Pavel Grabarnik

On Parameter Estimation for Pairwise Interaction Point Processes

Summary Pairwise interaction point processes form a useful class of models for spatial point patterns, especially patterns for which the spatial distribution of points is more regular than for a homogeneous planar Poisson process. Several authors have proposed methods for estimating the parameters of a pairwise interaction point process. However, there appears to be no general theory which provides grounds for preferring a particular method, nor have any extensive empirical comparisons been published. In this paper, we review three general methods of estimation which have been proposed in the literature and present the results of a comparative simulation study of the three methods.

Global envelope tests for spatial processes

Summary Envelope tests are a popular tool in spatial statistics, where they are used in goodness-of-fit testing. These tests graphically compare an empirical function T(r) with its simulated counterparts from the null model. However, the type I error probability α is conventionally controlled for a fixed distance r only, whereas the functions are inspected on an interval of distances I. In this study, we propose two approaches related to Barnards Monte Carlo test for building global envelope tests on I: ordering the empirical and simulated functions on the basis of their r-wise ranks among each other, and the construction of envelopes for a deviation test. These new tests allow the a priori choice of the global α and they yield p-values. We illustrate these tests by using simulated and real point pattern data.

Chapter Eighteen Uncertainty and Sensitivity Issues in Process-based Models of Carbon and Nitrogen Cycles in Terrestrial Ecosystems

Many process-based models of carbon (C) and nitrogen (N) cycles have been developed for northern forest ecosystems. These models are widely used to evaluate the long-term decisions in forest management dealing with effects like particulate pollution, productivity and climate change. Regarding climate change, one of the key questions that have sensitive political implications is whether northern forests will sequester atmospheric C or not. Whilst many process-based models have been tested for accuracy by evaluating or validating against observed data, few have dealt with the complexity of the incorporated procedures to estimate uncertainties associated with model predictions or the sensitivity of these predictions to input factors in a systematic, inter-model comparison fashion. In general, models differ in their underlying attempts to match natural complexities with assumed or imposed model structure and process formulations to estimate model parameters, to gather data and to address issues on scope, scale and natural variations. Uncertainties may originate from model structure, estimation of model parameters, data input, representation of natural variation and scaling exercises. Model structure relates to the mathematical representation of the processes modelled and the type of state variables that a model contains. The modelling of partitioning among above- and below-ground C and N pools and the interdependence among these pools remain a major source of uncertainty in model structure and error propagation. For example, most soil C models use at least three state variables to represent the different types of soil organic matter (SOM). This approach results in creating three artificial SOM pools, assuming that each one contains C compounds with the same turnover rate. In reality, SOM consists of many different types of C compounds with widely different turnover rates. Uncertainty in data and parameter estimates are closely linked. Data uncertainties are associated with high variations in estimating forest biomass, productivity and soil organic matter and their estimates may be incomplete for model initialisation, calibration, validation and sensitivity analysis of generalised predictor models. The scale at which a model is being used also affects the level of uncertainty, as the errors in the prediction of the C and N dynamics differ from site to landscape levels and across climatic regions. If the spatial or temporal scale of a model application is changed, additional uncertainty arises from neglecting natural variability in system variables in time and space. Uncertainty issues are also intimately related to model validation and sensitivity analysis. The estimation of uncertainties is needed to inform decision processes in order to detect the possible corridor of development. Uncertainty in this context is an essential measure of quality for stakeholder and decision makers.

Interacting neighbour point processes: Some models for clustering

We introduce a class of spatial point processesinteracting neighbour point (INP) processes, where the density of the process can be written by means of local interactions between a point and subsets of its neighbourhood but where the processes may not be Ripley-Kelly Markov processes with respect to this neighbourhood. We show that the processes are iterated Markov processes defined by Hayat and Gubner (1996). Furthermore, we pay special attention to a subclass of interacting neighbour processes, where the density belongs to the exponential family and all neighbours of a point affect it simultaneously. A simulation study is presented to show that some simple processes of this subclass can produce clustered patterns of great variety. Finally, an empirical example is given.

New procedure for the simulation of belowground competition can improve the performance of forest simulation models

The major part of existing models of belowground competition in mixed forest stands is limited in explaining the spatial distribution of roots as a response to competitive pressure from neighbours and heterogeneity of soil properties. We are presenting a new spatially explicit and multi-layered discrete model of belowground competition, RootInt (ROOTs INTake). It describes spatial distribution of belowground biomass and allows simulation of competition between trees for soil nutrients. The tree-specific area of root zone is calculated on the basis of stem diameter, with site-specific modifiers to account for the effect of soil fertility and moisture. The shape of root zone is dependent on the amount of available nitrogen in the current cell, distance between this cell and the stem base, and the mass of roots of other plants. RootInt was incorporated into ecosystem model EFIMOD to refine the existing description of belowground competition in forest stands with multiple cohorts and tree species. The results of simulation showed that bringing more complexity into structure of stand (including initial spatial locations of trees, species composition and age structure, vertical structure of canopy) resulted in higher spatial variation in competition intensity, as well as in higher rates of resource uptake. This indicates that stands with complex canopy structure had high plasticity in their root systems and were adapted to intensive competition for soil resources.

Modelling Tree Roots in Mixed Forest Stands by Inhomogeneous Marked Gibbs Point Processes

The aim of the paper is to apply point processes to root data modelling. We propose a new approach to parametric inference when the data are inhomogeneous replicated marked point patterns. We generalize Geyers saturation point process to a model, which combines inhomogeneity, marks and interaction between the marked points. Furthermore, the inhomogeneity influences the definition of the neighbourhood of points. Using the maximum pseudolikelihood method, this model is then fitted to root data from mixed stands of Norway spruce (Picea abies (L.) Karst.) and European beech (Fagus sylvatica L.) to quantify the degree of root aggregation in such mixed stands. According to the analysis there is no evidence that the two root systems are not independent.

20 - DLES: A Component-Based Framework for Ecological Modeling

Abstract Discrete Lattice Ecosystem Simulator (DLES) is a component-based framework which facilitates integration of stand-alone models with spatial interactions, and comparison between models. DLES allows coupling of models of different spatial and temporal resolution. The main features of our approach are (1) the system of models isn’t a “monolithic program” (single executable file), but a number of stand-alone modules that can be easily added to, and removed from, the system; (2) modules exchange data with each other via a shared area of memory controlled by a special system unit; and (3) a module can be either a submodel or another data provider (file, database, etc.).

Statistics for the stationary strauss model by the cusp point method

The eusp point methad yields estimates for the parameters of the stationary hard–core Strauss model using second–order characteristics.Monte Carlo experiments show that this method can compete with other more complicated estimation methods.