arXiv subjects

Simon Godsill

Publications and source records attributed to Simon Godsill.

At least 19 recordsLinked to original sources

arXiv2025

Integrated Gaussian Processes for Robust and Adaptive Multi-Object Tracking

This paper presents a computationally efficient multi-object tracking approach that can minimise track breaks (e.g., in challenging environments and against agile targets), learn the measurement model parameters on-line (e.g., in dynamically changing scenes) and infer the class of the tracked objects, if joint tracking and kinematic behaviour classification is sought. It capitalises on the flexibilities offered by the integrated Gaussian process as a motion model and the convenient statistical properties of non-homogeneous Poisson processes as a suitable observation model. This can be combined with the proposed effective track revival / stitching mechanism. We accordingly introduce the two robust and adaptive trackers, Gaussian and Poisson Process with Classification (GaPP-Class) and GaPP with Revival and Classification (GaPP-ReaCtion). They employ an appropriate particle filtering inference scheme that efficiently integrates track management and hyperparameter learning (including the object class, if relevant). GaPP-ReaCtion extends GaPP-Class with the addition of a Markov Chain Monte Carlo kernel applied to each particle permitting track revival and stitching (e.g., within a few time steps after deleting a trajectory). Performance evaluation and benchmarking using synthetic and real data show that GaPP-Class and GaPP-ReaCtion outperform other state-of-the-art tracking algorithms. For example, GaPP-ReaCtion significantly reduces track breaks (e.g., by around 30% from real radar data and markedly more from simulated data).

cs.CV↗

arXiv2025

Bayesian Non-Parametric Inference for L\'evy Measures in State-Space Models

L\'evy processes, known for their ability to model complex dynamics with skewness, heavy tails, and discontinuities, play a critical role in stochastic modeling across various domains. However, inference for most L\'evy processes, whether in parametric or non-parametric settings, remains a significant challenge. In this work, we present a novel Bayesian non-parametric inference framework for inferring the L\'evy measures of subordinators and normal variance-mean (NVM) processes within a linear state space model. A flexible random measure, the Independent Gamma-scaled Dirichlet Process (IGSDP), is introduced, for which the well-known Gamma process is a special case, leading to tractable conditional distributions for inference about both L\'evy measures. We further show that in the Gamma process special case, conjugacy can be achieved for hyper-parameter inference. An explicit characterization of the parameter contour for NVM processes is provided, enabling an identifiable parameterization of the model for effective Markov Chain Monte Carlo algorithms in posterior inference. The method is demonstrated on both synthetic and tick-level (high-frequency) financial datasets.

stat.ME↗

arXiv2024

Decentralised Variational Inference Frameworks for Multi-object Tracking on Sensor Networks: Additional Notes

This paper tackles the challenge of multi-sensor multi-object tracking by proposing various decentralised Variational Inference (VI) schemes that match the tracking performance of centralised sensor fusion with only local message exchanges among neighboring sensors. We first establish a centralised VI sensor fusion scheme as a benchmark and analyse the limitations of its decentralised counterpart, which requires sensors to await consensus at each VI iteration. Therefore, we propose a decentralised gradient-based VI framework that optimises the Locally Maximised Evidence Lower Bound (LM-ELBO) instead of the standard ELBO, which reduces the parameter search space and enables faster convergence, making it particularly beneficial for decentralised tracking. This proposed framework is inherently self-evolving, improving with advancements in decentralised optimisation techniques for convergence guarantees and efficiency. Further, we enhance the convergence speed of proposed decentralised schemes using natural gradients and gradient tracking strategies. Results verify that our decentralised VI schemes are empirically equivalent to centralised fusion in tracking performance. Notably, the decentralised natural gradient VI method is the most communication-efficient, with communication costs comparable to suboptimal decentralised strategies while delivering notably higher tracking accuracy.

