This comprehensive text gives an interesting and useful blend of the mathematical, probabilistic and statistical tools used in heavy-tail analysis. It is uniquely devoted to heavy-tails and emphasizes both probability modeling and statistical methods for fitting models. Prerequisites for the reader include a prior course in stochastic processes and probability, some statistical background, some familiarity with time series analysis, and ability to use a statistics package. This work will serve second-year graduate students and researchers in the areas of applied mathematics, statistics, operations research, electrical engineering, and economics.

An accessible yet rigorous package of probabilistic and statistical tools for anyone who must understand or model extreme events.

This book provides a comprehensive theory of mono- and multi-fractal traffic, including the basics of long-range dependent time series and 1/f noise, ergodicity and predictability of traffic, traffic modeling and simulation, stationarity tests of traffic, traffic measurement and the anomaly detection of traffic in communications networks. Proving that mono-fractal LRD time series is ergodic, the book exhibits that LRD traffic is stationary. The author shows that the stationarity of multi-fractal traffic relies on observation time scales, and proposes multi-fractional generalized Cauchy processes and modified multi-fractional Gaussian noise. The book also establishes a set of guidelines for determining the record length of traffic in measurement. Moreover, it presents an approach of traffic simulation, as well as the anomaly detection of traffic under distributed-denial-of service attacks. Scholars and graduates studying network traffic in computer science will find the book beneficial.

The goal of this thesis is to treat the temporal tail dependence and the cross-sectional tail dependence of heavy tailed functional time series. Functional time series are aimed at modelling spatio-temporal phenomena; for instance rain, temperature, pollution on a given geographical area, with temporally dependent observations. Heavy tails mean that the series can exhibit much higher spikes than with Gaussian distributions for instance. In such cases, second moments cannot be assumed to exist, violating the basic assumption in standard functional data analysis based on the sequence of autocovariance operators. As for random variables, regular variation provides the mathematical backbone for a coherent theory of extreme values. The main tools introduced in this thesis for a regularly varying functional time series are its tail process and its spectral process. These objects capture all the aspects of the probability distribution of extreme values jointly over time and space. The development of the tail and spectral process for heavy tailed functional time series is followed by three theoretical applications. The first application is a characterization of a variety of indices and objects describing the extremal behavior of the series: the extremal index, tail dependence coefficients, the extremogram and the point process of extremes. The second is the computation of an explicit expression of the tail and spectral processes for heavy tailed linear functional time series. The third and final application is the introduction and the study of a model for the spatio-temporal dependence for functional time series called maxima of moving maxima of continuous functions (CM3 processes), with the development of an estimation method.

This book constitutes the refereed proceedings of the 7th International Conference on Principles and Practice of Constraint Programming, CP 2001, held in Paphos, Cyprus, in November/December 2001. The 37 revised full papers, 9 innovative applications presentations, and 14 short papers presented were carefully reviewed and selected from a total of 135 submissions. All current issues in constraint processing are addressed, ranging from theoretical and foundational issues to advanced and innovative applications in a variety of fields.

by Peter J. Roussopoulos, Director, Southern Research Station The world and its ecosystems are repeatedly punctuated by natural disturbances, and human societies must learn to manage this reality Often severe and unp- dictable, dynamic natural forces disrupt human welfare and alter the structure and composition of natural systems Over the past century, land management ag- cies within the United States have relied on science to improve the sustainable management of natural resources Forest economics research can help advance this scientifc basis by integrating knowledge of forest disturbance processes with their economic causes and consequences As the twenty-frst century unfolds, people increasingly seek the goods and services provided by forest ecosystems, not only for wood supply, clean water, and leisure pursuits, but also to establish residential communities that are removed from the hustle and bustle of urban life As vividly demonstrated during the past few years, Santa Ana winds can blow wildfres down from the mountains of California, incinerating homes as readily as vegetation in the canyons below Hurricanes can fatten large swaths of forest land, while associated foods create havoc for urban and rural residents alike Less dramatic, but more insidious, trees and forest stands are succumbing to exotic insects and diseases, causing economic losses to private property values (including timber) as well as scenic and recreation values As human demands on public and private forests expand, science-based solutions need to be identifed so that social needs can be balanced with the vagaries of forest disturbance processes

No leading university department of mathematics or statistics, or library, can afford to be without this unique text. Leading authorities give a unique insight into a wide range of currently topical problems, from the mathematics of road networks to the genomics of cancer.

Multi-Chaos, Fractal and Multi-Fractional Artificial Intelligence of Different Complex Systems addresses different uncertain processes inherent in the complex systems, attempting to provide global and robust optimized solutions distinctively through multifarious methods, technical analyses, modeling, optimization processes, numerical simulations, case studies as well as applications including theoretical aspects of complexity. Foregrounding Multi-chaos, Fractal and Multi-fractional in the era of Artificial Intelligence (AI), the edited book deals with multi- chaos, fractal, multifractional, fractional calculus, fractional operators, quantum, wavelet, entropy-based applications, artificial intelligence, mathematics-informed and data driven processes aside from the means of modelling, and simulations for the solution of multifaceted problems characterized by nonlinearity, non-regularity and self-similarity, frequently encountered in different complex systems. The fundamental interacting components underlying complexity, complexity thinking, processes and theory along with computational processes and technologies, with machine learning as the core component of AI demonstrate the enabling of complex data to augment some critical human skills. Appealing to an interdisciplinary network of scientists and researchers to disseminate the theory and application in medicine, neurology, mathematics, physics, biology, chemistry, information theory, engineering, computer science, social sciences and other far-reaching domains, the overarching aim is to empower out-of-the-box thinking through multifarious methods, directed towards paradoxical situations, uncertain processes, chaotic, transient and nonlinear dynamics of complex systems. - Constructs and presents a multifarious approach for critical decision-making processes embodying paradoxes and uncertainty. - Includes a combination of theory and applications with regard to multi-chaos, fractal and multi-fractional as well as AI of different complex systems and many-body systems. - Provides readers with a bridge between application of advanced computational mathematical methods and AI based on comprehensive analyses and broad theories.

Fractional processes are widely found in science, technology and engineering systems. In Fractional Processes and Fractional-order Signal Processing, some complex random signals, characterized by the presence of a heavy-tailed distribution or non-negligible dependence between distant observations (local and long memory), are introduced and examined from the ‘fractional’ perspective using simulation, fractional-order modeling and filtering and realization of fractional-order systems. These fractional-order signal processing (FOSP) techniques are based on fractional calculus, the fractional Fourier transform and fractional lower-order moments. Fractional Processes and Fractional-order Signal Processing: presents fractional processes of fixed, variable and distributed order studied as the output of fractional-order differential systems; introduces FOSP techniques and the fractional signals and fractional systems point of view; details real-world-application examples of FOSP techniques to demonstrate their utility; and provides important background material on Mittag–Leffler functions, the use of numerical inverse Laplace transform algorithms and supporting MATLAB® codes together with a helpful survey of relevant webpages. Readers will be able to use the techniques presented to re-examine their signals and signal-processing methods. This text offers an extended toolbox for complex signals from diverse fields in science and engineering. It will give academic researchers and practitioners a novel insight into the complex random signals characterized by fractional properties, and some powerful tools to analyze those signals.