Spatio‐temporal Modelling of Weeds by Shot‐noise G Cox processes

Tractable space‐time point processes models are needed in various fields. For example in weed science for gaining biological knowledge, for prediction of weed development in order to optimize local treatments with herbicides or in epidemiology for prediction of the risk of a disease. Motivated by the spatio‐temporal point patterns for two weed species, we propose a spatio‐temporal Cox model with intensity based on gamma random fields. The model is an extension of Neyman–Scott and shot‐noise Cox processes to the space‐time domain and it allows spatial and temporal inhomogeneity. We use the weed example to give a first intuitive interpretation of the model and then show how the model is constructed more rigorously and how to estimate the parameters. The weed data are analysed using the proposed model, and both spatially and temporally the model shows a good fit to the data using classical goodness‐of‐fit tests.     

On statistical inference for the random set generated Cox process with set-marking

Cox point process is a process class for hierarchical modelling of systems of non-interacting points in Rd under environmental heterogeneity which is modelled through a random intensity function. In this work a class of Cox processes is suggested where the random intensity is generated by a random closed set. Such heterogeneity appears for example in forestry where silvicultural treatments like harvesting and site-preparation create geometrical patterns for tree density variation in two different phases. In this paper the second order property, important both in data analysis and in the context of spatial sampling, is derived. The usefulness of the random set generated Cox process is highly increased, if for each point it is observed whether it is included in the random set or not. This additional information is easy and economical to obtain in many cases and is hence of practical value; it leads to marks for the points. The resulting random set marked Cox process is a marked point process where the marks are intensity-dependent. The problem with set-marking is that the marks are not a representative sample from the random set. This paper derives the second order property of the random set marked Cox process and suggests a practical estimation method for area fraction and covariance of the random set and for the point densities within and outside the random set. A simulated example and a forestry example are given.     

Covariance density estimation for autoregressive spectral modelling of point processes

The use of autoregressive modelling has acquired great importance in time series analysis and in principle it may also be applicable in the spectral analysis of point processes with similar advantages over the nonparametric approach. Most of the methods used for autoregressive spectral analysis require positive semidefinite estimates for the covariance function, while current methods for the estimation of the covariance density function of a point process given a realization over the interval [0,T] do not guarantee a positive semidefinite estimate. This paper discusses methods for the estimation of the covariance density and conditional intensity function of point processes and present alternative computational efficient estimation algorithms leading always to positive semidefinite estimates, therefore adequate for autoregressive spectral analysis. Autoregressive spectral modelling of point processes from Yule-Walker type equations and Levinson recursion combined with the minimum AIC or CAT principle is illustrated with neurobiological data.     

Bayesian semiparametric intensity estimation for inhomogeneous spatial point processes

In this work we propose a fully Bayesian semiparametric method to estimate the intensity of an inhomogeneous spatial point process. The basic idea is to first convert intensity estimation into a Poisson regression setting via binning data points on a regular grid, and then model the log intensity semiparametrically using an adaptive version of Gaussian Markov random fields to smooth the corresponding counts. The inference is carried by an efficient Markov chain Monte Carlo simulation algorithm. Compared to existing methods for intensity estimation, for example, parametric modeling and kernel smoothing, the proposed estimator not only provides inference regarding the dependence of the intensity function on possible covariates, but also uses information from the data to adaptively determine the amount of smoothing at the local level. The effectiveness of using our method is demonstrated through simulation studies and an application to a rainforest dataset.     

