An application of Bayesian variable selection to spatial concurrent linear models

Spatial concurrent linear models, in which the model coefficients are spatial processes varying at a local level, are flexible and useful tools for analyzing spatial data. One approach places stationary Gaussian process priors on the spatial processes, but in applications the data may display strong nonstationary patterns. In this article, we propose a Bayesian variable selection approach based on wavelet tools to address this problem. The proposed approach does not involve any stationarity assumptions on the priors, and instead we impose a mixture prior directly on each wavelet coefficient. We introduce an option to control the priors such that high resolution coefficients are more likely to be zero. Computationally efficient MCMC procedures are provided to address posterior sampling, and uncertainty in the estimation is assessed through posterior means and standard deviations. Examples based on simulated data demonstrate the estimation accuracy and advantages of the proposed method. We also illustrate the performance of the proposed method for real data obtained through remote sensing.     

Time-varying models for extreme values

We propose a new approach for modeling extreme values that are measured in time and space. First we assume that the observations follow a Generalized Extreme Value (GEV) distribution for which the location, scale or shape parameters define the space–time structure. The temporal component is defined through a Dynamic Linear Model (DLM) or state space representation that allows to estimate the trend or seasonality of the data in time. The spatial element is imposed through the evolution matrix of the DLM where we adopt a process convolution form. We show how to produce temporal and spatial estimates of our model via customized Markov Chain Monte Carlo (MCMC) simulation. We illustrate our methodology with extreme values of ozone levels produced daily in the metropolitan area of Mexico City and with rainfall extremes measured at the Caribbean coast of Venezuela.     

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.     

A simple non-separable,non-stationary spatiotemporal model for ozone

The past two decades have witnessed an increasing interest in the use of space-time models for a wide range of environmental problems. The fundamental tool used to embody both the temporal and spatial components of the phenomenon in question is the covariance model. The empirical estimation of space-time covariance models can prove highly complex if simplifying assumptions are not employed. For this reason, many studies assume both spatiotemporal stationarity, and the separability of spatial and temporal components. This second assumption is often unrealistic from the empirical point of view. This paper proposes the use of a model in which non-separability arises from temporal non-stationarity. The model is used to analyze tropospheric ozone data from the Emilia-Romagna Region of Italy.     

Spatial and temporal modeling of community non-timber forest extraction

This paper examines the interaction of spatial and dynamic aspects of resource extraction from forests by local people. Highly cyclical and varied across space and time, the patterns of resource extraction resulting from the spatial–temporal model bear little resemblance to the patterns drawn from focusing either on spatial or temporal aspects of extraction alone. Ignoring this variability inaccurately depicts villagers’ dependence on different parts of the forest and could result in inappropriate policies. Similarly, the spatial links in extraction decisions imply that policies imposed in one area can have unintended consequences in other areas. Combining the spatial–temporal model with a measure of success in community forest management—the ability to avoid open-access resource degradation—characterizes the impact of incomplete property rights on patterns of resource extraction and stocks.     

Major PM10 source location by a spatial multivariate receptor model

We present a multivariate receptor model for identifying the spatial location of major PM₁₀ pollution sources through the concentrations at multiple monitoring stations. We build on a mixed multiplicative log-normal factor model adjusting the source contributions for meteorological covariates and for temporal correlation and considering source profiles as compositional Gaussian random fields, to account for the variability induced by the spatial distribution of the monitoring sites. Taking a Bayesian approach to estimation, the proposed hierarchical model is implemented and used to analyze average daily PM₁₀ concentration measurements from 13 monitoring sites in Taranto, Italy, for the period April–December 2005. Three major sources of pollution are identified and characterized in terms of their spatial and temporal behavior and in relation to meteorological data.     

