Learning on sequential data with evolution equations

Data which have a sequential structure are ubiquitous in many scientific domains such as physical sciences or mathematical finance. This motivates an important research effort in developing statistical and machine learning models for sequential data. Recently, the signature map, rooted in the theory of controlled differential equations, has emerged as a principled and systematic way to encode sequences into finite-dimensional vector representations. The signature kernel provides an interface with kernel methods which are recognized as a powerful class of algorithms for learning on structured data. Furthermore, the signature underpins the theory of neural controlled differential equations, neural networks which can handle sequential inputs, and more specifically the case of irregularly sampled time-series. This thesis is at the intersection of these three research areas and addresses key modelling and computational challenges for learning on sequential data. We make use of the well-established theory of reproducing kernels and the rich mathematical properties of the signature to derive an approximate inference scheme for Gaussian processes, Bayesian kernel methods, for learning with large datasets of multi-channel sequential data. Then, we construct new basis functions and kernel functions for regression problems where the inputs are sets of sequences instead of a single sequence. Finally, we use the modelling paradigm of stochastic partial differential equations to design a neural network architecture for learning functional relationships between spatio-temporal signals. The role of differential equations of evolutionary type is central in this thesis as they are used to model the relationship between independent and dependent signals, and provide tractable algorithms for kernel methods on sequential data

Financial predictions using intelligent systems

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.This thesis presents a collection of practical techniques for analysing various market properties in order to design advanced self-evolving trading systems based on neural networks combined with a genetic algorithm optimisation approach. Nonlinear multivariate statistical models have gained increasing importance in financial time series analysis, as it is very hard to fmd statistically significant market inefficiencies using standard linear modes. Nonlinear models capture more of the underlying dynamics of these high dimensional noisy systems than traditional models, whilst at the same time making fewer restrictive assumptions about them. These adaptive trading systems can extract information about associated time varying processes that may not be readily captured by traditional models. In order to characterise the fmancial time series in terms of its dynamic nature, this research employs various methods such as fractal analysis, chaos theory and dynamical recurrence analysis. These techniques are used for evaluating whether markets are stochastic and deterministic or nonlinear and chaotic, and to discover regularities that are completely hidden in these time series and not detectable using conventional analysis. Particular emphasis is placed on examining the feasibility of prediction in fmancial time series and the analysis of extreme market events. The market's fractal structure and log-periodic oscillations, typical of periods before extreme events occur, are revealed through recurrence plots. Recurrence qualification analysis indicated a strong presence of structure, recurrence and determinism in the fmancial time series studied. Crucial fmancial time series transition periods were also detected. This research performs several tests on a large number of US and European stocks using methodologies inspired by both fundamental analysis and technical trading rules. Results from the tests show that profitable trading models utilising advanced nonlinear trading systems can be created after accounting for realistic transaction costs. The return achieved by applying the trading model to a portfolio of real price series differs significantly from that achieved by applying it to a randomly generated price series. In some cases, these models are compared against simpler alternative approaches to ensure that there is an added value in the use of these more complex models. The superior performance of multivariate nonlinear models is also demonstrated. The long-short trading strategies performed well in both bull and bear markets, as well as in a sideways market, showing a great degree of flexibility and adjustability to changing market conditions. Empirical evidence shows that information is not instantly incorporated into market pnces and supports the claim that the fmancial time series studied, for the periods analysed, are not entirely random. This research clearly shows that equity markets are partially inefficient and do not behave along lines dictated by the efficient market hypothesis

Connecting Machine Learning to Causal Structure Learning with the Jacobian Matrix

In this thesis, a novel approach is proposed to connect machine learning to causal structure learning with the Jacobian matrix of neural networks w.r.t. input variables. Causal learning distinguishing causes and effects is the way human understanding and modeling the world. In the machine learning era, it also ensures that the model is more interpretable and sufficiently robust. Due to the enormous cost of the traditional intervention and randomized experimental methods, studies of causal learning have focused on passive observational data which can generally be divided into static data and time-series data. For different data types and different levels of causal modeling, different machine learning techniques are applied to do causal modeling and the causal structure can be read out by the Jacobian matrix. We focus on three aspects in this thesis. Firstly, a novel framework of neural networks to causal structure learning on static data under structural causal models assumptions is proposed and the results of various experiments show our method has achieved state-of-the-art performance. Secondly, we extend static data causal modeling to the highest level as the physical system which is usually in terms of ordinary differential equations. Lastly, our Jacobianbased causal modeling framework is applied to time series data with the ORE-RNN technique and the results show that the success of temporal causal structure learning in time series cases.Thesis (MPhil) -- University of Adelaide, School of Computer Science, 202

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

Financial predictions using intelligent systems : the application of advanced technologies for trading financial markets

This thesis presents a collection of practical techniques for analysing various market properties in order to design advanced self-evolving trading systems based on neural networks combined with a genetic algorithm optimisation approach. Nonlinear multivariate statistical models have gained increasing importance in financial time series analysis, as it is very hard to fmd statistically significant market inefficiencies using standard linear modes. Nonlinear models capture more of the underlying dynamics of these high dimensional noisy systems than traditional models, whilst at the same time making fewer restrictive assumptions about them. These adaptive trading systems can extract information about associated time varying processes that may not be readily captured by traditional models. In order to characterise the fmancial time series in terms of its dynamic nature, this research employs various methods such as fractal analysis, chaos theory and dynamical recurrence analysis. These techniques are used for evaluating whether markets are stochastic and deterministic or nonlinear and chaotic, and to discover regularities that are completely hidden in these time series and not detectable using conventional analysis. Particular emphasis is placed on examining the feasibility of prediction in fmancial time series and the analysis of extreme market events. The market's fractal structure and log-periodic oscillations, typical of periods before extreme events occur, are revealed through recurrence plots. Recurrence qualification analysis indicated a strong presence of structure, recurrence and determinism in the fmancial time series studied. Crucial fmancial time series transition periods were also detected. This research performs several tests on a large number of US and European stocks using methodologies inspired by both fundamental analysis and technical trading rules. Results from the tests show that profitable trading models utilising advanced nonlinear trading systems can be created after accounting for realistic transaction costs. The return achieved by applying the trading model to a portfolio of real price series differs significantly from that achieved by applying it to a randomly generated price series. In some cases, these models are compared against simpler alternative approaches to ensure that there is an added value in the use of these more complex models. The superior performance of multivariate nonlinear models is also demonstrated. The long-short trading strategies performed well in both bull and bear markets, as well as in a sideways market, showing a great degree of flexibility and adjustability to changing market conditions. Empirical evidence shows that information is not instantly incorporated into market pnces and supports the claim that the fmancial time series studied, for the periods analysed, are not entirely random. This research clearly shows that equity markets are partially inefficient and do not behave along lines dictated by the efficient market hypothesis.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

SIMULATING SEISMIC WAVE PROPAGATION IN TWO-DIMENSIONAL MEDIA USING DISCONTINUOUS SPECTRAL ELEMENT METHODS

We introduce a discontinuous spectral element method for simulating seismic wave in 2- dimensional elastic media. The methods combine the flexibility of a discontinuous finite element method with the accuracy of a spectral method. The elastodynamic equations are discretized using high-degree of Lagrange interpolants and integration over an element is accomplished based upon the Gauss-Lobatto-Legendre integration rule. This combination of discretization and integration results in a diagonal mass matrix and the use of discontinuous finite element method makes the calculation can be done locally in each element. Thus, the algorithm is simplified drastically. We validated the results of one-dimensional problem by comparing them with finite-difference time-domain method and exact solution. The comparisons show excellent agreement