Dynamical local models for segmentation and prediction of financial time series

In the analysis and prediction of many real-world time series, the assumption of stationarity is not valid. A special form of non-stationarity, where the underlying generator switches between (approximately) stationary regimes, seems particularly appropriate for financial markets. We introduce a new model which combines a dynamic switching (controlled by a hidden Markov model) and a non-linear dynamical system. We show how to train this hybrid model in a maximum likelihood approach and evaluate its performance on both synthetic and financial data.

Artefactual structure from least squares multidimensional scaling

We consider the problem of illusory or artefactual structure from the visualisation of high-dimensional structureless data. In particular we examine the role of the distance metric in the use of topographic mappings based on the statistical field of multidimensional scaling. We show that the use of a squared Euclidean metric (i.e. the SSTRESs measure) gives rise to an annular structure when the input data is drawn from a high-dimensional isotropic distribution, and we provide a theoretical justification for this observation.

Dynamical frustration of protein's environment at the nanoseconds time scale

A 21-residue peptide in explicit water has been simulated using classical molecular dynamics. The system's trajectory has been analysed with a novel approach that quantifies the process of how atom's environment trajectories are explored. The approach is based on the measure of Statistical Complexity that extracts complete dynamical information from the signal. The introduced characteristic quantifies the system's dynamics at the nanoseconds time scale. It has been found that the peptide exhibits nanoseconds long periods that significantly differ in the rates of the exploration of the dynamically allowed configurations of the environment. During these periods the rates remain the same but different from other periods and from the rate for water. Periods of dynamical frustration are detected when only limited routes in the space of possible trajectories of the surrounding atoms are realised.

Approximate Bayesian techniques for inference in stochastic dynamical systems

This thesis is concerned with approximate inference in dynamical systems, from a variational Bayesian perspective. When modelling real world dynamical systems, stochastic differential equations appear as a natural choice, mainly because of their ability to model the noise of the system by adding a variant of some stochastic process to the deterministic dynamics. Hence, inference in such processes has drawn much attention. Here two new extended frameworks are derived and presented that are based on basis function expansions and local polynomial approximations of a recently proposed variational Bayesian algorithm. It is shown that the new extensions converge to the original variational algorithm and can be used for state estimation (smoothing). However, the main focus is on estimating the (hyper-) parameters of these systems (i.e. drift parameters and diffusion coefficients). The new methods are numerically validated on a range of different systems which vary in dimensionality and non-linearity. These are the Ornstein-Uhlenbeck process, for which the exact likelihood can be computed analytically, the univariate and highly non-linear, stochastic double well and the multivariate chaotic stochastic Lorenz '63 (3-dimensional model). The algorithms are also applied to the 40 dimensional stochastic Lorenz '96 system. In this investigation these new approaches are compared with a variety of other well known methods such as the ensemble Kalman filter / smoother, a hybrid Monte Carlo sampler, the dual unscented Kalman filter (for jointly estimating the systems states and model parameters) and full weak-constraint 4D-Var. Empirical analysis of their asymptotic behaviour as a function of observation density or length of time window increases is provided.

Q parameter scaling in scalar and vector models with dominant signal-noise and noise-noise beating terms

The Q parameter scales differently with the noise power for the signal-noise and the noise-noise beating terms in scalar and vector models. Some procedures for including noise in the scalar model largely under-estimate the Q parameter.

Quantifying simulator discrepancy in discrete-time dynamical simulators

When making predictions with complex simulators it can be important to quantify the various sources of uncertainty. Errors in the structural specification of the simulator, for example due to missing processes or incorrect mathematical specification, can be a major source of uncertainty, but are often ignored. We introduce a methodology for inferring the discrepancy between the simulator and the system in discrete-time dynamical simulators. We assume a structural form for the discrepancy function, and show how to infer the maximum-likelihood parameter estimates using a particle filter embedded within a Monte Carlo expectation maximization (MCEM) algorithm. We illustrate the method on a conceptual rainfall-runoff simulator (logSPM) used to model the Abercrombie catchment in Australia. We assess the simulator and discrepancy model on the basis of their predictive performance using proper scoring rules. This article has supplementary material online. © 2011 International Biometric Society.

