Research on characteristics of noise-perturbed M–J sets based on equipotential point algorithm

AbstractAs the classical ones among the fractal sets, Julia set (abbreviated as J set) and Mandelbrot set (abbreviated as M set) have been explored widely in recent years. In this study, J set and M set under additive noise perturbation and multiplicative noise perturbation are created by equipotential point algorithm. Changes of the J set and M set under random noise perturbation as well as the close correlation between them are studied. Experimental results show that either additive noise perturbation or multiplicative noise perturbation may cause dramatic changes on J set. On the other hand, when the M set is perturbed by additive noise, it almost changes nothing but its position; when the M set is perturbed by multiplicative noise, its inner structures change with the stabilized areas shrinking, but it keeps the symmetry with respect to X axis. In addition, the J set and the M set still share the same stabilized periodic point in spite of noise perturbation

Methods and Measures for Analyzing Complex Street Networks and Urban Form

Complex systems have been widely studied by social and natural scientists in terms of their dynamics and their structure. Scholars of cities and urban planning have incorporated complexity theories from qualitative and quantitative perspectives. From a structural standpoint, the urban form may be characterized by the morphological complexity of its circulation networks - particularly their density, resilience, centrality, and connectedness. This dissertation unpacks theories of nonlinearity and complex systems, then develops a framework for assessing the complexity of urban form and street networks. It introduces a new tool, OSMnx, to collect street network and other urban form data for anywhere in the world, then analyze and visualize them. Finally, it presents a large empirical study of 27,000 street networks, examining their metric and topological complexity relevant to urban design, transportation research, and the human experience of the built environment.Comment: PhD thesis (2017), City and Regional Planning, UC Berkele

Use of wavelet-packet transforms to develop an engineering model for multifractal characterization of mutation dynamics in pathological and nonpathological gene sequences

This study uses dynamical analysis to examine in a quantitative fashion the information coding mechanism in DNA sequences. This exceeds the simple dichotomy of either modeling the mechanism by comparing DNA sequence walks as Fractal Brownian Motion (fbm) processes. The 2-D mappings of the DNA sequences for this research are from Iterated Function System (IFS) (Also known as the Chaos Game Representation (CGR)) mappings of the DNA sequences. This technique converts a 1-D sequence into a 2-D representation that preserves subsequence structure and provides a visual representation. The second step of this analysis involves the application of Wavelet Packet Transforms, a recently developed technique from the field of signal processing. A multi-fractal model is built by using wavelet transforms to estimate the Hurst exponent, H. The Hurst exponent is a non-parametric measurement of the dynamism of a system. This procedure is used to evaluate gene-coding events in the DNA sequence of cystic fibrosis mutations. The H exponent is calculated for various mutation sites in this gene. The results of this study indicate the presence of anti-persistent, random walks and persistent sub-periods in the sequence. This indicates the hypothesis of a multi-fractal model of DNA information encoding warrants further consideration.;This work examines the model\u27s behavior in both pathological (mutations) and non-pathological (healthy) base pair sequences of the cystic fibrosis gene. These mutations both natural and synthetic were introduced by computer manipulation of the original base pair text files. The results show that disease severity and system information dynamics correlate. These results have implications for genetic engineering as well as in mathematical biology. They suggest that there is scope for more multi-fractal models to be developed

