Janine Illian

Statistical Analysis and Modelling of Spatial Point Patterns: Illian/Statistical Analysis and Modelling of Spatial Point Patterns

Preface. List of Examples. 1. Introduction. 1.1 Point process statistics. 1.2 Examples of point process data. 1.2.1 A pattern of amacrine cells. 1.2.2 Gold particles. 1.2.3 A pattern of Western Australian plants. 1.2.4 Waterstriders. 1.2.5 A sample of concrete. 1.3 Historical notes. 1.3.1 Determination of number of trees in a forest. 1.3.2 Number of blood particles in a sample. 1.3.3 Patterns of points in plant communities. 1.3.4 Formulating the power law for the pair correlation function for galaxies. 1.4 Sampling and data collection. 1.4.1 General remarks. 1.4.2 Choosing an appropriate study area. 1.4.3 Data collection. 1.5 Fundamentals of the theory of point processes. 1.6 Stationarity and isotropy. 1.6.1 Model approach and design approach. 1.6.2 Finite and infinite point processes. 1.6.3 Stationarity and isotropy. 1.6.4 Ergodicity. 1.7 Summary characteristics for point processes. 1.7.1 Numerical summary characteristics. 1.7.2 Functional summary characteristics. 1.8 Secondary structures of point processes. 1.8.1 Introduction. 1.8.2 Random sets. 1.8.3 Random fields. 1.8.4 Tessellations. 1.8.5 Neighbour networks or graphs. 1.9 Simulation of point processes. 2. The Homogeneous Poisson point process. 2.1 Introduction. 2.2 The binomial point process. 2.2.1 Introduction. 2.2.2 Basic properties. 2.2.3 The periodic binomial process. 2.2.4 Simulation of the binomial process. 2.3 The homogeneous Poisson point process. 2.3.1 Introduction. 2.3.2 Basic properties. 2.3.3 Characterisations of the homogeneous Poisson process. 2.4 Simulation of a homogeneous Poisson process. 2.5 Model characteristics. 2.5.1 Moments and moment measures. 2.5.2 The Palm distribution of a homogeneous Poisson process. 2.5.3 Summary characteristics of the homogeneous Poisson process. 2.6 Estimating the intensity. 2.7 Testing complete spatial randomness. 2.7.1 Introduction. 2.7.2 Quadrat counts. 2.7.3 Distance methods. 2.7.4 The J-test. 2.7.5 Two index-based tests. 2.7.6 Discrepancy tests. 2.7.7 The L-test. 2.7.8 Other tests and recommendations. 3. Finite point processes. 3.1 Introduction. 3.2 Distributions of numbers of points. 3.2.1 The binomial distribution. 3.2.2 The Poisson distribution. 3.2.3 Compound distributions. 3.2.4 Generalised distributions. 3.3 Intensity functions and their estimation. 3.3.1 Parametric statistics for the intensity function. 3.3.2 Non-parametric estimation of the intensity function. 3.3.3 Estimating the point density distribution function. 3.4 Inhomogeneous Poisson process and finite Cox process. 3.4.1 The inhomogeneous Poisson process. 3.4.2 The finite Cox process. 3.5 Summary characteristics for finite point processes. 3.5.1 Nearest-neighbour distances. 3.5.2 Dilation function. 3.5.3 Graph-theoretic statistics. 3.5.4 Second-order characteristics. 3.6 Finite Gibbs processes. 3.6.1 Introduction. 3.6.2 Gibbs processes with fixed number of points. 3.6.3 Gibbs processes with a random number of points. 3.6.4 Second-order summary characteristics of finite Gibbs processes. 3.6.5 Further discussion. 3.6.6 Statistical inference for finite Gibbs processes. 4. Stationary point processes. 4.1 Basic definitions and notation. 4.2 Summary characteristics for stationary point processes. 4.2.1 Introduction. 4.2.2 Edge-correction methods. 4.2.3 The intensity lambda. 4.2.4 Indices as summary characteristics. 4.2.5 Empty-space statistics and other morphological summaries. 4.2.6 The nearest-neighbour distance distribution function. 4.2.7 The J-function. 4.3 Second-order characteristics. 4.3.1 The three functions: K, L and g. 4.3.2 Theoretical foundations of second-order characteristics. 4.3.3 Estimators of the second-order characteristics. 4.3.4 Interpretation of pair correlation functions. 4.4 Higher-order and topological characteristics. 4.4.1 Introduction. 4.4.2 Third-order characteristics. 