Asymptotic goodness-of-fit tests for the Palm mark distribution of stationary point processes with correlated marks

We consider spatially homogeneous marked point patterns in an unboundedly expanding convex sampling window. Our main objective is to identify the distribution of the typical mark by constructing an asymptotic $\chi^2$-goodness-of-fit test. The corresponding test statistic is based on a natural empirical version of the Palm mark distribution and a smoothed covariance estimator which turns out to be mean square consistent. Our approach does not require independent marks and allows dependences between the mark field and the point pattern. Instead we impose a suitable $\beta$-mixing condition on the underlying stationary marked point process which can be checked for a number of Poisson-based models and, in particular, in the case of geostatistical marking. In order to study test performance, our test approach is applied to detect anisotropy of specific Boolean models.Comment: Published in at http://dx.doi.org/10.3150/13-BEJ523 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm). arXiv admin note: substantial text overlap with arXiv:1205.504

A common goodness-of-fit framework for neural population models using marked point process time-rescaling

A critical component of any statistical modeling procedure is the ability to assess the goodness-of-fit between a model and observed data. For spike train models of individual neurons, many goodness-of-fit measures rely on the time-rescaling theorem and assess model quality using rescaled spike times. Recently, there has been increasing interest in statistical models that describe the simultaneous spiking activity of neuron populations, either in a single brain region or across brain regions. Classically, such models have used spike sorted data to describe relationships between the identified neurons, but more recently clusterless modeling methods have been used to describe population activity using a single model. Here we develop a generalization of the time-rescaling theorem that enables comprehensive goodness-of-fit analysis for either of these classes of population models. We use the theory of marked point processes to model population spiking activity, and show that under the correct model, each spike can be rescaled individually to generate a uniformly distributed set of events in time and the space of spike marks. After rescaling, multiple well-established goodness-of-fit procedures and statistical tests are available. We demonstrate the application of these methods both to simulated data and real population spiking in rat hippocampus. We have made the MATLAB and Python code used for the analyses in this paper publicly available through our Github repository at https://github.com/Eden-Kramer-Lab/popTRT.This work was supported by grants from the NIH (MH105174, NS094288) and the Simons Foundation (542971). (MH105174 - NIH; NS094288 - NIH; 542971 - Simons Foundation)Published versio

A space-time conditional intensity model for infectious disease occurence

A novel point process model continuous in space-time is proposed for infectious disease data. Modelling is based on the conditional intensity function (CIF) and extends an additive-multiplicative CIF model previously proposed for discrete space epidemic modelling. Estimation is performed by means of full maximum likelihood and a simulation algorithm is presented. The particular application of interest is the stochastic modelling of the transmission dynamics of the two most common meningococcal antigenic sequence types observed in Germany 2002–2008. Altogether, the proposed methodology represents a comprehensive and universal regression framework for the modelling, simulation and inference of self-exciting spatio-temporal point processes based on the CIF. Application is promoted by an implementation in the R package RLadyBug

Advances in point process modeling: feature selection, goodness-of-fit and novel applications

The research contained in this thesis extends multivariate marked point process modeling methods for neuroscience, generalizes goodness-of-fit techniques for the class of marked point processes, and introduces the use of a general history-dependent point process model to the domain of sleep apnea. Our first project involves further development of a modeling tool for spiking data from neural populations using the theory of marked point processes. This marked point process model uses features of spike waveforms as marks in order to estimate a state variable of interest. We examine the informational content of geometric features as well as principal components of the waveforms at hippocampal place cell activity by comparing decoding accuracies of a rat's position along a track. We determined that there was additional information available beyond that contained in traditional geometric features used for decoding in practice. The expanded use of this marked point process model in neuroscience necessitates corresponding goodness-of-fit protocols for the marked case. In our second project, we develop a generalized time-rescaling method for marked point processes that produces uniformly distributed spikes under a proper model. Once rescaled, the ground process then behaves as a Poisson process and can be analyzed using traditional point process goodness-of-fit methods. We demonstrate the method's ability to detect quality and manner of fit through both simulation and real neural data analysis. In the final project, we introduce history-dependent point process modeling as a superior method for characterizing severe sleep apnea over the current clinical standard known as the apnea-hypopnea index (AHI). We analyze model fits using combinations of both clinical covariates and event observations themselves through functions of history. Ultimately, apnea onset times were consistently estimated with significantly higher accuracy when history was incorporated alongside sleep stage. We present this method to the clinical audience as a means to gain detailed information on patterns of apnea and to provide more customized diagnoses and treatment prescriptions. These separate yet complementary projects extend existing point process modeling methods and further demonstrate their value in the neurosciences, sleep sciences, and beyond

