32nd Spring Lecture Series, University of Arkansas Spatial and Spatio-Temporal Statistics Fayetteville, April 12-14, 2007

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ABSTRACT Modeling and Testing Properties of Space-time Covariance Functions Marc G. Genton Department of Econometrics, University of Geneva Department of Statistics, Texas A&M University Modeling space-time data often relies on parametric covariance models and various assumptions such as full symmetry and separability. These assumptions are important because they simplify the structure of the model and its inference, and ease the possibly extensive computational burden associated with space-time data sets. We review various space-time covariance models and propose a unified framework for testing a variety of assumptions commonly made for covariance functions of stationary spatio-temporal random fields. The methodology is based on the asymptotic normality of space-time covariance estimators. We focus on tests for full symmetry and separability, but our framework naturally covers testing for isotropy, Taylor's hypothesis, and the structure of cross-covariances. The proposed test successfully detects the asymmetric and nonseparable features in two sets of wind speed data. We perform simulation experiments to evaluate our test and conclude that our method is reliable and powerful for assessing common assumptions on space-time covariance functions. The talk is based on joint work with Bo Li (NCAR) and Michael Sherman (TAMU). Extreme Precipitation Trends over the Continental United States Richard L Smith Department of Statistics and Operations Research University of North Carolina Chapel Hill, NC 27599-3260 rls@email.unc.edu In recent years there has been much climatological literature devoted to the apparent increase in extreme precipitation events that may be a consequence of broader trends in climate due to increased greenhouse gases. However the climatological literature (see e.g. Groisman et al, Journal of Climate, 2005) still uses rather simple statistical methods to study this phenomenon. Here, we apply state of the art extreme value methods based on exceedances over high thresholds, including covariates to represent trend and seasonality, to estimate 25-year return levels, and trends in those return levels over 1970-1999. This is done separately for nearly 5,000 stations in the US climatological network, then the results are combined across stations using spatial statistics. The results provide a detailed picture of how trends in rainfall extremes have varied across the United States, as well as summary statistics computed for 19 regions. They indeed confirm an overall increase in extreme rainfall levels, but it is by no mean homogeneous across the whole country. Separate analysis based on model runs from NCAR's Community Climate System Model provide a first insight into how such changes may project into the future. Space-Time Zero-Inflated Count Models of Harbor Seals Jay M. Ver Hoef1 and John Jansen1 1 National Marine Mammal Lab,, 7600 Sand Point Way NE, Bldg 4, Seattle, WA 98115-6349, Voice: (206) 526-4025, FAX: (206) 526-6615, E-mail: jay.verhoef@noaa.gov Environmental data are spatial, temporal, and often come with many zeros. In this paper, we take the standard formulation of a zero-inflated Poisson (ZIP) model, as well as an alternative parameterization, and develop a space-time model to investigate haulout patterns of harbor seals on glacial ice. The data consist of counts, for 18 dates on a lattice grid of samples, of harbor seals hauled out on glacial ice in Disenchantment Bay, a coastal bay near Yakutat, Alaska. A space-time ZIP was constructed by using spatial conditional autoregressive model (CAR) model and a temporal first-order autoregressive model (AR1) as random effects in ZIP regression model. Because seals are unlikely to be undetected, we consider another model that completely specifies and separates the binary from the count process, but still has an inflated number of zeros. We compare this model to the standard ZIP. Both models indicate that ice density plays a strong role in where seals haul out, with highest haulout probabilities and counts at intermediate ice densities. We create maps of smoothed prediction rates for harbor seal haulouts based on ice density and other covariates. Nonlinear Spatio-Temporal Dynamic Models Christopher K. Wikle Department of Statistics University of Missouri Dynamic spatio-temporal models (DSTMs) are of interest in many environmental problems as they provide a reasonable approximation of the underlying evolution mechanisms that describe real-world processes. The traditional focus in DSTMs has been on linear process evolution, at least in the statistical literature. As with linear evolution models, nonlinear evolution models are complicated in the spatio-temporal context by issues of dimensionality and efficient parameterization. Furthermore, estimation is necessarily