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.     

Spatial statistical models that use flow and stream distance

We develop spatial statistical models for stream networks that can estimate relationships between a response variable and other covariates, make predictions at unsampled locations, and predict an average or total for a stream or a stream segment. There have been very few attempts to develop valid spatial covariance models that incorporate flow, stream distance, or both. The application of typical spatial autocovariance functions based on Euclidean distance, such as the spherical covariance model, are not valid when using stream distance. In this paper we develop a large class of valid models that incorporate flow and stream distance by using spatial moving averages. These methods integrate a moving average function, or kernel, against a white noise process. By running the moving average function upstream from a location, we develop models that use flow, and by construction they are valid models based on stream distance. We show that with proper weighting, many of the usual spatial models based on Euclidean distance have a counterpart for stream networks. Using sulfate concentrations from an example data set, the Maryland Biological Stream Survey (MBSS), we show that models using flow may be more appropriate than models that only use stream distance. For the MBSS data set, we use restricted maximum likelihood to fit a valid covariance matrix that uses flow and stream distance, and then we use this covariance matrix to estimate fixed effects and make kriging and block kriging predictions. Received: July 2005 / Revised: March 2006     

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.     

Space-time stick-breaking processes for small area disease cluster estimation

We propose a space-time stick-breaking process for the disease cluster estimation. The dependencies for spatial and temporal effects are introduced by using space-time covariate dependent kernel stick-breaking processes. We compared this model with the space-time standard random effect model by checking each model’s ability in terms of cluster detection of various shapes and sizes. This comparison was made for simulated data where the true risks were known. For the simulated data, we have observed that space-time stick-breaking process performs better in detecting medium- and high-risk clusters. For the real data, county specific low birth weight incidences for the state of South Carolina for the years 1997–2007, we have illustrated how the proposed model can be used to find grouping of counties of higher incidence rate.     

Spatial statistics for modeling of abundance and distribution of wildlife species in the Masai Mara ecosystem, Kenya

This study illustrates the use of modern statistical procedures for better wildlife management by addressing three key issues: determination of abundance, modeling of animal distributions and variability of diversity in space and time. Prior information in Markov Chain Monte Carlo (MCMC) methods is used to improve estimates of abundance. Measures of autocorrelation are included when modeling distributions of animal counts, and a diversity index to indicate species abundance and richness for large herbivores is developed. Data from the Masai Mara ecosystem in Kenya are used to develop and demonstrate these procedures. The new abundance estimates are up to 35% more accurate than those obtained by existing methods. Significant temporal changes in spatial patterns are found from a space-time analysis of elephant counts over a 20-year period, with strong interactions over 5 km and 6 months space and time separations, respectively. The new diversity index is sensitive to both high abundance and species richness and is also able to capture year to year variation. It indicates an overall marginal decrease in diversity for large herbivores in the Mara ecosystem. The space-time analyses and diversity index can easily be computed thereby providing tools for rapid decision making.     

On spline-based approaches to spatial linear regression for geostatistical data

For spatial linear regression, the traditional approach to capture spatial dependence is to use a parametric linear mixed-effects model. Spline surfaces can be used as an alternative to capture spatial variability, giving rise to a semiparametric method that does not require the specification of a parametric covariance structure. The spline component in such a semiparametric method, however, impacts the estimation of the regression coefficients. In this paper, we investigate such an impact in spatial linear regression with spline-based spatial effects. Statistical properties of the regression coefficient estimators are established under the model assumptions of the traditional spatial linear regression. Further, we examine the empirical properties of the regression coefficient estimators under spatial confounding via a simulation study. A data example in precision agriculture research regarding soybean yield in relation to field conditions is presented for illustration.     

An ecological model of the habitat mosaic in estuarine nursery areas: Part I—Interaction of dispersal theory and habitat variability in describing juvenile fish distributions

Identification of critical habitat in estuarine nursery areas is an important conservation and management objective. Habitat can be viewed as a mosaic of both temporally variable environmental features and spatially variable structural features that combine to define optimal habitat. Effective models of juvenile distributions should account for individual movement, as well as the full suite of habitat variability including both spatial and temporal components. We have extended a terrestrial model of small-scale movement patterns to describe habitat choices of an index juvenile fish in an estuarine nursery system. Movement of small juvenile fishes was found to be influenced by both spatial and temporal patterns in habitat quality, and it was a balanced mix of both that resulted in an optimal distribution. Fishes that perceive habitat on a scale much smaller than the scale of spatial heterogeneity may respond to temporal change as a movement cue allowing for more deterministic outcomes at larger scales despite perceptual limitations. These model outcomes suggest a hierarchical approach is best for describing habitat choice in juvenile fishes and this approach will be used in the future to explore individual and population responses to predictable habitat change.     

