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The Editors are members of the research group "Data Analysis and Modeling of Dynamic Processes in the Life Sciences" at the Center for Data Analysis and Modeling of the University of Freiburg, Germany. In interdisciplinary projects, they develop, investigate and apply mathematical methods to elucidate properties of complex systems in the field of neurology based on multivariate time series.

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This handbook provides an up-to-date survey of current research topics and applications of time series analysis methods written by leading experts in their fields. It covers recent developments in linear and non-linear, as well as stochastic and deterministic analysis methods and their cross-fertilization between Mathematics, Physics, and Engineering. Each chapter comprises both methodological aspects and applications to time series from real world complex systems, such as the human brain or Earth's climate. Covering an exceptionally broad spectrum of topics, beginners, experts and practitioners in time series analysis who seek to understand the latest developments will profit from this handbook.

Foto von links nach rechts: M. Winterhalder, J. Timmer, B. Schelter

The Editors are working in the research group "Data Analysis and Modeling of Dynamic Processes in the Life Sciences" at the Center for Data Analysis and Modeling of the University of Freiburg, Germany. In interdisciplinary projects, they develop, investigate and apply mathematical methods to elucidate properties of complex systems in the field of neurology based on multivariate time series.

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This handbook provides an up-to-date survey of current research topics and applications of time series analysis methods written by leading experts in their fields. It covers recent developments in linear and non-linear, as well as stochastic and deterministic analysis methods and their cross-fertilization between Mathematics, Physics, and Engineering. Each chapter comprises both methodological aspects and applications to time series from real world complex systems, such as the human brain or Earth's climate. Covering an exceptionally broad spectrum of topics, beginners, experts and practitioners in time series analysis who seek to understand the latest developments will profit from this handbook.
 
Foto von links nach rechts: M. Winterhalder, J. Timmer, B. Schelter
 
The Editors are working in the research group "Data Analysis and Modeling of Dynamic Processes in the Life Sciences" at the Center for Data Analysis and Modeling of the University of Freiburg, Germany. In interdisciplinary projects, they develop, investigate and apply mathematical methods to elucidate properties of complex systems in the field of neurology based on multivariate time series.

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Handbook of Time Series Analysis

John Wiley & Sons

Chapter One

Bjrn Schelter, M. Winterhalder, and J. Timmer

Mathematics, Physics, and Engineering are very successful in understanding phenomena of the natural world and building technology upon this based on the first principle modeling. However, for complex systems like those appearing in the fields of biology and medicine, this approach is not feasible and an understanding of the behavior can only be based upon the analysis of the measured data of the dynamics, the so-called time series.

Time series analysis has different roots in Mathematics, Physics, and Engineering. The approaches differ by their basic assumptions. While in Mathematics linear stochastic systems were one of the centers of interest, in Physics nonlinear deterministic systems were investigated. While the different strains of the methodological developments and concepts evolved independently in different disciplines for many years, during the past decade, enhanced cross-fertilization between the different disciplines took place, for instance, by the development of methods for nonlinear stochastic systems.

This handbook written by leading experts in their fields provides an up-to-date survey of current research topics and applications of time series analysis. It covers univariate as well as bivariate and multivariate time series analysis techniques. The latter came into the focus of research when recording devices enabled more-dimensional simultaneous recordings. Even though bivariate analysis is basically multivariate analysis, there are some phenomena which can occur only in three or more dimensions, for instance, indirect interdependences between two processes.

The aim of this handbook is to present both theoretical concepts of various analysis techniques and the application of these techniques to real-world data. The applications cover a large variety of research areas ranging from electronic circuits to human electroencephalography. The interplay between challenges posed by empirical data and the possibilities offered by new analysis methods has been proven to be successful and stimulating.

