A $K-$function is presented for assessing second-order properties of inhomogeneous fiber patterns generated by marked point processes. The $K-$function takes into account the geometric features of the fibers, such as tangent directions. The $K-$function requires an estimate of the inhomogeneous density function of the fiber pattern. We introduce parametric estimates for the density function based on parametric models that represent large-scale features of the inhomogeneous fiber pattern. The proposed methodology is applied to simulated fiber patterns as well as a three-dimensional data set of steel fibers in concrete.

The development of statistical methods designed for analysing spatial point patterns has typically focused on Euclidean data and planar surfaces. However, with recent advances in 3D biological imaging technologies targeting protein molecules on cellular membranes, spatial point patterns are now being observed on complex shapes and manifolds whose geometry must be respected. Consequently, there is now a demand for tools that can analyse these data for important scientific studies in cellular and micro-biology. For this purpose, we extend the classical functional summary statistics for spatial point patterns to general convex bounded shapes. Using the Mapping Theorem, a Poisson process can be transformed from any convex shape to a Poisson process on the unit sphere where existing theory and methods can be leveraged. We present the first and second order properties of such summary statistics and demonstrate how they can be used to construct test statistics to determine whether an observed pattern exhibits complete spatial randomness on the original convex space. We will then present recent work in which we develop the kernel intensity estimator for manifolds.

The analysis of more complex spatial point process scenarios has gained immense attraction within the last decade. While there has been impressive progress in the theory and methodology of (marked) spatial point processes with points on e.g. nets, Euclidean graphs or the sphere, only a few extensions to the non-scalar (non-Euclidean) mark setting extist. In particular, apart from function-valued marks, the treatment of more general object-valued marks has just been introduced. This talk addresses a particular type of object-valued marks and focuses on the case where each point is augmented by a density-valued quantity.
References:
[1] Matthias Eckardt, Sonja Greven and Mari Myllymäki (2023): Composition-valued marked spatial point processes, Working paper
[2] Matthias Eckardt, Carles Comas, Jorge Mateu (2023): Summary characteristics for multivariate function-valued spatial point process attributes, Working paper
[3] Matthias Eckardt, Alessandra Menafoglio and Sonja Greven (2023): Spatial point processes with density-valued marks, Working paper

Aoristic data are interval-censored data depicting time spans in which an event occurred with certainty, but where the exact time of occurrence is unknown. To infer the occurrence times, we implement a Bayesian approach. For the forward model, we use a marked point process, with the marks representing the censored intervals. The prior is a Markov point process. Previously, approaches operate under the assumption that censoring occurs homogeneously in time when modelling aoristic data. In this work, we introduce a non-homogeneous, semi-Markov-inspired censoring mechanism that is able to capture seasonal variations.

Estimating the first-order intensity function in point pattern analysis is an important problem, and it has been approached so far from different perspectives. Motivated by eye-movement data, we introduce a convolution type model where the log-intensity is modelled as the convolution of a function $\beta(\cdot)$, to be estimated, and a single spatial covariate (the image an individual is looking at for eye-movement data). Based on a Fourier series expansion, we show that the proposed model can be viewed as a log-linear model with an infinite number of coefficients, which correspond to the spectral decomposition of $\beta(\cdot)$. After truncation, we estimate these coefficients through a penalized Poisson likelihood.

Parameter estimation for point processes is achieved via solving optimisation problems built using general strategies. Three well established strategies are enumerated. The first consists of considering contrast fuctions based on summary statistics. The second one uses the pseudo-likelihood. And the third approximates the likelihood function via Monte Carlo procedures. Each of these techniques has known advantages and drawbacks (Moler and Waagepetersen 2004, van Lieshout 2001, 2019). Sampling point process posterior densities is an inference approach deeply intertwinned wih the previous one, since it allows simultaneous parameter estimation and statistical tests based on observations. The auxiliary variable method (Moller et al.,2006) gives the mathematical solution to this problem, while pointing out the difficulties of its practical implementation due to poor mixing. The exchange algorithm proposed by (Murray et al. 2006), (Caimo and Friel, 2011) proposes a solution for the poor mixing induced by the auxiliary variable method. As its predecessor it requires exact simulation for the sampling of the auxiliary variable. This is not really a drawback, but it may explode the computational time for models exhibiting strong interactions (van Lieshout and Stoica, 2006). This talk presents the approximate ABC Shadow and SSA methods as complementary inference methods to the ones based on posterior density sampling. These methods do not require exact simulation, while providing the necessary theoretical control. The derived algorithms are applied on data from several application domains such as astronomy, geosciences and network sciences (Stoica et al.,17), (Stoica et al.,21), (Hurtado et al.,21), (Laporte et al.,22).

