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.     

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.     

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.     

Nonparametric spatial covariance functions: Estimation and testing

Spatial autocorrelation techniques are commonly used to describe genetic and ecological patterns. To improve statistical inference about spatial covariance, we propose a continuous nonparametric estimator of the covariance function in place of the spatial correlogram. The spline correlogram is an adaptation of a recent development in spatial statistics and is a generalization of the commonly used correlogram. We propose a bootstrap algorithm to erect a confidence envelope around the entire covariance function. The meaning of this envelope is discussed. Not all functions that can be drawn inside the envelope are candidate covariance functions, as they may not be positive semidefinite. However, covariance functions that do not fit, are not supported by the data. A direct estimate of the L₀ spatial correlation length with associated confidence interval is offered and its interpretation is discussed. The spline correlogram is found to have high precision when applied to synthetic data. For illustration, the method is applied to electrophoretic data of an alpine grass (Poa alpina).     

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.     

Insights on plant interaction between dominating species from patterns of plant association: expected covariance of pin-point cover measurements of two species

It has been suggested that in order to infer ecological processes from observed patterns of species abundance we need to investigate the covariance in species abundance. Consequently, an expression for the expected covariance of pin-point cover measurements of two species is developed. By comparing the observed covariance with the expected covariance we get a new type of information on the spatial arrangement of two species. Here the discrepancy between the observed and expected covariance may be thought of as a measure of the spatial configuration of the two species that may indicate underling ecological processes. The method is applied in a case study of Calluna vulgaris and Deschampsia flexuosa on dry heathland sites. The observed covariance of Calluna and Deschampsia at the level of the sites was positively and significantly correlated with the expected covariance. Negative covariance was observed on sites where both Calluna and Deschampsia had a high cover, which is in agreement with the notion that both species form distinct patches. Oppositely, at sites where both species have a low cover, we found that both the expected and observed covariance were positive. The proposed measure for the expected covariance of two species does capture information on the combined spatial configuration of the two species if the species are common. We show how this may be relevant for understanding the underlying ecological processes leading to the observed covariance.     

Spatial modeling and prediction under stationary non-geometric range anisotropy

For modeling spatial processes, we propose a rich parametric class of stationary range anisotropic covariance structures that, when applied in R², greatly increases the scope of variogram contors. Geometric anisotropy, which provides the most common generalization of isotropy within stationarity, is a special case. Our class is built from monotonic isotropic correlation functions and special cases include the Matérn and the general exponential functions. As a result, our range anisotropic correlation specification can be attached to a second order stationary spatial process model, unlike ad hoc approaches to range anisotropy in the literature. We adopt a Bayesian perspective to obtain full inference and demonstrate how to fit the resulting model using sampling-based methods. In the presence of measurement error/microscale effect, we can obtain both the usual predictive as well as the noiseless predictive distribution. We analyze a data set of scallop catches under the general exponential range anisotropic model, withholding ten sites to compare the accuracy and precision of the standard and noiseless predictive distributions.     

Hierarchical Bayesian Spatial Models for Multispecies Conservation Planning and Monitoring

Abstract: Biologists who develop and apply habitat models are often familiar with the statistical challenges posed by their data's spatial structure but are unsure of whether the use of complex spatial models will increase the utility of model results in planning. We compared the relative performance of nonspatial and hierarchical Bayesian spatial models for three vertebrate and invertebrate taxa of conservation concern (Church's sideband snails [Monadenia churchi], red tree voles [Arborimus longicaudus], and Pacific fishers [Martes pennanti pacifica]) that provide examples of a range of distributional extents and dispersal abilities. We used presence–absence data derived from regional monitoring programs to develop models with both landscape and site‐level environmental covariates. We used Markov chain Monte Carlo algorithms and a conditional autoregressive or intrinsic conditional autoregressive model framework to fit spatial models. The fit of Bayesian spatial models was between 35 and 55% better than the fit of nonspatial analogue models. Bayesian spatial models outperformed analogous models developed with maximum entropy (Maxent) methods. Although the best spatial and nonspatial models included similar environmental variables, spatial models provided estimates of residual spatial effects that suggested how ecological processes might structure distribution patterns. Spatial models built from presence–absence data improved fit most for localized endemic species with ranges constrained by poorly known biogeographic factors and for widely distributed species suspected to be strongly affected by unmeasured environmental variables or population processes. By treating spatial effects as a variable of interest rather than a nuisance, hierarchical Bayesian spatial models, especially when they are based on a common broad‐scale spatial lattice (here the national Forest Inventory and Analysis grid of 24 km² hexagons), can increase the relevance of habitat models to multispecies conservation planning.     

