arXiv subjects

John Maclean

Publications and source records attributed to John Maclean.

At least 19 recordsLinked to original sources

arXiv2025

Gaussian approximations for fast Bayesian inference of partially observed branching processes with applications to epidemiology

We consider the problem of inference for the states and parameters of a continuous-time multitype branching process from partially observed time series data. Exact inference for this class of models, typically using sequential Monte Carlo, can be computationally challenging when the populations that are being modelled grow exponentially or the time series is long. Instead, we derive a Gaussian approximation for the transition function of the process that leads to a Kalman filtering algorithm that runs in a time independent of the population sizes. We also develop a hybrid approach for when populations are smaller and the approximation is less applicable. We investigate the performance of our approximation and algorithms to both a simple and a complex epidemic model, finding good adherence to the true posterior distributions in both cases with large computational speed-ups in most cases. We also apply our method to a COVID-19 dataset with time dependent parameters where exact methods are intractable due to the population sizes involved.

stat.ME↗

arXiv2025

Data Denoising and Derivative Estimation for Data-Driven Modeling of Nonlinear Dynamical Systems

Data-driven modeling of nonlinear dynamical systems is often hampered by measurement noise. We propose a denoising framework, called Runge-Kutta and Total Variation Based Implicit Neural Representation (RKTV-INR), that represents the state trajectory with an implicit neural representation (INR) fitted directly to noisy observations. Runge-Kutta integration and total variation are imposed as constraints to ensure that the reconstructed state is a trajectory of a dynamical system that remains close to the original data. The trained INR yields a clean, continuous trajectory and provides accurate first-order derivatives via automatic differentiation. These denoised states and derivatives are then supplied to Sparse Identification of Nonlinear Dynamics (SINDy) to recover the governing equations. Experiments demonstrate effective noise suppression, precise derivative estimation, and reliable system identification.

cs.LG↗

arXiv2024

An insightful approach to bearings-only tracking in log-polar coordinates

The choice of coordinate system in a bearings-only (BO) tracking problem influences the methods used to observe and predict the state of a moving target. Modified Polar Coordinates (MPC) and Log-Polar Coordinates (LPC) have some advantages over Cartesian coordinates. In this paper, we derive closed-form expressions for the target state prior distribution after ownship manoeuvre: the mean, covariance, and higher-order moments in LPC. We explore the use of these closed-form expressions in simulation by modifying an existing BO tracker that uses the UKF. Rather than propagating sigma points, we directly substitute current values of the mean and covariance into the time update equations at the ownship turn. This modified UKF, the CFE-UKF, performs similarly to the pure UKF, verifying the closed-form expressions. The closed-form third and fourth central moments indicate non-Gaussianity of the target state when the ownship turns. By monitoring these metrics and appropriately initialising relative range error, we can achieve a desired output mean estimated range error (MRE). The availability of these higher-order moments facilitates other extensions of the tracker not possible with a standard UKF.

physics.data-an↗

arXiv2024

Unifying Lyapunov exponents with probabilistic uncertainty quantification

The Lyapunov exponent is well-known in deterministic dynamical systems as a measure for quantifying chaos and detecting coherent regions in physically evolving systems. In this Letter, we show how the Lyapunov exponent can be unified with stochastic sensitivity (which quantifies the uncertainty of an evolving uncertain system whose initial condition is certain) within a finite time uncertainty quantification framework in which both the dynamics and the initial condition of a continuously evolving $ n $-dimensional state variable are uncertain.

math.DS↗

arXiv2023

Optimal Proposal Particle Filters for Detecting Anomalies and Manoeuvres from Two Line Element Data

Detecting anomalous behaviour of satellites is an important goal within the broader task of space situational awareness. The Two Line Element (TLE) data published by NORAD is the only widely-available, comprehensive source of data for satellite orbits. We present here a filtering approach for detecting anomalies in satellite orbits from TLE data. Optimal proposal particle filters are deployed to track the state of the satellites' orbits. New TLEs that are unlikely given our belief of the current orbital state are designated as anomalies. The change in the orbits over time is modelled using the SGP4 model with some adaptations. A model uncertainty is derived to handle the errors in SGP4 around singularities in the orbital elements. The proposed techniques are evaluated on a set of 15 satellites for which ground truth is available and the particle filters are shown to be superior at detecting the subtle in-track and cross-track manoeuvres in the simulated dataset, as well as providing a measure of uncertainty of detections.