4.4.3 Delaunay tessellation characteristics. 4.4.4 The connectivity function. 4.5 Orientation analysis for stationary point processes. 4.5.1 Introduction. 4.5.2 Nearest-neighbour orientation distribution. 4.5.3 Second-order orientation analysis. 4.6 Outliers, gaps and residuals. 4.6.1 Introduction. 4.6.2 Simple outlier detection. 4.6.3 Simple gap detection. 4.6.4 Model-based outliers. 4.6.5 Residuals. 4.7 Replicated patterns. 4.7.1 Introduction. 4.7.2 Aggregation recipes. 4.8 Choosing appropriate observation windows. 4.8.1 General ideas. 4.8.2 Representative windows. 4.9 Multivariate analysis of series of point patterns. 4.10 Summary characteristics for the non-stationary case. 4.10.1 Formal application of stationary characteristics and estimators. 4.10.2 Intensity reweighting. 4.10.3 Local rescaling. 5. Stationary marked point processes. 5.1 Basic definitions and notation. 5.1.1 Introduction. 5.1.2 Marks and their properties. 5.1.3 Marking models. 5.1.4 Stationarity. 5.1.5 First-order characteristics. 5.1.6 Mark-sum measure. 5.1.7 Palm distribution. 5.2 Summary characteristics. 5.2.1 Introduction. 5.2.2 Intensity and mark-sum intensity. 5.2.3 Mean mark, mark d.f. and mark probabilities. 5.2.4 Indices for stationary marked point processes. 5.2.5 Nearest-neighbour distributions. 5.3 Second-order characteristics for marked point processes. 5.3.1 Introduction. 5.3.2 Definitions for qualitative marks. 5.3.3 Definitions for quantitative marks. 5.3.4 Estimation of second-order characteristics. 5.4 Orientation analysis for marked point processes. 5.4.1 Introduction. 5.4.2 Orientation analysis for non-isotropic processes with angular marks. 5.4.3 Orientation analysis for isotropic processes with angular marks. 5.4.4 Orientation analysis with constructed marks. 6. Modelling and simulation of stationary point processes. 6.1 Introduction. 6.2 Operations with point processes. 6.2.1 Thinning. 6.2.2 Clustering. 6.2.3 Superposition. 6.3 Cluster processes. 6.3.1 General cluster processes. 6.3.2 Neyman-Scott processes. 6.4 Stationary Cox processes. 6.4.1 Introduction. 6.4.2 Properties of stationary Cox processes. 6.5 Hard-core point processes. 6.5.1 Introduction. 6.5.2 Matern hard-core processes. 6.5.3 The dead leaves model. 6.5.4 The RSA model. 6.5.5 Random dense packings of hard spheres. 6.6 Stationary Gibbs processes. 6.6.1 Basic ideas and equations. 6.6.2 Simulation of stationary Gibbs processes. 6.6.3 Statistics for stationary Gibbs processes. 6.7 Reconstruction of point patterns. 6.7.1 Reconstructing point patterns without a specified model. 6.7.2 An example: reconstruction of Neyman-Scott processes. 6.7.3 Practical application of the reconstruction algorithm. 6.8 Formulas for marked point process models. 6.8.1 Introduction. 6.8.2 Independent marks. 6.8.3 Random field model. 6.8.4 Intensity-weighted marks. 6.9 Moment formulas for stationary shot-noise fields. 6.10 Space-time point processes. 6.10.1 Introduction. 6.10.2 Space-time Poisson processes. 6.10.3 Second-order statistics for completely stationary event processes. 6.10.4 Two examples of space-time processes. 6.11 Correlations between point processes and other random structures. 6.11.1 Introduction. 6.11.2 Correlations between point processes and random fields. 6.11.3 Correlations between point processes and fibre processes. 7. Fitting and testing point process models. 7.1 Choice of model. 7.2 Parameter estimation. 7.2.1 Maximum likelihood method. 7.2.2 Method of moments. 7.2.3 Trial-and-error estimation. 7.3 Variance estimation by bootstrap. 7.4 Goodness-of-fit tests. 7.4.1 Envelope test. 7.4.2 Deviation test. 7.5 Testing mark hypotheses. 7.5.1 Introduction. 7.5.2 Testing independent marking, test of association. 7.5.3 Testing geostatistical marking. 7.6 Bayesian methods for point pattern analysis. Appendix A Fundamentals of statistics. Appendix B Geometrical characteristics of sets. Appendix C Fundamentals of geostatistics. References. Notation index. Author index. Subject index.