While modeling the spatio-temporal intensity of the point pattern formed by lightning strikes in France over the period 2011-2015, we were faced with numerical problems which can be explained with the following facts: the number of impacts is huge, the spatial and spatio-temporal covariates are constant over a very fine spatial or spatio-temporal grid and last but not least the number of (spatio-temporal) cells over which no lighning strike is observed is huge. In this talk, I'll revisit standard parametric methods with constant covariates and will present an extension of these using zero-deflated subsampling to handle the excess of zeroes.

Birth-death-move processes with mutations are models for the spatio-temporal dynamics of a system of marked particles in motion where births, deaths and mutations can occur. We consider in this talk the parametric estimation of such processes. We obtain the expression of the likelihood function and we prove that under some regularity conditions, the model satisfies the LAN (local asymptotic normality) property, implying the optimality of regular estimators. We apply this result to the maximum likelihood estimation of parameters involved in the birth kernel of the process. After a short simulation study that confirms our theoretical findings, we analyse a real dataset representing the spatio-temporel dynamics of biomolecules observed in a living cell. Our estimation reveals and quantifies a colocalisation phenomenon between two types of proteins involved in the membrane trafficking of the cell. This work is part of the ongoing PhD study of Lisa Balsollier.

Chimney fires constitute one of the most commonly occurring fire types. Precise prediction and prompt prevention are crucial in reducing the harm they cause. We develop a combined machine learning and statistical modelling process to predict fire risk. Firstly, we use random forests and permutation importance techniques to identify the most informative explanatory variables. Secondly, we design a Poisson point process model and employ logistic regression estimation to estimate the parameters. Thirdly, we establish strong consistency and asymptotic normality for the parameters and propose a consistent estimator of the asymptotic covariance matrix that can be used to build confidence intervals for the predicted fire risk. Finally, we validate the Poisson model assumption using second-order summary statistics and residuals and implement the modelling process on data collected by the Twente Fire Brigade. This approach has two advantages: i) with random forests, we can select explanatory variables nonparametrically considering variable dependence; ii) using logistic regression estimation, we can fit our statistical model efficiently by tuning it to focus on regions and times that are salient for fire risk.

References:
C. Lu, M.N.M. van Lieshout, M. de Graaf and P.J. Visscher. Data-driven chimney fire risk prediction using machine learning and point process tools. Annals of Applied Statistics, to appear. M.N.M. van Lieshout and C. Lu. Infill asymptotics for logistic regression estimators for spatio-temporal point processes. ArXiv 2208.12080, August 2022.

Detecting change-points in multivariate settings is usually carried out by analyzing all marginals either independently, via univariate methods, or jointly, through multivariate approaches. The former discards any inherent dependencies between different marginals and the latter may suffer from domination/masking among different change-points of distinct marginals. As a remedy, we propose an approach which groups marginals with similar temporal behaviours and then performs group-wise multivariate change-point detection. Our approach groups marginals based on hierarchical clustering using distances which adjust for inherent dependencies.

Motivation comes from materials research, when observing particles on grain boundaries in the microstructure of polycrystals. Given a bounded Laguerre tessellation, the homogeneous Poisson process on the planar network of faces is second-order pseudostationary when the shortest path distance on the planar network is considered. Using the coarea formula we derive formula for the geometrically corrected K-function and suggest an algorithm for an approximate numerical evaluation of the correction factor. Finally a cluster process on the planar network is introduced and investigated.

To avoid measuring lots of zeroes, Finnish National Forest Inventory (NFI, Korhonen et al. doi.org/10.14214/sf.10662) distributes field sample plots in sparsely forested northernmost part of the country with stratified sampling. Strata with small predicted proportion of forests are sampled less intensively. This is based on maps produced using earlier NFI measurements and satellite images (Mäkisara et al. http://urn.fi/URN:ISBN:978-952-380-538-5). In this study, we evaluated methods to sample within the fragmented strata. Among spatially regular designs, local pivotal method (Grafström et al. doi.org/10.1111/j.1541-0420.2011.01699.x) was more efficient than systematic sampling in stratified sampling, but – somewhat surprisingly – also in unstratified sampling.

In classical statistics, there is usually an underlying i.i.d. assumption. However, in the field of spatial and temporal statistics we often have dependencies for example in forestry, seismology and epidemiology. A point process may be viewed as a generalized random sample where we allow dependence and a random sample size. Recently a prediction-based statistical theory was proposed called point process learning (PPL). It is based on the combination of two novel concepts for general point processes: cross-validation and prediction errors. We here explore how PPL can be exploited to model a range of common Gibbs point process models, like the hard-core process, the Strauss process and the Geyer saturation process. Through a simulation study, we compare the performance of PPL in terms of MSE and MAPE to pseudo-likelihood methods, which is the state of the art in the context of the studied models. We found that PPL outperforms pseudo-likelihood for all considered models. In addition, we propose a procedure to obtain resample-based confidence regions for the model parameters. The empirical coverage was studied for these so-called PPL-confidence intervals, and we found that there is merit to the confidence region idea.

