Hierarchical Bayesian space-time models

Space-time data are ubiquitous in the environmental sciences. Often, as is the case with atmo- spheric and oceanographic processes, these data contain many different scales of spatial and temporal variability. Such data are often non-stationary in space and time and may involve many observation/prediction locations. These factors can limit the effectiveness of traditional space- time statistical models and methods. In this article, we propose the use of hierarchical space-time models to achieve more flexible models and methods for the analysis of environmental data distributed in space and time. The first stage of the hierarchical model specifies a measurement- error process for the observational data in terms of some 'state' process. The second stage allows for site-specific time series models for this state variable. This stage includes large-scale (e.g. seasonal) variability plus a space-time dynamic process for the anomalies'. Much of our interest is with this anomaly proc ess. In the third stage, the parameters of these time series models, which are distributed in space, are themselves given a joint distribution with spatial dependence (Markov random fields). The Bayesian formulation is completed in the last two stages by speci- fying priors on parameters. We implement the model in a Markov chain Monte Carlo framework and apply it to an atmospheric data set of monthly maximum temperature.     

Fitting probability models to population dynamics data

Methods for modeling population dynamics in probability using the generalized point process approach are developed. The life history of these populations is such that seasonal reproduction occurs during a short time. Several models are developed and analyzed. Data about two species: colonial spiders (Stegodyphus dumicola) and a migratory bird (wood thrush, Hylocichla mustelina) are used to estimate model parameters with appropriate log maximum likelihood functions. For the spiders, the model is fitted to provide evolutionary feasible colony size based on maximum likelihood estimates of fecundity and survival data. For the migratory bird species, a maximum likelihood estimates are derived for the fecundity and survival rates of young and adult birds and immigration rate. The presented approach allows computation of quantities of interest such as probability of extinction and average time to extinction.     

A single-chain-based multidimensional Markov chain model for subsurface characterization

Multidimensional Markov chain models in geosciences were often built on multiple chains, one in each direction, and assumed these 1-D chains to be independent of each other. Thus, unwanted transitions (i.e., transitions of multiple chains to the same location with unequal states) inevitably occur and have to be excluded in estimating the states at unobserved locations. This consequently may result in unreliable estimates, such as underestimation of small classes (i.e., classes with smaller than average areas) in simulated realizations. This paper presents a single-chain-based multidimensional Markov chain model for estimation (i.e., prediction and conditional stochastic simulation) of spatial distribution of subsurface formations with borehole data. The model assumes that a single Markov chain moves in a lattice space, interacting with its nearest known neighbors through different transition probability rules in different cardinal directions. The conditional probability distribution of the Markov chain at the location to be estimated is formulated in an explicit form by following the Bayes’ Theorem and the conditional independence of sparse data in cardinal directions. Since no unwanted transitions are involved, the model can estimate all classes fairly. Transiogram models (i.e., 1-D continuous Markov transition probability diagrams) are used to provide transition probability input with needed lags to generalize the model. Therefore, conditional simulation can be conducted directly and efficiently. The model provides an alternative for heterogeneity characterization of subsurface formations.

Bayesian analysis under ordered functions of parameters

Ordered parameter problems arise in a wide variety of real world situations and are dealt with extensively in the literature. Traditional frequentist methods for dealing with these problems are rather complicated theoretically, especially when sample sizes are small. Bayesian methods are not widely used because high dimensional numerical integration is often required. However, Markov chain Monte Carlo methods provide alternatives to such numerical integration and also deal with ordered parameter problems in a straightforward manner. Little is known about the situation where functions of parameters are ordered. Such problems may seem to be of little practical concern initially, but one can readily see their importance in situations where ordering is placed on the means and variances of several normal or Gamma populations. For the Gamma distribution we will present real examples where we will analyze monthly precipitation data from San Francisco, California and Oakland Mills, Iowa. For the San Francisco data we will simultaneously order both monthly precipitation means and variances. For the Iowa data we will place ordering on seasonal average while still estimating monthly means. Our results show that we would obtain sharper, more accurate inference when order restrictions are employed.     

