Statistical Learning and Model Criticism for Networks and Point Processes

Point pattern data are ubiquitous in both the natural environment and human activities. Spatial and spatio-temporal point processes are widely utilized to model the occurrence of random events across space and time. These models aim to uncover the underlying patterns of such events by analyzing environmental variables that influence their probabilities, and by testing, estimating and simulating spatial or spatio-temporal correlations. Despite significant advances in point process theory and methodology in the last decades, classical approaches still require continuous refinement and extension. This need arises from the distinctive characteristics of point pattern data in practical applications, as well as the increasing amount and complexity of data available in contemporary research. This thesis explores spatial and spatio-temporal modelling in the context of fire risk prediction in the Netherlands. Fire occurrences exhibit unique features that pose specific challenges in point process modelling. Four critical research questions are identified based on real applications of point process methods to the data. To address these challenges, we developed novel statistical and machine learning approaches, proposing targeted solutions that advance the state-of-the-art in the research domain.<br/><br/>

Cellular networks have a great transformation during the last decade. The traditional cellular network system is often overlaid with well-deployed BSs of proper power. However, more and more complex elements such as picocells, femtocells and relays have been introduced to cellular network system. BSs in these elements are unplanned, user-installed and randomly deployed. All of these things compose heterogeneous cellular networks (HCNs). In order to evaluate the performance of HCNs, people usually model the distribution of BSs in HCNs as spatial stochastic point process model. One of the most popular spatial models for these HCNs is Poisson Point Process (PPP) model. However the PPP model is not exact for some cases where BSs in PPP model may be too close to each other. In this paper, we propose a more reasonable and practical model called Matern-like point process (MLPP) model which imposes a certain minimal distance between any two BSs to overcome this problem. Simulations show that the coverage probability of the proposed model noticeably outperforms the PPP model. This improvement depends on the value of minimal distance. We also evaluate the system efficiency defined by the ratio of coverage probability and total power. The results show that the MLPP model has a significant improvement compared to PPP model.

Matrix completion based collaborative filtering is considered scalable and effective for online service link prediction (e.g., movie recommendation) but does not meet the challenges of link prediction in ecological networks. A unique challenge of ecological networks is that the observed data are subject to systematic imperfect detection, due to the difficulty of accurate field sampling. In this work, we propose a new framework customized for ecological bipartite network link prediction. Our approach starts with incorporating the Poisson N-mixture model, a widely used framework in statistical ecology for modeling imperfect detection of a single species in field sampling. Despite its extensive use for single species analysis, this model has never been considered for link prediction between different species, perhaps because of the complex nature of both link prediction and N-mixture model inference. By judiciously combining the Poisson N-mixture model with a probabilistic nonnegative matrix factorization (NMF) model in latent space, we propose an intuitive statistical model for the problem of interest. We also offer a scalable and convergence-guaranteed optimization algorithm to handle the associated maximum likelihood identification problem. Experimental results on synthetic data and two real-world ecological networks data are employed to validate our proposed approach.

Goodness of fit (GOF) tests of logistic regression attempt to find out the suitability of the model to the data. The null hypothesis of all GOF tests is the model fit. R as a free software package has many GOF tests in different packages. A Monte Carlo simulation has been conducted to study two situations; the first, studying the ability of each test, under its default settings, to accept the null hypothesis when the model truly fitted. The second, studying the power of these tests when assumptions of sufficient linear combination of the explanatory variables are violated (by omitting linear covariate term, quadratic term, or interaction term). Moreover, checking whether the same test in different R packages had the same results or not. As the sample size supposed to affect simulation results, so the pattern of change of GOF tests results under different sample sizes as well as different model settings was estimated. All tests accept the null hypothesis (more than 95% of simulation trials) when the model truly fitted except modified Hosmer-Lemeshow test in "LogisticDx" package under all different model settings and Osius and Rojek’s (OsRo) test when the true model had an interaction term between binary and categorical covariates. In addition, le Cessie-van Houwelingen-Copas-Hosmer unweighted sum of squares (CHCH) test gave unexpected different results under different packages. Concerning the power study, all tests had a very low power when a departure of missing covariate existed. Generally, stukel’s test (package ’LogisticDX) and CHCH test (package "RMS") reached a power in detecting a missing quadratic term greater than 80% under lower sample size while OsRo test (package ’LogisticDX’) was better in detecting missing interaction term. Beside the simulation study, we evaluated the performance of GOF tests using the breast cancer dataset.

We propose and investigate dynamic spatial-temporal point process models for independent and interacting events. The models for independent events are dynamic spatial-temporal Poisson point process (DSTPPP) model that account for temporal and spatial clustering. The models proposed for events with interaction are Markov (Gibbs) space-time point process models. We model the intensity function of a DSTPPP via conditioning arguments that allow for additional interpretations and inclusion of well-known point process models as special cases. Depending on the nature of the questions to be answered, the in- tensity function of a DSTPPP can be modeled in several ways. First, we develop models via conditioning on the time component of the events and then consider conditioning on the location component of the events. Modeling, simulation and computation are accomplished in a fully hierarchical Bayesian framework. Finally, we focus on dynamic marked Markov space-time point processes, where the events are allowed to interact with each other across time and space. Once again the hierarchical Bayesian framework is invaluable in this case, since it allows us to introduce dynamic process models via conditioning. The methodolo- gies are illustrated using simulated data, and several applications, including modeling and inference for earthquake events in California, and tornado events in Missouri.