more complicated in the nonlinear framework. All of these issues limit the specification of general model classes. In this talk, we present an attempt at a general framework that does capture many of the realistic science-based nonlinear models that have been and could be considered for environmental processes. For example, this construct allows for structures motivated by partial differential equations such as found in ecological or atmospheric science. Markov Chain Monte Carlo for a Spatial-Temporal Autologistic Regression Model Jun Zhu, Department of Statistics, University of Wisconsin - Madison A spatial-temporal autologistic regression model developed by Zhu et al. (2005) relates a binary response variable to potential covariates while accounting for both dependence on a spatial lattice and dependence over discrete time points, which may be useful for analyzing spatial-temporal binary data. However the existing statistical inference is via maximum pseudolikelihood, which is statistically inefficient especially when the spatial and temporal dependence is strong. Here we propose a fully Bayesian approach for both model parameter inference and prediction at future time points using Markov chain Monte Carlo (MCMC). We demonstrate the methodology and compare the results with maximum pseudolikelihood and MCMC maximum likelihood approaches via a real data example concerning beetle outbreaks. Using a Multivariate Approach to Model Univariate Environmental Space Time Processes Yiping Dou*, Nhu Le** and Jim Zidek* * U British Columbia ** British Columbia Cancer Agency In this paper, I will describe how an empirically adapted multivariate hierarchical Bayes approach (MHM) can be used to model univariate space-time series. To assess its performance, that approach will be compared and contrasted with a popular univarite approach called dynamiclinear modelling (DLM), specifically for modeling an hourly ozone field over Chicago. The former, which is computationally straightforward and quick, using available software, follows the traditional approach of first prefiltering (to remove temporal components and autocorrlation), modelling the whitened series, and then defiltering to return to the original scales. Emphasis will be on how multivariate methods can be used to avoid the very challenging problem (implicit in DLM) of modeling the short term temporal autocorrelation typically found in hourly pollution concentrations. The approach allows "strength to be borrowed" across time and space for spatial prediction. And it sidesteps the "correlation leakage problem", a potential side effect of modeling the spatial component after prefiltering. Our empirical assessment of MHM and DLM shows the superiority of the former in terms of prediction accuracy. A Class of Kernel-Based Conditionally Autoregressive Models for Spatial Data Sujit K Ghosh and Minjung Kyung Department of Statistics, North Carolina State University. A spatial process observed over a lattice or a set of irregular regions is usually modeled using a conditionally autoregressive (CAR) model. The neighborhoods within a CAR model are generally formed deterministically using the inter-distances or boundaries between the regions. A new class of spatial models is proposed that adaptively determines the neighborhoods based on a bivariate kernel using the distances and angles between the centroid of the regions. The proposed model generalizes the usual CAR model by accounting for spatial anisotropy. Maximum likelihood estimators are derived and shown to be consistent under some regularity conditions. Simulation studies are presented to evaluate the finite sample performance of the new model as compared to CAR model. Finally the method is illustrated using a data set on the elevated blood lead levels of children under the age of 72 years observed in Virgina in the year of 2000. Objective Bayesian Analysis of Spatial Data With Measurement Error Victor De Oliveira Department of Management Science and Statistics The University of Texas at San Antonio San Antonio, Texas 78249 This work provides the basis for default Bayesian analysis of Gaussian random fields based on geostatistical data that contain measurement error. A reference prior and two versions of Jeffreys prior are derived for the model parameters, and properties of the resulting posteriors in terms of propriety and existence of relevant moments are provided. Existence of the mean and variance of the predictive distributions based on the these default priors is established. The reference prior is obtained from a representation of the integrated likelihood that is particularly convenient for computation and analysis. It is also shown that these default priors are not very sensitive to some aspects of the design and model, and have good frequentist properties. Finally, a dataset of carbon-nitrogen ratios from an agricultural field is used to illustrate the Bayesian analysis based on the reference prior.