Fragmentation profiles for real and simulated landscapes

When a natural landscape is represented by a series of categorical raster maps of varying resolution, a multiresolution characterization of spatial pattern can be obtained in which entropy is computed at each resolution conditional on the next coarser resolution. The series of entropy values is plotted as a function of resolution, resulting in a multiresolution profile of fragmentation pattern in the landscape. If a categorical raster map is available at a single resolution only, a series of degraded maps at increasingly coarser resolutions is generated and the fragmentation profile is computed for this series. An algorithm has been developed for obtaining the profile directly from the single resolution map without having to generate and store the coarser resolution maps. A hierarchical stochastic model is described for simulating categorical raster maps and the fragmentation profile of the generating process is obtained in terms of the model parameters. These process profiles provide benchmarks for assessing empirical profiles obtained from raster maps of actual landscapes. Methods of the paper are applied to several watersheds of Pennsylvania using landcover maps derived from satellite imagery. These examples indicate that characteristic landscape types induce characteristic features in their fragmentation profiles.     

Modeling and interpreting the accuracy assessment error matrix for a doubly classified map

This paper considers two maps having the same spatial extent and the same mapping categories but where each map is subject to classification error. An overlay of the maps yields a (dis)similarity matrix whose (i, j)-entry is the areal proportion placed into category i by the first map and into category j by the second map. A parametric model, called the latent truth model, is proposed which specifies the dissimilarity matrix in terms of the true (but unknown) proportions for the mapping categories as well as the unknown error rates for the two maps. The number of parameters in the model exceeds the degrees of freedom in the dissimilarity matrix. However, a method of regularization is applied to effectively reduce the dimension of the parameter space and to permit model fitting. From the fitted model, one obtains estimates for the true mapping proportions as well as estimated error matrices for each of the maps. Accuracy assessment characteristics for each map (such as user's accuracy, producer's accuracy, overall accuracy, and the kappa coefficient) can be computed from the estimated error matrices. Methods are illustrated with two landcover maps of Wicomico County, Maryland.     

Methods to summarize change among land categories across time intervals

Time-series maps have become more detailed in terms of numbers of categories and time points. Our paper proposes methods for raster datasets where detailed analysis of all categorical transitions would be initially overwhelming. We create two measurements: Incidents and States. The former is the number of times a pixel’s category changes across time intervals; the latter is the number of categories that a pixel represents across time points. The combinations of Incidents and States summarize change trajectories. We also describe categorical transitions in terms of annual flow matrices, which quantify the additional information generated by intermediate time points within the temporal extent. Our approach summarizes change at the pixel and landscape levels in ways that communicate where and how categories transition over time. These methods are useful to detect hotspots of change and to consider whether the apparent changes are real or due to map error.     

A geostatistical approach for mapping thematic classification accuracy and evaluating the impact of inaccurate spatial data on ecological model predictions

Spatial information in the form of geographical information system coverages and remotely sensed imagery is increasingly used in ecological modeling. Examples include maps of land cover type from which ecologically relevant properties, such as biomass or leaf area index, are derived. Spatial information, however, is not error-free: acquisition and processing errors, as well as the complexity of the physical processes involved, make remotely sensed data imperfect measurements of ecological attributes. It is therefore important to first assess the accuracy of the spatial information being used and then evaluate the impact of such inaccurate information on ecological model predictions. In this paper, the role of geostatistics for mapping thematic classification accuracy through integration of abundant image-derived (soft) and sparse higher accuracy (hard) class labels is presented. Such assessment leads to local indices of map quality, which can be used for guiding additional ground surveys. Stochastic simulation is proposed for generating multiple alternative realizations (maps) of the spatial distribution of the higher accuracy class labels over the study area. All simulated realizations are consistent with the available pieces of information (hard and soft labels) up to their validated level of accuracy. The simulated alternative class label representations can be used for assessing joint spatial accuracy, i.e., classification accuracy regarding entire spatial features read from the thematic map. Such realizations can also serve as input parameters to spatially explicit ecological models; the resulting distribution of ecological responses provides a model of uncertainty regarding the ecological model prediction. A case study illustrates the generation of alternative land cover maps for a Landsat Thematic Mapper (TM) subscene, and the subsequent construction of local map quality indices. Simulated land cover maps are then input into a biogeochemical model for assessing uncertainty regarding net primary production (NPP).     