Dynamic occupancy models for analyzing species' range dynamics across large geographic scales

Large‐scale biodiversity data are needed to predict species' responses to global change and to address basic questions in macroecology. While such data are increasingly becoming available, their analysis is challenging because of the typically large heterogeneity in spatial sampling intensity and the need to account for observation processes. Two further challenges are accounting for spatial effects that are not explained by covariates, and drawing inference on dynamics at these large spatial scales. We developed dynamic occupancy models to analyze large‐scale atlas data. In addition to occupancy, these models estimate local colonization and persistence probabilities. We accounted for spatial autocorrelation using conditional autoregressive models and autologistic models. We fitted the models to detection/nondetection data collected on a quarter‐degree grid across southern Africa during two atlas projects, using the hadeda ibis (Bostrychia hagedash) as an example. The model accurately reproduced the range expansion between the first (SABAP1: 1987–1992) and second (SABAP2: 2007–2012) Southern African Bird Atlas Project into the drier parts of interior South Africa. Grid cells occupied during SABAP1 generally remained occupied, but colonization of unoccupied grid cells was strongly dependent on the number of occupied grid cells in the neighborhood. The detection probability strongly varied across space due to variation in effort, observer identity, seasonality, and unexplained spatial effects. We present a flexible hierarchical approach for analyzing grid‐based atlas data using dynamical occupancy models. Our model is similar to a species' distribution model obtained using generalized additive models but has a number of advantages. Our model accounts for the heterogeneous sampling process, spatial correlation, and perhaps most importantly, allows us to examine dynamic aspects of species ranges.     

On Bayesian inference for the K function

The K function is a summary of spatial dependence in spatial point processes. In practice one observes a realization of the spatial point process, called a spatial point pattern. Although the K function of a spatial point process is typically unknown, several estimators of the process K function have been put forth. These estimators, however, are based upon empirical averages; the complicated distributional properties of the estimators unfortunately complicates interval estimation. In this paper, we propose a Bayesian inferential framework, allowing inference for the K function of the spatial point process (including interval estimation). Of particular interest is the unique use of the posterior predictive distribution to (efficiently) enable such inferences. To demonstrate our technique, the well known Swedish pine sapling data (Strand, 1972) is analyzed, including a discussion on evaluating model fit.     

A KPSS test for stationarity for spatial point processes

Summary .   We propose a formal method to test stationarity for spatial point processes. The proposed test statistic is based on the integrated squared deviations of observed counts of events from their means estimated under stationarity. We show that the resulting test statistic converges in distribution to a functional of a two-dimensional Brownian motion. To conduct the test, we compare the calculated statistic with the upper tail critical values of this functional. Our method requires only a weak dependence condition on the process but does not assume any parametric model for it. As a result, it can be applied to a wide class of spatial point process models. We study the efficacy of the test through both simulations and applications to two real data examples that were previously suspected to be nonstationary based on graphical evidence. Our test formally confirmed the suspected nonstationarity for both data.     

A Model-Based Approach for Making Ecological Inference from Distance Sampling Data

Summary .  We consider a fully model-based approach for the analysis of distance sampling data. Distance sampling has been widely used to estimate abundance (or density) of animals or plants in a spatially explicit study area. There is, however, no readily available method of making statistical inference on the relationships between abundance and environmental covariates. Spatial Poisson process likelihoods can be used to simultaneously estimate detection and intensity parameters by modeling distance sampling data as a thinned spatial point process. A model-based spatial approach to distance sampling data has three main benefits: it allows complex and opportunistic transect designs to be employed, it allows estimation of abundance in small subregions, and it provides a framework to assess the effects of habitat or experimental manipulation on density. We demonstrate the model-based methodology with a small simulation study and analysis of the Dubbo weed data set. In addition, a simple ad hoc method for handling overdispersion is also proposed. The simulation study showed that the model-based approach compared favorably to conventional distance sampling methods for abundance estimation. In addition, the overdispersion correction performed adequately when the number of transects was high. Analysis of the Dubbo data set indicated a transect effect on abundance via Akaike's information criterion model selection. Further goodness-of-fit analysis, however, indicated some potential confounding of intensity with the detection function.     

An estimating function approach to inference for inhomogeneous Neyman-Scott processes

This article is concerned with inference for a certain class of inhomogeneous Neyman-Scott point processes depending on spatial covariates. Regression parameter estimates obtained from a simple estimating function are shown to be asymptotically normal when the "mother" intensity for the Neyman-Scott process tends to infinity. Clustering parameter estimates are obtained using minimum contrast estimation based on the K-function. The approach is motivated and illustrated by applications to point pattern data from a tropical rain forest plot.     