We combine in situ environmental transmission electron microscopy (E-TEM) with automated image processing and statistical machine learning to uniquely formulate interpretable mathematical models and accurate simulation tools for complex nanoscale phenomena involving coupled physical and chemical processes and interactions that are otherwise hard to model. In particular, there is a need for a better understanding, characterization, and prediction of the proximity effects among dense populations of metal nanocatalysts as they form and evolve over time. Here, we leverage point process theory, a branch of statistical machine learning, to “learn” the spatial dependencies among ensembles of adjacent alumina-supported iron nanoparticles from a time sequence of E-TEM images. We construct a set of point process models to make statistical inferences about the nature of spatial dependencies that govern the rapid formation, or “popping” of nanoparticles during thin film dewetting, concomitant with metal reduction in the presence of acetylene at 750 °C. We show that nanoparticles exhibit strong dispersion behavior, i.e., new nanoparticles pop in dispersed locations at a predictable distance from their existing territorial neighbors. We also show that Gibbs point processes adequately describe the pairwise interactions underlying such time-dependent spatial variations. Further, we build on our machine-learned models to develop a computational simulation tool capable of producing accurate spatiotemporal simulations of nanoparticle formation at finer time resolutions and larger spatial domains than those of experimental observations. This is a much needed capability to overcome current limitations in computational methods supporting the design, analysis, and control of the collective behavior of nanocatalyst populations.

How is the progression of a virus influenced by properties intrinsic to individual cells? We address this question by studying the susceptibility of cells infected with two strains of the human respiratory syncytial virus (RSV-A and RSV-B) in an in vitro experiment. Spatial patterns of infected cells give us insight into how local conditions influence susceptibility to the virus. We observe a complicated attraction and repulsion behavior, a tendency for infected cells to lump together or remain apart. We develop a new spatial point process model to describe this behavior. Inference on spatial point processes is difficult because the likelihood functions of these models contain intractable normalizing constants; we adapt an MCMC algorithm called double Metropolis-Hastings to overcome this computational challenge. Our methods are computationally efficient even for large point patterns consisting of over 10,000 points. We illustrate the application of our model and inferential approach to simulated data examples and fit our model to various RSV experiments. Because our model parameters are easy to interpret, we are able to draw meaningful scientific conclusions from the fitted models.

Quenched invariance principle for simple random walk on discrete point processes

This paper presents a statistically based method for spotting target words in documents. The crux of the method is the representation of a word by a spatial (planar) point process evolving on a regular lattice of coordinate pairs. This is accomplished by extracting the coordinate pairs, i.e. pixel locations, where the binary bitmap values of the word are non-zero. With this representation the word is completely determined by the spatial intensity function, i.e. the unnormalized spatial probability density function, associated with the extracted set of coordinate pairs. In this work, we use a finite number of moments of the intensity function to characterize the word. Location and scale invariance is obtained by transforming the coordinate pairs to have zero mean and unit variance. Finally, optimal detection strategies are applied to the moments to make the decision. 1.

Self-exciting point processes are widely used to model events occurring in time and space whose rate depends on the past history of the process, such as earthquake<br>aftershocks, crime, and neural spike trains. By modeling the event rate as the sum of a background (or immigrant) process, often an inhomogeneous Poisson process, and<br>an offspring process consisting of events triggered by previous events, self-exciting point process models naturally account for complex clustering behavior. When the<br>model is physically motivated, as are models of earthquake aftershock sequences, model parameters have direct interpretations in terms of the generative mechanism.<br>In this thesis, I focus in particular on the application of self-exciting point processes to crime. Crime rates are known to vary greatly in space within a city, as a result of many demographic and economic factors, and crime often exhibits “nearrepeats,” when one crime is followed by another soon after, either from retaliation or because offenders tend to return to the same areas. Point process models have<br>been used to predict crime, but the available models can be improved: they cannot explicitly account for spatially varying covariates and estimate their effects, and there are no inference tools that could be used to test criminological theories or evaluate interventions. After extensively reviewing the literature on self-exciting point processes, I introduce<br>a new model which accounts for both spatial covariates and self-excitation, and explore its benefits over simple lagged regressions and other commonly used methods. After discussing computational issues in fitting the model, I use simulations to explore methods for parameter inference, review a set of residual diagnostics and animations, and use these diagnostics to explore the model’s behavior under<br>various forms of model misspecification, giving practical advice for the interpretation of model fits. To demonstrate the model’s utility, I then analyze large databases<br>of Pittsburgh and Baltimore crime records, linking crime rates to several relevant spatial covariate and leading indicator events, and comparing several model variations.<br>