Bayesian spatial prediction

This paper presents a complete Bayesian methodology for analyzing spatial data, one which employs proper priors and features diagnostic methods in the Bayesian spatial setting. The spatial covariance structure is modeled using a rich class of covariance functions for Gaussian random fields. A general class of priors for trend, scale, and structural covariance parameters is considered. In particular, we obtain analytic results that allow easy computation of the predictive distribution for an arbitrary prior on the parameters of the covariance function using importance sampling. The computations, as well as model diagnostics and sensitivity analysis, are illustrated with a set of precipitation data.     

A general model for analyzing Taylor's spatial scaling laws

Taylor's spatial scaling law concerns the relation between the variance and the mean population counts within areas of a given size. For a range of area sizes, the log of the variance often is an approximately linear function of the mean with a slope between 1 and 2, depending on the range of areas considered. In this paper, we investigate this relationship theoretically for random quadrat samples within a large area. The model makes a distinction between the local point process determining the position of each individual and the population density described by a spatial covariance function. The local point process and the spatial covariance of population density both contribute to the general relationship between the mean and the variance in which the slope may begin at 1, increase to 2, and decrease to 1 again. It is demonstrated by an example that the slope theoretically may exceed 2 by a small amount for very regular patterns that generate spatial covariance functions that increase in certain intervals. We also show how properties of population dynamics in space and time determine this relationship.     

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     

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.     

Calibration and uncertainty analysis of a simple model of silica-limited diatom growth in the Elbe River

A simple Lagrangian water quality model was designed to investigate the hypothesis of sporadic silica limitations of diatom growth in the lower Elbe River in Germany. For each fluid parcel a limited reservoir of silica was specified to be consumed by diatoms. The model's simplicity notwithstanding, a set of six selected model parameters could not be fully identified from existing observations at one station. After the introduction of prior knowledge of the ranges of meaningful parameter values, calibration of the over-parameterised model manifested itself primarily in the generation of posterior parameter covariances. Estimations of the covariance matrix based on (a) second order partial derivatives of a quadratic cost function at its optimum and (b) Monte Carlo simulations exploring the whole space of parameter values gave consistent results. Diagonalisation of the covariance matrix yielded two linear parameter combinations that were most effectively controlled by data from periods with and without lack of silica, respectively. The two parameter combinations were identified as the essential inputs that govern the successful simulation of intermittently decreasing chlorophyll a concentrations in summer. A satisfactory simulation of the pronounced chlorophyll a minimum in spring, by contrast, was found to be beyond the means of the simple model.     

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.

     

Geostatistical analysis of spatial distribution of Liocarcinus depurator,Macropipus tuberculatus and Polybius henslowii (Crustacea: Brachyura) over the Galician continental shelf (NW Spain)

Geostatistics was used to analyze the spatial structure and distribution of three species of brachyurans, Liocarcinus depurator, Macropipus tuberculatus and Polybius henslowii, from sets of data collected during three survey cruises (1983 and 1984) over the Galician continental shelf. The present study investigates the feasibility of using geostatistics with data collected according to traditional methods and of thereby improving the sampling methodology. In order to investigate the spatial structure of the species studied, experimental variograms were calculated and fitted to a spherical model. The spatial structure model was used to estimate and map abundance and distribution of the populations studied using the so-called Kriging technique. Geostatistical analysis enabled the determination of spatial density gradients as well as patch size (14 to 22 km, L. depurator; 10 to 28 km, M. tuberculatus; 7.5 to 28 km, P. henslowii) along the continental shelf. Depth was revealed as a limiting factor, restricting the distribution of L. depurator and M. tuberculatus on a large scale, whereas upwelling processes and nutrient-rich waters from the rías affected the spatial structure on a smaller scale, especially in the case of P. henslowii. A spatial segregation in the distribution of the three species also emerged; this probably arose from differences in physical and biological factors that result in different habitat-exploitation patterns. The study demonstrates the existence of spatial covariance and that the variograms vary as a function of population density and geographical area. This information will be useful in improving the design of future sampling cruises.     