Essays on the nonlinear and nonstochastic nature of stock market data

The nature and structure of stock-market price dynamics is an area of ongoing and rigourous scientific debate. For almost three decades, most emphasis has been given on upholding the concepts of Market Efficiency and rational investment behaviour. Such an approach has favoured the development of numerous linear and nonlinear models mainly of stochastic foundations. Advances in mathematics have shown that nonlinear deterministic processes i.e. "chaos" can produce sequences that appear random to linear statistical techniques. Till recently, investment finance has been a science based on linearity and stochasticity. Hence it is important that studies of Market Efficiency include investigations of chaotic determinism and power laws. As far as chaos is concerned, there are rather mixed or inconclusive research results, prone with controversy. This inconclusiveness is attributed to two things: the nature of stock market time series, which are highly volatile and contaminated with a substantial amount of noise of largely unknown structure, and the lack of appropriate robust statistical testing procedures. In order to overcome such difficulties, within this thesis it is shown empirically and for the first time how one can combine novel techniques from recent chaotic and signal analysis literature, under a univariate time series analysis framework. Three basic methodologies are investigated: Recurrence analysis, Surrogate Data and Wavelet transforms. Recurrence Analysis is used to reveal qualitative and quantitative evidence of nonlinearity and nonstochasticity for a number of stock markets. It is then demonstrated how Surrogate Data, under a statistical hypothesis testing framework, can be simulated to provide similar evidence. Finally, it is shown how wavelet transforms can be applied in order to reveal various salient features of the market data and provide a platform for nonparametric regression and denoising. The results indicate that without the invocation of any parametric model-based assumptions, one can easily deduce that there is more to linearity and stochastic randomness in the data. Moreover, substantial evidence of recurrent patterns and aperiodicities is discovered which can be attributed to chaotic dynamics. These results are therefore very consistent with existing research indicating some types of nonlinear dependence in financial data. Concluding, the value of this thesis lies in its contribution to the overall evidence on Market Efficiency and chaotic determinism in financial markets. The main implication here is that the theory of equilibrium pricing in financial markets may need reconsideration in order to accommodate for the structures revealed

Looking for a physical basis of rainfall multifractality

The study of rainfall arises from the necessity for knowing large and short-term climatic dynamics, as well as their affectations in the context of engineering practices. This research focus on the study of tropical rainfall and it was guided toward the conceptual exploration of the physical mechanism that explains how the multifractal scaling properties emerges in the rainfall field. On the basis of space-time rainfall records and model outputs analysis, it was possible to collect evidence that confirm rainfall multifractality exists and such a statistical property can be also identified in physically-based model outputs. The conceptual exploration that was developed in this research based on either classic--physics conservation principles or modern theories related to the study of the well-known critical phenomena. Among the findings, multifractality is understood as an essential reflection of the atmospheric instability by convection processes. Either instabilities or their resulting multifractality are sub-products of a diffusive mechanism which takes effect in the atmosphere. Under particular conditions of the dynamical system representing the convection processes, diffusion-driven instabilities give rise to the concentration of spatial structures in the rainfall field, and the organization of such structures is described by multifractality. Although open questions remain about the physics of rainfall multifractality, this work sets up a path for building a general theory and to promote innovative engineering design tools.El estudio de la precipitación responde a la necesidad inherente por conocer las dinámicas climáticas de corto y largo plazo, como también sus afectaciones en el contexto de las prácticas de ingeniería. La presente investigación se delimitó al estudio de la precipitación tropical y se orientó a la exploración conceptual del mecanismo físico que explica la emergencia de las propiedades de escalamiento multifractal del campo de precipitación. Partiendo del análisis de registros espacio-temporales de precipitación y de patrones simulados por computador se agruparon evidencias que ratifican la existencia de la multifractalidad en la precipitación y que tal propiedad estadística puede también ser identificada en modelo de base física. La exploración conceptual realizada en la investigación se apoyó en los principios de conservación provenientes de la física clásica y en las teorías modernas que han dado lugar a lo que hoy en día es conocido como fenómenos críticos. Entre los hallazgos encontrados, se concibe la multifractalidad como una manifestación inherente de la inestabilidad atmosférica por procesos de convección. Las inestabilidades y consecuentemente la multifractalidad son subproductos inducidos por un mecanismo difusivo en la atmósfera terrestre. Bajo condiciones especiales del sistema dinámico asociado a los procesos de convección, las inestabilidades inducidas por difusión dan lugar a la concentración de estructuras espaciales en el campo de precipitación y la organización de estas estructuras se describen a través de la multifractalidad. Aún cuando se mantienen algunas preguntas abiertas sobre la física de la multifractalidad en la precipitación, esta investigación establece una ruta para la consolidación de una teoría general y el desarrollo de nuevas herramientas de diseño en el marco de la ingeniería..Departamento Administrativo de Ciencia, Tecnología e Innovación (COLCIENCIAS)Crédito Educativo Condonable - Programa Nacional de Formación de InvestigadoresDoctor en Ingeniería - Recursos HidráulicosDoctorad

