arXiv subjects

Ian H. Jermyn

Publications and source records attributed to Ian H. Jermyn.

7 recordsLinked to original sources

arXiv2026

Bayesian inference of sparsity in stable vector autoregressive processes

Advances in sensing technology have made it possible to collect large volumes of high-dimensional time-series data. In fields like genetics and neuroscience, key questions concern whether directed relationships between variables can be learned from these data. To this end, graphical vector autoregressions are a popular tool because zeros among the autoregressive coefficients and error precision matrix have natural interpretations in terms of Granger non-causality and contemporaneous conditional independence. In applications where system dynamics are subject to functional or structural constraints, assuming the process is stable can be advantageous. However, enforcing stability demands restricting the autoregressive coefficients to lie in a constrained space with a complex geometry called the stationary region. The resulting inferential challenges are compounded when sparsity is also a requirement. Working in the Bayesian paradigm, we tackle the problem of developing a prior that simultaneously enforces stationarity and sparsity through parameter expansion, constructing a spike-and-slab prior with support constrained to the stationary region. A mixture of G-Wishart distributions provides a sparse prior for the error precision matrix. Computational inference is carried out using Metropolis-within-Gibbs, exploiting the No-U-Turn Sampler and reversible-jump steps. We demonstrate the inferential and predictive benefits of our approach through simulations and applications in macroeconomics and neuroscience.

stat.ME↗

arXiv2021

4D Atlas: Statistical Analysis of the Spatiotemporal Variability in Longitudinal 3D Shape Data

We propose a novel framework to learn the spatiotemporal variability in longitudinal 3D shape data sets, which contain observations of objects that evolve and deform over time. This problem is challenging since surfaces come with arbitrary parameterizations and thus, they need to be spatially registered. Also, different deforming objects, also called 4D surfaces, evolve at different speeds and thus they need to be temporally aligned. We solve this spatiotemporal registration problem using a Riemannian approach. We treat a 3D surface as a point in a shape space equipped with an elastic Riemannian metric that measures the amount of bending and stretching that the surfaces undergo. A 4D surface can then be seen as a trajectory in this space. With this formulation, the statistical analysis of 4D surfaces can be cast as the problem of analyzing trajectories embedded in a nonlinear Riemannian manifold. However, performing the spatiotemporal registration, and subsequently computing statistics, on such nonlinear spaces is not straightforward as they rely on complex nonlinear optimizations. Our core contribution is the mapping of the surfaces to the space of Square-Root Normal Fields where the L2 metric is equivalent to the partial elastic metric in the space of surfaces. Thus, by solving the spatial registration in the SRNF space, the problem of analyzing 4D surfaces becomes the problem of analyzing trajectories embedded in the SRNF space, which has a Euclidean structure. In this paper, we develop the building blocks that enable such analysis. These include: (1) the spatiotemporal registration of arbitrarily parameterized 4D surfaces in the presence of large elastic deformations and large variations in their execution rates; (2) the computation of geodesics between 4D surfaces; (3) the computation of statistical summaries; and (4) the synthesis of random 4D surfaces.

cs.CV↗

arXiv2020

A Bayesian approach to deconvolution in well test analysis

In petroleum well test analysis, deconvolution is used to obtain information about the reservoir system. This information is contained in the response function, which can be estimated by solving an inverse problem in the pressure and flow rate measurements. Our Bayesian approach to this problem is based upon a parametric physical model of reservoir behaviour, derived from the solution for fluid flow in a general class of reservoirs. This permits joint parametric Bayesian inference for both the reservoir parameters and the true pressure and rate values, which is essential due to the typical levels of observation error. Using a set of flexible priors for the reservoir parameters to restrict the solution space to physical behaviours, samples from the posterior are generated using MCMC. Summaries and visualisations of the reservoir parameters' posterior, response, and true pressure and rate values can be produced, interpreted, and model selection can be performed. The method is validated through a synthetic application, and applied to a field data set. The results are comparable to the state of the art solution, but through our method we gain access to system parameters, we can incorporate prior knowledge that excludes non-physical results, and we can quantify parameter uncertainty.