Space-time Bayesian small area disease risk models: development and evaluation with a focus on cluster detection

This paper extends the spatial local-likelihood model and the spatial mixture model to the space-time (ST) domain. For comparison, a standard random effect space-time (SREST) model is examined to allow evaluation of each model’s ability in relation to cluster detection. To pursue this evaluation, we use the ST counterparts of spatial cluster detection diagnostics. The proposed criteria are based on posterior estimates (e.g., misclassification rate) and some are based on post-hoc analysis of posterior samples (e.g., exceedance probability). In addition, we examine more conventional model fit criteria including mean square error (MSE). We illustrate the methodology with a real ST dataset, Georgia throat cancer mortality data for the years 1994–2005, and a simulated dataset where different levels and shapes of clusters are embedded. Overall, it is found that conventional SREST models fair well in ST cluster detection and in goodness-of-fit, while for extreme risk detection the local likelihood ST model does best.     

A fast Bayesian method for updating and forecasting hourly ozone levels

A Bayesian hierarchical space-time model is proposed by combining information from real-time ambient AIRNow air monitoring data, and output from a computer simulation model known as the Community Multi-scale Air Quality (Eta-CMAQ) forecast model. A model validation analysis shows that the model predicted maps are more accurate than the maps based solely on the Eta-CMAQ forecast data for a 2 week test period. These out-of sample spatial predictions and temporal forecasts also outperform those from regression models with independent Gaussian errors. The method is fully Bayesian and is able to instantly update the map for the current hour (upon receiving monitor data for the current hour) and forecast the map for several hours ahead. In particular, the 8 h average map which is the average of the past 4 h, current hour and 3 h ahead is instantly obtained at the current hour. Based on our validation, the exact Bayesian method is preferable to more complex models in a real-time updating and forecasting environment.     

A spatial-temporal optimization approach to watershed management: Acid mine drainage treatment in the Cheat River watershed, WV, USA

Most water management studies concentrate on the inter-temporal allocation problem or, more recently, spatial dynamics - but not both. While early spatial-temporal studies focused on the allocation of water quantity, this paper presents an approach to water quality analysis that incorporates both spatial and temporal dynamics in a watershed framework. The acid mine drainage (AMD) problem in the Cheat River watershed of West Virginia, USA, serves as a case study and provides an opportunity to test the modeling approach developed. The empirical models are written in General Algebraic Modeling System (GAMS) and solved using the CPLEX mixed integer programming package. The results suggest that available investments should be concentrated in heavily impaired stream segments. The model can be used to assess the economic implications of alternative watershed Total Maximum Daily Load (TMDL) implementation or other management strategies.     

Modelling abiotic indicators when obtaining spatial predictions of species richness

When conservation biologists formulate strategy used in decisions concerning the locations of new national parks, refuges and reserves, accurate information about species richness and spatialpatterns of species distributions can be critical. Recent research has demonstrated that spatial models and bioindicator taxa can be quite useful for determining generalized spatial patterns of unrelated taxa on a continental scale. In this research, I incorporate abiotic effects, in this case altitudinal relief, into both the mean and the covariance structures of the spatial prediction model. I use bird species data collected in the Indian subcontinent and cross-validation techniques to illustrate the degree of improvement in prediction accuracy engendered by using theabiotic factor and the modified spatial models.     

Space-time modeling of environmental monitoring data

This study describes and applies statistical methods for space-time modeling of data from environmental monitoring programs, e.g., within areas such as climate change, air pollution and aquatic environment. Such data are often characterized by sparse sampling in both the temporal and spatial dimensions. In order to improve the amount of information on the physical system in question we suggest using statistical modeling methods for monitoring data. Model predictions combined with observations could be analyzed directly to assess the environmental state or as forcing functions for time series models and deterministic, hydrodynamic models. To illustrate the approach we applied the proposed modeling methods to data from the Danish and Swedish marine monitoring programs. Time series with a weekly resolution were predicted from observations of dissolved inorganic nitrogen (DIN) from the Kattegat basin (1993–1997). DIN observations were sparse, irregularly distributed and comprised approximately 10% of the generated time series.     