astro-ph.EP↗

arXiv2023

Neural Likelihood Approximation for Integer Valued Time Series Data

Stochastic processes defined on integer valued state spaces are popular within the physical and biological sciences. These models are necessary for capturing the dynamics of small systems where the individual nature of the populations cannot be ignored and stochastic effects are important. The inference of the parameters of such models, from time series data, is challenging due to intractability of the likelihood. To work at all, current simulation based inference methods require the generation of realisations of the model conditional on the data, which can be both tricky to implement and computationally expensive. In this paper we instead construct a neural likelihood approximation that can be trained using unconditional simulation of the underlying model, which is much simpler. We demonstrate our method by performing inference on a number of ecological and epidemiological models, showing that we can accurately approximate the true posterior while achieving significant computational speed ups compared to current best methods.

stat.ML↗

arXiv2023

The convergence of stochastic differential equations to their linearisation in small noise limits

Prediction via deterministic continuous-time models will always be subject to model error, for example due to unexplainable phenomena, uncertainties in any data driving the model, or discretisation/resolution issues. In this paper, we build upon previous small-noise studies to provide an explicit bound for the error between a general class of stochastic differential equations and corresponding computable linearisations written in terms of a deterministic system. Our framework accounts for non-autonomous coefficients, multiplicative noise, and uncertain initial conditions. We demonstrate the predictive power of our bound on diverse numerical case studies. We confirm that our bound is sharp, in that it accurately predicts the error scaling in the moments of the linearised approximation as both the uncertainty in the initial condition and the magnitude of the noise in the differential equation are altered. This paper also provides an extension of stochastic sensitivity, a recently introduced tool for quantifying uncertainty in dynamical systems, to arbitrary dimensions and establishes the link to our characterisation of stochastic differential equation linearisations.

math.DS↗

arXiv2023

Computation of random time-shift distributions for stochastic population models

Even in large systems, the effect of noise arising from when populations are initially small can persist to be measurable on the macroscale. A deterministic approximation to a stochastic model will fail to capture this effect, but it can be accurately approximated by including an additional random time-shift to the initial conditions. We present a efficient numerical method to compute this time-shift distribution for a large class of stochastic models. The method relies on differentiation of certain functional equations, which we show can be effectively automated by deriving rules for different types of model rates that arise commonly when mass-action mixing is assumed. Explicit computation of the time-shift distribution can be used to build a practical tool for the efficient generation of macroscopic trajectories of stochastic population models, without the need for costly stochastic simulations. Full code is provided to implement this and we demonstrate our method on an epidemic model and a model of within-host viral dynamics.

q-bio.PE↗

arXiv2022

Wide-scale Monitoring of Satellite Lifetimes: Pitfalls and a Benchmark Dataset

An important task within the broader goal of Space Situational Awareness (SSA) is to observe changes in the orbits of satellites, where the data spans thousands of objects over long time scales (decades). The Two-Line Element (TLE) data provided by the North American Aerospace Defense Command is the most comprehensive and widely-available dataset cataloguing the orbits of satellites. This makes it a highly-attractive data source on which to perform this observation. However, when attempting to infer changes in satellite behaviour from TLE data, there are a number of potential pitfalls. These mostly relate to specific features of the TLE data which are not always clearly documented in the data sources or popular software packages for manipulating them. These quirks produce a particularly hazardous data type for researchers from adjacent disciplines (such as anomaly detection or machine learning). We highlight these features of TLE data and the resulting pitfalls in order to save future researchers from being trapped. A seperate, significant, issue is that existing contributions to manoeuvre detection from TLE data evaluate their algorithms on different satellites, making comparison between these methods difficult. Moreover, the ground-truth in these datasets is often poor quality, sometimes being based on subjective human assessment. We therefore release and describe in-depth an open, curated, benchmark dataset containing TLE data for 15 satellites alongside high-quality ground-truth manoeuvre timestamps.

