The discretely observed immigration-death process: likelihood inference and spatiotemporal applications

We consider a stochastic process, the homogeneous spatial immigration-death (HSID) process, which is a spatial birth-death process with as building blocks (i) an immigration-death (ID) process (a continuous-time Markov chain) and (ii) a probability distribution assigning iid spatial locations to all events. For the ID process, we derive the likelihood function, reduce the likelihood estimation problem to one dimension, and prove consistency and asymptotic normality for the maximum likelihood estimators (MLEs) under a discrete sampling scheme. We additionally prove consistency for the MLEs of HSID processes. In connection to the growth-interaction process, which has a HSID process as basis, we also fit HSID processes to Scots pine data

A cross-validation-based statistical theory for point processes

Motivated by cross-validation’s general ability to reduce overfitting and mean square error, we develop a cross-validation-based statistical theory for general point processes. It is based on the combination of two novel concepts for general point processes: cross-validation and prediction errors. Our cross-validation approach uses thinning to split a point process/pattern into pairs of training and validation sets, while our prediction errors measure discrepancy between two point processes. The new statistical approach, which may be used to model different distributional characteristics, exploits the prediction errors to measure how well a given model predicts validation sets using associated training sets. Having indicated that our new framework generalizes many existing statistical approaches, we then establish different theoretical properties for it, including large sample properties. We further recognize that non-parametric intensity estimation is an instance of Papangelou conditional intensity estimation, which we exploit to apply our new statistical theory to kernel intensity estimation. Using independent thinning-based cross-validation, we numerically show that the new approach substantially outperforms the state of the art in bandwidth selection. Finally, we carry out intensity estimation for a dataset in forestry (Euclidean domain) and a dataset in neurology (linear network)

Local Inhomogeneous Weighted Summary Statistics for Marked Point Processes

We introduce a family of local inhomogeneous mark-weighted summary statistics, of order two and higher, for general marked point processes. Depending on how the involved weight function is specified, these summary statistics capture different kinds of local dependence structures. We first derive some basic properties and show how these new statistical tools can be used to construct most existing summary statistics for (marked) point processes. We then propose a local test of random labeling. This procedure allows us to identify points, and consequently regions, where the random labeling assumption does not hold, for example, when the (functional) marks are spatially dependent. Through a simulation study we show that the test is able to detect local deviations from random labeling. We also provide an application to an earthquake point pattern with functional marks given by seismic waveforms

Bandwidth selection for kernel estimators of the spatial intensity function

We discuss and compare various approaches to the problem of bandwidth selection for kernel estimators of intensity functions of spatial point processes. We also propose a new method based on the Campbell formula applied to the reciprocal intensity function. The new method is fully non-parametric, does not require knowledge of the product densities, and is not restricted to a specific class of point process models

Spatiotemporal Modeling of Swedish Scots Pine Stands

The growth-interaction (GI) process is used for the spatiotemporal modeling of measurements of locations and radii at breast height made at three different time points of the individual trees in 10 Scots pine (Pinus sylvestris) plots from the Swedish National Forest Inventory. The GI process places trees at random locations in the study region and assigns sizes to the trees, which interact and grow with time. It has been used to model plots in previous studies and to improve the fit we suggest some modifications: a different location assignment strategy and a different open-growth (growth under negligible competition) function. We believe that the calibration data contain trees that are too small to reflect the open growth properly, which primarily affects the carrying capacity parameter. To better represent the open growth of Scots pines, we evaluate the open growth from a separate set of data (size and age measurements of older and larger single Scots pines). A linear relationship is found between the plot's estimated site indices and the sizes, and this is exploited in the estimation of the carrying capacity. We finally estimate the remaining GI process parameters and test the goodness of fit on simulated predictions from the fitted model

Bandwidth selection for kernel estimators of the spatial intensity function

We discuss and compare various approaches to the problem of bandwidth selection for kernel estimators of intensity functions of spatial point processes. We also propose a new method based on the Campbell formula applied to the reciprocal intensity function. The new method is fully non-parametric, does not require knowledge of the product densities, and is not restricted to a specific class of point process models