An evaluation of the state of spatial point pattern analysis in ecology

Over the last two decades spatial point pattern analysis (SPPA) has become increasingly popular in ecological research. To direct future work in this area we review studies using SPPA techniques in ecology and related disciplines. We first summarize the key elements of SPPA in ecology (i.e. data types, summary statistics and their estimation, null models, comparison of data and models, and consideration of heterogeneity); second, we review how ecologists have used these key elements; and finally, we identify practical difficulties that are still commonly encountered and point to new methods that allow current key questions in ecology to be effectively addressed. Our review of 308 articles published over the period 1992–2012 reveals that a standard canon of SPPA techniques in ecology has been largely identified and that most of the earlier technical issues that occupied ecologists, such as edge correction, have been solved. However, the majority of studies underused the methodological potential offered by modern SPPA. More advanced techniques of SPPA offer the potential to address a variety of highly relevant ecological questions. For example, inhomogeneous summary statistics can quantify the impact of heterogeneous environments, mark correlation functions can include trait and phylogenetic information in the analysis of multivariate spatial patterns, and more refined point process models can be used to realistically characterize the structure of a wide range of patterns. Additionally, recent advances in fitting spatially‐explicit simulation models of community dynamics to point pattern summary statistics hold the promise for solving the longstanding problem of linking pattern to process. All these newer developments allow ecologists to keep up with the increasing availability of spatial data sets provided by newer technologies, which allow point patterns and environmental variables to be mapped over large spatial extents at increasingly higher image resolutions.     

A new digital method of data collection for spatial point pattern analysis in grassland communities

A major objective of plant ecology research is to determine the underlying processes responsible for the observed spatial distribution patterns of plant species. Plants can be approximated as points in space for this purpose, and thus, spatial point pattern analysis has become increasingly popular in ecological research. The basic piece of data for point pattern analysis is a point location of an ecological object in some study region. Therefore, point pattern analysis can only be performed if data can be collected. However, due to the lack of a convenient sampling method, a few previous studies have used point pattern analysis to examine the spatial patterns of grassland species. This is unfortunate because being able to explore point patterns in grassland systems has widespread implications for population dynamics, community‐level patterns, and ecological processes. In this study, we developed a new method to measure individual coordinates of species in grassland communities. This method records plant growing positions via digital picture samples that have been sub‐blocked within a geographical information system (GIS). Here, we tested out the new method by measuring the individual coordinates of Stipa grandis in grazed and ungrazed S. grandis communities in a temperate steppe ecosystem in China. Furthermore, we analyzed the pattern of S. grandis by using the pair correlation function g(r) with both a homogeneous Poisson process and a heterogeneous Poisson process. Our results showed that individuals of S. grandis were overdispersed according to the homogeneous Poisson process at 0–0.16 m in the ungrazed community, while they were clustered at 0.19 m according to the homogeneous and heterogeneous Poisson processes in the grazed community. These results suggest that competitive interactions dominated the ungrazed community, while facilitative interactions dominated the grazed community. In sum, we successfully executed a new sampling method, using digital photography and a geographical information system, to collect experimental data on the spatial point patterns for the populations in this grassland community.     