cs.LG↗

arXiv2023

Joint Object Tracking and Intent Recognition

This paper presents a Bayesian framework for inferring the posterior of the augmented state of a target, incorporating its underlying goal or intent, such as any intermediate waypoints and/or the final destination. Thus, it is for joint object tracking and intent recognition. Several latent intent models are proposed here within a virtual leader formulation. They capture the influence of the target's hidden goal on its instantaneous behaviour. In this context, various motion models, including for highly maneuvering objects, are also considered. The a priori unknown target intent (e.g. destination) can dynamically change over time and take any value within the state space (e.g. a location or spatial region). A sequential Monte Carlo (particle filtering) approach is introduced for the simultaneous estimation of the target's (kinematic) state and its intent. Rao-Blackwellisation is employed to enhance the statistical performance of the inference routine. Simulated data and real radar measurements are used to demonstrate the efficacy of the proposed techniques.

stat.AP↗

arXiv2023

Generalised shot noise representations of stochastic systems driven by non-Gaussian Lévy processes

We consider the problem of obtaining effective representations for the solutions of linear, vector-valued stochastic differential equations (SDEs) driven by non-Gaussian pure-jump Lévy processes, and we show how such representations lead to efficient simulation methods. The processes considered constitute a broad class of models that find application across the physical and biological sciences, mathematics, finance and engineering. Motivated by important relevant problems in statistical inference, we derive new, generalised shot-noise simulation methods whenever a normal variance-mean (NVM) mixture representation exists for the driving Lévy process, including the generalised hyperbolic, normal-Gamma, and normal tempered stable cases. Simple, explicit conditions are identified for the convergence of the residual of a truncated shot-noise representation to a Brownian motion in the case of the pure Lévy process, and to a Brownian-driven SDE in the case of the Lévy-driven SDE. These results provide Gaussian approximations to the small jumps of the process under the NVM representation. The resulting representations are of particular importance in state inference and parameter estimation for Lévy-driven SDE models, since the resulting conditionally Gaussian structures can be readily incorporated into latent variable inference methods such as Markov chain Monte Carlo (MCMC), Expectation-Maximisation (EM), and sequential Monte Carlo.

math.PR↗

arXiv2023

Generalised Hyperbolic State-space Models for Inference in Dynamic Systems

In this work we study linear vector stochastic differential equation (SDE) models driven by the generalised hyperbolic (GH) Lévy process for inference in continuous-time non-Gaussian filtering problems. The GH family of stochastic processes offers a flexible framework for modelling of non-Gaussian, heavy-tailed characteristics and includes the normal inverse-Gaussian, variance-gamma and Student-t processes as special cases. We present continuous-time simulation methods for the solution of vector SDE models driven by GH processes and novel inference methodologies using a variant of sequential Markov chain Monte Carlo (MCMC). As an example a particular formulation of Langevin dynamics is studied within this framework. The model is applied to both a synthetically generated data set and a real-world financial series to demonstrate its capabilities.

stat.ME↗

arXiv2023

Driver Profiling and Bayesian Workload Estimation Using Naturalistic Peripheral Detection Study Data

Monitoring drivers' mental workload facilitates initiating and maintaining safe interactions with in-vehicle information systems, and thus delivers adaptive human machine interaction with reduced impact on the primary task of driving. In this paper, we tackle the problem of workload estimation from driving performance data. First, we present a novel on-road study for collecting subjective workload data via a modified peripheral detection task in naturalistic settings. Key environmental factors that induce a high mental workload are identified via video analysis, e.g. junctions and behaviour of vehicle in front. Second, a supervised learning framework using state-of-the-art time series classifiers (e.g. convolutional neural network and transform techniques) is introduced to profile drivers based on the average workload they experience during a journey. A Bayesian filtering approach is then proposed for sequentially estimating, in (near) real-time, the driver's instantaneous workload. This computationally efficient and flexible method can be easily personalised to a driver (e.g. incorporate their inferred average workload profile), adapted to driving/environmental contexts (e.g. road type) and extended with data streams from new sources. The efficacy of the presented profiling and instantaneous workload estimation approaches are demonstrated using the on-road study data, showing $F_{1}$ scores of up to 92% and 81%, respectively.

eess.SP↗

arXiv2023

Variational Tracking and Redetection for Closely-spaced Objects in Heavy Clutter: Supplementary Materials