Multiscale advanced raster map analysis system: Definition, design and development

This paper brings together a multidisciplinary initiative to develop advanced statistical and computational techniques for analyzing, assessing, and extracting information from raster maps. This information will provide a rigorous foundation to address a wide range of applications including disease mapping, emerging infectious diseases, landscape ecological assessment, land cover trends and change detection, watershed assessment, and map accuracy assessment. It will develop an advanced map analysis system that integrates these techniques with an advanced visualization toolbox, and use the system to conduct large case studies using rich sets of raster data, primarily from remotely sensed imagery. As a result, it will be possible to study and evaluate raster maps of societal, ecological, and environmental variables to facilitate quantitative characterization and comparative analysis of geospatial trends, patterns, and phenomena. In addition to environmental and ecological studies, these techniques and tools can be used for policy decisions at national, state, and local levels, crisis management, and protection of infrastructure. Geospatial data form the foundation of an information-based society. Remote sensing has been a vastly under-utilized resource involving a multi-million dollar investment at the national levels. Even when utilized, the credibility has been at stake, largely because of lack of tools that can assess, visualize, and communicate accuracy and reliability in timely manner and at desired confidence levels. Consider an imminent 21st century scenario: What message does a multi-categorical map have about the large landscape it represents? And at what scale, and at what level of detail? Does the spatial pattern of the map reveal any societal, ecological, environmental condition of the landscape? And therefore can it be an indicator of change? How do you automate the assessment of the spatial structure and behavior of change to discover critical areas, hot spots, and their corridors? Is the map accurate? How accurate is it? How do you assess the accuracy of the map? How do we evaluate a temporal change map for change detection? What are the implications of the kind and amount of change and accuracy on what matters, whether climate change, carbon emission, water resources, urban sprawl, biodiversity, indicator species, human health, or early warning? And with what confidence? The proposed research initiative is expected to find answers to these questions and a few more that involve multi-categorical raster maps based on remote sensing and other geospatial data. It includes the development of techniques for map modeling and analysis using Markov Random Fields, geospatial statistics, accuracy assessment and change detection, upper echelons of surfaces, advanced computational techniques for geospatial data mining, and advanced visualization techniques.     

Measuring q-bits in three-trophic level systems

The use of quantum information has been proposed as an approach to deal with biological data (Piqueira, J.R.C., Serboncini, F.A., Monteiro, L.H.A., 2006. Biological models: measuring variability with classical and quantum information. J. Theor. Biol. 242 (2), 309–313). Using three-trophic level systems as examples, we show how to model population data by expressing the system states with q-bits. The system time evolution is given by the state transition matrices which relate the states to successive time intervals. It is a complementary way of looking at the problem which is usually modeled with deterministic differential equations. This is possible because the dynamics of interacting populations in three-trophic level systems is a problem with several coupled variables and, consequently, complex dynamical behaviors seem to result. The non deterministic dynamics generated by the state transition matrices is supposed to model the biological system as a whole, with real data expressing even the global effects of small disturbances in the ecological parameters.     

Development of categorical mapping for quantitative assessment of eutrophication

Spatial distribution of nutrient and phytoplankton variables is often illustrated using categorical mapping for each variable. However, the assessment of eutrophication cannot be derived from a single parameter since a synthesis of the environmental variables related to eutrophication is required. These shortcomings are further complicated since it is difficult to discriminate between distinct trophic states along natural environmental gradients. In the present work, a methodological procedure for quantitative assessment of eutrophication at a spatial scale was examined in the Gulf of Saronicos, Greece, based on a thematic map generated from the synthesis of four variables characterising eutrophication. The categorical map of each variable was developed using the Kriging interpolation method and four trophic levels were indicated (eutrophic, upper-mesotrophic, lower-mesotrophic and oligotrophic) based on nutrient and phytoplankton concentration scaling. Multi-criteria choice methods were applied to generate a final categorical map showing the four trophic levels in the area. This synthesis of categorical maps for assessing eutrophication at a spatial scale is proposed as a methodological procedure appropriate for coastal management studies.     