Combined integral and particle model for describing the dispersion,dilution, terminal layer formation and influence area from a point source discharge into a water body

Contamination of coastal water is a persistent threat to ecosystems around the world. In this study, a novel model for describing the dispersion, dilution, terminal layer formation and influence area from a point source discharge into a water body is presented and compared with field measured data. The model is a Combined Integral and Particle model (CIPMO). In the initial stage, the motion, dispersion and dilution of a buoyant jet are calculated. The output from the buoyant jet model is then coupled with a Lagrangian Advection and Diffusion model describing the far-field. CIPMO ensures that both the near- and far-field processes are adequately resolved. The model either uses empirical data or collects environmental forcing data from open source hydrodynamic models with high spatial and temporal resolution. The method for coupling the near-field buoyant jet and the particle tracking model is described and the output is discussed. The model shows good results when compared with measurements from a field study.

     

Optimization of ecosystem model parameters using spatio-temporal soil moisture information

Parameters in process-based terrestrial ecosystem models are often nonlinearly related to the water flux to the atmosphere, and they also change temporally and spatially. Therefore, for estimating soil moisture, process-based terrestrial ecosystem models inevitably need to specify spatially and temporally variant model parameters. This study presents a two-stage data assimilation scheme (TSDA) to spatially and temporally optimize some key parameters of an ecosystem model which are closely related to soil moisture. At the first stage, a simplified ecosystem model, namely the Boreal Ecosystem Productivity Simulator (BEPS), is used to obtain the prior estimation of daily soil moisture. After the spatial distribution of 0–10 cm surface soil moisture is derived from remote sensing, an Ensemble Kalman Filter is used to minimize the difference between the remote sensing model results, through optimizing some model parameters spatially. At the second stage, BEPS is reinitialized using the optimized parameters to provide the updated model predictions of daily soil moisture. TSDA has been applied to an arid and semi-arid area of northwest China, and the performance of the model for estimating daily 0–10 cm soil moisture after parameter optimization was validated using field measurements. Results indicate that the TSDA developed in this study is robust and efficient in both temporal and spatial model parameter optimization. After performing the optimization, the correlation (r²) between model-predicted 0–10 cm soil moisture and field measurement increased from 0.66 to 0.75. It is demonstrated that spatial and temporal optimization of ecosystem model parameters can not only improve the model prediction of daily soil moisture but also help to understand the spatial and temporal variation of some key parameters in an ecosystem model and the corresponding ecological mechanisms controlling the variation.     

Stochastic gravity models for modeling lake invasions

Freshwater aquatic systems in North America are being invaded by many different species, ranging from fish, mollusks, cladocerans to various bacteria and viruses. These invasions have serious ecological and economic impacts. Human activities such as recreational boating are an important pathway for dispersal. Gravity models are used to quantify the dispersal effect of human activity. Gravity models currently used in ecology are deterministic. This paper proposes the use of stochastic gravity models in ecology, which provides new capabilities both in model building and in potential model applications. These models allow us to use standard statistical inference tools such as maximum likelihood estimation and model selection based on information criteria. To facilitate prediction, we use only those covariates that are easily available from common data sources and can be forecasted in future. This is important for forecasting the spread of invasive species in geographical and temporal domain. The proposed model is portable, that is it can be used for estimating relative boater traffic and hence relative propagule pressure for the lakes not covered by current boater surveys. This makes our results broadly applicable to various invasion prediction and management models.     

