http://bura.brunel.ac.uk/handle/2438/8629 2026-06-25T09:28:26Z 2026-06-25T09:28:26Z Zhang, Chuqiao http://bura.brunel.ac.uk/handle/2438/33509 2026-06-25T09:28:01Z 2025-01-01T00:00:00Z

Title: Learning random geometric graphs, and their applications Authors: Zhang, Chuqiao Abstract: In the Bayesian world, referring to a variable as “random” means that this vari-able – a scalar, tensor, a network, etc. – takes a value with a probability, such that it attains a values that live in the support of this probabilistic distribution. lies in a given interval, with uncertainties. We want to make inference on the unknown parameters of a model, using the modelled probability distribution of each such unknown, given the data. In my thesis, I discuss the motivation for undertaking inference via sampling unknowns given the data, and introduce a learning of real-isations of a random graph variable. This random graph is a Random Geometric Graph (RGG) that is drawn in a probabilistic metric space. Further learning of this graph variable has also been explored in the thesis, to identify the optimal cut-off value that is imposed on the posterior probability of any edge given the (multivari-ate) dataset at hand. Such an optimal cut-off then corresponds to graph realisations that produces the most robust - to changes in the cut-off values - graph. Theoretical illustrations of the learning of this graph and the applications of such graph learning are presented using real-world data, towards: (1) scoring of severity of a disease, by computing the distance between the posterior of the learnt random graph variables, given the time series data on the physiological parameters of two patients as they suffer from the disease; (2) a method for the learning of the individualised recovery trajectory of patients who are enrolled on a physical rehabilitation programme, with the aim of regaining their lost mobility; (3) a new way of recognising critical residues of an example protein, using static and dynamic nodal degree distributions of ran-dom graphs learnt using molecular dynamical simulations of the protein. Based on my PhD research, future work is discussed. Description: This thesis was submitted for the award of Doctor of Philosophy and was awarded by Brunel University London

2025-01-01T00:00:00Z Jiao, Dongchen http://bura.brunel.ac.uk/handle/2438/33226 2026-04-28T12:15:11Z 2025-01-01T00:00:00Z

Title: Mixed topics on geometry of varieties of Fano type Authors: Jiao, Dongchen Abstract: In this thesis, we investigate the deformation properties of Fano threefolds and the birational ge-ometry of foliations. First, we try to find compactification of several families of Fano threefolds. Then we give a description of the connections between foliated minimal models. Finally, we will discuss geometric properties of Fano foliations. This thesis contains results of (1), (26), (19) and some recent independent work. Description: This thesis was submitted for the award of Doctor of Philosophy and was awarded by Brunel University London

2025-01-01T00:00:00Z Penco, Vittorio http://bura.brunel.ac.uk/handle/2438/32341 2025-11-14T19:37:02Z 2025-01-01T00:00:00Z

Title: Application of factor models to risk premium estimation Authors: Penco, Vittorio Abstract: We provide a rigorous mathematical approach to the beta-pricing model, starting from the standard two-step cross-sectional regression, through Nonlinear Seemingly Unrelated Regression (NSUR) and Generalized Method of Moments (GMM), and finally compare the results with several linear approximation methods. The use of the linear approximation applied to a single-factor nonlinear system of equations is new in the literature and is one of the major contributions of this work. Our results show that, in the presence of heavy-tailed distributions, the L1-norm methods proposed in this study are more appropriate (exhibiting lower bias and variance) for risk price estimation than traditional L2-norm approaches. It is also the first time that the Capital Asset Pricing Model (CAPM) is applied systematically to compare the integration and segmentation between different markets and a given portfolio set. Our study, Penco and Lucas (2024), applies a two-factor integration model to the economies of Asia, Europe, Japan and North America, showing integration between the European and North American economies. We also extend the integration model to commodity markets. To capture more accurately the cross-sectional pricing of commodity risk we use the Cochrane factor mimicking approach and compare the results with alternative dependence-based integration measures using copulas. We show how the copula correlation between the Stochastic Discount Factor (SDF) and returns differentiates the contribution of joint dependence from the contribution of the risk prices. Finally, we introduce a penalised p-value Fama-MacBeth Generalized Least Squares (GLS) regularisation, which provides several advantages over other methods as it ensures that retained factors contribute not only to statistical fit but also to risk pricing. Unlike other approaches, this method regularises the pricing kernel directly. Factors that lack significance or explanatory power are penalised and removed, while priced and relevant sources of risk are preserved. Description: This thesis was submitted for the award of Doctor of Philosophy and was awarded by Brunel University London

2025-01-01T00:00:00Z Grublys, Mykolas http://bura.brunel.ac.uk/handle/2438/31718 2025-08-15T13:42:27Z 2024-01-01T00:00:00Z

Title: Echo state networks in forecasting chaotic dynamics and emergent universalities Authors: Grublys, Mykolas Abstract: Deep learning has become a popular way to accurately recognise, classify, generate and forecast data in various complex scenarios. It is commonly based on artificial neural networks, a type of machine learning architecture inspired by the human brain, that consist of multiple layers of neurons with non-linear activations. Their inside workings can be considered as a ‘black box’ because it is difficult, if not impossible, to truly understand “how” or “why” such networks work. Furthermore, the training relies on gradient decent methods which are inherently difficult and computationally expensive. Reservoir Computing has recently emerged as a new paradigm aimed to alleviate such difficulties. In this thesis, we study its particular realisation, known as Echo State Networks (ESNs), which show promise in many tasks, particularly in forecasting the dynamics of chaotic systems. The thesis will provide a detailed discussion of the ESN architecture, hyperparameters, and implementation. We introduce a new performance metric that highlights the networks maintaining small errors for the longest duration and use it in GridSearch to optimise ESN hyperparameters. We study the statistical properties of the correlation matrix that is formed during training, offering a novel approach not applied to ESNs or other types of recurrent networks. Our extensive numerical studies reveal universal results, consistent across tests and different chaotic training signals. Although analytical studies are limited, we introduce a simple ‘toy model’ to qualitatively describe some properties. We conclude our work with study of the statistics of trained output weights, which also exhibit universal characteristics. These universalities provide deeper insights into the inner workings and behaviour of ESNs, enhancing our understanding of how information spreads throughout the network. Description: This thesis was submitted for the award of Doctor of Philosophy and was awarded by Brunel University London

2024-01-01T00:00:00Z