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.     

Finite area smoothing with generalized distance splines

Most conventional spatial smoothers smooth with respect to the Euclidean distance between observations, even though this distance may not be a meaningful measure of spatial proximity, especially when boundary features are present. When domains have complicated boundaries leakage (the inappropriate linking of parts of the domain which are separated by physical barriers) can occur. To overcome this problem, we develop a method of smoothing with respect to generalized distances, such as within domain distances. We obtain the generalized distances between our points and then use multidimensional scaling to find a configuration of our observations in a Euclidean space of 2 or more dimensions, such that the Euclidian distances between points in that space closely approximate the generalized distances between the points. Smoothing is performed over this new point configuration, using a conventional smoother. To mitigate the problems associated with smoothing in high dimensions we use a generalization of thin plate spline smoothers proposed by Duchon (Constructive theory of functions of several variables, pp 85–100, 1977). This general method for smoothing with respect to generalized distances improves on the performance of previous within-domain distance spatial smoothers, and often provides a more natural model than the soap film approach of Wood et al. (J R Stat Soc Ser B Stat Methodol 70(5):931–955, 2008). The smoothers are of the linear basis with quadratic penalty type easily incorporated into a range of statistical models.     

Spatial methods for plot-based sampling of wildlife populations

Classical sampling methods can be used to estimate the mean of a finite or infinite population. Block kriging also estimates the mean, but of an infinite population in a continuous spatial domain. In this paper, I consider a finite population version of block kriging (FPBK) for plot-based sampling. The data are assumed to come from a spatial stochastic process. Minimizing mean-squared-prediction errors yields best linear unbiased predictions that are a finite population version of block kriging. FPBK has versions comparable to simple random sampling and stratified sampling, and includes the general linear model. This method has been tested for several years for moose surveys in Alaska, and an example is given where results are compared to stratified random sampling. In general, assuming a spatial model gives three main advantages over classical sampling: (1) FPBK is usually more precise than simple or stratified random sampling, (2) FPBK allows small area estimation, and (3) FPBK allows nonrandom sampling designs.     

Sampling designs on stream networks using the pseudo-Bayesian approach

Monitoring stream networks through time provides important ecological information. The sampling design problem is to choose locations where measurements are taken so as to maximise information gathered about physicochemical and biological variables on the stream network. This paper uses a pseudo-Bayesian approach, averaging a utility function over a prior distribution, in finding a design which maximizes the average utility. We use models for correlations of observations on the stream network that are based on stream network distances and described by moving average error models. Utility functions used reflect the needs of the experimenter, such as prediction of location values or estimation of parameters. We propose an algorithmic approach to design with the mean utility of a design estimated using Monte Carlo techniques and an exchange algorithm to search for optimal sampling designs. In particular we focus on the problem of finding an optimal design from a set of fixed designs and finding an optimal subset of a given set of sampling locations. As there are many different variables to measure, such as chemical, physical and biological measurements at each location, designs are derived from models based on different types of response variables: continuous, counts and proportions. We apply the methodology to a synthetic example and the Lake Eacham stream network on the Atherton Tablelands in Queensland, Australia. We show that the optimal designs depend very much on the choice of utility function, varying from space filling to clustered designs and mixtures of these, but given the utility function, designs are relatively robust to the type of response variable.     

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.     

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.     

Linking direct and indirect data on dispersal: isolation by slope in a headwater stream salamander

There is growing recognition of the need to incorporate information on movement behavior in landscape-scale studies of dispersal. One way to do this is by using indirect indices of dispersal (e.g., genetic differentiation) to test predictions derived from direct data on movement behavior. Mark-recapture studies documented upstream-biased movement in the salamander Gyrinophilus porphyriticus (Plethodontidae). Based on this information, we hypothesized that gene flow in G. porphyriticus is affected by the slope of the stream. Specifically, because the energy required for upstream dispersal is positively related to slope, we predicted gene flow to be negatively related to change in elevation between sampling sites. Using amplified DNA fragment length polymorphisms among tissue samples from paired sites in nine streams in the Hubbard Brook Watershed, New Hampshire, USA, we found that genetic distances between downstream and upstream sites were positively related to change in elevation over standardized 1-km distances. This pattern of isolation by slope elucidates controls on population connectivity in stream networks and underscores the potential for specific behaviors to affect the genetic structure of species at the landscape scale. More broadly, our results show the value of combining direct data on movement behavior and indirect indices to assess patterns and consequences of dispersal in spatially complex ecosystems.     