Spatial analysis of semi-arid patchy vegetation by the cumulative distribution of patch boundary spacings and transition probabilities

The vegetative cover in semi-arid lands typically occurs as patches of individual species more or less separated from one another by bare ground. We have adapted two methods to quantify the spatial pattern of such cover from measurements across patches on transects.Transects were laid in several directions across digital maps of the land surface or across the land itself, and the distances between successive patch boundaries were measured. The distances were ranked in order, and their cumulative distributions were computed and modeled with gamma functions. The parameters of the model provided estimates of the mean distance across patches. The means for different directions were further tested for anisotropy. Transitions between classes on the transects estimate the probabilities with which the different species occur next to others (and to bare ground) and so describe the arrangement of the patches occupied by the different species.The methods were tested with data from mosaic patterns at three semi-arid sites dominated by the tussock grass Stipa tenacissima. The differences in the estimated mean boundary spacings from site to site accorded with prior qualitative assessment, as did the estimated anisotropy. The transition matrices and the estimated proportions of cover showed the dominance of the bare soil with which all the individual species are intimately associated. The transitions also suggest the presence of both positive and negative relations among the main species. Those between Stipa tenacissima and Brachypodium retusum seem to be facilitative, as do those between this grass and the shrub Anthyllis cytisoides. In contrast, Globularia alypum seems to inhibit the other species.We also estimated transition probabilities geostatistically by summing the indicator variograms of the individual species. Standard variogram models were then fitted to describe the ordered series of values, and these again produced results that accorded with visual impressions.     

GIS and geostatistics: Essential partners for spatial analysis

Initially, geographical information systems (GIS) concentrated on two issues: automated map making, and facilitating the comparison of data on thematic maps. The first required high quality graphics, vector data models and powerful data bases, the second is based on grid cells that can be manipulated by suites of mathematical operators collectively termed map algebra. Both kinds of GIS are widely available and are taught in many universities and technical colleges. After more than 20 years of development, most standard GIS provide both kinds of functionality and good quality graphic display, but until recently they have not included the methods of statistics and geostatistics as tools for spatial analysis. Recently, standard statistical packages have been linked to GIS for both exploratory data analysis and statistical analysis and hypothesis testing. Standard statistical packages include methods for the analysis of random samples of cases or objects that are not necessarily co-located in space—if the results of statistical analysis display a spatial pattern then that is because the underlying data also share that pattern. Geostatistics addresses the need to make predictions of sampled attributes (i.e., maps) at unsampled locations from sparse, often expensive data. To make up for lack of hard data geostatistics has concentrated on the development of powerful methods based on stochastic theory. Though there have been recent moves to incorporate ancillary data in geostatistical analyses, insufficient attention has been paid to using modern methods of data display for the visualization of results. GIS can serve geostatistics by aiding geo-registration of data, facilitating spatial exploratory data analysis, providing a spatial context for interpolation and conditional simulation, as well as providing easy-to-use and effective tools for data display and visualization. The value of geostatistics for GIS lies in the provision of reliable interpolation methods with known errors, methods of upscaling and generalization, and for supplying multiple realizations of spatial patterns that can be used in environmental modeling. These stochastic methods are improving understanding of how errors in models of spatial processes accrue from errors in data or incompleteness in the structure of the models. New developments in GIS, based on ideas taken from map algebra, cellular automata and image analysis are providing high level programming languages for modeling dynamic processes such as erosion or the development of alluvial fans and deltas. Research has demonstrated that these models need stochastic inputs to yield realistic results. Non-stochastic tools such as fuzzy subsets have been shown to be useful for spatial analysis when probabilistic approaches are inappropriate or impossible. The conclusion is that in spite of differences in history and approach, the linkage of GIS, statistics and geostatistics provides a powerful, and complementary suite of tools for spatial analysis in the agricultural, earth and environmental sciences.     

Hierarchical modeling of count data with application to nuclear fall-out

We propose a method for a Bayesian hierarchical analysis of count data that are observed at irregular locations in a bounded domain of R². We model the data as having been observed on a fine regular lattice, where we do not have observations at all the sites. The counts are assumed to be independent Poisson random variables whose means are given by a log Gaussian process. In this article, the Gaussian process is assumed to be either a Markov random field (MRF) or a geostatistical model, and we compare the two models on an environmental data set. To make the comparison, we calibrate priors for the parameters in the geostatistical model to priors for the parameters in the MRF. The calibration is obtained empirically. The main goal is to predict the hidden Poisson-mean process at all sites on the lattice, given the spatially irregular count data; to do this we use an efficient MCMC. The spatial Bayesian methods are illustrated on radioactivity counts analyzed by Diggle et al. (1998).     