Contributions to statistical analysis methods for neural spiking activity

With the technical advances in neuroscience experiments in the past few decades, we have seen a massive expansion in our ability to record neural activity. These advances enable neuroscientists to analyze more complex neural coding and communication properties, and at the same time, raise new challenges for analyzing neural spiking data, which keeps growing in scale, dimension, and complexity. This thesis proposes several new statistical methods that advance statistical analysis approaches for neural spiking data, including sequential Monte Carlo (SMC) methods for efficient estimation of neural dynamics from membrane potential threshold crossings, state-space models using multimodal observation processes, and goodness-of-fit analysis methods for neural marked point process models. In a first project, we derive a set of iterative formulas that enable us to simulate trajectories from stochastic, dynamic neural spiking models that are consistent with a set of spike time observations. We develop a SMC method to simultaneously estimate the parameters of the model and the unobserved dynamic variables from spike train data. We investigate the performance of this approach on a leaky integrate-and-fire model. In another project, we define a semi-latent state-space model to estimate information related to the phenomenon of hippocampal replay. Replay is a recently discovered phenomenon where patterns of hippocampal spiking activity that typically occur during exploration of an environment are reactivated when an animal is at rest. This reactivation is accompanied by high frequency oscillations in hippocampal local field potentials. However, methods to define replay mathematically remain undeveloped. In this project, we construct a novel state-space model that enables us to identify whether replay is occurring, and if so to estimate the movement trajectories consistent with the observed neural activity, and to categorize the content of each event. The state-space model integrates information from the spiking activity from the hippocampal population, the rhythms in the local field potential, and the rat's movement behavior. Finally, we develop a new, general time-rescaling theorem for marked point processes, and use this to develop a general goodness-of-fit framework for neural population spiking models. We investigate this approach through simulation and a real data application

Spatial modeling of epidermal nerve fiber patterns

Peripheral neuropathy is a condition associated with poor nerve functionality. Epidermal nerve fiber (ENF) counts per epidermal surface are dramatically reduced and the two-dimensional (2D) spatial structure of ENFs tends to become more clustered as neuropathy progresses. Therefore, studying the spatial structure of ENFs is essential to fully understand the mechanisms that guide those morphological changes. In this article, we compare ENF patterns of healthy controls and subjects suffering from mild diabetic neuropathy by using suction skin blister specimens obtained from the right foot. Previous analysis of these data has focused on the analysis and modeling of the spatial ENF patterns consisting of the points where the nerves enter the epidermis, base points, and the points where the nerve fibers terminate, end points, projected on a 2D plane, regarding the patterns as realizations of spatial point processes. Here, we include the first branching points, the points where the nerve trees branch for the first time, and model the three-dimensional (3D) patterns consisting of these three types of points. To analyze the patterns, spatial summary statistics are used and a new epidermal active territory that measures the volume in the epidermis that is covered by the individual nerve fibers is constructed. We developed a model for both the 2D and the 3D patterns including the branching points. Also, possible competitive behavior between individual nerves is examined. Our results indicate that changes in the ENFs spatial structure can more easily be detected in the later parts of the ENFs