astro-ph.EP↗

arXiv2021

Adaptively detect and accurately resolve macro-scale shocks in an efficient Equation-Free multiscale simulation

The Equation-Free approach to efficient multiscale numerical computation marries trusted micro-scale simulations to a framework for numerical macro-scale reduction -- the patch dynamics scheme. A recent novel patch scheme empowered the Equation-Free approach to simulate systems containing shocks on the macro-scale. However, the scheme did not predict the formation of shocks accurately, and it could not simulate moving shocks. This article resolves both issues, as a first step in one spatial dimension, by embedding the Equation-Free, shock-resolving patch scheme within a classic framework for adaptive moving meshes. Our canonical micro-scale problems exhibit heterogeneous nonlinear advection and heterogeneous diffusion. We demonstrate many remarkable benefits from the moving patch scheme, including efficient and accurate macro-scale prediction despite the unknown macro-scale closure. Equation-free methods are here extended to simulate moving, forming and merging shocks without a priori knowledge of the existence or closure of the shocks. Whereas adaptive moving mesh equations are typically stiff, typically requiring small time-steps on the macro-scale, the moving macro-scale mesh of patches is typically not stiff given the context of the micro-scale time-steps required for the sub-patch dynamics.

math.DS↗

arXiv2021

Model and Data Reduction for Data Assimilation: Particle Filters Employing Projected Forecasts and Data with Application to a Shallow Water Model

The understanding of nonlinear, high dimensional flows, e.g, atmospheric and ocean flows, is critical to address the impacts of global climate change. Data Assimilation techniques combine physical models and observational data, often in a Bayesian framework, to predict the future state of the model and the uncertainty in this prediction. Inherent in these systems are noise (Gaussian and non-Gaussian), nonlinearity, and high dimensionality that pose challenges to making accurate predictions. To address these issues we investigate the use of both model and data dimension reduction based on techniques including Assimilation in Unstable Subspaces, Proper Orthogonal Decomposition, and Dynamic Mode Decomposition. Algorithms that take advantage of projected physical and data models may be combined with Data Analysis techniques such as Ensemble Kalman Filter and Particle Filter variants. The projected Data Assimilation techniques are developed for the optimal proposal particle filter and applied to the Lorenz'96 and Shallow Water Equations to test the efficacy of our techniques in high dimensional, nonlinear systems.

math.DS↗

arXiv2020

A surrogate-based approach to nonlinear, non-Gaussian joint state-parameter data assimilation

Many recent advances in sequential assimilation of data into nonlinear high-dimensional models are modifications to particle filters which employ efficient searches of a high-dimensional state space. In this work, we present a complementary strategy that combines statistical emulators and particle filters. The emulators are used to learn and offer a computationally cheap approximation to the forward dynamic mapping. This emulator-particle filter (Emu-PF) approach requires a modest number of forward-model runs, but yields well-resolved posterior distributions even in non-Gaussian cases. We explore several modifications to the Emu-PF that utilize mechanisms for dimension reduction to efficiently fit the statistical emulator, and present a series of simulation experiments on an atypical Lorenz-96 system to demonstrate their performance. We conclude with a discussion on how the Emu-PF can be paired with modern particle filtering algorithms.

math.DS↗

arXiv2020

Particle filters for data assimilation based on reduced order data models

We introduce a framework for Data Assimilation (DA) in which the data is split into multiple sets corresponding to low-rank projections of the state space. Algorithms are developed that assimilate some or all of the projected data, including an algorithm compatible with any generic DA method. The major application explored here is PROJ-PF, a projected Particle Filter. The PROJ-PF implementation assimilates highly informative but low-dimensional observations. The implementation considered here is based upon using projections corresponding to Assimilation in the Unstable Subspace (AUS). In the context of particle filtering, the projected approach mitigates the collapse of particle ensembles in high dimensional DA problems while preserving as much relevant information as possible, as the unstable and neutral modes correspond to the most uncertain model predictions. In particular we formulate and numerically implement a projected Optimal Proposal Particle Filter (PROJ-OP-PF) and compare to the standard optimal proposal and to the Ensemble Transform Kalman Filter.