The non-homogeneous Poisson process (NHPP) is a widely used measurement model that allows for an object to generate multiple measurements over time. However, it can be difficult to efficiently and reliably track multiple objects under this NHPP model in scenarios with a high density of closely-spaced objects and heavy clutter. Therefore, based on the general coordinate ascent variational filtering framework, this paper presents a variational Bayes association-based NHPP tracker (VB-AbNHPP) that can efficiently perform tracking, data association, and learning of target and clutter rates with a parallelisable implementation. In addition, a variational localisation strategy is proposed, which enables rapid rediscovery of missed targets from a large surveillance area under extremely heavy clutter. This strategy is integrated into the VB-AbNHPP tracker, resulting in a robust methodology that can automatically detect and recover from track loss. This tracker demonstrates improved tracking performance compared with existing trackers in challenging scenarios, in terms of both accuracy and efficiency.

eess.SP↗

arXiv2023

Consensus-based Distributed Variational Multi-object Tracker in Multi-Sensor Network

The growing need for accurate and reliable tracking systems has driven significant progress in sensor fusion and object tracking techniques. In this paper, we design two variational Bayesian trackers that effectively track multiple targets in cluttered environments within a sensor network. We first present a centralised sensor fusion scheme, which involves transmitting sensor data to a fusion center. Then, we develop a distributed version leveraging the average consensus algorithm, which is theoretically equivalent to the centralised sensor fusion tracker and requires only local message passing with neighbouring sensors. In addition, we empirically verify that our proposed distributed variational tracker performs on par with the centralised version with equal tracking accuracy. Simulation results show that our distributed multi-target tracker outperforms the suboptimal distributed sensor fusion strategy that fuses each sensor's posterior based on arithmetic sensor fusion and an average consensus strategy.

eess.SP↗

arXiv2023

Point process simulation of generalised hyperbolic Lévy processes

Generalised hyperbolic (GH) processes are a class of stochastic processes that are used to model the dynamics of a wide range of complex systems that exhibit heavy-tailed behavior, including systems in finance, economics, biology, and physics. In this paper, we present novel simulation methods based on subordination with a generalised inverse Gaussian (GIG) process and using a generalised shot-noise representation that involves random thinning of infinite series of decreasing jump sizes. Compared with our previous work on GIG processes, we provide tighter bounds for the construction of rejection sampling ratios, leading to improved acceptance probabilities in simulation. Furthermore, we derive methods for the adaptive determination of the number of points required in the associated random series using concentration inequalities. Residual small jumps are then approximated using an appropriately scaled Brownian motion term with drift. Finally the rejection sampling steps are made significantly more computationally efficient through the use of squeezing functions based on lower and upper bounds on the Lévy density. Experimental results are presented illustrating the strong performance under various parameter settings and comparing the marginal distribution of the GH paths with exact simulations of GH random variates. The new simulation methodology is made available to researchers through the publication of a Python code repository.

stat.ME↗

arXiv2022

Non-Gaussian Process Regression

Standard GPs offer a flexible modelling tool for well-behaved processes. However, deviations from Gaussianity are expected to appear in real world datasets, with structural outliers and shocks routinely observed. In these cases GPs can fail to model uncertainty adequately and may over-smooth inferences. Here we extend the GP framework into a new class of time-changed GPs that allow for straightforward modelling of heavy-tailed non-Gaussian behaviours, while retaining a tractable conditional GP structure through an infinite mixture of non-homogeneous GPs representation. The conditional GP structure is obtained by conditioning the observations on a latent transformed input space and the random evolution of the latent transformation is modelled using a Lévy process which allows Bayesian inference in both the posterior predictive density and the latent transformation function. We present Markov chain Monte Carlo inference procedures for this model and demonstrate the potential benefits compared to a standard GP.

stat.ML↗

arXiv2021

Point process simulation of generalised inverse Gaussian processes and estimation of the Jaeger integral