State space models with spatial deformation

Space deformation has been proposed to model space-time varying observation processes with non-stationary spatial covariance structure under the hypothesis of temporal stationarity. In real applications, however, the temporal stationarity assumption is inappropriate and unrealistic. In this work we propose a spatial-temporal model whose temporal trend is modeled through state space models and a spatially varying anisotropy is modeled through spatial deformation, under the Bayesian approach. A distinctive feature of our approach is the consideration of model uncertainty in an unified framework. Our model has a clear advantage over the ones proposed so far in the literature when the main objective of the study is to perform spatial interpolation for fixed points in time. Approximations of the posterior distributions of the model parameters are obtained via Markov chain Monte Carlo methods. This allows for prediction of the process values in space and time as well as handling of missing values. Two applications are presented: the first one to model concentrations of sulfur dioxide in the eastern United States and the second one to model monthly minimum temperatures in the State of Rio de Janeiro.     

Studying crop sequences with CarrotAge,a HMM-based data mining software

We have developed a knowledge discovery system based on high-order hidden Markov models for analyzing spatio-temporal data bases. This system, named CarrotAge , takes as input an array of discrete data – the rows represent the spatial sites and the columns the time slots – and builds a partition together with its a posteriori probability. CarrotAge has been developed for studying the cropping patterns of a territory. It uses therefore an agricultural drench database, named Ter-Uti , which records every year the land-use category of a set of sites regularly spaced. The results of CarrotAge are interpreted by agronomists and used in research works linking agricultural land use and water management. Moreover, CarrotAge can be used to find out and study crop sequences in large territories, that is a main question for agricultural and environmental research, as discussed in this paper.     

Optimal model averaging by data perturbation for spatial noisy data

Geostatistical models play an important role in spatial data analysis, in which model selection is inevitable. Model selection methods, such as AIC and BIC, are popular for selecting appropriate models. In recent years, some model averaging methods, such as smoothed AIC and smoothed BIC, are also applied to spatial data models. However, the corresponding averaging estimators are outperformed by optimal model averaging estimators (Hansen in Econometrica 75:1175–1189, 2007) for the ordinary linear models. Therefore, this paper focuses on the optimal model averaging method for geostatistical models. We propose a weight choice criterion for the model averaging estimator on the basis of the generalized degrees of freedom and data perturbation technique. We further theoretically prove the resultant estimator is asymptotically optimal in terms of the mean squared error, and numerically demonstrate its satisfactory performance. Finally, the proposed method is applied to a mercury data set.     

Photographic mark-recapture analysis of clustered mammal-eating killer whales around the Aleutian Islands and Gulf of Alaska

We used photographic mark-recapture methods to estimate the number of mammal-eating “transient” killer whales using the coastal waters from the central Gulf of Alaska to the central Aleutian Islands, around breeding rookeries of endangered Steller sea lions. We identified 154 individual killer whales from 6,489 photographs collected between July 2001 and August 2003. A Bayesian mixture model estimated seven distinct clusters (95% probability interval = 7–10) of individuals that were differentially covered by 14 boat-based surveys exhibiting varying degrees of association in space and time. Markov Chain Monte Carlo methods were used to sample identification probabilities across the distribution of clusters to estimate a total of 345 identified and undetected whales (95% probability interval = 255–487). Estimates of covariance between surveys, in terms of their coverage of these clusters, indicated spatial population structure and seasonal movements from these near-shore waters, suggesting spatial and temporal variation in the predation pressure on coastal marine mammals.     

A process-convolution approach to modelling temperatures in the North Atlantic Ocean

This paper develops a process-convolution approach for space-time modelling. With this approach, a dependent process is constructed by convolving a simple, perhaps independent, process. Since the convolution kernel may evolve over space and time, this approach lends itself to specifying models with non-stationary dependence structure. The model is motivated by an application from oceanography: estimation of the mean temperature field in the North Atlantic Ocean as a function of spatial location and time. The large amount of this data poses some difficulties; hence computational considerations weigh heavily in some modelling aspects. A Bayesian approach is taken here which relies on Markov chain Monte Carlo for exploring the posterior distribution.     