Hierarchical spatial point process analysis for a plant community with high biodiversity

A complex multivariate spatial point pattern of a plant community with high biodiversity is modelled using a hierarchical multivariate point process model. In the model, interactions between plants with different post-fire regeneration strategies are of key interest. We consider initially a maximum likelihood approach to inference where problems arise due to unknown interaction radii for the plants. We next demonstrate that a Bayesian approach provides a flexible framework for incorporating prior information concerning the interaction radii. From an ecological perspective, we are able both to confirm existing knowledge on species’ interactions and to generate new biological questions and hypotheses on species’ interactions.

Multispecies coexistence of trees in tropical forests: spatial signals of topographic niche differentiation increase with environmental heterogeneity

Neutral and niche theories give contrasting explanations for the maintenance of tropical tree species diversity. Both have some empirical support, but methods to disentangle their effects have not yet been developed. We applied a statistical measure of spatial structure to data from 14 large tropical forest plots to test a prediction of niche theory that is incompatible with neutral theory: that species in heterogeneous environments should separate out in space according to their niche preferences. We chose plots across a range of topographic heterogeneity, and tested whether pairwise spatial associations among species were more variable in more heterogeneous sites. We found strong support for this prediction, based on a strong positive relationship between variance in the spatial structure of species pairs and topographic heterogeneity across sites. We interpret this pattern as evidence of pervasive niche differentiation, which increases in importance with increasing environmental heterogeneity.

Atmospheric forcing on chlorophyll concentration in the Mediterranean

Recent research suggests the coupling of climatic fluctuations and changes in biological indices that describe species richness, abundance and spatiotemporal distribution. In this study, large-scale modes of atmospheric variability over the northern hemisphere are associated with chlorophyll-a concentration in the Mediterranean. Sea level atmospheric pressure, air temperature, wind speed and precipitation are used to account for climatic and local weather effects, whereas sea surface temperature, sea surface height and salinity are employed to describe oceanic variation. Canonical Correlation Analysis was applied to relate chlorophyll concentration to the above-mentioned environmental variables, while correlation maps were also built to distinguish between localized and distant effects. Spectral analysis was used to identify common temporal cycles between chlorophyll concentration and each environmental variable. These cycles could be interpreted as mechanistic links between chlorophyll and large-scale atmospheric variability. Known teleconnection patterns such as the East Atlantic/Western Russian pattern, the North Atlantic Oscillation, the Polar/Eurasian pattern, the East Pacific/North Pacific, the East Atlantic jet and the Mediterranean Oscillation are found to be the most important modes of atmospheric variability related to chlorophyll-a concentration and distribution. The areas that are mostly affected are near the coasts and areas of upwelling and gyre formation. The results also suggest that this influence may arise either through local effects of teleconnection patterns on oceanic features or large-scale changes superimposed onto the general circulation in the Mediterranean.

