Derivations of variational gaussian process approximation framework

Recently, within the VISDEM project (EPSRC funded EP/C005848/1), a novel variational approximation framework has been developed for inference in partially observed, continuous space-time, diffusion processes. In this technical report all the derivations of the variational framework, from the initial work, are provided in detail to help the reader better understand the framework and its assumptions.

Gaussian process approximations of stochastic differential equations

Stochastic differential equations arise naturally in a range of contexts, from financial to environmental modeling. Current solution methods are limited in their representation of the posterior process in the presence of data. In this work, we present a novel Gaussian process approximation to the posterior measure over paths for a general class of stochastic differential equations in the presence of observations. The method is applied to two simple problems: the Ornstein-Uhlenbeck process, of which the exact solution is known and can be compared to, and the double-well system, for which standard approaches such as the ensemble Kalman smoother fail to provide a satisfactory result. Experiments show that our variational approximation is viable and that the results are very promising as the variational approximate solution outperforms standard Gaussian process regression for non-Gaussian Markov processes.

Le prix de l'individualité. Maine de Brian, Samuel Taylor Coleridge et la volonté: Deux postétités distinctes

This thesis is concerned with Maine de Biran’s and Samuel Taylor Coleridge’s conceptions of will, and the way in which both thinkers’ posterities have been affected by the central role of these very conceptions in their respective bodies of thought. The research question that animates this work can therefore be divided into two main parts, one of which deals with will, while the other deals with its effects on posterity. In the first pages of the Introduction, I make the case for a comparison between two philosophers, and show how this comparison can bring one closer to truth, understood not in objective, but in subjective terms. I then justify my choice by underlining that, in spite of their many differences, Maine de Biran and Samuel Taylor Coleridge followed comparable paths, intellectually and spiritually, and came to similar conclusions concerning the essential activity of the human mind. Finally, I ask whether it is possible that this very focus on the human will may have contributed to the state of both thinkers’ works and of the reception of those works. This prologue is followed by five parts. In the first part, the similarities and differences between the two thinkers are explored further. In the second part, the connections between philosophy and singularity are examined, in order to show the ambivalence of the will as a foundation for truth. The third part is dedicated to the traditional division between subject and object in psychology, and its relevance in history and in moral philosophy. The fourth part tackles the complexity of the question of influence, with respect to both Maine de Biran’s and Coleridge’s cases, both thinkers being indebted to many philosophers of all times and places, and having to rely heavily on others for the publication, or the interpretation of their own works. The fifth part is concerned with the different aspects of the faculty of will, and primarily its relationship with interiority, as incommensurability, and actual, conditioned existence in a certain historical and spatial context. It ends with a return to the question of will and posterity and an announcement of what will be covered in the main body of the thesis. The main body is divided into three parts:‘L’émancipation’, ‘L’affirmation, and ‘La projection’. The first part is devoted to the way Maine de Biran and Samuel Taylor Coleridge extricated themselves from one epistemological paradigm to contribute to the foundation of another. It is divided in four chapters. The first chapter deals with the aforementioned change of paradigm, as corresponding to the emergence of two separate but associated movements, Romanticism and what the French philosopher refers to as ‘The Age of History’. The second chapter concerns the movement that preceded them, i.e. the Enlightenment, its main features according to both of our thinkers, and the two epistemological models that prevailed under it and influenced them heavily in their early years: Sensationism (Maine de Biran) and Associationism (Coleridge). The third chapter is about the probable influence of Immanuel Kant and his followers on Maine de Biran and Coleridge, and the various facts that allow us to claim originality for both thinkers’ works. In the fourth chapter, I contrast Maine de Biran and Coleridge with other movements and thinkers of their time, showing that, contrary to their respective thoughts, Maine de Biran and Coleridge could not but break free from the then prevailing systematic approach to truth. The second part of the thesis is concerned with the first part of its research question, namely, Maine de Biran’s and Coleridge’s conceptions of the will. It is divided into four chapters. The first chapter is a reflection on the will as a paradox: on the one hand, the will cannot be caused by any other phenomenon, or it is no longer a will; but it cannot be left purely undetermined, as if it is, it is then not different from chance. It thus needs, in order to be, to be contradictorily already moral. The second chapter is a comparison between Maine de Biran’s and Coleridge’s accounts of the origin of the will, where it is found that the French philosopher only observes that he has a will, whereas the English philosopher postulates the existence of this will. The comparison between Maine de Biran’s and Coleridge’s conceptions of the will is pursued in the third chapter, which tackles the question of the coincidence between the will and the self, in both thinkers’ works. It ends with the fourth chapter, which deals with the question of the relationship between the will and what is other to it, i.e. bodily sensations, passions and desires. The third part of the thesis focuses on the second part of its research question, namely the posterity of Maine de Biran’s and Coleridge’s works. It is divided into four chapters. The first chapter constitutes a continuation of the last chapter of the preceding part, in that that it deals with Maine de Biran’s and Coleridge’s relations to the ‘other’, and particularly their potential and actual audience, and with the way these relations may have affected their writing and publishing practices. The second chapter is a survey of both thinkers’ general reception, where it is found that, while Maine de Biran has been claimed by two important movements of thoughts as their initiator, Coleridge has been neglected by the only real movement he could have, or may indeed have pioneered. The third chapter is more directly concerned with the posterities of Maine de Biran’s and Coleridge’s conceptions of will, and attempts to show that the approach to, and the meaning of the will have evolved throughout the nineteenth century, and in the French Spiritualist and the British Idealist movements, from an essentially personal one to a more impersonal one. The fourth chapter is a partial conclusion, whose aim is to give a precise idea of where Maine de Biran and Coleridge stand, in relation to their century and to the philosophical movements and matters we are concerned with. The conclusion is a recapitulation of what has been found, with a particular emphasis on the dialogue initiated between Maine de Biran and Coleridge on the will, and the relation between will and posterity. It suggests that both thinkers have to pay the price of a problematic reception for the individuality that pervades their respective works, and goes further in suggesting that s/he who chooses to found his individuality on the will is bound to feel this incompleteness in his/her own personal life more acutely than s/he who does not. It ends with a reflection on fixedness and movement, as the two antagonistic states that the theoretician of the will paradoxically aspires to.