Dealing with virtual aggregation – a new index for analysing heterogeneous point patterns

Point pattern analyses such as the estimation of Ripley's K‐function or the pair‐correlation function g are commonly used in ecology to characterise ecological patterns in space. However, a major disadvantage of these methods is their missing ability to deal with spatial heterogeneity. A heterogeneous intensity of points causes a systematic bias in estimates of the K‐ and g‐functions, a phenomenon termed “virtual aggregation” in the recent literature. To address this problem, we derive a new index, called K2‐index, as an extension of existing point pattern characteristics. The K2‐index has a heuristic interpretation as an approximation to the first derivative of the g‐function. We estimate the K‐, g‐ and K2‐functions for six different types of simulated point patterns and show that the K2‐index may provide important information on point patterns that the other methods fail to detect. The results indicate that particularly the small‐scale distributions of points are better represented by the K2‐index. This might be important for testing hypotheses on ecological processes, because most of these processes, such as direct neighbour interactions, occur very locally. When applied to empirical patterns of molehill distribution, the results of the K2‐analysis show regularity up to distances between 0.1 and 0.4 m in most of the study areas, and aggregation of molehills up to distances between 0.2 and 1.1 m. The type and scale of these deviations from randomness agree with a priori expectations on the hill‐building behaviour of moles. In contrast, the estimated g‐functions merely indicate aggregation at the full range from 0 to 7 m (or even above). Considering the advantages and disadvantages of the different methods, we suggest that the K2‐index should be used as a complement to existing approaches, particularly for point patterns generated by processes that act on more than one scale.     

Unsupervised noise removal algorithms for three-dimensional confocal fluorescence microscopy

Algorithms are presented for effective suppression of the quantum noise artifact that is inherent to three-dimensional confocal fluorescence microscopy images of extended spatial objects such as neurons. The specific advances embodied in these algorithms are as follows: (i) they incorporate an automatic and pattern-constrained three-dimensional segmentation of the image field, and use it to limit any smoothing to the interiors of specified image regions and hence avoid the blurring that is inevitably associated with conventional noise removal algorithms; (ii) they are ‘unsupervised’ in the sense that they automatically estimate and adapt to the unknown spatially and temporally varying noise level in the microscopy data. Fast computation is achieved by parallel computation methods, rather than by algorithmic or modelling compromises.The quantum noise artifact is modelled using a mixture of spatially non-homogeneous Poisson point processes. The intensity of each component process is constrained to lie in specific intervals. A set of segmentation and edge-site variables are used to determine the intensity of the mixture process. Using this model, the noise-removal process is formulated as the joint optimal estimation of the segmentation labels, edge-sites and intensity of the mixture Poisson point process, subject to a combination of stochastic priors and syntactic pattern constraints. The computations proceed iteratively, starting from a set of approximate user-supplied, or default initial guesses of the underlying random process parameters. An Expectation Maximization algorithm is used to obtain a more precise characterization of these parameters, that are then input to a joint estimation algorithm.Stereoscopic renderings of processed three-dimensional images of murine hippocampal neurons are presented to demonstrate the effectiveness of the method. The processed images exhibit increased contrast and significant smoothing and reduction of the background intensity while avoiding any blurring of the foreground neuronal structures.     

Bayesian Wombling for Spatial Point Processes

Summary In many applications involving geographically indexed data, interest focuses on identifying regions of rapid change in the spatial surface, or the related problem of the construction or testing of boundaries separating regions with markedly different observed values of the spatial variable. This process is often referred to in the literature as boundary analysis or wombling. Recent developments in hierarchical models for point‐referenced (geostatistical) and areal (lattice) data have led to corresponding statistical wombling methods, but there does not appear to be any literature on the subject in the point‐process case, where the locations themselves are assumed to be random and likelihood evaluation is notoriously difficult. We extend existing point‐level and areal wombling tools to this case, obtaining full posterior inference for multivariate spatial random effects that, when mapped, can help suggest spatial covariates still missing from the model. In the areal case we can also construct wombled maps showing significant boundaries in the fitted intensity surface, while the point‐referenced formulation permits testing the significance of a postulated boundary. In the computationally demanding point‐referenced case, our algorithm combines Monte Carlo approximants to the likelihood with a predictive process step to reduce the dimension of the problem to a manageable size. We apply these techniques to an analysis of colorectal and prostate cancer data from the northern half of Minnesota, where a key substantive concern is possible similarities in their spatial patterns, and whether they are affected by each patient's distance to facilities likely to offer helpful cancer screening options.     