Linking atmospheric circulation to daily precipitation patterns over the territory of Bulgaria

An eight states non-homogeneous hidden Markov model is developed for linking daily precipitation amounts data at a network of 32 stations broadly covering the territory of Bulgaria to large-scale atmospheric patterns. At each site a 40-year record 1960?C2000 of daily October?CMarch precipitation amounts is modeled. The atmospheric data consists of daily sea-level pressure, geopotential height at 500 and 850?hPa, air temperature at 850?hPa and relative humidity at 700 and 850?hPa on a 2.5?×?2.5 grid based on NCEP-NCAR reanalysis data set covering the Europe-Atlantic sector 30W?C60E, 20N?C70N for the same period. The first 30?years data are used for model fitting purposes while the remaining 10?years are used for model evaluation. Detailed model validation is carried out on various aspects. The proposed model reproduces well the rainfall statistics for the observed and reserved data whereas the identified states are found to be physically interpretable in terms of regional climatology.     

A stochastic weather generation model for Alaska

A specific problem encountered in ecosystem-level simulation of Arctic ecosystems is the depth and extent of the driving variable record. Often, climate records are of short duration, gathered at locations different from the area to be simulated, or do not contain all the variables required by a given model. This paper addresses this problem for ecosystem simulation in Alaska with the development of a weather generator. The generator, called WGENAL, is based on the WGEN climate generator developed and validated in the 48 conterminous states. Because of the extreme variability of weather in Alaska that is not accommodated by the statistical metrics in the earlier model, a new climate generator was developed. WGENAL generates daily values of precipitation, maximum temperature, minimum temperature, solar radiation, and wind run. Precipitation is generated using a Markov chain-gamma model. A two-parameter gamma distribution is used to generate wind run. Temperatures and solar radiation are generated using procedures developed in the earlier study. Validation of the generator shows it provides adequate diurnal and seasonal weather records for Fairbanks. Other comparisons of synthetic weather with observed weather for sites north of the Brooks Range in Alaska are also within the error of the original data.     

Ecology of a caribbean coral reef. The Porites reef-flat biotope: Part I. Meteorology and hydrography

Observations on certain conditions of the physical environment and plankton ecology of a Caribbean coral reef form the subjects of a two-part study. Here, physical factors are considered, with special attention directed to their influence on the Porites reef-flat biotope. Meteorologic (temperature, precipitation, wind) and hydrographic (temperature, salinity, tide, sea level, current) conditions are examined in order to determine their influence on water movement over the reef and correlation with seasonal variations in plankton abundance. Shoal-water circulation is characterized with reference to patterns of movement, origin, and volume flow. A relationship between wind velocity and direction to volume flow is examined in order to describe the interaction of these parameters. The effects of low tidal exposures and storms on the dominant coral species Porites furcata Lamarck are also examined. Observed mortalities and physical alterations due to these factors are shown to be significant, resulting in relatively rapid modifications of the reef-flat habitat. A chief overall objective of this study is to obtain a quantitative assessment of drifting net plankton crossing the reef-flat environment, and to evaluate its contribution as a food source to the shoal-reef biota. Integration of the physical observations with the plankton ecology will form the subject of a forthcoming publication.     

Improved statistical downscaling models based on canonical correlation analysis, for generating temperature scenarios over Greece

This study attempts to improve upon statistical downscaling (Sd) models based on the classical approach which uses canonical correlation analysis, in order to generate temperature scenarios over Greece. Considering the long-term trends of the predictor variables (1,000–500 hPa thickness field geopotential heights—using NCEP data) and the predictand variables (observed mean maximum summer temperatures over Greece), a new Sd model is constructed. Regression models using generalized least square estimators are developed in order to eliminate the trends within the time series. The advantages of the suggested method compared to the classical method are quantified in terms of a number of distinct performance criteria, e.g., Mean squared error which is the basic criterion of the estimated downscaled values relative to the observed. Finally, the suggested Sd models are used to evaluate the effects of a future climate scenario (IPCC-SRES: A2) on mean maximum summer temperatures over Greece. The results from the climate projection indicate a temperature increase for the period 2070–2100 which is smaller than the corresponding increase from the classical approach.     