Phase transitions and memory effects in the dynamics of Boolean networks

The generating functional method is employed to investigate the synchronous dynamics of Boolean networks, providing an exact result for the system dynamics via a set of macroscopic order parameters. The topology of the networks studied and its constituent Boolean functions represent the system's quenched disorder and are sampled from a given distribution. The framework accommodates a variety of topologies and Boolean function distributions and can be used to study both the noisy and noiseless regimes; it enables one to calculate correlation functions at different times that are inaccessible via commonly used approximations. It is also used to determine conditions for the annealed approximation to be valid, explore phases of the system under different levels of noise and obtain results for models with strong memory effects, where existing approximations break down. Links between Boolean networks and general Boolean formulas are identified and results common to both system types are highlighted. © 2012 Copyright Taylor and Francis Group, LLC.

A new variational radial basis function approximation for inference in multivariate diffusions

In this paper we present a radial basis function based extension to a recently proposed variational algorithm for approximate inference for diffusion processes. Inference, for state and in particular (hyper-) parameters, in diffusion processes is a challenging and crucial task. We show that the new radial basis function approximation based algorithm converges to the original algorithm and has beneficial characteristics when estimating (hyper-)parameters. We validate our new approach on a nonlinear double well potential dynamical system.

Approximation of photonic crystal fibres with large air holes by the step index fibre model

An equivalent step index fibre with a silica core and air cladding is used to model photonic crystal fibres with large air holes. We model this fibre for linear polarisation (we focus on the lowest few transverse modes of the electromagnetic field). The equivalent step index radius is obtained by equating the lowest two eigenvalues of the model to those calculated numerically for the photonic crystal fibres. The step index parameters thus obtained can then be used to calculate nonlinear parameters like the nonlinear effective area of a photonic crystal fibre or to model nonlinear few-mode interactions using an existing model.