Evolutionary diversity and spatiotemporal dynamics of a spatial game

In this paper, the evolution of cooperation is studied by a spatially structured evolutionary game model in which the players are located on a two-dimensional square lattice. Each player can choose one of the following strategies: “always defect” (ALLD), “tit-for-tat” (TFT), and “always cooperate” (ALLC). Players merely interact with four immediate neighbors at first and adjust strategies according to their rewards. First, the evolutionary dynamics of the three strategies in non-spatial population is investigated, and the results indicate that cooperation is not favored in most settings without spatial structure. Next, an analytical method, which is based on comparing the local payoff structures, is introduced for the spatial game model. Using the conditions derived from the method as criteria, the parameter plane for two major parameters of the spatial game model is divided and nine representative regions are identified. In each parameter region, a distinct spatiotemporal dynamics is characterized. The spatiotemporal dynamics not only verify that the spatial structure promote the evolution of cooperation but also reveal how cooperation is favored. Our results show that spatial structure is the keystone of the evolution of intraspecific diversity.     

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.     

Nitrogen source apportionment—a comparison between a dynamic and a statistical model

A dynamic model, HBV-N, and a statistical model, MESAW, for nitrogen source apportionment were compared regarding model performance, model uncertainty and user applicability. The HBV-N model simulates continuous series of nitrogen concentrations with meteorological data and sub-basin characteristics as input. Diffuse nitrogen emissions are defined as regional model parameters which are calibrated by comparison of observed and simulated nitrogen data. The MESAW model uses nitrogen loads for a fixed time interval at each monitoring site as response variable and sub-basin characteristics as explanatory variables to estimate diffuse nitrogen emissions through non-linear regression analysis. The two models were applied in the Matsalu Bay watershed (3640 km²) in Estonia and the same land use and point sources data were used as input. Both models gave similar levels of diffuse total nitrogen emissions and retention rates, which also fit well with previous estimates made in Estonia and Scandinavia. A sensitivity analysis of the model parameters also showed similar uncertainty levels, which indicated that the model uncertainty was more dependent on the availability of nitrogen data and land cover distribution than the choice of model. Furthermore, the sensitivity analysis showed a parameter interdependency in both models, which implied the risk of compensation between estimated diffuse emissions and retention. In conclusion, however, the study showed that both models were capable of estimating nitrogen leakage from the dominating land classes and giving reliable source apportionment from the available input data. The study indicated that the HBV-N model has its advantage in assessments where detailed outputs are needed and when run-off data are limited, while the statistical MESAW model has its advantage in extensive studies since it is easily applied to large watersheds that have dense monitoring networks.     

The maximum likelihood based intensity estimate for circular point processes

Identifying the geometrical nature of spatial point patterns plays an important role in many areas of scientific research. Common types of spatial point processes involve random, regular, and cluster patterns. However, some point patterns suggest identifiable geometrical shapes such as a circular or other conic patterns. These patterns may be recognized as either a specific clustered shape or an inhomogeneous point pattern. Less noisy conic shapes, including circular patterns, are heavily discussed in the pattern recognition literature, but the goodness-of-fit of conic-fitting algorithms is rarely discussed for very noisy data. This study addresses a parameter estimation technique for noisy circular point patterns using the maximum likelihood principle. Additionally, a spatial statistical tool known as the L-function is used to investigate whether the fitted location pattern is reasonably attributable to a circular shape. A novel quantity named ‘relative log-error’ (\(\gamma \)) is introduced to quantify the goodness-of-fit for circular model fits. An iteratively re-weighted least squares procedure is introduced and robustness is evaluated under several error structures. Computational efficiency of the current and novel circle-fitting methods is also discussed. The findings are applied to two environmental science data sets.     

The Scotia Sea krill fishery and its possible impacts on dependent predators: modeling localized depletion of prey

The nature and impact of fishing on predators that share a fished resource is an important consideration in ecosystem-based fisheries management. Krill (Euphausia superba) is a keystone species in the Antarctic, serving as a fundamental forage source for predators and simultaneously being subject to fishing. We developed a spatial multispecies operating model (SMOM) of krill-predator fishery dynamics to help advise on allocation of the total krill catch among 15 small-scale management units (SSMUs) in the Scotia Sea, with a goal to reduce the potential impact of fishing on krill predators. The operating model describes the underlying population dynamics and is used in simulations to compare different management options for adjusting fishing activities (e.g., a different spatial distribution of catches). The numerous uncertainties regarding the choice of parameter values pose a major impediment to constructing reliable ecosystem models. The pragmatic solution proposed here involves the use of operating models that are composed of alternative combinations of parameters that essentially try to bound the uncertainty in, for example, the choice of survival rate estimates as well as the functional relationships between predators and prey. Despite the large uncertainties, it is possible to discriminate the ecosystem impacts of different spatial fishing allocations. The spatial structure of the model is fundamental to addressing concerns of localized depletion of prey in the vicinity of land-based predator breeding colonies. Results of the model have been considered in recent management deliberations for spatial allocations of krill catches in the Scotia Sea and their associated impacts on dependent predator species.