Bayesian entropy for spatial sampling design of environmental data

We develop a spatial statistical methodology to design national air pollution monitoring networks with good predictive capabilities while minimizing the cost of monitoring. The underlying complexity of atmospheric processes and the urgent need to give credible assessments of environmental risk create problems requiring new statistical methodologies to meet these challenges. In this work, we present a new method of ranking various subnetworks taking both the environmental cost and the statistical information into account. A Bayesian algorithm is introduced to obtain an optimal subnetwork using an entropy framework. The final network and accuracy of the spatial predictions is heavily dependent on the underlying model of spatial correlation. Usually the simplifying assumption of stationarity, in the sense that the spatial dependency structure does not change location, is made for spatial prediction. However, it is not uncommon to find spatial data that show strong signs of nonstationary behavior. We build upon an existing approach that creates a nonstationary covariance by a mixture of a family of stationary processes, and we propose a Bayesian method of estimating the associated parameters using the technique of Reversible Jump Markov Chain Monte Carlo. We apply these methods for spatial prediction and network design to ambient ozone data from a monitoring network in the eastern US.     

Estimating the area affected by phosphorus runoff in an Everglades wetland: a comparison of universal kriging and Bayesian kriging

Phosphorus-enriched agriculture runoff is believed to be the leading cause of ecosystem changes of Everglades wetlands. To study this effect, it is necessary to estimate the area of the affected region. In this study, Bayesian kriging and universal kriging were used to estimate the area by analysing the data collected by Reddy et al. (1991). The background level of the soil's total phosphorus concentration is usedto determine whether the region is affected by the agriculture runoff, through an indicator function. The area of the affected region was represented by the integration of the indicator function over the entire wetland. The expected value of the affected area was calculated using the results derived from Bayesian and universal kriging. The outcome indicates that universal kriging is sensitive to specification of thecovariance model. It was observed that universal kriging and Bayesian kriging yield comparable results, if the specified covariance structures are of similar nature.     

A method for assigning species into groups based on generalized Mahalanobis distance between habitat model coefficients

Habitat association models are commonly developed for individual animal species using generalized linear modeling methods such as logistic regression. We considered the issue of grouping species based on their habitat use so that management decisions can be based on sets of species rather than individual species. This research was motivated by a study of western landbirds in northern Idaho forests. The method we examined was to separately fit models to each species and to use a generalized Mahalanobis distance between coefficient vectors to create a distance matrix among species. Clustering methods were used to group species from the distance matrix, and multidimensional scaling methods were used to visualize the relations among species groups. Methods were also discussed for evaluating the sensitivity of the conclusions because of outliers or influential data points. We illustrate these methods with data from the landbird study conducted in northern Idaho. Simulation results are presented to compare the success of this method to alternative methods using Euclidean distance between coefficient vectors and to methods that do not use habitat association models. These simulations demonstrate that our Mahalanobis-distance-based method was nearly always better than Euclidean-distance-based methods or methods not based on habitat association models. The methods used to develop candidate species groups are easily explained to other scientists and resource managers since they mainly rely on classical multivariate statistical methods.     