Computational fluid dynamical studies of structured distillation packings

In recent years structured packings have become more widely used in the process industries because of their improved volumetric efficiency. Most structured packings consist of corrugated sheets placed in the vertical plane The corrugations provide a regular network of channels for vapour liquid contact. Until recently it has been necessary to develop new packings by trial and error, testing new shapes in the laboratory. The orderly repetitive nature of the channel network produced by a structured packing suggests it may be possible to develop improved structured packings by the application of computational fluid dynamics (CFD) to calculate the packing performance and evaluate changes in shape so as to reduce the need for laboratory testing. In this work the CFD package PHOENICS has been used to predict the flow patterns produced in the vapour phase as it passes through the channel network. A particular novelty of the approach is to set up a method of solving the Navier Stokes equations for any particular intersection of channels. The flow pattern of the streams leaving the intersection is then made the input to the downstream intersection. In this way the flow pattern within a section of packing can be calculated. The resulting heat or mass transfer performance can be calculated by other standard CFD procedures. The CFD predictions revealed a circulation developing within the channels which produce a loss in mass transfer efficiency The calculations explained and predicted a change in mass transfer efficiency with depth of the sheets. This effect was also shown experimentally. New shapes of packing were proposed to remove the circulation and these were evaluated using CFD. A new shape was chosen and manufactured. This was tested experimentally and found to have a higher mass transfer efficiency than the standard packing.

Approximating differentiable relationships between delay embedded dynamical systems with radical basis functions

This thesis is about the study of relationships between experimental dynamical systems. The basic approach is to fit radial basis function maps between time delay embeddings of manifolds. We have shown that under certain conditions these maps are generically diffeomorphisms, and can be analysed to determine whether or not the manifolds in question are diffeomorphically related to each other. If not, a study of the distribution of errors may provide information about the lack of equivalence between the two. The method has applications wherever two or more sensors are used to measure a single system, or where a single sensor can respond on more than one time scale: their respective time series can be tested to determine whether or not they are coupled, and to what degree. One application which we have explored is the determination of a minimum embedding dimension for dynamical system reconstruction. In this special case the diffeomorphism in question is closely related to the predictor for the time series itself. Linear transformations of delay embedded manifolds can also be shown to have nonlinear inverses under the right conditions, and we have used radial basis functions to approximate these inverse maps in a variety of contexts. This method is particularly useful when the linear transformation corresponds to the delay embedding of a finite impulse response filtered time series. One application of fitting an inverse to this linear map is the detection of periodic orbits in chaotic attractors, using suitably tuned filters. This method has also been used to separate signals with known bandwidths from deterministic noise, by tuning a filter to stop the signal and then recovering the chaos with the nonlinear inverse. The method may have applications to the cancellation of noise generated by mechanical or electrical systems. In the course of this research a sophisticated piece of software has been developed. The program allows the construction of a hierarchy of delay embeddings from scalar and multi-valued time series. The embedded objects can be analysed graphically, and radial basis function maps can be fitted between them asynchronously, in parallel, on a multi-processor machine. In addition to a graphical user interface, the program can be driven by a batch mode command language, incorporating the concept of parallel and sequential instruction groups and enabling complex sequences of experiments to be performed in parallel in a resource-efficient manner.

Synchronization processes in nonlinear systems and their relation to cortical oscillatory dynamics

