Paul Fearnhead
Lancaster University, Mathematics and Statistics, Faculty Member
- I am currently Professor of Statistics at Lancaster University. I work on computational statistical methods, scalabil... moreI am currently Professor of Statistics at Lancaster University. I work on computational statistical methods, scalability of algorithms, and methods for detecting changepoints. I am currently editor of Biometrika.edit
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This paper considers the problem of bearings only tracking of manoeuvring targets. A learning particle filtering algorithm is proposed which can estimate both the unknown target states and unknown model parameters. The algorithm... more
This paper considers the problem of bearings only tracking of manoeuvring targets. A learning particle filtering algorithm is proposed which can estimate both the unknown target states and unknown model parameters. The algorithm performance is validated and tested over a challenging scenario with abrupt manoeuvres. A comparison of the proposed algorithm with the Interacting Multiple Model (IMM) filter is presented. The learning particle filter has shown accurate estimation results and improved accuracy compared with the ...
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Time-series data often experiences abrupt changes in structure. If the time-series is to be modelled accurately then these changepoints must be taken into account. Many methods for detecting these changepoints, both heuristic and exact,... more
Time-series data often experiences abrupt changes in structure. If the time-series is to be modelled accurately then these changepoints must be taken into account. Many methods for detecting these changepoints, both heuristic and exact, have been developed and these are reviewed in this report. The report in particular looks at dynamic programming approaches and pruning methods which can be used to speed these methods up. The report concludes by discussing a few of the open questions which are still unresolved and ...
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Generalized autoregressive conditionally heteroskedastic (GARCH) processes are widely used for modelling financial returns, with their extremal properties being of interest for market risk management. For GARCH($$p,q$$ p , q ) processes... more
Generalized autoregressive conditionally heteroskedastic (GARCH) processes are widely used for modelling financial returns, with their extremal properties being of interest for market risk management. For GARCH($$p,q$$ p , q ) processes with $$\max (p,q) = 1$$ max ( p , q ) = 1 all extremal features have been fully characterised, but when $$\max (p,q)\ge 2$$ max ( p , q ) ≥ 2 much remains to be found. Previous research has identified that both marginal and dependence extremal features of strictly stationary GARCH($$p,q$$ p , q ) processes are determined by a multivariate regular variation property and tail processes. Currently there are no reliable methods for evaluating these characterisations, or even assessing the stationarity, for the classes of GARCH($$p,q$$ p , q ) processes that are used in practice, i.e., with unbounded and asymmetric innovations. By developing a mixture of new limit theory and particle filtering algorithms for fixed point distributions we produce novel and ...
Research Interests: Statistics and Algorithm
Consider the problem of searching a large set of items, such as emails, for a small set which are relevant to a given query. This can be implemented in a sequential manner whereby we use knowledge from earlier items that we have screened... more
Consider the problem of searching a large set of items, such as emails, for a small set which are relevant to a given query. This can be implemented in a sequential manner whereby we use knowledge from earlier items that we have screened to help us choose future items in an informed way. Often the items we are searching have an underlying network structure: for example emails can be related to a network of participants, where an edge in the network relates to the presence of a communication between those two participants. Recent work by Dimitrov, Kress and Nevo has shown that using the information about the network structure together with a modelling assumption that relevant items and participants are likely to cluster together, can greatly increase the rate of screening relevant items. However their approach is computationally expensive and thus limited in applicability to small networks. Here we show how Bayes Linear methods provide a natural approach to modelling such data; that ...
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One of the contemporary challenges in anomaly detection is the ability to detect, and differentiate between, both point and collective anomalies within a data sequence or time series. The anomaly package has been developed to provide... more
One of the contemporary challenges in anomaly detection is the ability to detect, and differentiate between, both point and collective anomalies within a data sequence or time series. The anomaly package has been developed to provide users with a choice of anomaly detection methods and, in particular, provides an implementation of the recently proposed CAPA family of anomaly detection algorithms. This article describes the methods implemented whilst also highlighting their application to simulated data as well as real data examples contained in the package.
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This article focuses on the challenging problem of efficiently detecting changes in mean within multivariate data sequences. Multivariate changepoints can be detected by projecting a multivariate series to a univariate one using a... more
This article focuses on the challenging problem of efficiently detecting changes in mean within multivariate data sequences. Multivariate changepoints can be detected by projecting a multivariate series to a univariate one using a suitable projection direction that preserves a maximal proportion of signal information. However, for some existing approaches the computation of such a projection direction can scale unfavourably with the number of series and might rely on additional assumptions on the data sequences, thus limiting their generality. We introduce BayesProject, a computationally inexpensive Bayesian approach to compute a projection direction in such a setting. The proposed approach allows the incorporation of prior knowledge of the changepoint scenario, when such information is available, which can help to increase the accuracy of the method. A simulation study shows that BayesProject is robust, yields projections close to the oracle projection direction and, moreover, that...