Graphical spatial models: a new view on interpreting spatial pattern

Graphical models provide an important tool for facilitating communication between scientists, decision-makers, and statisticians—many complicated ecological processes can be described in terms of “box-and-arrow” conceptual diagrams (e.g., Shipley in Cause and correlation in biology: a user’s guide to path analysis, structural equations and causal inferences, Cambridge Universtiy Press, Cambridge, 2000; Clark and Gelfand TRENDS in Ecology and Evolution 21:375–380, 2006). In particular, problems in landscape ecology often involve modeling relationships among multiple physical and/or biological variables that may operate on differing spatial scales (e.g., Rossi et al. in Ecol Monographs 62:277–314, 1992; Legendre et al. in Ecography 25:601–615, 2002; Overmars et al. in Ecol Model 164:257–270, 2003; Brown and Spector in J Appl Ecol 45:1639–1648, 2008; Koniak and Noy-Meir in Ecol Model 220:1148–1158, 2008). These problems are inherently multivariate, though researchers commonly rely on univariate methods, such as spatial regression models, to address them. In this paper, we introduce a multivariate method—graphical spatial models—that extends path analysis to incorporate spatial autocorrelation in one or more variables in a directed graph. We show how both exogenous and endogenous ecological processes as defined by Legendre et al. (Ecography 25:601–615, 2002) and Lichstein et al. (Ecol Monographs 72:445–463, 2002) can be represented in a graph. Most importantly, we show how to translate graphs representing these ecological processes into statistically estimable models. We motivate our theoretical results using an example of stream health data from the Willamette Valley, Oregon. For these data we are interested in the spatial pattern within both riparian land use and an index of stream health, and whether there is an association between land use and stream health, after accounting for these spatial patterns. We use a graphical spatial model to address these ecological questions simultaneously. We find that the health of a stream decreases as the percent of developed land within a 120-m riparian buffer increases; interestingly, there is only evidence of spatial pattern within land use.     

Spatial modelling of socioeconomic data to understand patterns of human-caused wildfire ignition risk in the SW of Madrid (central Spain)

The majority of wildfires in the Mediterranean Basin are caused directly or indirectly by human activity. Many biophysical and socioeconomic factors have been used in quantitative analyses of wildfire risk. However, the importance and effects of socioeconomic factors in spatial modelling have been given inadequate attention. In this paper, we use different approaches to spatially model our data to examine the influence of human activity on wildfire ignition in the south west of the Madrid region, central Spain. We examine the utility of choropleth and dasymetric mapping with both Euclidean and functional distance surfaces for two differently defined wildfire seasons. We use a method from Bayesian statistics, the Weights of Evidence model, and produce ten predictive maps of wildfire risk: (1) five maps for a two-month fire season combining datasets of evidence variables and (2) five maps for the four-month fire season using the same dataset combinations. We find that the models produced from a choropleth mapping approach with spatial variables using Euclidian and functional distance surfaces are the best of the ten models. Results indicate that spatial patterns of wildfire ignition are strongly associated with human access to the natural landscape. We suggest the methods and results presented will be useful to optimize wildfire prevention resources in areas where human activity and the urban-forest interface are important factors for wildfire ignition.     

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.     

Monitoring and estimating the flow conditions and fish presence probability under various flow conditions at reach scale using genetic algorithms and kriging methods

The combination of current velocity and water depth influences stream flow conditions, and fish activities prefer particular flow conditions. This study develops a novel optimal flow classification method for identifying types of stream flow based on the current velocity and the water depth using a genetic algorithm. It is applied to the Datuan stream in northern Taiwan. Fish were sampled and their habitat investigated at the study site during the spring, summer, fall and winter of 2008-2009. The current velocity, water depth and maps of the presence probability of fish were estimated by ordinary and indicator kriging. The optimal classification results were compared with the classification results obtained using the Froude number and empirical methods. The flow classification results demonstrate that the proposed optimal flow classification method that considers depth-velocity and optimally identified criteria for classifying flow types, yields a current velocity and water depth of 0.32 (m/s) and 0.29 (m), respectively, and classifies the flow conditions in the study area as pool, run, riffle and slack. The variography results of the current velocity and the water depth data reveal that seasonal flows are not spatially stationary among seasons in the study area. Kriging methods and a two-dimensional hydrodynamic model (River 2D) with empirical and optimal flow classification methods are more effective than the Froude number method in classifying flow conditions in the study area. The flow condition classifications and probability maps were generated by River 2D, ordinary kriging and indicator kriging, to quantify the flow conditions preferred by Sicyopterus japonicus in the study area. However, the proposed optimal classification method with kriging and River 2D is an effective alternative method for mapping flow conditions and determining the relationship between flow and the presence probability of target fish in support of stream restoration.     