Scaling-up spatially-explicit ecological models using graphics processors

How the properties of ecosystems relate to spatial scale is a prominent topic in current ecosystem research. Despite this, spatially explicit models typically include only a limited range of spatial scales, mostly because of computing limitations. Here, we describe the use of graphics processors to efficiently solve spatially explicit ecological models at large spatial scale using the CUDA language extension. We explain this technique by implementing three classical models of spatial self-organization in ecology: a spiral-wave forming predator-prey model, a model of pattern formation in arid vegetation, and a model of disturbance in mussel beds on rocky shores. Using these models, we show that the solutions of models on large spatial grids can be obtained on graphics processors with up to two orders of magnitude reduction in simulation time relative to normal pc processors. This allows for efficient simulation of very large spatial grids, which is crucial for, for instance, the study of the effect of spatial heterogeneity on the formation of self-organized spatial patterns, thereby facilitating the comparison between theoretical results and empirical data. Finally, we show that large-scale spatial simulations are preferable over repetitions at smaller spatial scales in identifying the presence of scaling relations in spatially self-organized ecosystems. Hence, the study of scaling laws in ecology may benefit significantly from implementation of ecological models on graphics processors.     

Risk spreading, connectivity, and optimal reserve spacing

Two important processes determining the dynamics of spatially structured populations are dispersal and the spatial covariance of demographic fluctuations. Spatially explicit approaches to conservation, such as reserve networks, must consider the tension between these two processes and reach a balance between distances near enough to maintain connectivity, but far enough to benefit from risk spreading. Here, we model this trade-off. We show how two measures of metapopulation persistence depend on the shape of the dispersal kernel and the shape of the distance decay in demographic covariance, and we consider the implications of this trade-off for reserve spacing. The relative rates of distance decay in dispersal and demographic covariance determine whether the long-run metapopulation growth rate, and quasi-extinction risk, peak for adjacent patches or intermediately spaced patches; two local maxima in metapopulation persistence are also possible. When dispersal itself fluctuates over time, the trade-off changes. Temporal variation in mean distance that propagules are dispersed (i.e., propagule advection) decreases metapopulation persistence and decreases the likelihood that persistence will peak for adjacent patches. Conversely, variation in diffusion (the extent of random spread around mean dispersal) increases metapopulation persistence overall and causes it to peak at shorter inter-patch distances. Thus, failure to consider temporal variation in dispersal processes increases the risk that reserve spacings will fail to meet the objective of ensuring metapopulation persistence. This study identifies two phenomena that receive relatively little attention in empirical work on reserve spacing, but that can qualitatively change the effectiveness of reserve spacing strategies: (1) the functional form of the distance decay in covariance among patch-specific demographic rates and (2) temporal variation in the shape of the dispersal kernel. The sensitivity of metapopulation recovery and persistence to how covariance of vital rates decreases with distance suggests that estimating the shape of this function is likely to be as important for effective reserve design as estimating connectivity. Similarly, because temporal variation in dispersal dynamics influences the effect of reserve spacing, approaches to reserve design that ignore such variation, and rely instead on long-term average dispersal patterns, are likely to lead to lower metapopulation viability than is actually achievable.     

Estimating the spatial covariance structure using the geoadditive model

In geostatistics, both kriging and smoothing splines are commonly used to generate an interpolated map of a quantity of interest. The geoadditive model proposed by Kammann and Wand (J R Stat Soc: Ser C (Appl Stat) 52(1):1–18, 2003) represents a fusion of kriging and penalized spline additive models. Complex data issues, including non-linear covariate trends, multiple measurements at a location and clustered observations are easily handled using the geoadditive model. We propose a likelihood based estimation procedure that enables the estimation of the range (spatial decay) parameter associated with the penalized splines of the spatial component in the geoadditive model. We present how the spatial covariance structure (covariogram) can be derived from the geoadditive model. In a simulation study, we show that the underlying spatial process and prediction of the spatial map are estimated well using the proposed likelihood based estimation procedure. We present several applications of the proposed methods on real-life data examples.     

Aphid population response to agricultural landscape change: A spatially explicit,individual-based model

A spatially explicit individual-based simulation model has been developed to represent aphid population dynamics in agricultural landscapes. The application of the model to Rhopalosiphum padi (L.) population dynamics is detailed, including an outline of the construction of the model, its parameterisation and validation. Over time, the aphids interact with the landscape and with one another. The landscape is modified by varying a simple pesticide regime, and the multi-scale spatial and temporal implications for a population of aphids is analysed. The results show that a spatial modelling approach that considers the effects on the individual of landscape properties and factors such as wind speed and wind direction provides novel insight into aphid population dynamics both spatially and temporally. This forms the basis for the development of further simulation models that can be used to analyse how changes in landscape structure impact upon important species distributions and population dynamics.