Deterministic Annealing: A Variant of Simulated Annealing and its Application to Fuzzy Clustering

Deterministic annealing (DA) is a deterministic variant of simulated annealing. In this chapter, after briefly introducing DA, we explain how DA is combined with the fuzzy c-means (FCM) clustering by employing the entropy maximization method, especially the Tsallis entropy maximization. The Tsallis entropy is a q parameter extension of the Shannon entropy. Then, we focus on Tsallis-entropy-maximized FCM (Tsallis-DAFCM), and examine effects of cooling functions for DA on accuracy and convergence. A shape of a membership function of Tsallis-DAFCM depends on both a system temperature and q. Accordingly, a relationship between the temperature and q is quantitatively investigated

Analysis of signals related to the generation process of extreme events: towards a unified approach

This thesis was submitted for the award of Doctor of Philosophy and was awarded by Brunel University London.In the last decades, although the scientific community has attempted to explain a series of complex phenomena, ranging from natural hazards to physical conditions and economic crises, aspects of their generation process still escape our full understanding. The present thesis intends to promote our understanding of the spatiotemporal behavior and the generation mechanisms that govern large and strong earthquakes, employing a broad multidisciplinary perspective for the interpretation of catastrophic events. Two main questions are debated. The first question concentrates on “whether the generation process of an extreme event has more than one facets prior to its final appearance”. In the scientific study of earthquakes, attention is drawn to the predictive capability and monitoring of different precursory observations. Among them preseismic electromagnetic emissions have been also observed indicating that the science of earthquake prediction should be from the start multidisciplinary. Drawing on recently introduced models for earthquake dynamics, that address issues such as long-range correlations, self-affinity, complexity-organization and fractal structures, the present work endeavors to further penetrate on the analysis of preseismic electromagnetic emissions and elucidate their link with the generation process of large and strong earthquakes. A second question deals with “whether there is a unified approach for the study of catastrophic events”. This question implies the possibility for common statistical behavior of diverse extreme events and the potential for transferability of methods from the study of earthquake dynamics across other fields. On these grounds, the present work extends the focus of inquiry to the analysis of electroencephalogram recordings related to epileptic seizures, in the prospect to identify common mechanisms that may explain the nature and the generation process of both phenomena, and to open up different directions for future research. Finally, with a view to consider alternative ways of studying key theoretical principles associated with the generation process of catastrophic phenomena, a relevant framework based on proposed algorithms is presented, focusing on parameters such as: the energy of earthquakes, the mean and maximum magnitude of the sample, the probability that two samples may come from the same population. Such an attempt aims to contribute to the knowledge of natural phenomena, by extending the existing theory and models and providing a few more ways for their interpretation.Greek State Scholarships Foundation (IKY

Robust Inference via Lq-Likelihood

Robust inference is of great importance in modern statistics. In this dissertation, we introduce a series of robust statistical procedures based on the concept of Lq-likelihood. The Lq-likelihood function partially preserves the desired properties of the log-likelihood function. Moreover, it provides remarkable robustness, on which we can develop robust statistical procedures. The tuning parameter q of the Lq-likelihood makes our robust statistical procedures more flexible; because when q tends to 1, the Lq-likelihood reduces to the traditional log-likelihood. Therefore, we can use q to adjust the efficiency-robustness trade off as well as the bias-variance trade off. In this dissertation, we first introduce a new robust estimator called maximum Lq-likelihood estimate (MLqE) and derive its properties from a robust statistics point of view. We also develop a robust testing procedure --- the Lq-likelihood ratio test (LqLR) --- and demonstrate its effectiveness on contaminated data. We further move to the problem of robust estimation of mixture models and propose an expectation maximization algorithm for Lq-likelihood (EM-Lq). Finally, we develop a robust clustering technique and provide an application of our technique to brain graph data

A statistical mechanical model of economics

Statistical mechanics pursues low-dimensional descriptions of systems with a very large number of degrees of freedom. I explore this theme in two contexts. The main body of this dissertation explores and extends the Yard Sale Model (YSM) of economic transactions using a combination of simulations and theory. The YSM is a simple interacting model for wealth distributions which has the potential to explain the empirical observation of Pareto distributions of wealth. I develop the link between wealth condensation and the breakdown of ergodicity due to nonlinear diffusion effects which are analogous to the geometric random walk. Using this, I develop a deterministic effective theory of wealth transfer in the YSM that is useful for explaining many quantitative results. I introduce various forms of growth to the model, paying attention to the effect of growth on wealth condensation, inequality, and ergodicity. Arithmetic growth is found to partially break condensation, and geometric growth is found to completely break condensation. Further generalizations of geometric growth with growth in- equality show that the system is divided into two phases by a tipping point in the inequality parameter. The tipping point marks the line between systems which are ergodic and systems which exhibit wealth condensation. I explore generalizations of the YSM transaction scheme to arbitrary betting functions to develop notions of universality in YSM-like models. I find that wealth condensation is universal to a large class of models which can be divided into two phases. The first exhibits slow, power-law condensation dynamics, and the second exhibits fast, finite-time condensation dynamics. I find that the YSM, which exhibits exponential dynamics, is the critical, self-similar model which marks the dividing line between the two phases. The final chapter develops a low-dimensional approach to materials microstructure quantification. Modern materials design harnesses complex microstructure effects to develop high-performance materials, but general microstructure quantification is an unsolved problem. Motivated by statistical physics, I envision microstructure as a low-dimensional manifold, and construct this manifold by leveraging multiple machine learning approaches including transfer learning, dimensionality reduction, and computer vision breakthroughs with convolutional neural networks

Dynamical Systems

Complex systems are pervasive in many areas of science integrated in our daily lives. Examples include financial markets, highway transportation networks, telecommunication networks, world and country economies, social networks, immunological systems, living organisms, computational systems and electrical and mechanical structures. Complex systems are often composed of a large number of interconnected and interacting entities, exhibiting much richer global scale dynamics than the properties and behavior of individual entities. Complex systems are studied in many areas of natural sciences, social sciences, engineering and mathematical sciences. This special issue therefore intends to contribute towards the dissemination of the multifaceted concepts in accepted use by the scientific community. We hope readers enjoy this pertinent selection of papers which represents relevant examples of the state of the art in present day research. [...

Image similarity in medical images

Recent experiments have indicated a strong influence of the substrate grain orientation on the self-ordering in anodic porous alumina. Anodic porous alumina with straight pore channels grown in a stable, self-ordered manner is formed on (001) oriented Al grain, while disordered porous pattern is formed on (101) oriented Al grain with tilted pore channels growing in an unstable manner. In this work, numerical simulation of the pore growth process is carried out to understand this phenomenon. The rate-determining step of the oxide growth is assumed to be the Cabrera-Mott barrier at the oxide/electrolyte (o/e) interface, while the substrate is assumed to determine the ratio β between the ionization and oxidation reactions at the metal/oxide (m/o) interface. By numerically solving the electric field inside a growing porous alumina during anodization, the migration rates of the ions and hence the evolution of the o/e and m/o interfaces are computed. The simulated results show that pore growth is more stable when β is higher. A higher β corresponds to more Al ionized and migrating away from the m/o interface rather than being oxidized, and hence a higher retained O:Al ratio in the oxide. Experimentally measured oxygen content in the self-ordered porous alumina on (001) Al is indeed found to be about 3% higher than that in the disordered alumina on (101) Al, in agreement with the theoretical prediction. The results, therefore, suggest that ionization on (001) Al substrate is relatively easier than on (101) Al, and this leads to the more stable growth of the pore channels on (001) Al