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Non UBCUnreviewedAuthor affiliation: Lancaster UniversityFacult
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Abstract We review work on how to perform exact online inference for a class of multiple changepoint models. These models have a conditional independence structure, and require you to be able to integrate out (either analytically or... more
Abstract We review work on how to perform exact online inference for a class of multiple changepoint models. These models have a conditional independence structure, and require you to be able to integrate out (either analytically or numerically) the parameters associated within each segment. The computational cost per observation increases linearly with the number of observations. This algorithm is closely related to a particle filter algorithm, and we describe how efficient resampling algorithms can be used to produce an accurate particle ...
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We have performed simulations to assess the performance of three population genetics approximate-likelihood methods in estimating the population-scaled recombination rate from sequence data. We measured performance in two ways: accuracy... more
We have performed simulations to assess the performance of three population genetics approximate-likelihood methods in estimating the population-scaled recombination rate from sequence data. We measured performance in two ways: accuracy when the sequence data were simulated according to the (simplistic) standard model underlying the methods and robustness to violations of many different aspects of the standard model. Although we found some differences between the methods, performance tended to be similar for all three methods. Despite the fact that the methods are not robust to violations of the underlying model, our simulations indicate that patterns of relative recombination rates should be inferred reasonably well even if the standard model does not hold. In addition, we assess various techniques for improving the performance of approximate-likelihood methods. In particular we find that the composite-likelihood method of Hudson (2001) can be improved by including log-likelihood c...
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Particle MCMC involves using a particle filter within an MCMC algorithm. For inference of a model which involves an unobserved stochastic process, the standard implementation uses the particle filter to propose new values for the... more
Particle MCMC involves using a particle filter within an MCMC algorithm. For inference of a model which involves an unobserved stochastic process, the standard implementation uses the particle filter to propose new values for the stochastic process, and MCMC moves to propose new values for the parameters. We show how particle MCMC can be generalised beyond this. Our key idea is to introduce new latent variables. We then use the MCMC moves to update the latent variables, and the particle filter to propose new values for the parameters and stochastic process given the latent variables. A generic way of defining these latent variables is to model them as pseudo-observations of the parameters or of the stochastic process. By choosing the amount of information these latent variables have about the parameters and the stochastic process we can often improve the mixing of the particle MCMC algorithm by trading off the Monte Carlo error of the particle filter and the mixing of the MCMC moves...
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State-space models can be used to incorporate subject knowledge on the underlying dynamics of a time series by the introduction of a latent Markov state-process. A user can specify the dynamics of this process together with how the state... more
State-space models can be used to incorporate subject knowledge on the underlying dynamics of a time series by the introduction of a latent Markov state-process. A user can specify the dynamics of this process together with how the state relates to partial and noisy observations that have been made. Inference and prediction then involves solving a challenging inverse problem: calculating the conditional distribution of quantities of interest given the observations. This article reviews Monte Carlo algorithms for solving this inverse problem, covering methods based on the particle filter and the ensemble Kalman filter. We discuss the challenges posed by models with high-dimensional states, joint estimation of parameters and the state, and inference for the history of the state process. We also point out some potential new developments which will be important for tackling cutting-edge filtering applications.
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Many modern applications of online changepoint detection require the ability to process high-frequency observations, sometimes with limited available computational resources. Online algorithms for detecting a change in mean often involve... more
Many modern applications of online changepoint detection require the ability to process high-frequency observations, sometimes with limited available computational resources. Online algorithms for detecting a change in mean often involve using a moving window, or specifying the expected size of change. Such choices affect which changes the algorithms have most power to detect. We introduce an algorithm, Functional Online CuSUM (FOCuS), which is equivalent to running these earlier methods simultaneously for all sizes of window, or all possible values for the size of change. Our theoretical results give tight bounds on the expected computational cost per iteration of FOCuS, with this being logarithmic in the number of observations. We show how FOCuS can be applied to a number of different change in mean scenarios, and demonstrate its practical utility through its state-of-the art performance at detecting anomalous behaviour in computer server data.
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We consider (approximate) likelihood methods for estimating the population-scaled recombination rate from population genetic data. We show that the dependence between the data from two regions of a chromosome decays inversely with the... more
We consider (approximate) likelihood methods for estimating the population-scaled recombination rate from population genetic data. We show that the dependence between the data from two regions of a chromosome decays inversely with the amount of recombination between the two regions. We use this result to show that the maximum likelihood estimator (mle) for the recombination rate, based on the composite likelihood of Fearnhead and Donnelly, is consistent. We also consider inference based on the pairwise likelihood of Hudson. We consider two approximations to this likelihood, and prove that the mle based on one of these approximations is consistent, while the mle based on the other approximation (which is used by McVean, Awadalla and Fearnhead) is not.