In the first chapter, Henry D. I. Abarbanel and Ulrich Parlitz present different approaches to nonlinear systems. By means of a real-world example of a recording from a single neuron, they discuss how to analyze these data. Concepts such as the Lyapunov exponent, i.e., a measure for chaos, prediction, and modeling in nonlinear systems, are introduced with a critical focus on their limitations. Ready to apply procedures are given allowing an immediate application to one's own data.

Local modeling is being dealt with by David Engster and Ulrich Parlitz. Local models are amongst the most precise methods for time series prediction. This chapter describes the basic parameters of local modeling. To show the efficiency of this procedure, several artificial and real-world data, for instance experimental friction data sets, are predicted using local models. As an alternative to strict local modeling, cluster weighted modeling is also discussed using an expectation-maximization (EM) algorithm as a parameter optimization procedure.

Holger Kantz and Eckehard Olbrich present concepts, methods, and algorithms for predicting time series from the knowledge of the past. Thereby, they especially concentrate on nonlinear stochastic processes which have to be dealt with by probabilistic predictions. They calculate a certain prediction range in which future values are going to fall. They complete their chapter by discussing verification techniques for their forecasted values, which is very important when dealing with real-world data.

Noise and randomness in biological systems have often been treated as an unwelcome byproduct. Patrick Celka and co-workers identify different noise sources and their impact on dynamical systems. This contribution discusses the concept of randomness and how to best access the information one wants to retrieve. Different time series analysis techniques are presented. The applications govern speech enhancement, evoked potentials, cardiovascular system, and brain-machine interface.

The chapter of Ursula Gather and co-workers is dedicated to robust filtering procedures for signal extraction from noisy time series. The authors present various filter techniques with their specific properties and extensions in order to process noisy data or data contaminated with outliers. They point to the variety of different approaches and compare the advantages and disadvantages. By means of simulated data they demonstrate the different conceptual properties.

Dealing with bivariate time series analysis techniques, the chapter of Michael Rosenblum and co-workers is dedicated to the phenomenon of phase synchronization and the detection of coupling in nonlinear dynamical systems. The authors discuss the usage of model-based and nonmodel-based techniques and introduce novel ideas to detect weak interactions between two processes, together with the corresponding strength and direction of interactions. They illustrate their analysis techniques by application to data characterizing the cardiorespiratory interaction.

An approach to detect directional coupling between oscillatory systems from short time series based on empirical modeling of their phase dynamics is introduced by Dmitry Smirnov and Boris Petrovich Bezruchko. This time series analysis technique is utilized to analyze electroencephalography recordings with the purpose of epileptic focus localization and climatic data representing the dynamics of the North Atlantic Oscillation and El Nio/Southern Oscillation processes.

Phase synchronization analysis of brain signals, for instance intracranial electroencephalography data recorded from epilepsy patients, has come into the focus of neuroscience research. Mario Chavez and co-workers suggest a data-driven time series analysis technique to select the important contents in a signal with multiple frequencies, the empirical mode decomposition. They summarize this concept and demonstrate its applicability to model systems and apply it to the analysis of human epilepsy data.

For cases where the definition of the phase used by common approaches is impossible, Mamen Romano and co-workers present a way to detect and quantify phase synchronization using the concept of recurrences. Furthermore, to test for phase synchronization, an algorithm to generate surrogate time series based on recurrences is discussed. An application to fixational eye movement data complements the results for model systems.

Theoden I. Netoff and co-workers dedicated their work to infer coupling and interaction in weakly coupled systems, especially in the presence of noise and nonlinearity. To this end, they applied several analysis techniques to model data and to data obtained from an electronic circuit. They explored advantages and disadvantages of the methods in specific cases. The conclusion of their chapter is that nonlinear methods are more sensitive to detect coupling under ideal conditions. However, in the presence of noise, linear techniques are more robust.

Dealing with multivariate systems, the chapter of Manfred Deistler is dedicated to state space and autoregressive moving average models. He summarizes the basic ideas about state space models and autoregressive moving average models including...