Simulating property exchange in estuarine ecosystem models at ecologically appropriate scales

A computational scheme has been developed and tested to simulate property exchange by advection and dispersion in estuaries at time and space scales that are well suited to ecological and management simulations, but are coarse relative to the demands of physical hydrodynamic models. An implementation of the Regional Ocean Model System (ROMS) for the Providence River and Narragansett Bay (RI, USA) was used to determine property exchanges between the spatial elements of an ecological box model. The basis for the method is the statistical tabulation of numerical dye experiments done with the full ROMS physical model. The ROMS model domain was subdivided into fifteen coarse boxes, each with two vertical layers, defining 30 elements that were used for the box model simulations. Dye concentrations were set to arbitrary initial concentrations for all ROMS grids in the large elements, and the ROMS model was run for 24 h. The final distribution of the dye among the elements was used as a tracer for property exchange over that day and was used to develop an exchange matrix. Box model predictions of salinity over 77 days in each element compared favorably with ROMS simulated salinity averaged over the same spatial elements, although the disparity was greater in areas where large river inflows caused strong gradients in ROMS within elements assumed to be homogeneous in the box model. The 77-day simulation included periods of high and low river flow. Despite the large size of the spatial elements, dispersion artifacts were small, much less than the modeled daily exchanges. While others have taken a similar approach, we found a number of theoretical and practical considerations deserved careful attention for this approach to perform satisfactorily. Whereas the full ROMS model takes 9 days on a powerful computing cluster to compute the physics simulation for 77 days, the box model simulates physics and biology for the same interval in 5 s on a personal computer, and a full year in under 1 min. The exchange matrix mixing model is a fast, cost effective, and convenient way to simulate daily variation of complex estuarine physics in ecological modeling at appropriate scales of space and time.     

Modeling species-habitat relationships with spatially autocorrelated observation data.

Spatial autocorrelation in wildlife observation data arises when extrinsic environmental processes and patterns that influence the spatial distribution of wildlife are themselves spatially structured, or when species are subject to intrinsic population processes, causing contagion or dispersion effects. Territoriality, Allee effects, dispersal limitations, and social clustering are examples of intrinsic processes. Both forms of autocorrelation can violate the assumptions of generalized linear regression models, resulting in biased estimation of model coefficients and diminished predictive performance. Such consequences may be avoided for extrinsic autocorrelation when autocorrelated environmental variables are available for use as model covariates, whereas intrinsic spatial autocorrelation requires an alternative modeling approach. The autologistic model provides an approach suited to the binary observations often obtained in wildlife surveys, but its performance has not been tested across widely varying sampling intensities or strengths of intrinsic spatial structure. Here we use simulated data to test the autologistic model under a range of sampling conditions. The autologistic model obtains better fits and substantially better predictive performance than the standard logistic regression model over the full range of sampling designs and intensities tested. We provide a simple Bayesian implementation of the autologistic model, which until now has not been achieved with standard statistical software alone. A step-by-step procedure is given for characterizing and modeling spatial autocorrelation in binary observation data, along with computer code for fitting autologistic models in WinBUGS, a freeware Bayesian analysis package. This approach avoids normal approximations to the pseudo-likelihood, in contrast to previous Bayesian applications of the autologistic model. We provide a sample application of the autologistic model, fitted to survey data for a gliding marsupial in southeastern Australia.     

Forecasting population trend from the scaling pattern of occupancy

Forecasting the temporal trend of a focal species, its range expansion or retraction, provides crucial information regarding population viability. To this end, we require the accumulation of temporal records which is evidently time consuming. Progress in spatial data capturing has enabled rapid and accurate assessment of species distribution across large scales. Therefore, it would be appealing to infer the temporal trends of populations from the spatial structure of their distributions. Based on a combination of models from the fields of range dynamics, occupancy scaling and spatial autocorrelation, here I present a model for forecasting the population trend solely from its spatial distribution. Numerical tests using cellular automata confirm a positive correlation, as inferred from the model, between the temporal change in species range sizes and the exponent of the power-law scaling pattern of occupancy. The model is thus recommended for rapid estimation of species range dynamics from a single snapshot of its current distribution. Further applications in biodiversity conservation could provide a swift risk assessment, especially, for endangered and invasive species.     