Numerical approximation of the one-dimensional inverse Cauchy–Stefan problem using a method of fundamental solutions

We investigate an application of the method of fundamental solutions (MFS) to the one-dimensional parabolic inverse Cauchy–Stefan problem, where boundary data and the initial condition are to be determined from the Cauchy data prescribed on a given moving interface. In [B.T. Johansson, D. Lesnic, and T. Reeve, A method of fundamental solutions for the one-dimensional inverse Stefan Problem, Appl. Math Model. 35 (2011), pp. 4367–4378], the inverse Stefan problem was considered, where only the boundary data is to be reconstructed on the fixed boundary. We extend the MFS proposed in Johansson et al. (2011) and show that the initial condition can also be simultaneously recovered, i.e. the MFS is appropriate for the inverse Cauchy-Stefan problem. Theoretical properties of the method, as well as numerical investigations, are included, showing that accurate results can be efficiently obtained with small computational cost.

Generalized mean field approximation for parallel dynamics of the Ising model

The dynamics of the non-equilibrium Ising model with parallel updates is investigated using a generalized mean field approximation that incorporates multiple two-site correlations at any two time steps, which can be obtained recursively. The proposed method shows significant improvement in predicting local system properties compared to other mean field approximation techniques, particularly in systems with symmetric interactions. Results are also evaluated against those obtained from Monte Carlo simulations. The method is also employed to obtain parameter values for the kinetic inverse Ising modeling problem, where couplings and local field values of a fully connected spin system are inferred from data. © 2014 IOP Publishing Ltd and SISSA Medialab srl.

A survey of methods of numerical approximation to functions

DUE TO COPYRIGHT RESTRICTIONS ONLY AVAILABLE FOR CONSULTATION AT ASTON UNIVERSITY LIBRARY AND INFORMATION SERVICES WITH PRIOR ARRANGEMENT

Hierarchized block wise image approximation by greedy pursuit strategies

An approach for effective implementation of greedy selection methodologies, to approximate an image partitioned into blocks, is proposed. The method is specially designed for approximating partitions on a transformed image. It evolves by selecting, at each iteration step, i) the elements for approximating each of the blocks partitioning the image and ii) the hierarchized sequence in which the blocks are approximated to reach the required global condition on sparsity. © 2013 IEEE.

Hierarchized block wise image approximation by greedy pursuit strategies

An approach for effective implementation of greedy selection methodologies, to approximate an image partitioned into blocks, is proposed. The method is specially designed for approximating partitions on a transformed image. It evolves by selecting, at each iteration step, i) the elements for approximating each of the blocks partitioning the image and ii) the hierarchized sequence in which the blocks are approximated to reach the required global condition on sparsity. © 2013 IEEE.

Foveation time measure in Congenital Nystagmus through second order approximation of the slow phases

Congenital Nystagmus (CN) is an ocular-motor disorder characterised by involuntary, conjugated ocular oscillations, and its pathogenesis is still unknown. The pathology is de fined as "congenital" from the onset time of its arise which could be at birth or in the first months of life. Visual acuity in CN subjects is often diminished due to nystagmus continuous oscillations, mainly on the horizontal plane, which disturb image fixation on the retina. However, during short periods in which eye velocity slows down while the target image is placed onto the fovea (called foveation intervals) the image of a given target can still be stable, allowing a subject to reach a higher visual acuity. In CN subjects, visual acuity is usually assessed both using typical measurement techniques (e.g. Landolt C test) and with eye movement recording in different gaze positions. The offline study of eye movement recordings allows physicians to analyse nystagmus main features such as waveform shape, amplitude and frequency and to compute estimated visual acuity predictors. This analytical functions estimates the best corrected visual acuity using foveation time and foveation position variability, hence a reliable estimation of this two parameters is a fundamental factor in assessing visual acuity. This work aims to enhance the foveation time estimation in CN eye movement recording, computing a second order approximation of the slow phase components of nystag-mus oscillations. About 19 infraredoculographic eye-movement recordings from 10 CN subjects were acquired and the visual acuity assessed with an acuity predictor was compared to the one measured in primary position. Results suggest that visual acuity measurements based on foveation time estimation obtained from interpolated data are closer to value obtained during Landolt C tests. © 2010 IEEE.

Organized crime legislation in the European Union. Harmonization and approximation of criminal law, national legislations and the EU framework decision on the fight against organised crime, by Francesco Calderoni

Book review: Heidelberg, Dordrecht, London, and New York, Springer, 2010, 189 pp., £93.55 (hardcover), ISBN 978-3-642-04330-7, e-ISBN 978-3-642-04331-4