Heidelberg, Universiẗat, Diss., 2004.

We consider the problem of predicting the direction of daily volatility changes in the Dow Jones Industrial Average (DJIA). This is accomplished by quantizing a series of historic volatility changes into a symbolic stream over 2 or 4 symbols. We compare predictive performance of the classical fixed-order Markov models with that of a novel approach to variable memory length prediction (called prediction fractal machine, or PFM) which is able to select very specific deep prediction contexts (whenever there is a sufficient support for such contexts in the training data). We learn that daily volatility changes of the DJIA only exhibit rather shallow finite memory structure. On the other hand, a careful selection of quantization cut values can strongly enhance predictive power of symbolic schemes. Results on 12 non-overlapping epochs of the DJIA strongly suggest that PFMs can outperform both traditional Markov models and (continuous-valued) GARCH models in the task of predicting volatility one time-step ahead. (author's abstract) / Series: Working Papers SFB "Adaptive Information Systems and Modelling in Economics and Management Science"

We study cellular automata as discrete dynamical systems and in particular investigate under which conditions two cellular automata are topologically conjugate. Based on work of McKinsey, Tarski, Pierce and Head we introduce derivative algebras to study the topological structure of sofic shifts in dimension one. This allows us to classify periodic cellular automata on sofic shifts up to topological conjugacy based on the structure of their periodic points. We also get new conjugacy invariants in the general case. Based on a construction by Hanf and Halmos, we construct a pair of non-homeomorphic subshifts whose disjoint sums with themselves are homeomorphic. From this we can construct two cellular automata on homeomorphic state spaces for which all points have minimal period two, which are, however, not topologically conjugate. We apply our methods to classify the 256 elementary cellular automata with radius one over the binary alphabet up to topological conjugacy. By means of linear algebra over the field with two elements and identities between Fibonacci-polynomials we show that every conjugacy between rule 90 and rule 150 cannot have only a finite number of local rules. Finally, we look at the sequences of finite dynamical systems obtained by restricting cellular automata to spatially periodic points. If these sequences are termwise conjugate, we call the cellular automata conjugate on all tori. We then study the invariants under this notion of isomorphism. By means of an appropriately defined entropy, we can show that surjectivity is such an invariant.

We study cellular automata as discrete dynamical systems and in particular investigate under which conditions two cellular automata are topologically conjugate. Based on work of McKinsey, Tarski, Pierce and Head we introduce derivative algebras to study the topological structure of sofic shifts in dimension one. This allows us to classify periodic cellular automata on sofic shifts up to topological conjugacy based on the structure of their periodic points. We also get new conjugacy invariants in the general case. Based on a construction by Hanf and Halmos, we construct a pair of non-homeomorphic subshifts whose disjoint sums with themselves are homeomorphic. From this we can construct two cellular automata on homeomorphic state spaces for which all points have minimal period two, which are, however, not topologically conjugate. We apply our methods to classify the 256 elementary cellular automata with radius one over the binary alphabet up to topological conjugacy. By means of linear algebra over the field with two elements and identities between Fibonacci-polynomials we show that every conjugacy between rule 90 and rule 150 cannot have only a finite number of local rules. Finally, we look at the sequences of finite dynamical systems obtained by restricting cellular automata to spatially periodic points. If these sequences are termwise conjugate, we call the cellular automata conjugate on all tori. We then study the invariants under this notion of isomorphism. By means of an appropriately defined entropy, we can show that surjectivity is such an invariant.

Die Untersuchung kausaler Zusammenhänge in komplexen dynamischen Systemen spielt in der Wissenschaft eine immer wichtigere Rolle. Ziel dieses aktuellen, interdisziplinären Forschungsbereiches ist ein grundlegendes, tiefes Verständnis der vorherrschenden Prozesse und deren Wechselwirkungen in solchen Systemen. Die Untersuchung von Zeitreihen mithilfe moderner Kopplungsanalysemethoden liefert dabei Möglichkeiten zur Modellierung der betreffenden Systeme und somit bessere Vorhersagemethoden und fortgeschrittene Interpretationsmöglichkeiten der Ergebnisse. In der vorliegenden Arbeit werden zunächst einige existierende Kopplungsmaße mit ihren jeweiligen Anwendungsgebieten vorgestellt. Eine Gemeinsamkeit dieser Maße liegt in der Voraussetzung stationärer Zeitreihen, um die Anwendbarkeit zu gewährleisten. Daher wird im Verlauf der Dissertation eine Möglichkeit zur Erweiterung solcher Maße vorgestellt, die eine Kopplungsanalyse mit einer sehr hohen Zeitauflösung und somit auch die Untersuchung nichtstationärer, transienter Ereignisse ermöglicht. Die Erweiterung basiert auf der Verwendung von Ensembles von Messreihen und der Schätzung der jeweiligen Maße über das Ensemble anstatt über die Zeit. Dies ermöglicht eine Zeitauflösung bei der Analyse in der Größenordnung der Abtastrate des ursprünglichen Signals, die nur von der Art der verwendeten Kopplungsmaße abhängt. Der Ensemble-Ansatz wird auf verschiedene Kopplungsmaße angewandt. Zunächst werden die Methoden ausführlich an verschiedenen theoretischen Modellen und unter verschiedenen Bedingungen getestet. Anschließend erfolgt eine zeitaufgelöste Kopplungsanalyse kardiovaskulärer Zeitreihen, die während transienter Ereignisse aufgenommen wurden. Die Ergebnisse dieser Analyse bestätigen zum einen aktuelle Studienresultate, liefern aber auch neue Erkenntnisse, die es in Zukunft ermöglichen können, Modelle des Herz-Kreislauf-Systems zu erweitern und zu verbessern. / The analysis of causal relationships in complex dynamic systems plays a more and more important role in various scientific fields. The aim of this current, interdisciplinary field of research is a fundamental, deep understanding of predominant processes and their interactions in such systems. The study of time series using modern coupling analysis tools allows the modelling of the respective systems and thus better prediction methods and advanced interpretation possibilities for the results. In this work, initially some existing coupling measures and their fields of application are introduced. One trait these measures have in common is the requirement of stationary time series to ensure their applicability. Therefore, in the course of this thesis a possibility to extend these measures is presented, which allows a coupling analysis with a high temporal resolution and thus also the analysis of transient, nonstationary events. The extension is based on the use of ensembles of time series and the calculation of the respective measures across these ensembles instead of across time. This allows for a temporal resolution of the same order of magnitude as the sampling rate in the original signal. The resolution only depends on the kind of coupling analysis method employed. The ensemble extension is applied to different coupling measures. To begin with, the regarded tools are tested on various theoretical models and under different conditions. This is followed by a coupling analysis of cardiovascular time series recorded during transient events. The results on the one hand confirm topical study outcomes and on the other hand deliver new insights, which will allow to extend and improve cardiovascular system models in the future.