Estimating species – area relationships by modeling abundance and frequency subject to incomplete sampling

Models and data used to describe species–area relationships confound sampling with ecological process as they fail to acknowledge that estimates of species richness arise due to sampling. This compromises our ability to make ecological inferences from and about species–area relationships. We develop and illustrate hierarchical community models of abundance and frequency to estimate species richness. The models we propose separate sampling from ecological processes by explicitly accounting for the fact that sampled patches are seldom completely covered by sampling plots and that individuals present in the sampling plots are imperfectly detected. We propose a multispecies abundance model in which community assembly is treated as the summation of an ensemble of species‐level Poisson processes and estimate patch‐level species richness as a derived parameter. We use sampling process models appropriate for specific survey methods. We propose a multispecies frequency model that treats the number of plots in which a species occurs as a binomial process. We illustrate these models using data collected in surveys of early‐successional bird species and plants in young forest plantation patches. Results indicate that only mature forest plant species deviated from the constant density hypothesis, but the null model suggested that the deviations were too small to alter the form of species–area relationships. Nevertheless, results from simulations clearly show that the aggregate pattern of individual species density–area relationships and occurrence probability–area relationships can alter the form of species–area relationships. The plant community model estimated that only half of the species present in the regional species pool were encountered during the survey. The modeling framework we propose explicitly accounts for sampling processes so that ecological processes can be examined free of sampling artefacts. Our modeling approach is extensible and could be applied to a variety of study designs and allows the inclusion of additional environmental covariates.     

Do species distribution models explain spatial structure within tree species ranges?

Aim To evaluate the ability of species distribution models (SDMs) to predict the spatial structure of tree species within their geographical ranges (how trees are distributed within their ranges). Location Continental Spain. Methods We used an extensive dataset consisting of c. 90,000 plots (1 plot km^?2) where presence/absence data for 23 common Mediterranean and Atlantic tree species had been surveyed. We first generated SDMs relating the presence or absence of each species to a set of 16 environmental predictors, following a stepwise modelling process based on maximum likelihood methods. Superimposing spatial correlograms generated from the predictions of the SDMs over those generated from the raw data allowed a model–observation comparison of the nature, scale and intensity (level of aggregation) of spatial structure with the species ranges. Results SDMs predicted accurately the nature and scale of the spatial structure of trees. However, for most species, the observed intensity of spatial structure (level of aggregation of species in space) was substantially greater than that predicted by the SDMs. On average, the intensity of spatial aggregation was twice that predicted by SDMs. In addition, we also found a negative correlation between intensity of aggregation and species range size. Main conclusions Standard SDM predictions of spatial structure patterns differ among species. SDMs are apparently able to reproduce both the scale and intensity of species spatial structure within their ranges. However, one or more missing processes not included in SDMs results in species being substantially more aggregated in space than can be captured by the SDMs. This result adds to recent calls for a new generation of more biologically realistic SDMs. In particular, future SDMs should incorporate ecological processes that are likely to increase the intensity of spatial aggregation, such as source–sink dynamics, fine‐scale environmental heterogeneity and disequilibrium.     