Sensitivity of population viability to spatial and nonspatial parameters using GRIP.

Metapopulation dynamics are influenced by spatial parameters including the amount and arrangement of suitable habitat, yet these parameters may be uncertain when deciding how to manage species or their habitats. Sensitivity analyses of population viability analysis (PVA) models can help measure relative parameter influences on predictions, identify research priorities for reducing uncertainty, and evaluate management strategies. Few spatial PVAs, however, include sensitivity analyses of both spatial and nonspatial parameters, perhaps because computationally efficient tools for such analyses are lacking or inaccessible. We developed GRIP, a program to facilitate sensitivity analysis of spatial and nonspatial input parameters for PVAs created in RAMAS Metapop, a widely applied software program. GRIP creates random sets of input files by varying parameters specified in the PVA model including vital rates and their correlations among populations, the number and configuration of populations, dispersal rates, dispersal survival, initial population abundances, carrying capacities, and the probability, intensity, and spatial extent of catastrophes, while drawing on specified parameter distributions. We evaluated GRIP's performance as a tool for sensitivity analysis of spatial PVAs and explored the consequences of varying spatial input parameters for predictions of a published PVA model of the sand lizard (Lacerta agilis). We used GRIP output to generate standardized regression coefficients (SRCs) and nonparametric correlation coefficients as indices of the relative sensitivity of predicted conservation status to input parameters. GRIP performed well; with a single analysis we were able to rank the relative influence of input parameters identified as influential by the PVA's original author, S. A. Berglind, who used three separate forms of sensitivity analysis. Our analysis, however, also underscored the value of exploring the relative influence of spatial parameters on PVA predictions; both SRCs and correlation coefficients indicated that the most influential parameters in the sand lizard model were spatial in nature. We provide annotated code so that GRIP may be modified to reflect particular species biology, customized for more complex spatial PVA models, upgraded to incorporate features added in newer versions of RAMAS Metapop, used as a template to develop similar programs, or used as it is for computationally efficient sensitivity analyses in support of conservation planning.     

Effect of Planning for Connectivity on Linear Reserve Networks

Although the concept of connectivity is decades old, it remains poorly understood and defined, and some argue that habitat quality and area should take precedence in conservation planning instead. However, fragmented landscapes are often characterized by linear features that are inherently connected, such as streams and hedgerows. For these, both representation and connectivity targets may be met with little effect on the cost, area, or quality of the reserve network. We assessed how connectivity approaches affect planning outcomes for linear habitat networks by using the stock‐route network of Australia as a case study. With the objective of representing vegetation communities across the network at a minimal cost, we ran scenarios with a range of representation targets (10%, 30%, 50%, and 70%) and used 3 approaches to account for connectivity (boundary length modifier, Euclidean distance, and landscape‐value [LV]). We found that decisions regarding the target and connectivity approach used affected the spatial allocation of reserve systems. At targets ≥50%, networks designed with the Euclidean distance and LV approaches consisted of a greater number of small reserves. Hence, by maximizing both representation and connectivity, these networks compromised on larger contiguous areas. However, targets this high are rarely used in real‐world conservation planning. Approaches for incorporating connectivity into the planning of linear reserve networks that account for both the spatial arrangement of reserves and the characteristics of the intervening matrix highlight important sections that link the landscape and that may otherwise be overlooked. El Efecto de la Planeación para la Conectividad en Redes de Reservas Lineales     

Heterogeneity shapes invasion: host size and environment influence susceptibility to a nonnative pathogen.