In this paper novel simulation methods are provided for the generalised inverse Gaussian (GIG) Lévy process. Such processes are intractable for simulation except in certain special edge cases, since the Lévy density associated with the GIG process is expressed as an integral involving certain Bessel Functions, known as the Jaeger integral in diffusive transport applications. We here show for the first time how to solve the problem indirectly, using generalised shot-noise methods to simulate the underlying point processes and constructing an auxiliary variables approach that avoids any direct calculation of the integrals involved. The resulting augmented bivariate process is still intractable and so we propose a novel thinning method based on upper bounds on the intractable integrand. Moreover our approach leads to lower and upper bounds on the Jaeger integral itself, which may be compared with other approximation methods. The shot noise method involves a truncated infinite series of decreasing random variables, and as such is approximate, although the series are found to be rapidly convergent in most cases. We note that the GIG process is the required Brownian motion subordinator for the generalised hyperbolic (GH) Lévy process and so our simulation approach will straightforwardly extend also to the simulation of these intractable proceses. Our new methods will find application in forward simulation of processes of GIG and GH type, in financial and engineering data, for example, as well as inference for states and parameters of stochastic processes driven by GIG and GH Lévy processes.

stat.ME↗

arXiv2021

Autonomous Tracking and State Estimation with Generalised Group Lasso

We address the problem of autonomous tracking and state estimation for marine vessels, autonomous vehicles, and other dynamic signals under a (structured) sparsity assumption. The aim is to improve the tracking and estimation accuracy with respect to classical Bayesian filters and smoothers. We formulate the estimation problem as a dynamic generalised group Lasso problem and develop a class of smoothing-and-splitting methods to solve it. The Levenberg--Marquardt iterated extended Kalman smoother-based multi-block alternating direction method of multipliers (LM-IEKS-mADMM) algorithms are based on the alternating direction method of multipliers (ADMM) framework. This leads to minimisation subproblems with an inherent structure to which three new augmented recursive smoothers are applied. Our methods can deal with large-scale problems without pre-processing for dimensionality reduction. Moreover, the methods allow one to solve nonsmooth nonconvex optimisation problems. We then prove that under mild conditions, the proposed methods converge to a stationary point of the optimisation problem. By simulated and real-data experiments including multi-sensor range measurement problems, marine vessel tracking, autonomous vehicle tracking, and audio signal restoration, we show the practical effectiveness of the proposed methods.

stat.ME↗

arXiv2020

Hidden Markov Models Applied To Intraday Momentum Trading With Side Information

A Hidden Markov Model for intraday momentum trading is presented which specifies a latent momentum state responsible for generating the observed securities' noisy returns. Existing momentum trading models suffer from time-lagging caused by the delayed frequency response of digital filters. Time-lagging results in a momentum signal of the wrong sign, when the market changes trend direction. A key feature of this state space formulation, is no such lagging occurs, allowing for accurate shifts in signal sign at market change points. The number of latent states in the model is estimated using three techniques, cross validation, penalized likelihood criteria and simulation-based model selection for the marginal likelihood. All three techniques suggest either 2 or 3 hidden states. Model parameters are then found using Baum-Welch and Markov Chain Monte Carlo, whilst assuming a single (discretized) univariate Gaussian distribution for the emission matrix. Often a momentum trader will want to condition their trading signals on additional information. To reflect this, learning is also carried out in the presence of side information. Two sets of side information are considered, namely a ratio of realized volatilities and intraday seasonality. It is shown that splines can be used to capture statistically significant relationships from this information, allowing returns to be predicted. An Input Output Hidden Markov Model is used to incorporate these univariate predictive signals into the transition matrix, presenting a possible solution for dealing with the signal combination problem. Bayesian inference is then carried out to predict the securities $t+1$ return using the forward algorithm. Simple modifications to the current framework allow for a fully non-parametric model with asynchronous prediction.

q-fin.TR↗

arXiv2020

A Multi-Target Track-Before-Detect Particle Filter Using Superpositional Data in Non-Gaussian Noise