Upscaling leaf area index in an Arctic landscape through multiscale observations

Monitoring and understanding global change requires a detailed focus on upscaling, the process for extrapolating from the site‐specific scale to the smallest scale resolved in regional or global models or earth observing systems. Leaf area index (LAI) is one of the most sensitive determinants of plant production and can vary by an order of magnitude over short distances. The landscape distribution of LAI is generally determined by remote sensing of surface reflectance (e.g. normalized difference vegetation index, NDVI) but the mismatch in scales between ground and satellite measurements complicates LAI upscaling. Here, we describe a series of measurements to quantify the spatial distribution of LAI in a sub‐Arctic landscape and then describe the upscaling process and its associated errors. Working from a fine‐scale harvest LAI–NDVI relationship, we collected NDVI data over a 500 m × 500 m catchment in the Swedish Arctic, at resolutions from 0.2 to 9.0 m in a nested sampling design. NDVI scaled linearly, so that NDVI at any scale was a simple average of multiple NDVI measurements taken at finer scales. The LAI–NDVI relationship was scale invariant from 1.5 to 9.0 m resolution. Thus, a single exponential LAI–NDVI relationship was valid at all these scales, with similar prediction errors. Vegetation patches were of a scale of ～0.5 m and at measurement scales coarser than this, there was a sharp drop in LAI variance. Landsat NDVI data for the study catchment correlated significantly, but poorly, with ground‐based measurements. A variety of techniques were used to construct LAI maps, including interpolation by inverse distance weighting, ordinary Kriging, External Drift Kriging using Landsat data, and direct estimation from a Landsat NDVI–LAI calibration. All methods produced similar LAI estimates and overall errors. However, Kriging approaches also generated maps of LAI estimation error based on semivariograms. The spatial variability of this Arctic landscape was such that local measurements assimilated by Kriging approaches had a limited spatial influence. Over scales >50 m, interpolation error was of similar magnitude to the error in the Landsat NDVI calibration. The characterisation of LAI spatial error in this study is a key step towards developing spatio‐temporal data assimilation systems for assessing C cycling in terrestrial ecosystems by combining models with field and remotely sensed data.     

Geographic range estimates and environmental requirements for the harpy eagle derived from spatial models of current and past distribution

Understanding species–environment relationships is key to defining the spatial structure of species distributions and develop effective conservation plans. However, for many species, this baseline information does not exist. With reliable presence data, spatial models that predict geographic ranges and identify environmental processes regulating distribution are a cost‐effective and rapid method to achieve this. Yet these spatial models are lacking for many rare and threatened species, particularly in tropical regions. The harpy eagle (Harpia harpyja) is a Neotropical forest raptor of conservation concern with a continental distribution across lowland tropical forests in Central and South America. Currently, the harpy eagle faces threats from habitat loss and persecution and is categorized as Near‐Threatened by the International Union for the Conservation of Nature (IUCN). Within a point process modeling (PPM) framework, we use presence‐only occurrences with climatic and topographical predictors to estimate current and past distributions and define environmental requirements using Ecological Niche Factor Analysis. The current PPM prediction had high calibration accuracy (Continuous Boyce Index = 0.838) and was robust to null expectations (pROC ratio = 1.407). Three predictors contributed 96% to the PPM prediction, with Climatic Moisture Index the most important (72.1%), followed by minimum temperature of the warmest month (15.6%) and Terrain Roughness Index (8.3%). Assessing distribution in environmental space confirmed the same predictors explaining distribution, along with precipitation in the wettest month. Our reclassified binary model estimated a current range size 11% smaller than the current IUCN range polygon. Paleoclimatic projections combined with the current model predicted stable climatic refugia in the central Amazon, Guyana, eastern Colombia, and Panama. We propose a data‐driven geographic range to complement the current IUCN range estimate and that despite its continental distribution, this tropical forest raptor is highly specialized to specific environmental requirements.     

Estimating animal behavior and residency from movement data

We present a process‐based approach to estimate residency and behavior from uncertain and temporally correlated movement data collected with electronic tags. The estimation problem is formulated as a hidden Markov model (HMM) on a spatial grid in continuous time, which allows straightforward implementation of barriers to movement. Using the grid to explicitly resolve space, location estimation can be supplemented by or based entirely on environmental data (e.g. temperature, daylight). The HMM method can therefore analyze any type of electronic tag data. The HMM computes the joint posterior probability distribution of location and behavior at each point in time. With this, the behavioral state of the animal can be associated to regions in space, thus revealing migration corridors and residence areas. We demonstrate the inferential potential of the method by analyzing satellite‐linked archival tag data from a southern bluefin tuna Thunnus maccoyii where longitudinal coordinates inferred from daylight are supplemented by latitudinal information in recorded sea surface temperatures.