Location‐only and use‐availability data: analysis methods converge

First I would like to thank the authors for providing a thorough review of Goodnessof-Fit (GoF) tests for regression models. It is truly an achievement to cover such a wide spectrum of methods of GoF within a paper of reasonable length. With their expertise in GoF, Professors Gonzalez-Manteiga and Crujeiras (GMC) have succeeded in providing an up-to-date review of methods and corresponding theory of tests dealing mostly with the structure of the regression function, and in doing so manage to cover the basic parametric regression model as well as alternative models which may be generally described as nonparametric or semiparametric. They also consider, apart from the typical scenario of independent observations, GoF tests with dependence, and tests under more complex data structures. In what follows I will try to discuss in more detail specific alternative aspects of the regression model. For these aspects one may formulate corresponding hypotheses that may also be included under the general heading ‘GoF for regression’. In addition, I will try to provide some insight on methods of GoF which utilize the characteristic function (CF).

Objective. Information encoding in neurons can be described through their response fields. The spatial response field of a neuron is the region of space in which a sensory stimulus or a behavioral event causes that neuron to fire. Neurons can also exhibit temporal response fields (TRFs), which characterize a transient response to stimulus or behavioral event onsets. These neurons can thus be described by a spatio-temporal response field (STRF). The activity of neurons with STRFs can be well-described with point process models that characterize binary spike trains with an instantaneous firing rate that is a function of both time and space. However, developing decoders for point process models of neurons that exhibit TRFs is challenging because it requires prior knowledge of event onset times, which are unknown. Indeed, point process filters (PPF) to date have largely focused on decoding neuronal activity without considering TRFs. Also, neural classifiers have required data to be behavior- or stimulus-aligned, i.e. event times to be known, which is often not possible in real-world applications. Our objective in this work is to develop a viable decoder for neurons with STRFs when event times are unknown. Approach. To enable decoding of neurons with STRFs, we develop a novel point-process matched filter (PPMF) that can detect events and estimate their onset times from population spike trains. We also devise a PPF for neurons with transient responses as characterized by STRFs. When neurons exhibit STRFs and event times are unknown, the PPMF can be combined with the PPF or with discrete classifiers for continuous and discrete brain state decoding, respectively. Main results. We validate our algorithm on two datasets: simulated spikes from neurons that encode visual saliency in response to stimuli, and prefrontal spikes recorded in a monkey performing a delayed-saccade task. We show that the PPMF can estimate the stimulus times and saccade times accurately. Further, the PPMF combined with the PPF can decode visual saliency maps without knowing the stimulus times. Similarly, the PPMF combined with a point process classifier can decode the saccade direction without knowing the saccade times. Significance. These event detection and decoding algorithms can help develop neurotechnologies to decode cognitive states from neural responses that exhibit STRFs.

Point processes are a useful mathematical tool for describing events over time, and so there are many recent approaches for representing and learning them. One notable open question is how to precisely describe the flexibility of point process models and whether there exists a general model that can represent all point processes. Our work bridges this gap. Focusing on the widely used event intensity function representation of point processes, we provide a proof that a class of learnable functions can universally approximate any valid intensity function. The proof connects the well known Stone-Weierstrass Theorem for function approximation, the uniform density of non-negative continuous functions using a transfer functions, the formulation of the parameters of a piece-wise continuous functions as a dynamic system, and a recurrent neural network implementation for capturing the dynamics. Using these insights, we design and implement UNIPoint, a novel neural point process model, using recurrent neural networks to parameterise sums of basis function upon each event. Evaluations on synthetic and real world datasets show that this simpler representation performs better than Hawkes process variants and more complex neural network-based approaches. We expect this result will provide a practical basis for selecting and tuning models, as well as furthering theoretical work on representational complexity and learnability.

We discuss Bayesian modelling and computational methods in analysis of indirectly observed spatial point processes. The context involves noisy measurements on an underlying point process that provide indirect and noisy data on locations of point outcomes. We are interested in problems in which the spatial intensity function may be highly heterogenous, and so is modelled via flexible nonparametric Bayesian mixture models. Analysis aims to estimate the underlying intensity function and the abundance of realized but unobserved points. Our motivating applications involve immunological studies of multiple fluorescent intensity images in sections of lymphatic tissue where the point processes represent geographical configurations of cells. We are interested in estimating intensity functions and cell abundance for each of a series of such data sets to facilitate comparisons of outcomes at different times and with respect to differing experimental conditions. The analysis is heavily computational, utilizing recently introduced MCMC approaches for spatial point process mixtures and extending them to the broader new context here of unobserved outcomes. Further, our example applications are problems in which the individual objects of interest are not simply points, but rather small groups of pixels; this implies a need to work at an aggregate pixel region level and we develop the resulting novel methodology for this. Two examples with with immunofluorescence histology data demonstrate the models and computational methodology.