arXiv subjects

Paul Marriott

Publications and source records attributed to Paul Marriott.

13 recordsLinked to original sources

arXiv2025

Using Information Geometry to Characterize Higher-Order Interactions in EEG

In neuroscience, methods from information geometry (IG) have been successfully applied in the modelling of binary vectors from spike train data, using the orthogonal decomposition of the Kullback-Leibler divergence and mutual information to isolate different orders of interaction between neurons. While spike train data is well-approximated with a binary model, here we apply these IG methods to data from electroencephalography (EEG), a continuous signal requiring appropriate discretization strategies. We developed and compared three different binarization methods and used them to identify third-order interactions in an experiment involving imagined motor movements. The statistical significance of these interactions was assessed using phase-randomized surrogate data that eliminated higher-order dependencies while preserving the spectral characteristics of the original signals. We validated our approach by implementing known second- and third-order dependencies in a forward model and quantified information attenuation at different steps of the analysis. This revealed that the greatest loss in information occurred when going from the idealized binary case to enforcing these dependencies using oscillatory signals. When applied to the real EEG dataset, our analysis detected statistically significant third-order interactions during the task condition despite the relatively sparse data (45 trials per condition). This work demonstrates that IG methods can successfully extract genuine higher-order dependencies from continuous neural recordings when paired with appropriate binarization schemes.

q-bio.NC↗

arXiv2023

Local behavior of critical points of isotropic Gaussian random fields

In this paper we examine isotropic Gaussian random fields defined on $\mathbb R^N$ satisfying certain conditions. Specifically, we investigate the type of a critical point situated within a small vicinity of another critical point, with both points surpassing a given threshold. It is shown that the Hessian of the random field at such a critical point is equally likely to have a positive or negative determinant. Furthermore, as the threshold tends to infinity, almost all the critical points above the threshold are local maxima and the saddle points with index $N-1$. Consequently, we conclude that the closely paired critical points above a high threshold must comprise one local maximum and one saddle point with index $N-1$.

math.PR↗

arXiv2022

A Multivariate Point Process Model for Simultaneously Recorded Neural Spike Trains

The current state-of-the-art in neurophysiological data collection allows for simultaneous recording of tens to hundreds of neurons, for which point processes are an appropriate statistical modelling framework. However, existing point process models lack multivariate generalizations which are both flexible and computationally tractable. This paper introduces a multivariate generalization of the Skellam process with resetting (SPR), a point process tailored to model individual neural spike trains. The multivariate SPR (MSPR) is biologically justified as it mimics the process of neural integration. Its flexible dependence structure and a fast parameter estimation method make it well-suited for the analysis of simultaneously recorded spike trains from multiple neurons. The strengths and weaknesses of the MSPR are demonstrated through simulation and analysis of experimental data.

stat.ME↗

arXiv2017

Asymptotic coverage probabilities of bootstrap percentile confidence intervals for constrained parameters

The asymptotic behaviour of the commonly used bootstrap percentile confidence interval is investigated when the parameters are subject to linear inequality constraints. We concentrate on the important one- and two-sample problems with data generated from general parametric distributions in the natural exponential family. The focus of this paper is on quantifying the coverage probabilities of the parametric bootstrap percentile confidence intervals, in particular their limiting behaviour near boundaries. We propose a local asymptotic framework to study this subtle coverage behaviour. Under this framework, we discover that when the true parameters are on, or close to, the restriction boundary, the asymptotic coverage probabilities can always exceed the nominal level in the one-sample case; however, they can be, remarkably, both under and over the nominal level in the two-sample case. Using illustrative examples, we show that the results provide theoretical justification and guidance on applying the bootstrap percentile method to constrained inference problems.

math.ST↗

arXiv2015

Mixture Models: Building a Parameter Space

Despite the flexibility and popularity of mixture models, their associated parameter spaces are often difficult to represent due to fundamental identification problems. This paper looks at a novel way of representing such a space for general mixtures of exponential families, where the parameters are identifiable, interpretable, and, due to a tractable geometric structure, the space allows fast computational algorithms to be constructed.

stat.ME↗

arXiv2015

Generalizing the Frailty Assumptions in Survival Analysis

This paper studies Cox's regression hazard model with an unobservable random frailty where no specific distribution is postulated for the frailty variable, and the marginal lifetime distribution allows both parametric and non-parametric models. Laplace's approximation method and gradient search on smooth manifolds embedded in Euclidean space are applied, and a non-iterative profile likelihood optimization method is proposed for estimating the regression coefficients. The proposed method is compared with the Expected-Maximization method developed based on a gamma frailty assumption, and also in the case when the frailty model is misspecified.

