Processes in ocean physics, air–sea interaction and ocean biogeochemistry span enormous ranges in spatial and temporal scales, that is, from molecular to planetary and from seconds to millennia. Identifying and implementing sustainable human practices depend critically on our understandings of key aspects of ocean physics and ecology within these scale ranges. The set of all ocean data is distorted such that three- and four-dimensional (i.e., time-dependent) in situ data are very sparse, while observations of surface and upper ocean properties from space-borne platforms have become abundant in the past few decades. Precisions in observations of all types vary as well. In the face of these challenges, the interface between Statistics and Oceanography has proven to be a fruitful area for research and the development of useful models. With the recognition of the key importance of identifying, quantifying and managing uncertainty in data and models of ocean processes, a hierarchical perspective has become increasingly productive. As examples, we review a heterogeneous mix of studies from our own work demonstrating Bayesian hierarchical model applications in ocean physics, air–sea interaction, ocean forecasting and ocean ecosystem models. This review is by no means exhaustive and we have endeavored to identify hierarchical modeling work reported by others across the broad range of ocean-related topics reported in the statistical literature. We conclude by noting relevant ocean-statistics problems on the immediate research horizon, and some technical challenges they pose, for example, in terms of nonlinearity, dimensionality and computing.

This article concerns a systemic manifestation of small scale interfacial heterogeneities in large scale quantities of interest to a variety of diverse applications spanning the earth, biological and ecological sciences. Beginning with formulations in terms of partial differential equations governing the conservative, advective-dispersive transport of mass concentrations in divergence form, the specific interfacial heterogeneities are introduced in terms of (spatial) discontinuities in the diffusion coefficient across a lower-dimensional hypersurface. A pathway to an equivalent stochastic formulation is then developed with special attention to the interfacial effects in various functionals such as first passage times, occupation times and local times. That an appreciable theory is achievable within a framework of applications involving one-dimensional models having piecewise constant coefficients greatly facilitates our goal of a gentle introduction to some rather dramatic mathematical consequences of interfacial effects that can be used to predict structure and to inform modeling.

Models for forecasting earthquakes are currently tested prospectively in well-organized testing centers, using data collected after the models and their parameters are completely specified. The extent to which these models agree with the data is typically assessed using a variety of numerical tests, which unfortunately have low power and may be misleading for model comparison purposes. Promising alternatives exist, especially residual methods such as super-thinning and Voronoi residuals. This article reviews some of these tests and residual methods for determining the goodness of fit of earthquake forecasting models.

Earthquakes occur because of abrupt slips on faults due to accumulated stress in the Earth’s crust. Because most of these faults and their mechanisms are not readily apparent, deterministic earthquake prediction is difficult. For effective prediction, complex conditions and uncertain elements must be considered, which necessitates stochastic prediction. In particular, a large amount of uncertainty lies in identifying whether abnormal phenomena are precursors to large earthquakes, as well as in assigning urgency to the earthquake. Any discovery of potentially useful information for earthquake prediction is incomplete unless quantitative modeling of risk is considered. Therefore, this manuscript describes the prospect of earthquake predictability research to realize practical operational forecasting in the near future.

In this paper we first describe the class of log-Gaussian Cox processes (LGCPs) as models for spatial and spatio-temporal point process data. We discuss inference, with a particular focus on the computational challenges of likelihood-based inference. We then demonstrate the usefulness of the LGCP by describing four applications: estimating the intensity surface of a spatial point process; investigating spatial segregation in a multi-type process; constructing spatially continuous maps of disease risk from spatially discrete data; and real-time health surveillance. We argue that problems of this kind fit naturally into the realm of geostatistics, which traditionally is defined as the study of spatially continuous processes using spatially discrete observations at a finite number of locations. We suggest that a more useful definition of geostatistics is by the class of scientific problems that it addresses, rather than by particular models or data formats.

Renewable energy sources, especially wind energy, are to play a larger role in providing electricity to industrial and domestic consumers. This is already the case today for a number of European countries, closely followed by the US and high growth countries, for example, Brazil, India and China. There exist a number of technological, environmental and political challenges linked to supplementing existing electricity generation capacities with wind energy. Here, mathematicians and statisticians could make a substantial contribution at the interface of meteorology and decision-making, in connection with the generation of forecasts tailored to the various operational decision problems involved. Indeed, while wind energy may be seen as an environmentally friendly source of energy, full benefits from its usage can only be obtained if one is able to accommodate its variability and limited predictability. Based on a short presentation of its physical basics, the importance of considering wind power generation as a stochastic process is motivated. After describing representative operational decision-making problems for both market participants and system operators, it is underlined that forecasts should be issued in a probabilistic framework. Even though, eventually, the forecaster may only communicate single-valued predictions. The existing approaches to wind power forecasting are subsequently described, with focus on single-valued predictions, predictive marginal densities and space–time trajectories. Upcoming challenges related to generating improved and new types of forecasts, as well as their verification and value to forecast users, are finally discussed.

