Huiyan Sang

Hierarchical modeling for extreme values observed over space and time

We propose a hierarchical modeling approach for explaining a collection of spatially referenced time series of extreme values. We assume that the observations follow generalized extreme value (GEV) distributions whose locations and scales are jointly spatially dependent where the dependence is captured using multivariate Markov random field models specified through coregionalization. In addition, there is temporal dependence in the locations. There are various ways to provide appropriate specifications; we consider four choices. The models can be fitted using a Markov Chain Monte Carlo (MCMC) algorithm to enable inference for parameters and to provide spatio–temporal predictions. We fit the models to a set of gridded interpolated precipitation data collected over a 50-year period for the Cape Floristic Region in South Africa, summarizing results for what appears to be the best choice of model.

Improving the performance of predictive process modeling for large datasets

Advances in Geographical Information Systems (GIS) and Global Positioning Systems (GPS) enable accurate geocoding of locations where scientific data are collected. This has encouraged collection of large spatial datasets in many fields and has generated considerable interest in statistical modeling for location-referenced spatial data. The setting where the number of locations yielding observations is too large to fit the desired hierarchical spatial random effects models using Markov chain Monte Carlo methods is considered. This problem is exacerbated in spatial-temporal and multivariate settings where many observations occur at each location. The recently proposed predictive process, motivated by kriging ideas, aims to maintain the richness of desired hierarchical spatial modeling specifications in the presence of large datasets. A shortcoming of the original formulation of the predictive process is that it induces a positive bias in the non-spatial error term of the models. A modified predictive process is proposed to address this problem. The predictive process approach is knot-based leading to questions regarding knot design. An algorithm is designed to achieve approximately optimal spatial placement of knots. Detailed illustrations of the modified predictive process using multivariate spatial regression with both a simulated and a real dataset are offered.

Covariance approximation for large multivariate spatial data sets with an application to multiple climate model errors

This paper investigates the cross-correlations across multiple climate model errors. We build a Bayesian hierarchical model that accounts for the spatial dependence of individual models as well as cross-covariances across different climate models. Our method allows for a nonseparable and nonstationary cross-covariance structure. We also present a covariance approximation approach to facilitate the computation in the modeling and analysis of very large multivariate spatial data sets. The covariance approximation consists of two parts: a reduced-rank part to capture the large-scale spatial dependence, and a sparse covariance matrix to correct the small-scale dependence error induced by the reduced rank approximation. We pay special attention to the case that the second part of the approximation has a block-diagonal structure. Simulation results of model fitting and prediction show substantial improvement of the proposed approximation over the predictive process approximation and the independent blocks analysis. We then apply our computational approach to the joint statistical modeling of multiple climate model errors.

Adaptive Bayesian Nonstationary Modeling for Large Spatial Datasets Using Covariance Approximations

Gaussian process models have been widely used in spatial statistics but face tremendous modeling and computational challenges for very large nonstationary spatial datasets. To address these challenges, we develop a Bayesian modeling approach using a nonstationary covariance function constructed based on adaptively selected partitions. The partitioned nonstationary class allows one to knit together local covariance parameters into a valid global nonstationary covariance for prediction, where the local covariance parameters are allowed to be estimated within each partition to reduce computational cost. To further facilitate the computations in local covariance estimation and global prediction, we use the full-scale covariance approximation (FSA) approach for the Bayesian inference of our model. One of our contributions is to model the partitions stochastically by embedding a modified treed partitioning process into the hierarchical models that leads to automated partitioning and substantial computational benefits. We illustrate the utility of our method with simulation studies and the global Total Ozone Matrix Spectrometer (TOMS) data. Supplementary materials for this article are available online.