stat.ME↗

arXiv2015

Local and global robustness in conjugate Bayesian analysis

This paper studies the influence of perturbations of conjugate priors in Bayesian inference. A perturbed prior is defined inside a larger family, local mixture models, and the effect on posterior inference is studied. The perturbation, in some sense, generalizes the linear perturbation studied in \cite{Gustafson1996}. It is intuitive, naturally normalized and is flexible for statistical applications. Both global and local sensitivity analyses are considered. A geometric approach is employed for optimizing the sensitivity direction function, the difference between posterior means and the divergence function between posterior predictive models. All the sensitivity measure functions are defined on a convex space with non-trivial boundary which is shown to be a smooth manifold.

stat.ME↗

arXiv2014

Detection of Phase Shift Events

We consider the problem of change-point estimation of the instantaneous phase of an observed time series. Such change points, or phase shifts, can be markers of information transfer in complex systems; their analysis occurring in geology, biology and physics, but most notably in neuroscience. We develop two non-parametric approaches to this problem: the cumulative summation (CUSUM) and phase derivative (PD) estimators. In general the CUSUM estimator has higher power for identifying single shift events, while the PD estimator has better temporal resolution for multiple ones. A system of weakly coupled Rossler attractors provides an application in which there are high levels of systematic and time-dependent noise. Shift identification is also performed on beta-band activity from electroencephalogram recordings of a visual attention task, an unsupervised application which requires high temporal resolution.

stat.AP↗

arXiv2013

Variational Bayes inference and Dirichlet process priors

This paper shows how the variational Bayes method provides a computational efficient technique in the context of hierarchical modelling using Dirichlet process priors, in particular without requiring conjugate prior assumption. It shows, using the so called parameter separation parameterization, a simple criterion under which the variational method works well. Based on this framework, its provides a full variational solution for the Dirichlet process. The numerical results show that the method is very computationally efficient when compared to MCMC. Finally, we propose an empirical method to estimate the truncation level for the truncated Dirichlet process.

stat.CO↗

arXiv2013

Diagnostics for Variational Bayes approximations

Variational Bayes (VB) has shown itself to be a powerful approximation method in many application areas. This paper describes some diagnostics methods which can assess how well the VB approximates the true posterior, particularly with regards to its covariance structure. The methods proposed also allow us to generate simple corrections when the approximation error is large. It looks at joint, marginal and conditional aspects of the approximate posterior and shows how to apply these techniques in both simulated and real data examples.

stat.CO↗

arXiv2013

Logistic regression geometry

This paper looks at effects, due to the boundary, on inference in logistic regression. It shows that first -- and, indeed, higher -- order asymptotic results are not uniform across the model. Near the boundary, effects such as high skewness, discreteness and collinearity dominate, any of which could render inference based on asymptotic normality suspect. A highly interpretable diagnostic tool is proposed allowing the analyst to check if the boundary is going to have an appreciable effect on standard inferential techniques.

stat.ME↗

arXiv2012

Computational information geometry: theory and practice

This paper lays the foundations for a unified framework for numerically and computationally applying methods drawn from a range of currently distinct geometrical approaches to statistical modelling. In so doing, it extends information geometry from a manifold based approach to one where the simplex is the fundamental geometrical object, thereby allowing applications to models which do not have a fixed dimension or support. Finally, it starts to build a computational framework which will act as a proxy for the 'space of all distributions' that can be used, in particular, to investigate model selection and model uncertainty. A varied set of substantive running examples is used to illustrate theoretical and practical aspects of the discussion. Further developments are briefly indicated.

math.ST↗

arXiv2007

Local mixture models of exponential families

Exponential families are the workhorses of parametric modelling theory. One reason for their popularity is their associated inference theory, which is very clean, both from a theoretical and a computational point of view. One way in which this set of tools can be enriched in a natural and interpretable way is through mixing. This paper develops and applies the idea of local mixture modelling to exponential families. It shows that the highly interpretable and flexible models which result have enough structure to retain the attractive inferential properties of exponential families. In particular, results on identification, parameter orthogonality and log-concavity of the likelihood are proved.

math.ST↗