Hierarchical modeling for extreme values observed over space and time

We propose a hierarchical modeling approach for explaining a collection of spatially referenced time series of extreme values. We assume that the observations follow generalized extreme value (GEV) distributions whose locations and scales are jointly spatially dependent where the dependence is captured using multivariate Markov random field models specified through coregionalization. In addition, there is temporal dependence in the locations. There are various ways to provide appropriate specifications; we consider four choices. The models can be fitted using a Markov Chain Monte Carlo (MCMC) algorithm to enable inference for parameters and to provide spatio–temporal predictions. We fit the models to a set of gridded interpolated precipitation data collected over a 50-year period for the Cape Floristic Region in South Africa, summarizing results for what appears to be the best choice of model.

Estimating equations for separable spatial-temporal binary data

For binary data with correlation across space and over time, the literature concerning the estimation of fixed effects in marginal models is limited. In this paper, we model the marginal probability of binary responses in terms of parameters of interest by a logistic function. An estimating equation based on the quasi-likelihood concept is developed to estimate parameters. Under separable correlation models, we show that the quasi-likelihood estimate is asymptotically optimal. A series of simulations is conducted to evaluate how the efficiency varies with the regression coefficients. We also compare the relative efficiency with another estimating equation by simulations. The proposed method is applied to an ecological study of forest decline to test independence of two spatial-temporal binary outcomes.     

Evaluating the importance of demographic connectivity in a marine metapopulation

Recently researchers have gone to great lengths to measure marine metapopulation connectivity via tagging, genetic, and trace-elemental fingerprinting studies. These empirical estimates of larval dispersal are key to assessing the significance of metapopulation connectivity within a demographic context, but the life-history data required to do this are rarely available. To evaluate the demographic consequences of connectivity we constructed seasonal, size-structured metapopulation matrix models for two species of mytilid mussel in San Diego County, California, USA. The self-recruitment and larval exchange terms were produced from a time series of realized connectivities derived from trace-elemental fingerprinting of larval shells during spring and fall from 2003 to 2008. Both species exhibited a strong seasonal pattern of southward movement of recruits in spring and northward movement in fall. Growth and mortality terms were estimated using mark-recapture data from representative sites for each species and subpopulation, and literature estimates of juvenile mortality. Fecundity terms were estimated using county-wide settlement data from 2006-2008; these data reveal peak reproduction and recruitment in fall for Mytilus californianus, and spring for M. galloprovincialis. Elasticity and life-stage simulation analyses were employed to identify the season- and subpopulation-specific vital rates and connectivity terms to which the metapopulation growth rate (lambda) was most sensitive. For both species, metapopulation growth was most sensitive to proportional changes in adult fecundity, survival and growth of juvenile stages, and population connectivity, in order of importance, but relatively insensitive to adult growth or survival. The metapopulation concept was deemed appropriate for both Mytilus species as exchange between the subpopulations was necessary for subpopulation persistence. However, highest metapopulation growth occurred in years when a greater proportion of recruits was retained within the predominant source subpopulation. Despite differences in habitat and planktonic duration, both species exhibited similar overall metapopulation dynamics with respect to key life stages and processes. However, different peak reproductive periods in an environment of seasonal current reversals led to different regional (subpopulation) contributions to metapopulation maintenance; this result emphasizes the importance of connectivity analysis for spatial management of coastal resources.     

Spatial extremes of wildfire sizes: Bayesian hierarchical models for extremes

In Portugal, due to the combination of climatological and ecological factors, large wildfires are a constant threat and due to their economic impact, a big policy issue. In order to organize efficient fire fighting capacity and resource management, correct quantification of the risk of large wildfires are needed. In this paper, we quantify the regional risk of large wildfire sizes, by fitting a Generalized Pareto distribution to excesses over a suitably chosen high threshold. Spatio-temporal variations are introduced into the model through model parameters with suitably chosen link functions. The inference on these models are carried using Bayesian Hierarchical Models and Markov chain Monte Carlo methods.     