Predicting catastrophes: the role of criticality

Is prediction feasible in systems at criticality? While conventional scale-invariant arguments suggest a negative answer, evidence from simulation of driven-dissipative systems and real systems such as ruptures in material and crashes in the financial market have suggested otherwise. In this dissertation, I address the question of predictability at criticality by investigating two non-equilibrium systems: a driven-dissipative system called the OFC model which is used to describe earthquakes and damage spreading in the Ising model. Both systems display a phase transition at the critical point. By using machine learning, I show that in the OFC model, scaling events are indistinguishable from one another and only the large, non-scaling events are distinguishable from the small, scaling events. I also show that as the critical point is approached, predictability falls. For damage spreading in the Ising model, the opposite behavior is seen: the accuracy of predicting whether damage will spread or heal increases as the critical point is approached. I will also use machine learning to understand what are the useful precursors to the prediction problem

A chaos theory and nonlinear dynamics approach to the analysis of financial series : a comparative study of Athens and London stock markets

This dissertation presents an effort to implement nonlinear dynamic tools adapted from chaos theory in financial applications. Chaos theory might be useful in explaining the dynamics of financial markets, since chaotic models are capable of exhibiting behaviour similar to that observed in empirical financial data. In this context, the scope of this research is to provide an insight into the role that nonlinearities and, in particular, chaos theory may play in explaining the dynamics of financial markets. From a theoretical point of view, the basic features of chaos theory, as well as, the rationales for bringing chaos theory to the attention of financial researchers are discussed. Empirically, the fundamental issue of determining whether chaos can be observed in financial time series is addressed. Regarding the latter, empirical literature has been controversial. A quite exhaustive analysis of the existing literature is provided, revealing the inadequacies in terms of methodology and the testing framework adopted, so far. A new "multiple testing" methodology is developed combining methods and techniques from the fields of both Natural Sciences and the Economics, most of which have not been applied to financial data before. A serious effort has been made to fill, as much as possible, the gap which results from the lack of a proper statistical framework for the chaotic methods. To achieve this the bootstrap methodology is adopted. The empirical part of this work focuses on the comparison of two markets with different levels of maturity; the Athens Stock Exchange (ASE), an emerging market, and London Stock Exchange (LSE). Our aim is to determine whether structural differences exist in these markets in terms of chaotic dynamics. In the empirical level we find nonlinearities in both markets by the use of the BDS test. R/S analysis reveals fractality and long term memory for the ASE series only. Chaotic methods, such as the correlation dimension (and related methods and techniques) and the largest Lyapunov exponent estimation, cannot rule out a chaotic explanation for the ASE market, but no such indication could be found for the LSE market. Noise filtering by the SVD method does not alter these findings. Alternative techniques based on nonlinear nearest neighbour forecasting methods, such as the "piecewise polynomial approximation" and the "simplex" methods, support our aforementioned conclusion concerning the ASE series. In all, our results suggest that, although nonlinearities are present, chaos is not a widespread phenomenon in financial markets and it is more likely to exist in less developed markets such as the ASE. Even then, chaos is strongly mixed with noise and the existence of low-dimensional chaos is highly unlikely. Finally, short-term forecasts trying to exploit the dependencies found in both markets seem to be of no economic importance after accounting for transaction costs, a result which supports further our conclusions about the limited scope and practical implications of chaos in Finance