Using hidden Markov chains and empirical Bayes change-point estimation for transect data

Consider a lattice of locations in one dimension at which data are observed. We model the data as a random hierarchical process. The hidden process is assumed to have a (prior) distribution that is derived from a two-state Markov chain. The states correspond to the mean values (high and low) of the observed data. Conditional on the states, the observations are modelled, for example, as independent Gaussian random variables with identical variances. In this model, there are four free parameters: the Gaussian variance, the high and low mean values, and the transition probability in the Markov chain. A parametric empirical Bayes approach requires estimation of these four parameters from the marginal (unconditional) distribution of the data and we use the EM-algorithm to do this. From the posterior of the hidden process, we use simulated annealing to find the maximum a posteriori (MAP) estimate. Using a Gibbs sampler, we also obtain the maximum marginal posterior probability (MMPP) estimate of the hidden process. We use these methods to determine where change-points occur in spatial transects through grassland vegetation, a problem of considerable interest to plant ecologists.     

Influence of grain size on species-habitat models

High resolution remote sensing data facilitate the use of small-scale habitat features such as trees or hedges in the analysis of species-habitat relationships. Such data potentially enable more accurate species-habitat mapping than lower resolution data. Here, for the first time, we systematically investigated this hypothesis by altering the spatial resolution from 1 m up to 1000 m grain size in species-habitat models of 13 bird species. The study area covered the Nidda river catchment in central Germany, a large heterogeneous landscape of 1620 km². A high resolution habitat map of the area was converted to coarser spatial and thematic resolutions in seven steps. We investigated how model performance responded to grain size, and we compared the differential effects of spatial resolution and thematic resolution on model performance. Explained deviance (D²) of the bird models generally decreased with coarser spatial resolution of the data, although it did not decrease monotonically in all species. On average across all species, model D² decreased from 41.5 at 1 m grain size to 15.9 at 1000 m grain size. Ten species were best modelled at 1 m, two species at 3 m and one species at 32 m grain size. Model performance degraded continuously with increasing grain size, both in habitat generalist and habitat specialist bird species, and was systematically lower in habitat generalists. The higher model performance observed at finer grain sizes was most likely caused by the combination of three factors: (1) high spatial accuracy of bird records and (2) a more precise location and delineation of habitat features and, (3) to a lesser degree, by more habitat types differentiated in maps of finer resolution. We conclude that higher spatial and thematic resolution data can be essential for deriving accurate predictions on bird distribution patterns from species-habitat models. Especially for bird species that are sensitive to specific land-use types or to small-scaled habitat features, a grain size of 1-3 m seems most promising.     

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.     

Crustacean habitat potential mapping in a tidal flat using remote sensing and GIS

To make a macrofaunal (crustacean) habitat potential map, the spatial distribution of ecological variables in the Hwangdo tidal flat, Korea, was explored. Spatial variables were mapped using remote sensing and a geographic information system (GIS) combined with field observations. A frequency ratio (FR) and logistic regression (LR) model were employed to map the macrofauna potential area for the Ilyoplax dentimerosa, a crustacean species. Spatial variables affecting the tidal macrofauna distribution were selected based on abundance and biomass and used within a spatial database derived from remotely sensed data of various types of sensors. The spatial variables included the intertidal digital elevation model (DEM), slope, distance from a tidal channel, tidal channel density, surface sediment facies, spectral reflectance of the near infrared (NIR) bands and the tidal exposure duration. The relation between the I. dentimerosa and each spatial variable was calculated using the FR and LR. The species was randomly divided into a training set (70%) to analyse habitat potential using FR and LR and a test set (30%) to validate the predicted habitat potential map. The relations were overlaid to produce a habitat potential map with the species potential index (SPI) value for each pixel. The potential habitat maps were compared with the surveyed habitat locations such as validation data set. The comparison results showed that the LR model (accuracy is 85.28%) is better in prediction than the FR (accuracy is 78.96%) model. The performance of models gave satisfactory accuracies. The LR provides the quantitative influence of variables on a potential habitat of species; otherwise, the FR shows the quantitative influence of a class in each variable. The combination of a GIS-based frequency ratio and logistic regression models and remote sensing with field observations is an effective method to determine locations favorable for macrofaunal species occurrences in a tidal flat.     

