Smoothing and filtering with a class of outer measures

Filtering and smoothing with a generalised representation of uncertainty is considered. Here, uncertainty is represented using a class of outer measures. It is shown how this representation of uncertainty can be propagated using outer-measure-type versions of Markov kernels and generalised Bayesian-like update equations. This leads to a system of generalised smoothing and filtering equations where integrals are replaced by supremums and probability density functions are replaced by positive functions with supremum equal to one. Interestingly, these equations retain most of the structure found in the classical Bayesian filtering framework. It is additionally shown that the Kalman filter recursion can be recovered from weaker assumptions on the available information on the corresponding hidden Markov model

An Introduction to Wishart Matrix Moments

These lecture notes provide a comprehensive, self-contained introduction to the analysis of Wishart matrix moments. This study may act as an introduction to some particular aspects of random matrix theory, or as a self-contained exposition of Wishart matrix moments. Random matrix theory plays a central role in statistical physics, computational mathematics and engineering sciences, including data assimilation, signal processing, combinatorial optimization, compressed sensing, econometrics and mathematical finance, among numerous others. The mathematical foundations of the theory of random matrices lies at the intersection of combinatorics, non-commutative algebra, geometry, multivariate functional and spectral analysis, and of course statistics and probability theory. As a result, most of the classical topics in random matrix theory are technical, and mathematically difficult to penetrate for non-experts and regular users and practitioners. The technical aim of these notes is to review and extend some important results in random matrix theory in the specific context of real random Wishart matrices. This special class of Gaussian-type sample covariance matrix plays an important role in multivariate analysis and in statistical theory. We derive non-asymptotic formulae for the full matrix moments of real valued Wishart random matrices. As a corollary, we derive and extend a number of spectral and trace-type results for the case of non-isotropic Wishart random matrices. We also derive the full matrix moment analogues of some classic spectral and trace-type moment results. For example, we derive semi-circle and Marchencko-Pastur-type laws in the non-isotropic and full matrix cases. Laplace matrix transforms and matrix moment estimates are also studied, along with new spectral and trace concentration-type inequalities

Controlled Sequential Monte Carlo

Sequential Monte Carlo methods, also known as particle methods, are a popular set of techniques for approximating high-dimensional probability distributions and their normalizing constants. These methods have found numerous applications in statistics and related fields; e.g. for inference in non-linear non-Gaussian state space models, and in complex static models. Like many Monte Carlo sampling schemes, they rely on proposal distributions which crucially impact their performance. We introduce here a class of controlled sequential Monte Carlo algorithms, where the proposal distributions are determined by approximating the solution to an associated optimal control problem using an iterative scheme. This method builds upon a number of existing algorithms in econometrics, physics, and statistics for inference in state space models, and generalizes these methods so as to accommodate complex static models. We provide a theoretical analysis concerning the fluctuation and stability of this methodology that also provides insight into the properties of related algorithms. We demonstrate significant gains over state-of-the-art methods at a fixed computational complexity on a variety of applications

A Multilevel Approach for Stochastic Nonlinear Optimal Control

We consider a class of finite time horizon nonlinear stochastic optimal control problem, where the control acts additively on the dynamics and the control cost is quadratic. This framework is flexible and has found applications in many domains. Although the optimal control admits a path integral representation for this class of control problems, efficient computation of the associated path integrals remains a challenging Monte Carlo task. The focus of this article is to propose a new Monte Carlo approach that significantly improves upon existing methodology. Our proposed methodology first tackles the issue of exponential growth in variance with the time horizon by casting optimal control estimation as a smoothing problem for a state space model associated with the control problem, and applying smoothing algorithms based on particle Markov chain Monte Carlo. To further reduce computational cost, we then develop a multilevel Monte Carlo method which allows us to obtain an estimator of the optimal control with $\mathcal{O}(\epsilon^2)$ mean squared error with a computational cost of $\mathcal{O}(\epsilon^{-2}\log(\epsilon)^2)$. In contrast, a computational cost of $\mathcal{O}(\epsilon^{-3})$ is required for existing methodology to achieve the same mean squared error. Our approach is illustrated on two numerical examples, which validate our theory