This thesis was focused on theoretical models of synchronization to cortical dynamics as measured by magnetoencephalography (MEG). Dynamical systems theory was used in both identifying relevant variables for brain coordination and also in devising methods for their quantification. We presented a method for studying interactions of linear and chaotic neuronal sources using MEG beamforming techniques. We showed that such sources can be accurately reconstructed in terms of their location, temporal dynamics and possible interactions. Synchronization in low-dimensional nonlinear systems was studied to explore specific correlates of functional integration and segregation. In the case of interacting dissimilar systems, relevant coordination phenomena involved generalized and phase synchronization, which were often intermittent. Spatially-extended systems were then studied. For locally-coupled dissimilar systems, as in the case of cortical columns, clustering behaviour occurred. Synchronized clusters emerged at different frequencies and their boundaries were marked through oscillation death. The macroscopic mean field revealed sharp spectral peaks at the frequencies of the clusters and broader spectral drops at their boundaries. These results question existing models of Event Related Synchronization and Desynchronization. We re-examined the concept of the steady-state evoked response following an AM stimulus. We showed that very little variability in the AM following response could be accounted by system noise. We presented a methodology for detecting local and global nonlinear interactions from MEG data in order to account for residual variability. We found crosshemispheric nonlinear interactions of ongoing cortical rhythms concurrent with the stimulus and interactions of these rhythms with the following AM responses. Finally, we hypothesized that holistic spatial stimuli would be accompanied by the emergence of clusters in primary visual cortex resulting in frequency-specific MEG oscillations. Indeed, we found different frequency distributions in induced gamma oscillations for different spatial stimuli, which was suggestive of temporal coding of these spatial stimuli. Further, we addressed the bursting character of these oscillations, which was suggestive of intermittent nonlinear dynamics. However, we did not observe the characteristic-3/2 power-law scaling in the distribution of interburst intervals. Further, this distribution was only seldom significantly different to the one obtained in surrogate data, where nonlinear structure was destroyed. In conclusion, the work presented in this thesis suggests that advances in dynamical systems theory in conjunction with developments in magnetoencephalography may facilitate a mapping between levels of description int he brain. this may potentially represent a major advancement in neuroscience.

Analysis of magnetoencephaoographic data as a nonlinear dynamical system

This thesis presents the results from an investigation into the merits of analysing Magnetoencephalographic (MEG) data in the context of dynamical systems theory. MEG is the study of both the methods for the measurement of minute magnetic flux variations at the scalp, resulting from neuro-electric activity in the neocortex, as well as the techniques required to process and extract useful information from these measurements. As a result of its unique mode of action - by directly measuring neuronal activity via the resulting magnetic field fluctuations - MEG possesses a number of useful qualities which could potentially make it a powerful addition to any brain researcher's arsenal. Unfortunately, MEG research has so far failed to fulfil its early promise, being hindered in its progress by a variety of factors. Conventionally, the analysis of MEG has been dominated by the search for activity in certain spectral bands - the so-called alpha, delta, beta, etc that are commonly referred to in both academic and lay publications. Other efforts have centred upon generating optimal fits of "equivalent current dipoles" that best explain the observed field distribution. Many of these approaches carry the implicit assumption that the dynamics which result in the observed time series are linear. This is despite a variety of reasons which suggest that nonlinearity might be present in MEG recordings. By using methods that allow for nonlinear dynamics, the research described in this thesis avoids these restrictive linearity assumptions. A crucial concept underpinning this project is the belief that MEG recordings are mere observations of the evolution of the true underlying state, which is unobservable and is assumed to reflect some abstract brain cognitive state. Further, we maintain that it is unreasonable to expect these processes to be adequately described in the traditional way: as a linear sum of a large number of frequency generators. One of the main objectives of this thesis will be to prove that much more effective and powerful analysis of MEG can be achieved if one were to assume the presence of both linear and nonlinear characteristics from the outset. Our position is that the combined action of a relatively small number of these generators, coupled with external and dynamic noise sources, is more than sufficient to account for the complexity observed in the MEG recordings. Another problem that has plagued MEG researchers is the extremely low signal to noise ratios that are obtained. As the magnetic flux variations resulting from actual cortical processes can be extremely minute, the measuring devices used in MEG are, necessarily, extremely sensitive. The unfortunate side-effect of this is that even commonplace phenomena such as the earth's geomagnetic field can easily swamp signals of interest. This problem is commonly addressed by averaging over a large number of recordings. However, this has a number of notable drawbacks. In particular, it is difficult to synchronise high frequency activity which might be of interest, and often these signals will be cancelled out by the averaging process. Other problems that have been encountered are high costs and low portability of state-of-the- art multichannel machines. The result of this is that the use of MEG has, hitherto, been restricted to large institutions which are able to afford the high costs associated with the procurement and maintenance of these machines. In this project, we seek to address these issues by working almost exclusively with single channel, unaveraged MEG data. We demonstrate the applicability of a variety of methods originating from the fields of signal processing, dynamical systems, information theory and neural networks, to the analysis of MEG data. It is noteworthy that while modern signal processing tools such as independent component analysis, topographic maps and latent variable modelling have enjoyed extensive success in a variety of research areas from financial time series modelling to the analysis of sun spot activity, their use in MEG analysis has thus far been extremely limited. It is hoped that this work will help to remedy this oversight.

