Hierarchical Bayesian Models for Predicting The Spread of Ecological Processes

This is the pre-print version of the article found in Ecology (http://www.esajournals.org/loi/ecol).There is increasing interest in predicting ecological processes. Methods to accomplish such predictions must account for uncertainties in observation, sampling, models, and parameters. Statistical methods for spatio-temporal processes are powerful, yet difficult to implement in complicated, high-dimensional settings. However, recent advances in hierarchical formulations for such processes can be utilized for ecological prediction. These formulations are able to account for the various sources of uncertainty, and can incorporate scientific judgment in a probabilistically consistent manner. In particular, analytical diffusion models can serve as motivation for the hierarchical model for invasive species. We demonstrate by example that such a framework can be utilized to predict spatially and temporally, the house finch relative population abundance over the eastern United States.This research has been supported by a grant from the U.S. Environmental Protection Agency's Science to Achieve Results (STAR) program, Assistance Agreement No. R827257-01-0

Multiresolution Models for Nonstationary Spatial Covariance Functions

This is the pre-print version of the article found in Statistical Modelling (http://smj.sagepub.com/).Many geophysical and environmental problems depend on estimating a spatial process that has nonstationary structure. A nonstationary model is proposed based on the spatial field being a linear combination of a multiresolution (wavelet) basis functions and random coefficients. The key is to allow for a limited some number of correlations among coefficients and also to use a wavelet basis that is smooth. When approximately 6 % nonzero correlations are enforced, this representation gives a good approximation to a family of Matern covariance functions. This sparseness is important not only for model parsimony but also has implications for the efficient analysis of large spatial data sets. The covariance model is successfully applied to ozone model output and results in a nonstationary but smooth estimate.This work was supported by National Science Foundation grants DMS-93122686 and DMS-9815344.This work was supported by National Science Foundation grants DMS-93122686 and DMS-9815344

Population Influences on Tornado Reports in the United States

The number of tornadoes reported in the United States is believed to be less than the actual incidence of tornadoes, especially prior to the 1990s, because tornadoes may be undetectable by human witnesses in sparsely populated areas. We use a hierarchical Bayesian model to simultaneously correct for population-based sampling bias and estimate tornado density using historical tornado report data. The expected result is that F2-F5 compared to F0-F1 tornado reports would vary less with population density. The results agree with this hypothesis for the following population centers: Atlanta, GA; Champaign, IL; Des Moines, IA. However, the results indicated just the opposite in Oklahoma. We speculate the result is explained by misclassification of tornadoes that were worthy of F2-F5 Fujita scale rating but were classified as F0-F1 tornadoes, thereby artificially decreasing the number of F2-F5 and increasing the number of F0-F1 reports in rural Oklahoma.Wikle and Zhou acknowledge the support of NSF grant DMS 0139903. Anderson acknowledges the support of NSF grant ATM-9911417

Hierarchical Bayesian Approach to Boundary Value Problems with Stochastic Boundary Conditions

This is the pre-print version of the article found in the Monthly Weather Review (http://journals.ametsoc.org/toc/mwre/138/10).Boundary value problems are ubiquitous in the atmospheric and ocean sciences. Typical settings include bounded, partially bounded, global and limited area domains, discretized for applications of numerical models of the relevant fluid equations. Often, limited area models are constructed to interpret intensive datasets collected over a specific region, from a variety of observational platforms. These data are noisy and they typically do not span the domain of interest uniformly in space and time. Traditional numerical procedures cannot easily account for these uncertainties. A hierarchical Bayesian modeling framework is developed for solving boundary value problems in such settings. By allowing the boundary process to be stochastic, and conditioning the interior process on this boundary, one can account for the uncertainties in the boundary process in a reasonable fashion. In the presence of data and all its uncertainties, this idea can be related through Bayes' Theorem to produce distributions of the interior process given the observational data. The method is illustrated with an example of obtaining atmospheric streamfunction fields in the Labrador Sea region, given scatterometer-derived observations of the surface wind field

Spatio-Temporal Hierarchical Bayesian Modeling: Tropical Ocean Surface Winds

This is the author's version of the article found in the Journal of the American Statistical Association. The publisher's version can be found at http://pubs.amstat.org/loi/jasa.Spatio-temporal processes are ubiquitous in the environmental and physical sciences. This is certainly true of atmospheric and oceanic processes, which typically exhibit many different scales of spatial and temporal variability. The complexity of these processes and large number of observation/prediction locations preclude the use of traditional covariance-based space-time statistical methods. Alternatively, we focus on conditionally-specified (i.e., hierarchical) spatio-temporal models. These methods offer several advantages over traditional approaches. Primarily, physical and dynamical constraints are easily incorporated into the conditional formulation, so that the series of relatively simple, yet physically realistic, conditional models leads to a much more complicated space-time covariance structure than can be specified directly. Furthermore, by making use of the sparse structure inherent in the hierarchical approach, as well as multiresolution (wavelet) bases, the models are computable with very large datasets. This modeling approach was necessitated by a scientifically meaningful problem in the geosciences. Satellite-derived wind estimates have high spatial resolution but are limited in global coverage. In contrast, wind fields provided by the major weather centers provide complete coverage but have low spatial resolution. The goal is to combine these data in a manner that incorporates the space-time dynamics inherent in the surface wind field. This is an essential task to enable meteorological research as no complete high resolution surface wind datasets exist over the world oceans. High resolution datasets of this kind are crucial for improving our understanding of: global air-sea interactions affecting climate, tropical disturbances, and for driving large-scale ocean circulation models.Support for this research was provided for CKW, DN, and LMB by the NCAR Geophysical Statistics Project, sponsored by the National Science Foundation (NSF) under Grant DMS93-12686. Support for RFM and CKW is provided by the NCAR NSCAT Science Working Team cooperative agreement with NASA JPL. NCAR is supported in part by the NSF