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We present an informal review of recent work on the asymptotics of Approximate Bayesian Computation (ABC). In particular we focus on how does the ABC posterior, or point estimates obtained by ABC, behave in the limit as we have more data?... more
We present an informal review of recent work on the asymptotics of Approximate Bayesian Computation (ABC). In particular we focus on how does the ABC posterior, or point estimates obtained by ABC, behave in the limit as we have more data? The results we review show that ABC can perform well in terms of point estimation, but standard implementations will over-estimate the uncertainty about the parameters. If we use the regression correction of Beaumont et al. then ABC can also accurately quantify this uncertainty. The theoretical results also have practical implications for how to implement ABC.
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Many statistical applications involve models for which it is difficult to evaluate the likelihood, but from which it is relatively easy to sample. Approximate Bayesian computation is a likelihood-free method for implementing Bayesian... more
Many statistical applications involve models for which it is difficult to evaluate the likelihood, but from which it is relatively easy to sample. Approximate Bayesian computation is a likelihood-free method for implementing Bayesian inference in such cases. We present results on the asymptotic variance of estimators obtained using approximate Bayesian computation in a large-data limit. Our key assumption is that the data are summarized by a fixed-dimensional summary statistic that obeys a central limit theorem. We prove asymptotic normality of the mean of the approximate Bayesian computation posterior. This result also shows that, in terms of asymptotic variance, we should use a summary statistic that is the same dimension as the parameter vector, p; and that any summary statistic of higher dimension can be reduced, through a linear transformation, to dimension p in a way that can only reduce the asymptotic variance of the posterior mean. We look at how the Monte Carlo error of an ...
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Whilst there are many approaches to detecting changes in mean for a univariate time-series, the problem of detecting multiple changes in slope has comparatively been ignored. Part of the reason for this is that detecting changes in slope... more
Whilst there are many approaches to detecting changes in mean for a univariate time-series, the problem of detecting multiple changes in slope has comparatively been ignored. Part of the reason for this is that detecting changes in slope is much more challenging. For example, simple binary segmentation procedures do not work for this problem, whilst efficient dynamic programming methods that work well for the change in mean problem cannot be directly used for detecting changes in slope. We present a novel dynamic programming approach, CPOP, for finding the "best" continuous piecewise-linear fit to data. We define best based on a criterion that measures fit to data using the residual sum of squares, but penalises complexity based on an L_0 penalty on changes in slope. We show that using such a criterion is more reliable at estimating changepoint locations than approaches that penalise complexity using an L_1 penalty. Empirically CPOP has good computational properties, and c...
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The Knowledge Gradient (KG) policy was originally proposed for online ranking and selection problems but has recently been adapted for use in online decision making in general and multi-armed bandit problems (MABs) in particular. We study... more
The Knowledge Gradient (KG) policy was originally proposed for online ranking and selection problems but has recently been adapted for use in online decision making in general and multi-armed bandit problems (MABs) in particular. We study its use in a class of exponential family MABs and identify weaknesses, including a propensity to take actions which are dominated with respect to both exploitation and exploration. We propose variants of KG which avoid such errors. These new policies include an index heuristic which deploys a KG approach to develop an approximation to the Gittins index. A numerical study shows this policy to perform well over a range of MABs including those for which index policies are not optimal. While KG does not make dominated actions when bandits are Gaussian, it fails to be index consistent and appears not to enjoy a performance advantage over competitor policies when arms are correlated to compensate for its greater computational demands.
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Calcium imaging data promises to transform the field of neuroscience by making it possible to record from large populations of neurons simultaneously. However, determining the exact moment in time at which a neuron spikes, from a calcium... more
Calcium imaging data promises to transform the field of neuroscience by making it possible to record from large populations of neurons simultaneously. However, determining the exact moment in time at which a neuron spikes, from a calcium imaging data set, amounts to a non-trivial deconvolution problem which is of critical importance for downstream analyses. While a number of formulations have been proposed for this task in the recent literature, in this paper we focus on a formulation recently proposed in Jewell and Witten (2017) which has shown initial promising results. However, this proposal is slow to run on fluorescence traces of hundreds of thousands of timesteps. Here we develop a much faster online algorithm for solving the optimization problem of Jewell and Witten (2017) that can be used to deconvolve a fluorescence trace of 100,000 timesteps in less than a second. Furthermore, this algorithm overcomes a technical challenge of Jewell and Witten (2017) by avoiding the occurr...