Environmental drivers of the anchovy/sardine complex in the Eastern Mediterranean

The anchovy/sardine complex is an important fishery resource in some of the largest upwelling systems in the world. Synchronous, but out of phase, fluctuations of the two species in distant parts of the oceans have prompted a number of studies dedicated to determining the phenomena, atmospheric and oceanic, responsible for the observed synchronicity and the biological mechanisms behind the population changes of the two species. Anchovy and sardine are of high commercial value for the fishing sector in Greece; this study investigates the impact of large-scale climatic indices on the anchovy/sardine complex in the Greek seas using fishery catches as a proxy for fish productivity. Time series of catches for both species were analysed for relationships with teleconnection indices and local environmental variability. The connection between the teleconnection indices and local weather/oceanic variation was also examined in an effort to describe physical mechanisms that link large-scale atmospheric patterns with anchovy and sardine. The West African Summer Monsoon, East Atlantic Jet and Pacific–North American (PNA) pattern exhibit coherent relationships with the catches of the two species. The first two aforementioned patterns are prominent atmospheric modes of variability during the summer months when sardine is spawning and anchovy juveniles are growing. PNA is related with El Niño Southern Oscillation events. Sea Surface Temperature (SST) appears as a significant link between atmospheric and biological variability either because higher temperatures seem to be favouring sardine growth or because lower temperatures, characteristic of productivity-enhancing oceanic features, exert a positive influence on both species. However at a local scale, other parameters such as wind and mesoscale circulation describe air–sea variability affecting the anchovy/sardine complex. These relationships are non-linear and in agreement with results of previous studies stressing the importance of optimal environmental windows. The results also show differences in the response of the two species to environmental forcing and possible interactions between the two species. The nature of these phenomena, e.g., if the species interactions are direct through competition or indirect through the food web, remains to be examined.

Goodness‐of‐fit measures of evenness: a new tool for exploring changes in community structure

Growing concern about the fate of biodiversity, highlighted by the Convention on Biological Diversitys 2010 and 2020 targets for stemming biodiversity loss, has intensified interest in methods of assessing change in ecological communities through time. Biodiversity is a multivariate concept, which cannot be well-represented by a single measure. However, diversity profiles summarize the multivariate nature of multi-species datasets, and allow a more nuanced interpretation of biodiversity trends than unitary metrics. Here we introduce a new approach to diversity profiling. Our method is based on the knowledge that an ecological community is never completely even and uses this departure from perfect evenness as a novel and insightful way of measuring diversity. We plot our measure of departure as a function of a free parameter, to generate “evenness profiles”. These profiles allow us to separate changes due to dominant species from those due to rare species, and relate these patterns to shifts in overall diversity. This separation of the influence of dominance and rarity on overall diversity enables the user to uncover changes in diversity that would be masked in other methods. We discuss profiling techniques based on this parametric family, and explore its connections with existing diversity indices. Next, we evaluate our approach in terms of predicted community structure (following Tokeshis niche models) and present an example assessing temporal trends in diversity of British farmland birds. We conclude that this method is an informative and tractable parametric approach for quantifying evenness. It provides novel insights into community structure, revealing the contributions of both rare and common species to biodiversity trends.

A family of spatial biodiversity measures based on graphs

While much research in ecology has focused on spatially explicit modelling as well as on measures of biodiversity, the concept of spatial (or local) biodiversity has been discussed very little. This paper generalises existing measures of spatial biodiversity and introduces a family of spatial biodiversity measures by flexibly defining the notion of the individuals’ neighbourhood within the framework of graphs associated to a spatial point pattern. We consider two non-independent aspects of spatial biodiversity, scattering, i.e. the spatial arrangement of the individuals in the study area and exposure, the local diversity in an individual’s neighbourhood. A simulation study reveals that measures based on the most commonly used neighbourhood defined by the geometric graph do not distinguish well between scattering and exposure. This problem is much less pronounced when other graphs are used. In an analysis of the spatial diversity in a rainforest, the results based on the geometric graph have been shown to spuriously indicate a decrease in spatial biodiversity when no such trend was detected by the other types of neighbourhoods. We also show that the choice of neighbourhood markedly impacts on the classification of species according to how strongly and in what way different species spatially structure species diversity. Clearly, in an analysis of spatial or local diversity an appropriate choice of local neighbourhood is crucial in particular in terms of the biological interpretation of the results. Due to its general definition, the approach discussed here offers the necessary flexibility that allows suitable and varying neighbourhood structures to be chosen.

