L. Mark Berliner

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.

Hierarchical Bayesian Time Series Models

Notions of Bayesian analysis are reviewed, with emphasis on Bayesian modeling and Bayesian calculation. A general hierarchical model for time series analysis is then presented and discussed. Both discrete time and continuous time formulations are discussed. An brief overview of generalizations of the fundamental hierarchical time series model concludes the article.

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...

Statistical Design for Adaptive Weather Observations

Abstract Suppose that one has the freedom to adapt the observational network by choosing the times and locations of observations. Which choices would yield the best analysis of the atmospheric state or the best subsequent forecast? Here, this problem of “adaptive observations” is formulated as a problem in statistical design. The statistical framework provides a rigorous mathematical statement of the adaptive observations problem and indicates where the uncertainty of the current analysis, the dynamics of error evolution, the form and errors of observations, and data assimilation each enter the calculation. The statistical formulation of the problem also makes clear the importance of the optimality criteria (for instance, one might choose to minimize the total error variance in a given forecast) and identifies approximations that make calculation of optimal solutions feasible in principle. Optimal solutions are discussed and interpreted for a variety of cases. Selected approaches to the adaptive observati...

Subsampling the Gibbs Sampler

Abstract This article provides a justification of the ban against sub-sampling the output of a stationary Markov chain that is suitable for presentation in undergraduate and beginning graduate-level courses. The justification does not rely on reversibility of the chain as does Geyers (1992) argument and so applies to the usual implementation of the Gibbs sampler.

Bayesian Climate Change Assessment

Abstract A Bayesian fingerprinting methodology for assessing anthropogenic impacts on climate was developed. This analysis considers the effect of increased CO2 on near-surface temperatures. A spatial CO2 fingerprint based on control and forced model output from the National Center for Atmospheric Research Climate System Model was developed. The Bayesian approach is distinguished by several new facets. First, the prior model for the amplitude of the fingerprint is a mixture of two distributions: one reflects prior uncertainty in the anticipated value of the amplitude under the hypothesis of “no climate change.” The second reflects behavior assuming“climate change forced by CO2.” Second, within the Bayesian framework, a new formulation of detection and attribution analyses based on practical significance of impacts rather than traditional statistical significance was presented. Third, since Bayesian analyses can be very sensitive to prior inputs, a robust Bayesian approach, which investigates the ranges of...

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.

Markov switching time series models with application to a daily runoff series

We consider a class of Bayesian dynamic models that involve switching among various regimes. As an example we produce a model for a runoff time series exhibiting pulsatile behavior. This model is a mixture of three autoregressive models which accommodate “rising,” “falling,” and “normal” states in the runoff process. The mechanism for switching among regimes is given by a three-state Markov chain whose transition probabilities are modeled on the basis both of past runoff values and of a time series of rainfall data. We adopt the Bayesian approach and use the Gibbs sampler in the numerical analyses. A study of a daily runoff series from Lake Taupo, New Zealand, is given.

Likelihood and Bayesian Prediction of Chaotic Systems

Abstract There has recently been considerable interest in both applied disciplines and in mathematics, as well as in the popular science literature, in the areas of nonlinear dynamical systems and chaotic processes. By a nonlinear, deterministic dynamical system, we mean a time series in which, starting at some initial condition, the values of the series are some fixed, nonlinear function of the previous states. One of the more intriguing aspects of these models is their propensity for displaying very complex, apparently random behavior, even when simple models are analyzed. A consequence of such chaotic behavior is that it is difficult to predict the exact behavior of a chaotic system. The difficulty in prediction stems from the fact that even the tiniest of errors, including computer roundoff, in either the specification of the function or the initial condition, can lead to huge errors in prediction. After a brief review of dynamical systems and the role of probability in dealing with uncertainty, a com...

Bayesian Design and Analysis for Superensemble-Based Climate Forecasting

Abstract The authors develop statistical data models to combine ensembles from multiple climate models in a fashion that accounts for uncertainty. This formulation enables treatment of model specific means, biases, and covariance matrices of the ensembles. In addition, the authors model the uncertainty in using computer model results to estimate true states of nature. Based on these models and principles of decision making in the presence of uncertainty, this paper poses the problem of superensemble experimental design in a quantitative fashion. Simple examples of the resulting optimal designs are presented. The authors also provide a Bayesian climate modeling and forecasting analysis. The climate variables of interest are Northern and Southern Hemispheric monthly averaged surface temperatures. A Bayesian hierarchical model for these quantities is constructed, including time-varying parameters that are modeled as random variables with distributions depending in part on atmospheric CO2 levels. This allows ...