Spatial scaling in human peripheral vision

The observation that performance in many visual tasks can be made independent of eccentricity by increasing the size of peripheral stimuli according to the cortical magnification factor has dominated studies of peripheral vision for many years. However, it has become evident that the cortical magnification factor cannot be successfully applied to all tasks. To find out why, several tasks were studied using spatial scaling, a method which requires no pre-determined scaling factors (such as those predicted from cortical magnification) to magnify the stimulus at any eccentricity. Instead, thresholds are measured at the fovea and in the periphery using a series of stimuli, all of which are simply magnified versions of one another. Analysis of the data obtained in this way reveals the value of the parameter E2, the eccentricity at which foveal stimulus size must double in order to maintain performance equivalent to that at the fovea. The tasks investigated include hyperacuities (vernier acuity, bisection acuity, spatial interval discrimination, referenced displacement detection, and orientation discrimination), unreferenced instantaneous and gradual movement, flicker sensitivity, and face discrimination. In all cases tasks obeyed the principle of spatial scaling since performance in the periphery could be equated to that at the fovea by appropriate magnification. However, E2 values found for different spatial tasks varied over a 200-fold range. In spatial tasks (e.g. bisection acuity and spatial interval discrimination) E2 values were low, reaching about 0.075 deg, whereas in movement tasks the values could be as high as 16 deg. Using a method of spatial scaling it has been possible to equate foveal and peripheral perfonnance in many diverse visual tasks. The rate at which peripheral stimulus size had to be increased as a function of eccentricity was dependent upon the stimulus conditions and the task itself. Possible reasons for these findings are discussed.

Scaling the Advantages of Intra-channel Nonlinearity Compensation in Future Flexible Optical Networks

We report that, contrary to common perception, intra-channel nonlinearity compensation offers significant improvements of up to 4dB, in nonlinear tolerance (Q-factor), in a flexible traffic scenario, and further improvements with increasing local link dispersion, for an optical transport network employing flexible 28Gbaud PM-mQAM transponders.

Scaling up question-answering to linked data

Linked Data semantic sources, in particular DBpedia, can be used to answer many user queries. PowerAqua is an open multi-ontology Question Answering (QA) system for the Semantic Web (SW). However, the emergence of Linked Data, characterized by its openness, heterogeneity and scale, introduces a new dimension to the Semantic Web scenario, in which exploiting the relevant information to extract answers for Natural Language (NL) user queries is a major challenge. In this paper we discuss the issues and lessons learned from our experience of integrating PowerAqua as a front-end for DBpedia and a subset of Linked Data sources. As such, we go one step beyond the state of the art on end-users interfaces for Linked Data by introducing mapping and fusion techniques needed to translate a user query by means of multiple sources. Our first informal experiments probe whether, in fact, it is feasible to obtain answers to user queries by composing information across semantic sources and Linked Data, even in its current form, where the strength of Linked Data is more a by-product of its size than its quality. We believe our experiences can be extrapolated to a variety of end-user applications that wish to scale, open up, exploit and re-use what possibly is the greatest wealth of data about everything in the history of Artificial Intelligence. © 2010 Springer-Verlag.

Dynamical embedding and feature extraction of electroencephalographic data

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