On the Mathematical Theory of Ensemble (Linear-Gaussian) Kalman-Bucy Filtering

The purpose of this review is to present a comprehensive overview of the theory of ensemble Kalman-Bucy filtering for linear-Gaussian signal models. We present a system of equations that describe the flow of individual particles and the flow of the sample covariance and the sample mean in continuous-time ensemble filtering. We consider these equations and their characteristics in a number of popular ensemble Kalman filtering variants. Given these equations, we study their asymptotic convergence to the optimal Bayesian filter. We also study in detail some non-asymptotic time-uniform fluctuation, stability, and contraction results on the sample covariance and sample mean (or sample error track). We focus on testable signal/observation model conditions, and we accommodate fully unstable (latent) signal models. We discuss the relevance and importance of these results in characterising the filter's behaviour, e.g. it's signal tracking performance, and we contrast these results with those in classical studies of stability in Kalman-Bucy filtering. We provide intuition for how these results extend to nonlinear signal models and comment on their consequence on some typical filter behaviours seen in practice, e.g. catastrophic divergence

Dual Energy X-Ray Absorptiometry Body Composition Reference Values from NHANES

In 2008 the National Center for Health Statistics released a dual energy x-ray absorptiometry (DXA) whole body dataset from the NHANES population-based sample acquired with modern fan beam scanners in 15 counties across the United States from 1999 through 2004. The NHANES dataset was partitioned by gender and ethnicity and DXA whole body measures of %fat, fat mass/height2, lean mass/height2, appendicular lean mass/height2, %fat trunk/%fat legs ratio, trunk/limb fat mass ratio of fat, bone mineral content (BMC) and bone mineral density (BMD) were analyzed to provide reference values for subjects 8 to 85 years old. DXA reference values for adults were normalized to age; reference values for children included total and sub-total whole body results and were normalized to age, height, or lean mass. We developed an obesity classification scheme by using estabbody mass index (BMI) classification thresholds and prevalences in young adults to generate matching classification thresholds for Fat Mass Index (FMI; fat mass/height2). These reference values should be helpful in the evaluation of a variety of adult and childhood abnormalities involving fat, lean, and bone, for establishing entry criteria into clinical trials, and for other medical, research, and epidemiological uses

Targeting patient recovery priorities in degenerative cervical myelopathy:design and rationale for the RECEDE-Myelopathy trial-study protocol

Introduction: Degenerative cervical myelopathy (DCM) is a common and disabling condition of symptomatic cervical spinal cord compression secondary to degenerative changes in spinal structures leading to a mechanical stress injury of the spinal cord. RECEDE-Myelopathy aims to test the disease-modulating activity of the phosphodiesterase 3/phosphodiesterase 4 inhibitor Ibudilast as an adjuvant to surgical decompression in DCM. Methods and analysis: RECEDE-Myelopathy is a multicentre, double-blind, randomised, placebo-controlled trial. Participants will be randomised to receive either 60-100 mg Ibudilast or placebo starting within 10 weeks prior to surgery and continuing for 24 weeks after surgery for a maximum of 34 weeks. Adults with DCM, who have a modified Japanese Orthopaedic Association (mJOA) score 8-14 inclusive and are scheduled for their first decompressive surgery are eligible for inclusion. The coprimary endpoints are pain measured on a visual analogue scale and physical function measured by the mJOA score at 6 months after surgery. Clinical assessments will be undertaken preoperatively, postoperatively and 3, 6 and 12 months after surgery. We hypothesise that adjuvant therapy with Ibudilast leads to a meaningful and additional improvement in either pain or function, as compared with standard routine care. Study design: Clinical trial protocol V.2.2 October 2020. Ethics and dissemination: Ethical approval has been obtained from HRA - Wales. The results will be presented at an international and national scientific conferences and in a peer-reviewed journals.Trial registration number: ISRCTN Number: ISRCTN16682024.</p