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This paper considers approximate smoothing for discretely observed non-linear stochastic differential equations. The problem is tackled by developing methods for linearising stochastic differential equations with respect to an arbitrary Gaussian process. Two methods are developed based on (1) taking the limit of statistical linear regression of the discretised process and (2) minimising an upper b

In this article, we propose an efficient method for solving analysis-l1-TV regularization problems with a multi-step alternating direction method of multipliers (ADMM) approach as the fast solver. Additionally, we apply it to a real-data magnetoen-cephalography (MEG) brain imaging problem as well as to signal reconstruction. In our approach, the inverse problem arising in MEG or signal reconstruct

We show how probabilistic numerics can be used to convert an initial value problem into a Gauss–Markov process parametrised by the dynamics of the initial value problem. Consequently, the often difficult problem of parameter estimation in ordinary differential equations is reduced to hyper-parameter estimation in Gauss–Markov regression, which tends to be considerably easier. The method’s relation

This article studies the maximum likelihood estimates of magnitude and scale parameters for a Gaussian process of Ornstein-Uhlenbeck type used to model a deterministic function that does not have to be a realisation of an Ornstein- Uhlenbeck process. Specifically, we derive explicit expressions for the limiting values of the maximum likelihood estimates as the number of observations increases. The

Probabilistic numerical solvers for ordinary differential equations compute posterior distributions over the solution of an initial value problem via Bayesian inference. In this paper, we leverage their probabilistic formulation to seamlessly include additional information as general likelihood terms. We show that second-order differential equations should be directly provided to the solver, inste

This paper is concerned with the estimation of unknown drift functions of stochastic differential equations (SDEs) from observations of their sample paths. We propose to formulate this as a non-parametric Gaussian process regression problem and use an Ito-Taylor expansion for approximating the SDE. To address the computational complexity problem of Gaussian process regression, we cast the model in

In this letter, we consider Gaussian approximations of the optimal importance density in sequential importance sampling for nonlinear, non-Gaussian state-space models. The proposed method is based on generalized statistical linear regression and posterior linearization using conditional expectations. Simulation results show that the method outperforms the compared methods in terms of the effective

In this article, we address the problem of estimating the state and learning of the parameters in a linear dynamic system with generalized L 1 -regularization. Assuming a sparsity prior on the state, the joint state estimation and parameter learning problem is cast as an unconstrained optimization problem. However, when the dimensionality of state or parameters is large, memory requirements and co

In this paper, we develop a novel method for approximate continuous-discrete Bayesian filtering. The projection filtering framework is exploited to develop accurate approximations of posterior distributions within parametric classes of probability distributions. This is done by formulating an ordinary differential equation for the posterior distribution that has the prior as initial value and hits

In this paper, the connection between the Matérn kernel and scale mixtures of squared exponential kernels is explored. It is shown that the Matérn kernel can be approximated by a finite scale mixture of squared exponential kernels through a quadrature approximation which in turn allows for (i) state space approximations of the Matérn kernel for arbitrary smoothness parameters using established sta

In this paper, we propose a Gaussian process-based nonlinear, time-varying drift model for stochastic differential equations. In particular, we combine eigenfunction expansion of the Gaussian process’ covariance kernel in the spatial input variables with spectral decomposition in the time domain to obtain a reduced rank state space representation of the drift model, which avoids the growing comple

In this paper the issue of filtering and smoothing in continuous discrete time is studied when the state variable evolves in some submanifold of Euclidean space, which may not have the usual Lebesgue measure. Formal expressions for prediction and smoothing problems are reviewed, which agree with the classical results except that the formal adjoint of the generator is different in general. These re

Many problems in science and engineering involve estimating a dynamic signal from indirect measurements subject to noise, where points can either evolve in continuous time or in discrete time. These problems are often formalised as inference in probabilistic state-space models, which are also frequently assumed to be Markovian. For inferring the value of the signal at a particular point in time, m

Microelectromechanical-systems-based inertial sensors and magnetometers are low-cost, off-the-shelf sensors that are widely used in both consumer and industrial applications. However, these sensors suffer from biases and effects such as axis misalignment or scale errors, which require careful system design and periodic sensor calibration. In this paper, we propose a fast calibration method for joi

This paper is concerned with inferring the state of a Itô stochastic differential equation (SDE) from noisy discrete-time measurements. The problem is approached by considering basis function expansions of Brownian motion, that as a consequence give approximations to the underlying stochastic differential equation in terms of an ordinary differential equation with random coefficients. This allows

Alla blir vi rädda någon gång, både människor och djur. Ibland är det samma saker som skrämmer och ibland är det något helt annat som djuren tycker är läskigt.Vad gör djuren när de blir skrämda? Kanske gömmer de sig? Försöker de springa därifrån? Eller försöker djuren lura den som skräms att titta åt ett helt annat håll? Ska vi kolla hur djuren gör?Vad gör djuren när de blir rädda? är en bok som g

This extended abstract compiles statistics and information regarding the process of gathering water damage statistics. A questionnaire wasused to determine similarities, differences, and tendencies in the water damage statistics, in the Nordic countries Sweden, Norway, Denmark,Finland and Iceland. The study aimed to answer what lessons were learned and what knowledge could be shared between the No

Third Party Funding (TPF) is presented as a tool to help fund the cost of expensive litigation. In the context of Investment Arbitration, however, TPF has instead led to the commodification of justice, and raises concerns around its assetization. Arbitration often comes at a net loss for States, and the extraordinary expenditures required my pique the interest of third party funders who wish to pr

This article explores how football players and scouts in Sweden narrate their stories of transfers within the Swedish football system. This article presents first-person narrations as male players as well as scouts from elite clubs were interviewed in connection to transfer patters to, from and within Sweden. The aim is to analyse how the transfer system takes shape in stories presented by scouts