Improving inverse model fitting in trees—Anisotropy,multiplicative effects,and Bayes estimation

Model fitting for individual-based effects in forests has some problems. Because samples measuring the separate influence of each individual are rarely available, the measured value in the sample represents the influence of all surrounding individual trees. Therefore, it is helpful to build inverse models that use the spatial pattern of the variable as well as that of the source trees. For example, since seed dispersal is influenced by wind effects, a model is discussed describing anisotropic effects to ensure an unbiased estimate of the total fruit number. Further, we present a model describing the absorption of radiation by trees. In this case a multiplicative combination of individual effects yields the total effect. Our approach uses logarithmic transformations of the original data to model multiplicative combinations as sum of transformed single effects. For fitting model parameters we propose an approach based on Bayesian statistics, to ensure ecologically interpretable parameters.     

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

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

Bayesian 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 GIS-based approach to spatio-temporal analysis of environmental pollution in urban areas: A case study of Prague's environment extended by LIDAR data

Environmental pollution of urban areas is one of key factors that state authorities and local agencies have to consider in the decision-making process. To find a compromise among many criteria, spatial analysis extended by geostatistical methods and dynamic models has to be carried out. In this case, spatial analysis includes processing of a wide range of air, water and soil pollution data and possibly noise assessment and waste management data. Other spatial inputs consist of data from remote sensing and GPS field measurements. Integration and spatial data management are carried out within the framework of a geographic information system (GIS). From a modeling point of view, GIS is used mainly for the preprocessing and postprocessing of data to be displayed in digital map layers and visualized in 3D scenes. Moreover, for preprocessing and postprocessing, deterministic and geostatistical methods (IDW, ordinary kriging) are used for spatial interpolation; geoprocessing and raster algebra are used in multi-criteria evaluation and risk assessment methods. GIS is also used as a platform for spatio-temporal analyses or for building relationships between the GIS database and stand-alone modeling tools. A case study is presented illustrating the application of spatial analysis to the urban areas of Prague. This involved incorporating environmental data from monitoring networks and field measurements into digital map layers. Extra data inputs were used to represent the 3D concentration fields of air pollutants (ozone, NO₂) measured by differential absorption LIDAR. ArcGIS was used to provide spatial data management and analysis, extended by modeling tools developed internally in the ArcObjects environment and external modules developed with MapObjects. Ordinary kriging methods were employed to predict ozone concentrations in selected 3D locations together with estimates of variability. Higher ozone concentrations were found above crossroads with their heavy traffic than above the surrounding areas. Ozone concentrations also varied with height above the digital elevation model. Processed data, spatial analysis and models are integrated within the framework of the GIS project, providing an approach that state and local authorities can use to address environmental protection issues.     

Revisiting Kappa to account for change in the accuracy assessment of land-use change models

Land-use change models are typically calibrated to reproduce known historic changes. Calibration results can then be assessed by comparing two datasets: the simulated land-use map and the actual land-use map at the same time. A common method for this is the Kappa statistic, which expresses the agreement between two categorical datasets corrected for the expected agreement. This expected agreement is based on a stochastic model of random allocation given the distribution of class sizes. However, when a model starts from an initial land-use map and makes changes to it, that stochastic model does not pose a meaningful reference level. This paper introduces K_Simulation, a statistic that is identical in form to the Kappa statistic but instead applies a more appropriate stochastic model of random allocation of class transitions relative to the initial map. The new method is illustrated on a simple example and then the results of the Kappa statistic and K_Simulation are compared using the results of a land-use model. It is found that only K_Simulation truly tests models in their capacity to explain land-use changes over time, and unlike Kappa it does not inflate results for simulations where little change takes place over time.     

Lotka-Volterra competition models for sessile organisms

Markov models are widely used to describe the dynamics of communities of sessile organisms, because they are easily fitted to field data and provide a rich set of analytical tools. In typical ecological applications, at any point in time, each point in space is in one of a finite set of states (e.g., species, empty space). The models aim to describe the probabilities of transitions between states. In most Markov models for communities, these transition probabilities are assumed to be independent of state abundances. This assumption is often suspected to be false and is rarely justified explicitly. Here, we start with simple assumptions about the interactions among sessile organisms and derive a model in which transition probabilities depend on the abundance of destination states. This model is formulated in continuous time and is equivalent to a Lotka-Volterra competition model. We fit this model and a variety of alternatives in which transition probabilities do not depend on state abundances to a long-term coral reef data set. The Lotka-Volterra model describes the data much better than all models we consider other than a saturated model (a model with a separate parameter for each transition at each time interval, which by definition fits the data perfectly). Our approach provides a basis for further development of stochastic models of sessile communities, and many of the methods we use are relevant to other types of community. We discuss possible extensions to spatially explicit models.