Wildfire is an important system process of the earth that occurs across a wide range of spatial and temporal scales. A variety of methods have been used to predict wildfire phenomena during the past century to better our understanding of fire processes and to inform fire and land management decision-making. Statistical methods have an important role in wildfire prediction due to the inherent stochastic nature of fire phenomena at all scales.

Predictive models have exploited several sources of data describing fire phenomena. Experimental data are scarce; observational data are dominated by statistics compiled by government fire management agencies, primarily for administrative purposes and increasingly from remote sensing observations. Fires are rare events at many scales. The data describing fire phenomena can be zero-heavy and nonstationary over both space and time. Users of fire modeling methodologies are mainly fire management agencies often working under great time constraints, thus, complex models have to be efficiently estimated.

We focus on providing an understanding of some of the information needed for fire management decision-making and of the challenges involved in predicting fire occurrence, growth and frequency at regional, national and global scales.

Critical decisions frequently rely on high-dimensional output from complex computer simulation models that show intricate cross-variable, spatial and temporal dependence structures, with weather and climate predictions being key examples. There is a strongly increasing recognition of the need for uncertainty quantification in such settings, for which we propose and review a general multi-stage procedure called ensemble copula coupling (ECC), proceeding as follows:

1. Generate a raw ensemble, consisting of multiple runs of the computer model that differ in the inputs or model parameters in suitable ways.

2. Apply statistical postprocessing techniques, such as Bayesian model averaging or nonhomogeneous regression, to correct for systematic errors in the raw ensemble, to obtain calibrated and sharp predictive distributions for each univariate output variable individually.

3. Draw a sample from each postprocessed predictive distribution.

4. Rearrange the sampled values in the rank order structure of the raw ensemble to obtain the ECC postprocessed ensemble.

The use of ensembles and statistical postprocessing have become routine in weather forecasting over the past decade. We show that seemingly unrelated, recent advances can be interpreted, fused and consolidated within the framework of ECC, the common thread being the adoption of the empirical copula of the raw ensemble. Depending on the use of Quantiles, Random draws or Transformations at the sampling stage, we distinguish the ECC-Q, ECC-R and ECC-T variants, respectively. We also describe relations to the Schaake shuffle and extant copula-based techniques. In a case study, the ECC approach is applied to predictions of temperature, pressure, precipitation and wind over Germany, based on the 50-member European Centre for Medium-Range Weather Forecasts (ECMWF) ensemble.

Historically, matrix projection models (MPMs) have been employed to study population dynamics with regard to size, age or structure. To work with continuous traits, in the past decade, integral projection models (IPMs) have been proposed. Following the path for MPMs, currently, IPMs are handled first with a fitting stage, then with a projection stage. Model fitting has, so far, been done only with individual-level transition data. These data are used in the fitting stage to estimate the demographic functions (survival, growth, fecundity) that comprise the kernel of the IPM specification. The estimated kernel is then iterated from an initial trait distribution to obtain what is interpreted as steady state population behavior. Such projection results in inference that does not align with observed temporal distributions. This might be expected; a model for population level projection should be fitted with population level transitions.

Ghosh, Gelfand and Clark [J. Agric. Biol. Environ. Stat. 17 (2012) 641–699] offer a remedy by viewing the observed size distribution at a given time as a point pattern over a bounded interval, driven by an operating intensity. They propose a three-stage hierarchical model. At the deepest level, demography is driven by an unknown deterministic IPM. The operating intensities are allowed to vary around this deterministic specification. Further uncertainty arises in the realization of the point pattern given the operating intensities. Such dynamic modeling, optimized by fitting data observed over time, is better suited to projection.

Here, we address scaling of population IPM modeling, with the objective of moving from projection at plot level to projection at the scale of the eastern U.S. Such scaling is needed to capture climate effects, which operate at a broader geographic scale, and therefore anticipated demographic response to climate change at larger scales. We work with the Forest Inventory Analysis (FIA) data set, the only data set currently available to enable us to attempt such scaling. Unfortunately, this data set has more than 80% missingness; less than 20% of the 43,396 plots are inventoried each year. We provide a hierarchical modeling approach which still enables us to implement the desired scaling at annual resolution. We illustrate our methodology with a simulation as well as with an analysis for two tree species, one generalist, one specialist.