Fine-tuning the assessment of large-scale temporal trends in biodiversity using the example of British breeding birds

Summary The current headline indicator for ecosystem health and sustainability incorporates a geometric mean of relative abundances of breeding birds. Recently, a family of diversity measures (λ-measures) has been proposed as a novel instrument to separate diversity trends in dominant and rare species. This makes them an ecologically informative complement to current composite diversity indices. Using both a geometric mean and the set of λ-measures, we study habitat-specific temporal trends in the diversity of British breeding birds. The analysis employs abundance estimates corrected for variation in detectability between individuals from different species to reduce bias. Applying generalized additive models, we predict long-term trends. We locate significant changes in these diversity trends. While the geometric mean reveals overall diversity trends by habitat type, supplementing these by the λ-measures provides a more nuanced picture of trends: a positive trend in the geometric mean may hide predominantly declining trends among the rarer species, which is then revealed by trends in the λ-measures. Synthesis and Applications. Bird populations are seen as useful indicators of the health of wildlife and the countryside because they occupy a range of habitats, they tend to be towards the top of the food chain, and data is provided by long-term surveys. Hence, many countries apply wild bird indicators (WBIs), quantifying trends in biodiversity, to monitor environmental health. The UKs WBI, for example, has become one of the governments headline indicators of sustainable development. Understanding the population changes underlying the estimated trends is indispensable if we are to allocate limited resources more effectively. Employing a novel set of measures alongside the traditional geometric mean index, we analyse diversity trends among British breeding birds. It reveals that species that are scarce, but not yet in the focus of conservation action, may be the ‘losers’ in biodiversity action plans. This suggests that additional resources should be devoted to species showing long-term decline before they reach the low population levels that currently trigger large-scale species-specific rescue projects.

Success of spatial statistics in determining underlying process in simulated plant communities

Spatial statistics are widely used in studies of ecological processes in plant communities, especially to provide evidence of neutral or non‐neutral mechanisms that might support species coexistence. The contribution of such statistics has been substantial, but their ability to identify any links between underlying processes and emergent patterns is not certain. We investigate the ability of a number of spatial statistics to distinguish theorized mechanisms of species coexistence (spatial and temporal niche differentiation, neutrality, the Janzen–Connell effect and heteromyopia) in a simulated plant community. We find that individual statistics differ substantially in their sensitivity to these mechanisms, with those based on nearest neighbour species identities being the most sensitive. These differences are largely robust to changes in the strength of the modelled mechanisms when simulated independently and in combination. The spatial signal of niche differentiation is always distinct in simulations that combine mechanisms. Synthesis. We describe full spatial signals of modelled coexistence mechanisms that are observed consistently across statistics and simulated strengths and combinations of mechanisms, and identify a set of spatial statistics that holds particular promise for empirical studies designed to investigate mechanisms of these kinds.

Lianas and soil nutrients predict fine‐scale distribution of above‐ground biomass in a tropical moist forest

Acknowledgements. This study was supported by the FP7-PEOPLE-2013-IEF Marie-Curie Action – SPATFOREST. Tree data from BCI were provided by the Center for Tropical Forest Science of the Smithsonian Tropical Research Institute and the primary granting agencies that have supported the BCI plot tree census. Data for the liana censuses were supported by the US National Science Foundation grants: DEB-0613666, DEB-0845071, and DEB-1019436 (to SAS). Soil data was funded by the National Science Foundation grants DEB021104, DEB021115, DEB0212284 and DEB0212818 supporting soils mapping in the BCI plot. We thank Helene Muller-Landau for providing some data on tree height for some BCI trees. We also thank all the people that contributed to obtain the data.