stat.AP↗

arXiv2018

Shape-Constrained Univariate Density Estimation

While the problem of estimating a probability density function (pdf) from its observations is classical, the estimation under additional shape constraints is both important and challenging. We introduce an efficient, geometric approach for estimating pdfs given the number of its modes. This approach explores the space of constrained pdf's using an action of the diffeomorphism group that preserves their shapes. It starts with an initial template, with the desired number of modes and arbitrarily chosen heights at the critical points, and transforms it via: (1) composition by diffeomorphisms and (2) normalization to obtain the final density estimate. The search for optimal diffeomorphism is performed under the maximum-likelihood criterion and is accomplished by mapping diffeomorphisms to the tangent space of a Hilbert sphere, a vector space whose elements can be expressed using an orthogonal basis. This framework is first applied to shape-constrained univariate, unconditional pdf estimation and then extended to conditional pdf estimation. We derive asymptotic convergence rates of the estimator and demonstrate this approach using a synthetic dataset involving speed distribution for different traffic flow on Californian driveways.

stat.ME↗

arXiv2018

Bayesian shape modelling of cross-sectional geological data

Shape information is of great importance in many applications. For example, the oil-bearing capacity of sand bodies, the subterranean remnants of ancient rivers, is related to their cross-sectional shapes. The analysis of these shapes is therefore of some interest, but current classifications are simplistic and ad hoc. In this paper, we describe the first steps towards a coherent statistical analysis of these shapes by deriving the integrated likelihood for data shapes given class parameters. The result is of interest beyond this particular application.

stat.ME↗

arXiv2016

Numerical Inversion of SRNF Maps for Elastic Shape Analysis of Genus-Zero Surfaces

Recent developments in elastic shape analysis (ESA) are motivated by the fact that it provides comprehensive frameworks for simultaneous registration, deformation, and comparison of shapes. These methods achieve computational efficiency using certain square-root representations that transform invariant elastic metrics into Euclidean metrics, allowing for applications of standard algorithms and statistical tools. For analyzing shapes of embeddings of $\mathbb{S}^2$ in $\mathbb{R}^3$, Jermyn et al. introduced square-root normal fields (SRNFs) that transformed an elastic metric, with desirable invariant properties, into the $\mathbb{L}^2$ metric. These SRNFs are essentially surface normals scaled by square-roots of infinitesimal area elements. A critical need in shape analysis is to invert solutions (deformations, averages, modes of variations, etc) computed in the SRNF space, back to the original surface space for visualizations and inferences. Due to the lack of theory for understanding SRNFs maps and their inverses, we take a numerical approach and derive an efficient multiresolution algorithm, based on solving an optimization problem in the surface space, that estimates surfaces corresponding to given SRNFs. This solution is found effective, even for complex shapes, e.g. human bodies and animals, that undergo significant deformations including bending and stretching. Specifically, we use this inversion for computing elastic shape deformations, transferring deformations, summarizing shapes, and for finding modes of variability in a given collection, while simultaneously registering the surfaces. We demonstrate the proposed algorithms using a statistical analysis of human body shapes, classification of generic surfaces and analysis of brain structures.

cs.GR↗

arXiv2005

Invariant Bayesian estimation on manifolds

A frequent and well-founded criticism of the maximum a posteriori (MAP) and minimum mean squared error (MMSE) estimates of a continuous parameter γtaking values in a differentiable manifold Γis that they are not invariant to arbitrary ``reparameterizations'' of Γ. This paper clarifies the issues surrounding this problem, by pointing out the difference between coordinate invariance, which is a sine qua non for a mathematically well-defined problem, and diffeomorphism invariance, which is a substantial issue, and then provides a solution. We first show that the presence of a metric structure on Γcan be used to define coordinate-invariant MAP and MMSE estimates, and we argue that this is the natural way to proceed. We then discuss the choice of a metric structure on Γ. By imposing an invariance criterion natural within a Bayesian framework, we show that this choice is essentially unique. It does not necessarily correspond to a choice of coordinates. In cases of complete prior ignorance, when Jeffreys' prior is used, the invariant MAP estimate reduces to the maximum likelihood estimate. The invariant MAP estimate coincides with the minimum message length (MML) estimate, but no discretization or approximation is used in its derivation.

math.ST↗