Predicting the Spatial Distribution of Ground Flora on Large Domains Using a Hierarchical Bayesian Model

This is the pre-print version of the article found in Landscape Ecology. The original publication is available at www.springerlink.com.Accomodation of important sources of uncertainty in ecological models is essential to realistically predicting ecological processes. The purpose of this project is to develop a robust methodology for modeling natural processes on a landscape while accounting for the variability in a process by utilizing environmental and spatial random effects. A hierarchical Bayesian framework has allowed the simultaneous integration of these effects. This framework naturally assumes variables to be random and the posterior distribution of the model provides probabilistic information about the process. Two species in the genus Desmodium were used as examples to illustrate the utility of the model in Southeast Missouri. In addition, two validation techniques were applied to evaluate the qualitative and quantitative characteristics of the predictions.NASA and the University of Montana provided funding through the EOS Training Center Project. Wikle's research was supported by a grant from the U.S. Environmental Protection Agency's Science to Achieve Results (STAR) program, Assistance Agreement No. R827257-01-0

Accounting for Uncertainty in Ecological Analysis: The Strengths and Limitations of Hierarchical Statistical Modeling

Copyright by the Ecological Society of America.Analyses of ecological data should account for the uncertainty in the process(es) that generated the data. However, accounting for these uncertainties is a difficult task, since ecology is known for its complexity. Measurement and/or process errors are often the only sources of uncertainty modeled when addressing complex ecological problems, yet analyses should also account for uncertainty in sampling design, in model specification, in parameters governing the specified model, and in initial and boundary conditions. Only then can we be confident in the scientific inferences and forecasts made from an analysis. Probability and statistics provide a framework that accounts for multiple sources of uncertainty. Given the complexities of ecological studies, the hierarchical statistical model is an invaluable tool. This approach is not new in ecology, and there are many examples (both Bayesian and non-Bayesian) in the literature illustrating the benefits of this approach. In this article, we provide a baseline for concepts, notation, and methods, from which discussion on hierarchical statistical modeling in ecology can proceed. We have also planted some seeds for discussion and tried to show where the practical difficulties lie. Our thesis is that hierarchical statistical modeling is a powerful way of approaching ecological analysis in the presence of inevitable but quantifiable uncertainties, even if practical issues sometimes require pragmatic compromises

Efficient Statistical Mapping of Avian Count Data

This is the pre-print version of the article found in Environmental and Ecological Statistics. The original publication is available at www.springerlink.com.We develop a spatial modeling framework for count data that is efficient to implement in high-dimensional prediction problems. We consider spectral parameterizations for the spatially varying mean of the Poisson model. The spectral parameterization of the spatial process is very computationally efficient, enabling effective estimation and prediction in large problems using Markov chain Monte Carlo techniques. We apply this model to creating avian relative abundance maps from the North American Breeding Bird Survey (BBS) data. Variation in the ability of observers to count birds is modeled as spatially-independent noise, resulting in over-dispersion relative to the Poisson assumption. This approach represents an improvement over existing approaches used for spatial modeling of BBS data which are either inefficient for continental scale modeling and prediction or fail to accommodate important distributional features of count data thus leading to inaccurate accounting of prediction uncertainty

A Hierarchical Bayesian Non-linear Spatio-temporal Model for the Spread of Invasive Species with Application to the Eurasian Collared-Dove

This is the pre-print of the article found in Environmental and Ecological Statistics. The original publication is available at www.springerlink.com.Differential equation based advection-diffusion models have been used in atmospheric science to mimic complex processes such as weather and climate. Differential and partial-differential equations (PDE's) have become popular in biological and ecological fields as well. In many cases, these models are considered in a strictly deterministic framework even though many sources of uncertainty in the process, the model, and the measurements may exist. Many deterministic PDE models are well-equiped to represent the theoretical spread of organisms, but have no mechanism to account for the various sources of uncertainty related to the inadequacies of the model as well as the process itself and our knowledge of it. However, the use of a PDE within the framework of a hierarchical Bayesian model can provide a useful link between scientifically based deterministic models and statistical models that accurately portray variability (Wikle 2003). Specifically we model the spread of Eurasian Collared-Dove (ECD; Streptopelia decaocto) in the United States using a reaction-diffusion PDE (Fisher 1937, Skellam 1951) within a hierarchical model

Shifts in the Spatio-Temporal Growth Dynamics of Shortleaf Pine

This is the pre-print version of the article found in Environmental and Ecological Statistics. The original publication is available at www.springerlink.com.Previous studies focusing on the growth history of pinus echinata at the edge of its geographical range have suggested that changes in growth correspond to climatic and non-climatic (e.g., anthropogenic) factors. We employ a regime-dependent state-space model that allows us to detect and characterize the changes in tree growth dynamics over space and time using readily available dendrochronological and climatic data in the presence of various sources of uncertainty.This research was funded by NSF grant DMS 0139903