Christopher K. Wikle

Accounting for uncertainty in ecological analysis: the strengths and limitations of hierarchical statistical modeling

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.

HIERARCHICAL BAYESIAN MODELS FOR PREDICTING THE SPREAD OF ECOLOGICAL PROCESSES

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 spatiotemporal 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 relative population abundance of House Finches over the eastern United States. Corresponding Editor (ad hoc): J. S. Clark.

Spatiotemporal Hierarchical Bayesian Modeling Tropical Ocean Surface Winds

Spatiotemporal 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 the large number of observation/prediction locations preclude the use of traditional covariance-based spatiotemporal statistical methods. Alternatively, we focus on conditionally specified (i.e., hierarchical) spatiotemporal models. These methods offer several advantages over traditional approaches. Primarily, physical and dynamical constraints can be easily incorporated into the conditional formulation, so that the series of relatively simple yet physically realistic conditional models leads to a much more complicated spatiotemporal 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 can be computed 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 limited 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, because no complete high-resolution surface wind datasets exist over the world oceans. High-resolution datasets of this type are crucial for improving our understanding of global air–sea interactions affecting climate and tropical disturbances, and for driving large-scale ocean circulation models.

Multiresolution models for nonstationary spatial covariance functions

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 multiresolution (wavelet) basis functions and random coefficients. The key is to allow for a limited 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.

Long-Lead Prediction of Pacific SSTs via Bayesian Dynamic Modeling

Abstract Tropical Pacific sea surface temperatures (SSTs) and the accompanying El Nino–Southern Oscillation phenomenon are recognized as significant components of climate behavior. The atmospheric and oceanic processes involved display highly complicated variability over both space and time. Researchers have applied both physically derived modeling and statistical approaches to develop long-lead predictions of tropical Pacific SSTs. The comparative successes of these two approaches are a subject of substantial inquiry and some controversy. Presented in this article is a new procedure for long-lead forecasting of tropical Pacific SST fields that expresses qualitative aspects of scientific paradigms for SST dynamics in a statistical manner. Through this combining of substantial physical understanding and statistical modeling and learning, this procedure acquires considerable predictive skill. Specifically, a Markov model, applied to a low-order (empirical orthogonal function–based) dynamical system of tropi...

A kernel-based spectral model for non-Gaussian spatio-temporal processes

Spatio-temporal processes can often be written as hierarchical state-space processes. In situations with complicated dynamics such as wave propagation, it is difficult to parameterize state transition functions for high-dimensional state processes. Although in some cases prior understanding of the physical process can be used to formulate models for the state transition, this is not always possible. Alternatively, for processes where one considers discrete time and continuous space, complicated dynamics can be modeled by stochastic integro-difference equations in which the associated redistribution kernel is allowed to vary with space and/or time. By considering a spectral implementation of such models, one can formulate a spatio-temporal model with relatively few parameters that can accommodate complicated dynamics. This approach can be developed in a hierarchical framework for non-Gaussian processes, as demonstrated on cloud intensity data.

A Kernel-Based Spatio-Temporal Dynamical Model for Nowcasting Weather Radar Reflectivities

A good short-period forecast of heavy rainfall is essential for many meteorological and hydrological applications. Traditional deterministic and stochastic nowcasting methodologies have been inadequate in their characterization of pixelwise rainfall reflectivity propagation, intensity, and uncertainty. The methodology presented herein uses an approach that efficiently parameterizes spatio-temporal dynamic models in terms of integro-difference equations within a hierarchical framework. The approach accounts for the uncertainty in the prediction and provides relevant distributional information concerning the nowcast. An application is presented that shows the effectiveness of the technique and its potential for nowcasting weather radar reflectivities.

Combining Information Across Spatial Scales

Spatial and spatiotemporal processes in the physical, environmental, and biological sciences often exhibit complicated and diverse patterns across different space–time scales. Both scientific understanding and observational data vary in form and content across scales. We develop and examine a Bayesian hierarchical framework by which the combination of such information sources can be accomplished. Our approach is targeted to settings in which various special spatial scales arise. These scales may be dictated by the data collection methods, availability of prior information, and/or goals of the analysis. The approach restricts to a few essential scales. Hence we avoid the challenging problem of constructing a model that can be used at all scales. This means that we can provide inferences only at the preselected special scales. However, problems involving special scales are sufficiently common to justify the trade-off between our comparatively simple modeling and analysis strategy with the formidable task of forming models valid at all scales. Specifically, our approach is based on a simple idea of conditioning the spatially continuous process on an areal average of the process at some resolution of interest. In addition, the data at prescribed resolutions are then conditioned on this areal-averaged true process. These conditioning arguments fit nicely into the hierarchical Bayesian framework. The methodology is demonstrated for the spatial prediction of an important quantity known as streamfunction based on wind information from satellite observations and weather center, computer model output.

Predicting the spatial distribution of ground flora on large domains using a hierarchical Bayesian model

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, USA. In addition, two validation techniques were applied to evaluate the qualitative and quantitative characteristics of the predictions.

Efficient Statistical Mapping of Avian Count Data

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 a 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 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.