stat.CO↗

arXiv2020

A toolbox of Equation-Free functions in Matlab\Octave for efficient system level simulation

The `equation-free toolbox' empowers the computer-assisted analysis of complex, multiscale systems. Its aim is to enable you to immediately use microscopic simulators to perform macro-scale system level tasks and analysis, because micro-scale simulations are often the best available description of a system. The methodology bypasses the derivation of macroscopic evolution equations by computing the micro-scale simulator only over short bursts in time on small patches in space, with bursts and patches well-separated in time and space respectively. We introduce the suite of coded equation-free functions in an accessible way, link to more detailed descriptions, discuss their mathematical support, and introduce a novel and efficient algorithm for Projective Integration. Some facets of toolbox development of equation-free functions are then detailed. Download the toolbox functions (https://github.com/uoa1184615/EquationFreeGit) and use to empower efficient and accurate simulation in a wide range of your science and engineering problems.

cs.MS↗

arXiv2020

A multiscale scheme accurately simulates macroscale shocks in an equation-free framework

Scientists and engineers often create accurate, trustworthy, computational simulation schemes - but all too often these are too computationally expensive to execute over the time or spatial domain of interest. The equation-free approach is to marry such trusted simulations to a framework for numerical macroscale reduction - the patch dynamics scheme. This article extends the patch scheme to scenarios in which the trusted simulation resolves abrupt state changes on the microscale that appear as shocks on the macroscale. Accurate simulation for problems in these scenarios requires extending the patch scheme by capturing the shock within a novel patch, and also modifying the patch coupling rules in the vicinity in order to maintain accuracy. With these two extensions to the patch scheme, straightforward arguments derive consistency conditions that match the usual order of accuracy for patch schemes. The new scheme is successfully tested on four archetypal problems. This technique will empower scientists and engineers to accurately and efficiently simulate, over large spatial domains, multiscale multiphysics systems that have rapid transition layers on the microscale.

math.NA↗

arXiv2017

A coherent structure approach for parameter estimation in Lagrangian Data Assimilation

We introduce a data assimilation method to estimate model parameters with observations of passive tracers by directly assimilating Lagrangian Coherent Structures. Our approach differs from the usual Lagrangian Data Assimilation approach, where parameters are estimated based on tracer trajectories. We employ the Approximate Bayesian Computation (ABC) framework to avoid computing the likelihood function of the coherent structure, which is usually unavailable. We solve the ABC by a Sequential Monte Carlo (SMC) method, and use Principal Component Analysis (PCA) to identify the coherent patterns from tracer trajectory data. Our new method shows remarkably improved results compared to the bootstrap particle filter when the physical model exhibits chaotic advection.

physics.ao-ph↗

arXiv2015

A note on implementations of the Boosting Algorithm and Heterogeneous Multiscale Methods

We present improved convergence results for the Boosting Algorithm (BA), and demonstrate that an existing formulation of the Heterogeneous Multiscale Methods (HMM) is accurate to first order only in the macro time step, regardless of the order of the numerical solvers employed. These results are obtained by considering the BA and two other formulations of HMM as special cases of a general formulation of HMM applied to dissipative stiff ordinary differential equations.

math.DS↗

arXiv2015

On convergence of higher order schemes for the projective integration method for stiff ordinary differential equations

We present a convergence proof for higher order implementations of the projective integration method (PI) for a class of deterministic multi-scale systems in which fast variables quickly settle on a slow manifold. The error is shown to contain contributions associated with the length of the microsolver, the numerical accuracy of the macrosolver and the distance from the slow manifold caused by the combined effect of micro- and macrosolvers, respectively. We also provide stability conditions for the PI methods under which the fast variables will not diverge from the slow manifold. We corroborate our results by numerical simulations.

math.NA↗