This paper proposes a novel particle filter for tracking time-varying states of multiple targets jointly from superpositional data, which depend on the sum of contributions of all targets. Many conventional tracking methods rely on preprocessing for detection (e.g., thresholding), which severely limits tracking performance at a low signal-to-noise ratio (SNR). In contrast, the proposed method operates directly on raw sensor signals without requiring such preprocessing. Though there also exist methods applicable to raw sensor signals called track-before-detect, the proposed method has significant advantages over them. First, it is general without any restrictions on observation/process noise statistics (e.g., Gaussian) or the functional form of each target's contribution to the sensors (e.g., linear, separable, binary). Especially, it includes Salmond et al.'s track-before-detect particle filter for a single target as a particular example up to some implementation details. Second, it can track an unknown, time-varying number of targets without knowing their initial states owing to a target birth/death model. We present a simulation example of radio-frequency tomography, where it significantly outperformed Nannuru et al.'s state-of-the-art method based on random finite sets in terms of the optimal subpattern assignment (OSPA) metric.

eess.SP↗

arXiv2020

The Lévy State Space Model

In this paper we introduce a new class of state space models based on shot-noise simulation representations of non-Gaussian Lévy-driven linear systems, represented as stochastic differential equations. In particular a conditionally Gaussian version of the models is proposed that is able to capture heavy-tailed non-Gaussianity while retaining tractability for inference procedures. We focus on a canonical class of such processes, the $α$-stable Lévy processes, which retain important properties such as self-similarity and heavy-tails, while emphasizing that broader classes of non-Gaussian Lévy processes may be handled by similar methodology. An important feature is that we are able to marginalise both the skewness and the scale parameters of these challenging models from posterior probability distributions. The models are posed in continuous time and so are able to deal with irregular data arrival times. Example modelling and inference procedures are provided using Rao-Blackwellised sequential Monte Carlo applied to a two-dimensional Langevin model, and this is tested on real exchange rate data.

math.PR↗

arXiv2020

Nonasymptotic Gaussian Approximation for Inference with Stable Noise

The results of a series of theoretical studies are reported, examining the convergence rate for different approximate representations of $α$-stable distributions. Although they play a key role in modelling random processes with jumps and discontinuities, the use of $α$-stable distributions in inference often leads to analytically intractable problems. The LePage series, which is a probabilistic representation employed in this work, is used to transform an intractable, infinite-dimensional inference problem into a conditionally Gaussian parametric problem. A major component of our approach is the approximation of the tail of this series by a Gaussian random variable. Standard statistical techniques, such as Expectation-Maximization, Markov chain Monte Carlo, and Particle Filtering, can then be applied. In addition to the asymptotic normality of the tail of this series, we establish explicit, nonasymptotic bounds on the approximation error. Their proofs follow classical Fourier-analytic arguments, using Esséen's smoothing lemma. Specifically, we consider the distance between the distributions of: $(i)$~the tail of the series and an appropriate Gaussian; $(ii)$~the full series and the truncated series; and $(iii)$~the full series and the truncated series with an added Gaussian term. In all three cases, sharp bounds are established, and the theoretical results are compared with the actual distances (computed numerically) in specific examples of symmetric $α$-stable distributions. This analysis facilitates the selection of appropriate truncations in practice and offers theoretical guarantees for the accuracy of resulting estimates. One of the main conclusions obtained is that, for the purposes of inference, the use of a truncated series together with an approximately Gaussian error term has superior statistical properties and is likely a preferable choice in practice.

math.PR↗

arXiv2017

Bayesian Compressive Sensing Approaches for Direction of Arrival Estimation with Mutual Coupling Effects

The problem of estimating the dynamic direction of arrival of far field signals impinging on a uniform linear array, with mutual coupling effects, is addressed. This work proposes two novel approaches able to provide accurate solutions, including at the endfire regions of the array. Firstly, a Bayesian compressive sensing Kalman filter is developed, which accounts for the predicted estimated signals rather than using the traditional sparse prior. The posterior probability density function of the received source signals and the expression for the related marginal likelihood function are derived theoretically. Next, a Gibbs sampling based approach with indicator variables in the sparsity prior is developed. This allows sparsity to be explicitly enforced in different ways, including when an angle is too far from the previous estimate. The proposed approaches are validated and evaluated over different test scenarios and compared to the traditional relevance vector machine based method. An improved accuracy in terms of average root mean square error values is achieved (up to 73.39% for the modified relevance vector machine based approach and 86.36% for the Gibbs sampling based approach). The proposed approaches prove to be particularly useful for direction of arrival estimation when the angle of arrival moves into the endfire region of the array.

cs.IT↗