INTERPRETING SELF-ORGANIZING MAPS THROUGH SPACE-TIME DATA MODELS

Self-organizing maps (SOMs) are a technique that has been used with high-dimensional data vectors to develop an archetypal set of states (nodes) that span, in some sense, the high-dimensional space. Noteworthy applications include weather states as described by weather variables over a region and speech patterns as characterized by frequencies in time. The SOM approach is essentially a neural network model that implements a nonlinear projection from a high-dimensional input space to a low-dimensional array of neurons. In the process, it also becomes a clustering technique, assigning to any vector in the high-dimensional data space the node (neuron) to which it is closest (using, say, Euclidean distance) in the data space. The number of nodes is thus equal to the number of clusters. However, the primary use for the SOM is as a representation technique, that is, finding a set of nodes which representatively span the high-dimensional space. These nodes are typically displayed using maps to enable visualization of the continuum of the data space. The technique does not appear to have been discussed in the statistics literature so it is our intent here to bring it to the attention of the community. The technique is implemented algorithmically through a training set of vectors. However, through the introduction of stochasticity in the form of a space-time process model, we seek to illuminate and interpret its performance in the context of application to daily data collection. That is, the observed daily state vectors are viewed as a time series of multivariate process realizations which we try to understand under the dimension reduction achieved by the SOM procedure. The application we focus on here is to synoptic climatology where the goal is to develop an array of atmospheric states to capture a collection of distinct circulation patterns. In particular, we have daily weather data observed in the form of 11 variables measured for each of 77 grid cells yielding an 847 x 1 vector for each day. We have such daily vectors for a period of 31 years (11,315 days). Twelve SOM nodes have been obtained by the meteorologists to represent the space of these data vectors. Again, we try to enhance our understanding of dynamic SOM node behavior arising from this dataset.

Full scale multi-output Gaussian process emulator with nonseparable auto-covariance functions

Gaussian process emulator with separable covariance function has been utilized extensively in modeling large computer model outputs. The assumption of separability imposes constraints on the emulator and may negatively affect its performance in some applications where separability may not hold. We propose a multi-output Gaussian process emulator with a nonseparable auto-covariance function to avoid limitations of using separable emulators. In addition, to facilitate the computation of nonseparable emulator, we introduce a new computational method, referred to as the Full-Scale approximation method with block modulating function (FSA-Block) approach. The FSA-Block is an effective and accurate covariance approximation method to reduce computations for Gaussian process models, which applies to both nonseparable and partially separable covariance models. We illustrate the effectiveness of our method through simulation studies and compare it with emulators with separable covariances. We also apply our method to a real computer code of the carbon capture system.

Sensing Statistical Primary Network Patterns via Bayesian Network Structure Learning

In cognitive radio (CR) technology, the trend of sensing is no longer to only detect the presence of active primary users, as a large number of applications demand for more comprehensive knowledge on primary network behaviors in spatial, temporal, and frequency domains. To satisfy such requirements, we study the statistical relationship among primary nodes by introducing a Bayesian network (BN)-based framework. How to efficiently learn such a BN structure is a long-standing issue that is not fully understood even in the statistical learning community. To address such an issue in CR, this paper proposes a BN structure learning scheme consisting of a concise directional dependence checking function and a regular BN graph, which achieves significantly lower computational complexity compared with existing approaches. With this result, cognitive users could efficiently understand the statistical behavior patterns in the primary networks, such that more efficient cognitive protocols could be designed across different network layers.

An adaptive spatial model for precipitation data from multiple satellites over large regions

Satellite measurements have of late become an important source of information for climate features such as precipitation due to their near-global coverage. In this article, we look at a precipitation dataset during a 3-hour window over tropical South America that has information from two satellites. We develop a flexible hierarchical model to combine instantaneous rainrate measurements from those satellites while accounting for their potential heterogeneity. Conceptually, we envision an underlying precipitation surface that influences the observed rain as well as absence of it. The surface is specified using a mean function centered at a set of knot locations, to capture the local patterns in the rainrate, combined with a residual Gaussian process to account for global correlation across sites. To improve over the commonly used pre-fixed knot choices, an efficient reversible jump scheme is used to allow the number of such knots as well as the order and support of associated polynomial terms to be chosen adaptively. To facilitate computation over a large region, a reduced rank approximation for the parent Gaussian process is employed.

Efficient learning of statistical primary patterns via Bayesian network

In cognitive radio (CR) technology, the trend of sensing is no longer to only detect the presence of active primary users. A large number of applications demand for primary user behavior correlation in spatial, temporal, and frequency domains. To satisfy such requirements, we study the statistical relationship of primary users by introducing a Bayesian network (BN) based framework. How to learn such a BN structure is a long standing issue, not fully understood even in the statistical learning community. To solve such an issue in CR, this paper proposes a BN structure learning scheme which incurs significantly lower computational complexity compared with previous ones. Thus, with this scheme, cognitive users could efficiently understand the statistical pattern of primary networks.