Ordinary kriging for function-valued spatial data

In various scientific fields properties are represented by functions varying over space. In this paper, we present a methodology to make spatial predictions at non-data locations when the data values are functions. In particular, we propose both an estimator of the spatial correlation and a functional kriging predictor. We adapt an optimization criterion used in multivariable spatial prediction in order to estimate the kriging parameters. The curves are pre-processed by a non-parametric fitting, where the smoothing parameters are chosen by cross-validation. The approach is illustrated by analyzing real data based on soil penetration resistances.     

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.     

Phytoplankton patchiness and their role in the modelled productivity of a large, seasonally stratified lake

Phytoplankton concentration in Lake Kinneret (Israel) has varied up to 10-fold in space and time, with horizontal patches ranging from a couple of kilometres to a basin scale. Previous studies have used a 1D model to reproduce the temporal evolution of physical and biogeochemical variables in this lake. The question that arises then is how appropriate is a 1D approach to represent the dynamic of a spatially heterogeneous system, where there are non-linear dependencies between variables. Field data, a N-P-Z model coupled to both a 1D and a 3D hydrodynamic model, a 1D diffusion-reaction equation and scaling analysis are used to understand the role of spatial variability, expressed as phytoplankton patchiness, in the modelling of primary production. The analysis and results are used to investigate the effect of horizontal variability in the forcing and in the free mechanisms that affect the growth of patterns. The study shows that the use of averaged properties in a 1D approach may produce misleading results in the presence of localised patches, in terms of both concentration and composition of phytoplankton. The reason lies in the fact that the calibration process of ecological parameters in the 1D model appears to be site and process specific. That is, it depends on the pattern's characteristics and the underlying physical processes causing them. And this is a critical point for the success of numerical simulations under spatial variability. In this study, it is also shown that a length scale based on diffusion and growth rate of phytoplankton could be used as a criterion to assess the appropriateness of the 1D assumption.     

Spatially-balanced adaptive web sampling

The paper deals with sampling from a finite population that is distributed over space and has a highly uneven spatial distribution. It suggests a sampling design that allocates a portion of the sample units that are well spread over the population and sequentially selects the remaining units in sub-areas that appear to be of more interest according to the study variable values observed during the survey. In order to estimate the population mean while using this sampling design, a computationally intense estimator, obtained via the Rao–Blackwell approach, is proposed and a resampling method is used that makes the inference computationally feasible. The whole sampling strategy is evaluated through several Monte Carlo experiments.     

Space-time estimation of grid-cell hourly ozone levels for assessment of a deterministic model

We present an approach to estimate hourly grid-cell surface ozone concentrations based on observations from point monitoring sites in space, for comparison with grid-based results from the SARMAP photochemical air-quality model for a region of northern California. Statistical estimation is carried out on a transformed (square root) scale, followed by back-transforming to the original scale of ozone in parts per billion, adjusting for bias and variance. We estimate a spatially-varying diurnal mean structure and a non-separable space-time correlation structure on the transformed scale. Temporal pre-whitening is followed by modelling of a spatially non-stationary, diurnally-varying spatial correlation structure using a spatial deformation approach. Comparisons of SARMAP model results with the estimated grid-cell ozone levels are presented.     

Comparison between remote-sensed data and in situ measurements in coastal waters: The taranto sea case

Monitoring and managing small coastal ecosystems requires a considerable understanding of the temporal dynamics of biophysical factors describing the coastal water systems. For this reason, daily observation from space could be a very efficient tool. The objective of the work described in this paper is to evaluate the contribution of remote sensing to the continuous monitoring of coastal areas. It is well known that in coastal areas, the presence of inorganic suspended sediments and coloured dissolved organic matter can make chlorophyll-concentration measurements from remote sensing difficult. To overcome these difficulties, an alternative approach to the SeaWiFS standard chlorophyll algorithm is presented, based on a semi-analytic model for sea water and on the use of MODIS data as input in a model for atmospheric effects removal. Moreover, land contamination (mixed sea–land pixels) can introduce ambiguities in sea-surface temperature measurements from remote sensing. This paper proposes the use of a hydrodynamic model as a time–space interpolator of in situ campaign data, to extensively validate the temperature values extracted from AVHRR sensor. We validated the proposed approach, using experimental field data collected over a two-year campaign in the Taranto Gulf. The results seem to indicate a good agreement between remote-sensed and in situ data.     