The poster concerns a new statistical method for assessing similarity of two random sets based on one realisation of each of them. The method focuses on shapes of the components in the given realisations, namely on curvature of their boundaries together with ratios of their perimeters and areas. Brief theoretical background is introduced, the method is described, justified by a simulation study and applied to real data.

Permutation-based global envelope tests allow us to perform simultaneous inference of the quantile regression process, even in the presence of nuisance covariates. When the covariate of interest is categorical, graphical n-sample tests for comparing distributions can be constructed. The global tests are applied to nerve fiber point patterns to compare the nerve tree size distributions between patients with diabetes and healthy controls.

We use the tools of topological data analysis to detect some features of random sets in order to do the classification of random sets and for detecting outliers. We track the birth and death of connected components and cycles that emerge and disappear as the realization of the random set erodes and dilates by the disks of increasing radii. Persistence diagram keeps that information and we view it as an empirical measure. Statistical depth can be defined on that (random) measure, for example by using support functions of corresponding lift zonoid and applying methods for assigning depth to functional data. We apply it to real data concerning mastopatic and mammary cancer breast tissue histological images. We also use the persistence diagram for testing goodness of fit to Boolean model using so called accumulated persistence functions as test functions for global envelope test and compare the results with other test functions such as capacity functional, spherical contact distribution function and support function of lift zonoid. It seems like it is very useful tool in recognizing whether clustering or repulsiveness occurs and can be used for classification of random sets.

The estimation of forest attributes representing aspects of structural biodiversity from vast areas is difficult not only because of the lack of agreement on what constitutes a structurally diverse forest, but also due to difficulties regarding the detection of small trees. However, there is a consensus that a structurally diverse forest should have large variation in tree size (i.e., height and diameter), as well as some clustering of trees, as regular tree pattern is typical for managed forests. We developed a framework for building structurally representative tree maps using airborne laser scanning data combined with a limited number of ground measurements using data from two locations in Finland. In the proposed method, an individual tree detection algorithm is optimized so that the number of missing trees and false discoveries is minimized. The ground measurements are then used to train models for the number and location of missing trees and false discoveries. This model can then be applied to other areas in the proximity of the ground measurements to build tree maps, with the location, height, and diameter for each tree. The methodology was shown to reproduce the number of trees, the height distribution, and the spatial pattern of the trees with sufficient accuracy for practical use in large-scale mapping of forest attributes. Our next steps are to account for tree species when building tree maps.

We model cloud-to-ground lightning strikes collected over a period covering 2011-2021 for the French Alps using a spatio-temporal point process. This results to the observation of more than a million of points. With this characteristic in mind, we investigate the non-parametric estimation of the spatio-temporal intensity function as well as the intensity functions of the point processes aggregated over space and time. Then, we present a non-parametric approach to test the first-order separability of the spatio-temporal point process. Negative answer to this question leads to natural perspectives presented in the end of the extended summary.

Geometric modelling of materials microstructures allows to generate synthetic samples, e.g. for training neural networks or simulating material properties. For fitting and comparing models, a measure of general similarity between realizations of geometric models is beneficial. Ideally, similarity should be measured based on features describing, e.g., directionality, porosity, surface texture or shape. We propose such similarity measures and evaluate them on Boolean models of spheres, ellipsoids, cylinders, cubes and cuboids.

Gradient boosting regression trees (GBRT), and in particular extreme gradient boosting trees (XGBoost), are widely adopted machine learning tools for classification and regression problems. We consider their utilities for improving first- and second-order estimation of spatial point processes. For intensity estimation, classic approaches include kernel estimators in the spatial or covariate domain and log-linear models that parametrically model covariate information. Compared to them, our XGBoost model is flexible, nonparametric, exploits both spatial and covariate information, avoids the adaptive bandwidth selection problem and is still computationally fast. Specifically, we consider customised loss functions related to Poisson and logistic likelihoods, and, in the spirit of the quasi-likelihood approach, we propose (dynamic) weighted loss functions to involve spatial correlations. For second-order estimation, in the recent literature, non-stationarity depending on covariate information to pair correlation functions was considered. Following that, we propose an XGBoost model to estimate them nonparametrically based on both covariates and inter-point distances by designing a loss function of composite likelihood. Several simulation studies were conducted to validate our models.

Polycrystalline materials are widely used in various applications and are often modeled using stochastic methods to simulate their 3D microstructure. We propose a methodology that models the joint distribution of crystallographic orientations conditioned on a 3D Laguerre tessellation, accounting for the interaction between neighboring grains' orientations. Estimation of the model parameters and methods for model comparison are discussed. Finally, our approach is demonstrated using a nickel-titanium shape memory alloy dataset.