecast accuracy that makes these models appealing, by presenting a Bayesian RNN model for nonlinear spatio-temporal forecasting. Additionally, we make simple modifications to the basic RNN to help accommodate the unique nature of nonlinear spatio-temporal data. The proposed model is applied to a Lorenz simulation and two real-world nonlinear spatio-temporal forecasting applications