arXiv subjects

Adrian N. Bishop

Publications and source records attributed to Adrian N. Bishop.

22 records · Page 2Linked to original sources

arXiv2018

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.

stat.ME↗

arXiv2017

Matrix product moments in normal variables

Let ${\cal X }=XX^{\prime}$ be a random matrix associated with a centered $r$-column centered Gaussian vector $X$ with a covariance matrix $P$. In this article we compute expectations of matrix-products of the form $\prod_{1\leq i\leq n}({\cal X } P^{v_i})$ for any $n\geq 1$ and any multi-index parameters $v_i\in\mathbb{N}$. We derive closed form formulae and a simple sequential algorithm to compute these matrices w.r.t. the parameter $n$. The second part of the article is dedicated to a non commutative binomial formula for the central matrix-moments $\mathbb{E}\left(\left[{\cal X }-P\right]^n\right)$. The matrix product moments discussed in this study are expressed in terms of polynomial formulae w.r.t. the powers of the covariance matrix, with coefficients depending on the trace of these matrices. We also derive a series of estimates w.r.t. the Loewner order on quadratic forms. For instance we shall prove the rather crude estimate $\mathbb{E}\left(\left[{\cal X }-P\right]^n\right)\leq \mathbb{E}\left({\cal X }^n-P^n\right)$, for any $n\geq 1$

math.ST↗

arXiv2014

Adaptively Blocked Particle Filtering with Spatial Smoothing in Large-Scale Dynamic Random Fields

The typical particle filtering approximation error is exponentially dependent on the dimension of the model. Therefore, to control this error, an enormous number of particles are required, which means a heavy computational burden that is often so great it is simply prohibitive. Rebeschini and van Handel (2013) consider particle filtering in a large-scale dynamic random field. Through a suitable localisation operation, they prove that a modified particle filtering algorithm can achieve an approximation error that is mostly independent of the problem dimension. To achieve this feat, they inadvertently introduce a systematic bias that is spatially dependent (in that the bias at one site is dependent on the location of that site). This bias consequently varies throughout field. In this work, a simple extension to the algorithm of Rebeschini and van Handel is introduced which acts to average this bias term over each site in the field through a kind of spatial smoothing. It is shown that for a certain class of random field it is possible to achieve a completely spatially uniform bound on the bias and that in any general random field the spatial inhomogeneity is significantly reduced when compared to the case in which spatial smoothing is not considered. While the focus is on spatial averaging in this work, the proposed algorithm seemingly exhibits other advantageous properties such as improved robustness and accuracy in those cases in which the underlying dynamic field is time varying.

math.OC↗

arXiv2014

Reducing the Bias in Blocked Particle Filtering for High-Dimensional Systems

Particle filtering is a powerful approximation method that applies to state estimation in nonlinear and non-Gaussian dynamical state-space models. Unfortunately, the approximation error depends exponentially on the system dimension. This means that an incredibly large number of particles may be needed to appropriately control the error in very large scale filtering problems. The computational burden required is often prohibitive in practice. Rebeschini and Van Handel (2013) analyse a new approach for particle filtering in large-scale dynamic random fields. Through a suitable localisation operation they reduce the dependence of the error to the size of local sets, each of which may be considerably smaller than the dimension of the original system. The drawback is that this localisation operation introduces a bias. In this work, we propose a modified version of Rebeschini and Van Handel's blocked particle filter. We introduce a new degree of freedom allowing us to reduce the bias. We do this by enlarging the space during the update phase and thus reducing the amount of dependent information thrown away due to localisation. By designing an appropriate tradeoff between the various tuning parameters it is possible to reduce the total error bound via allowing a temporary enlargement of the update operator without really increasing the overall computational burden.

math.OC↗