Interpolation performance of a spatio-temporal model with spatially varying coefficients: application to PM<Subscript>10</Subscript> concentrations in Rio de Janeiro

In this work we present a Bayesian analysis in linear regression models with spatially varying coefficients for modeling and inference in spatio-temporal processes. This kind of model is particularly appealing in situations where the effect of one or more explanatory processes on the response present substantial spatial heterogeneity. We describe for this model how to make inference about the regression coefficients and response processes under two scenarios: when the explanatory processes are known throughout the study region, and when they are known only at the sampling locations. Using a simulation experiment we investigate how parameter inference and interpolation performance are affected by some features of the data and prior distribution that is used. The proposed methodology is used to model the dataset on PM₁₀ levels in the metropolitan region of Rio de Janeiro presented in Paez and Gamerman (2003).     

A new algorithm for estimating the parameters of the spatial generalized linear mixed models

Non-Gaussian spatial responses are usually modeled using a spatial generalized linear mixed model with location specific latent variables. The likelihood function of this model cannot usually be given in a closed form, thus the maximum likelihood approach is very challenging. So far, several numerical algorithms to solve the problem of calculating maximum likelihood estimates of this model have been presented. In this paper to estimate the parameters an approximate method is considered and a new algorithm is introduced that is much faster than existing algorithms but just as accurate. This is called the Approximate Expectation Maximization Gradient algorithm. The performance of the proposed algorithm and is illustrated with a simulation study and on a real data set.     

Composite likelihood approach to the regression analysis of spatial multivariate ordinal data and spatial compositional data with exact zero values

In many environmental and ecological studies, it is of interest to model compositional data. One approach is to consider positive random vectors that are subject to a unit-sum constraint. In landscape ecological studies, it is common that compositional data are also sampled in space with some elements of the composition absent at certain sampling sites. In this paper, we first propose a practical spatial multivariate ordered probit model for multivariate ordinal data, where the response variables can be viewed as the discretized non-negative compositions without the unit-sum constraint. We then propose a novel two-stage spatial mixture Dirichlet regression model. The first stage models the spatial dependence and the presence of exact zero values, and the second stage models all the non-zero compositional data. A maximum composite likelihood approach is developed for parameter estimation and inference in both the spatial multivariate ordered probit model and the two-stage spatial mixture Dirichlet regression model. The standard errors of the parameter estimates are computed by an estimate of the Godambe information matrix. A simulation study is conducted to evaluate the performance of the proposed models and methods. A land cover data example in landscape ecology further illustrates that accounting for spatial dependence can improve the accuracy in the prediction of presence/absence of different land covers as well as the magnitude of land cover compositions.     

High Spatial and Temporal Variability of Dry Deposition in a Coastal Region

A real meteorological situation characterized by strong spatial and temporal variability of the meteorological fields in a coastal region of eastern Denmark is examined in view of the transport of a passive tracer and dry deposition. Model simulations using a full mesoscale NWP model (COAMPS^TM) at different horizontal resolutions are performed. A realistic simulation showed that the differences in the amount of dry deposited matter can reach one order of magnitude and larger, during the period of one afternoon, depending on the model horizontal resolution. In addition, the results of an idealized experiment with straight coastline indicate that the horizontal model resolution alone is responsible for most of the differences. This study confirms the importance of high spatial and temporal resolution modelling for environmental applications.