Theoretical study of invasion dynamics has suggested that spatial heterogeneity should strongly influence the rate and extent of spreading organisms. However, empirical support for this prediction is scant, and the importance of understanding heterogeneity for real-world systems has remained ambiguous. This study quantified the influence of host and environmental heterogeneity on the dynamics of a 19-year disease invasion by the exotic and fatal pathogen, Phytophthora lateralis, within a stream population of its host tree, Port Orford cedar (Chamaecyparis lawsoniana). Using dendrochronology, we reconstructed the invasion history along a 1350-m length of infected stream, which serves as the only route of pathogen dispersal. Contrary to theoretical predictions, the temporal progression of the disease invasion was not related to a host's downstream spatial position, but instead was determined by two sources of heterogeneity: host size and proximity to the stream channel. These sources of heterogeneity influenced both the epidemic and endemic dynamics of this pathogen invasion. This analysis provides empirical support for the influence of heterogeneity on the invasion dynamics of a commercially important forest pathogen and highlights the need to incorporate such natural variability into both invasion theory and methods aimed at controlling future spread.     

Accounting for rate instability and spatial patterns in the boundary analysis of cancer mortality maps

Boundary analysis of cancer maps may highlight areas where causative exposures change through geographic space, the presence of local populations with distinct cancer incidences, or the impact of different cancer control methods. Too often, such analysis ignores the spatial pattern of incidence or mortality rates and overlooks the fact that rates computed from sparsely populated geographic entities can be very unreliable. This paper proposes a new methodology that accounts for the uncertainty and spatial correlation of rate data in the detection of significant edges between adjacent entities or polygons. Poisson kriging is first used to estimate the risk value and the associated standard error within each polygon, accounting for the population size and the risk semivariogram computed from raw rates. The boundary statistic is then defined as half the absolute difference between kriged risks. Its reference distribution, under the null hypothesis of no boundary, is derived through the generation of multiple realizations of the spatial distribution of cancer risk values. This paper presents three types of neutral models generated using methods of increasing complexity: the common random shuffle of estimated risk values, a spatial re-ordering of these risks, or p-field simulation that accounts for the population size within each polygon. The approach is illustrated using age-adjusted pancreatic cancer mortality rates for white females in 295 US counties of the Northeast (1970–1994). Simulation studies demonstrate that Poisson kriging yields more accurate estimates of the cancer risk and how its value changes between polygons (i.e., boundary statistic), relatively to the use of raw rates or local empirical Bayes smoother. When used in conjunction with spatial neutral models generated by p-field simulation, the boundary analysis based on Poisson kriging estimates minimizes the proportion of type I errors (i.e., edges wrongly declared significant) while the frequency of these errors is predicted well by the p-value of the statistical test.

Estimation of the spatial covariance in Universal Kriging: Application to forest inventory

This paper presents a simple least squares procedure for estimating the spatial covariance and compares it with the numerically more difficult restricted maximum likelihood procedure. Thereafter, it compares design-based and kriging techniques for the estimation of spatial averages in the context of double sampling, as used in forest inventory, where terrestrial sample plots are combined with auxiliary information based on aerial photographs. A case study illustrates the theory.     

Dynamic environ analysis of compartmental systems: A computational approach

Ecosystems are often modeled as stocks of matter or energy connected by flows. Network environ analysis (NEA) is a set of mathematical methods for using powers of matrices to trace energy and material flows through such models. NEA has revealed several interesting properties of flow–storage networks, including dominance of indirect effects and the tendency for networks to create mutually positive interactions between species. However, the applicability of NEA is greatly limited by the fact that it can only be applied to models at constant steady states. In this paper, we present a new, computationally oriented approach to environ analysis called dynamic environ approximation (DEA). As a test of DEA, we use it to compute compartment throughflow in two implementations of a model of energy flow through an oyster reef ecosystem. We use a newly derived equation to compute model throughflow and compare its output to that of DEA. We find that DEA approximates the exact results given by this equation quite closely – in this particular case, with a mean Euclidean error ranging between 0.0008 and 0.21 – which gives a sense of how closely it reproduces other NEA-related quantities that cannot be exactly computed and discuss how to reduce this error. An application to calculating indirect flows in ecosystems is also discussed and dominance of indirect effects in a nonlinear model is demonstrated.