Multivariate Bayesian analysis of atmosphere–ocean general circulation models

Numerical experiments based on atmosphere–ocean general circulation models (AOGCMs) are one of the primary tools in deriving projections for future climate change. Although each AOGCM has the same underlying partial differential equations modeling large scale effects, they have different small scale parameterizations and different discretizations to solve the equations, resulting in different climate projections. This motivates climate projections synthesized from results of several AOGCMs’ output. We combine present day observations, present day and future climate projections in a single highdimensional hierarchical Bayes model. The challenging aspect is the modeling of the spatial processes on the sphere, the number of parameters and the amount of data involved. We pursue a Bayesian hierarchical model that separates the spatial response into a large scale climate change signal and an isotropic process representing small scale variability among AOGCMs. Samples from the posterior distributions are obtained with computer-intensive MCMC simulations. The novelty of our approach is that we use gridded, high resolution data covering the entire sphere within a spatial hierarchical framework. The primary data source is provided by the Coupled Model Intercomparison Project (CMIP) and consists of 9 AOGCMs on a 2.8 by 2.8 degree grid under several different emission scenarios. In this article we consider mean seasonal surface temperature and precipitation as climate variables. Extensions to our model are also discussed.     

Modeling seasonal course of carbon fluxes and evapotranspiration in response to low temperature and moisture in a boreal Scots pine ecosystem

Environmental conditions act above and below ground, and regulate carbon fluxes and evapotranspiration. The productivity of boreal forest ecosystems is strongly governed by low temperature and moisture conditions, but the understanding of various feedbacks between vegetation and environmental conditions is still unclear. In order to quantify the seasonal responses of vegetation to environmental factors, the seasonality of carbon and heat fluxes and the corresponding responses for temperature and moisture in air and soil were simulated by merging a process-based model (CoupModel) with detailed measurements representing various components of a forest ecosystem in Hyytiälä, southern Finland. The uncertainties in parameters, model assumptions, and measurements were identified by generalized likelihood uncertainty estimation (GLUE). Seasonal and diurnal courses of sensible and latent heat fluxes and net ecosystem exchange (NEE) of CO₂ were successfully simulated for two contrasting years. Moreover, systematic increases in efficiency of photosynthesis, water uptake, and decomposition occurred from spring to summer, demonstrating the strong coupling between processes. Evapotranspiration and NEE flux both showed a strong response to soil temperature conditions via different direct and indirect ecosystem mechanisms. The rate of photosynthesis was strongly correlated with the corresponding water uptake response and the light use efficiency. With the present data and model assumptions, it was not possible to precisely distinguish the various regulating ecosystem mechanisms. Our approach proved robust for modeling the seasonal course of carbon fluxes and evapotranspiration by combining different independent measurements. It will be highly interesting to continue using long-term series data and to make additional tests of optional stomatal conductance models in order to improve our understanding of the boreal forest ecosystem in response to climate variability and environmental conditions.     

Lifting a veil on diversity: a Bayesian approach to fitting relative-abundance models.

Bayesian methods incorporate prior knowledge into a statistical analysis. This prior knowledge is usually restricted to assumptions regarding the form of probability distributions of the parameters of interest, leaving their values to be determined mainly through the data. Here we show how a Bayesian approach can be applied to the problem of drawing inference regarding species abundance distributions and comparing diversity indices between sites. The classic log series and the lognormal models of relative- abundance distribution are apparently quite different in form. The first is a sampling distribution while the other is a model of abundance of the underlying population. Bayesian methods help unite these two models in a common framework. Markov chain Monte Carlo simulation can be used to fit both distributions as small hierarchical models with shared common assumptions. Sampling error can be assumed to follow a Poisson distribution. Species not found in a sample, but suspected to be present in the region or community of interest, can be given zero abundance. This not only simplifies the process of model fitting, but also provides a convenient way of calculating confidence intervals for diversity indices. The method is especially useful when a comparison of species diversity between sites with different sample sizes is the key motivation behind the research. We illustrate the potential of the approach using data on fruit-feeding butterflies in southern Mexico. We conclude that, once all assumptions have been made transparent, a single data set may provide support for the belief that diversity is negatively affected by anthropogenic forest disturbance. Bayesian methods help to apply theory regarding the distribution of abundance in ecological communities to applied conservation.