Recurrence-based time series analysis by means of complex network methods

Complex networks are an important paradigm of modern complex systems sciences which allows quantitatively assessing the structural properties of systems composed of different interacting entities. During the last years, intensive efforts have been spent on applying network-based concepts also for the analysis of dynamically relevant higher-order statistical properties of time series. Notably, many corresponding approaches are closely related with the concept of recurrence in phase space. In this paper, we review recent methodological advances in time series analysis based on complex networks, with a special emphasis on methods founded on recurrence plots. The potentials and limitations of the individual methods are discussed and illustrated for paradigmatic examples of dynamical systems as well as for real-world time series. Complex network measures are shown to provide information about structural features of dynamical systems that are complementary to those characterized by other methods of time series analysis and, hence, substantially enrich the knowledge gathered from other existing (linear as well as nonlinear) approaches.Comment: To be published in International Journal of Bifurcation and Chaos (2011

Reconstruction and Parameter Estimation of Dynamical Systems using Neural Networks

Dynamical systems can be loosely regarded as systems whose dynamics is entirely determined by en evolution function and an initial condition, being therefore completely deterministic and a priori predictable. Nevertheless, their phenomenology is surprisingly rich, including intriguing phenomena such as chaotic dynamics, fractal dimensions and entropy production. In Climate Science for example, the emergence of chaos forbids us to have meteorological forecasts going beyond fourteen days in the future in the current epoch and therefore building predictive systems that overcome this limitation, at least partially, are of the extreme importance since we live in fast-changing climate world, as proven by the recent not-so-extreme-anymore climate phenomena. At the same time, Machine Learning techniques have been widely applied to practically every field of human knowledge starting from approximately ten years ago, when essentially two factors contributed to the so-called rebirth of Deep Learning: the availability of larger datasets, putting us in the era of Big Data, and the improvement of computational power. However, the possibility to apply Neural Networks to chaotic systems have been widely debated, since these models are very data hungry and rely thus on the availability of large datasets, whereas often Climate data are rare and sparse. Moreover, chaotic dynamics should not rely much on past statistics, which these models are built on. In this thesis, we explore the possibility to study dynamical systems, seen as simple proxies of Climate models, by using Neural Networks, possibly adding prior knowledge on the underlying physical processes in the spirit of Physics Informed Neural Networks, aiming to the reconstruction of the Weather (short term dynamics) and Climate (long term dynamics) of these dynamical systems as well as the estimation of unknown parameters from Data.Dynamical systems can be loosely regarded as systems whose dynamics is entirely determined by en evolution function and an initial condition, being therefore completely deterministic and a priori predictable. Nevertheless, their phenomenology is surprisingly rich, including intriguing phenomena such as chaotic dynamics, fractal dimensions and entropy production. In Climate Science for example, the emergence of chaos forbids us to have meteorological forecasts going beyond fourteen days in the future in the current epoch and therefore building predictive systems that overcome this limitation, at least partially, are of the extreme importance since we live in fast-changing climate world, as proven by the recent not-so-extreme-anymore climate phenomena. At the same time, Machine Learning techniques have been widely applied to practically every field of human knowledge starting from approximately ten years ago, when essentially two factors contributed to the so-called rebirth of Deep Learning: the availability of larger datasets, putting us in the era of Big Data, and the improvement of computational power. However, the possibility to apply Neural Networks to chaotic systems have been widely debated, since these models are very data hungry and rely thus on the availability of large datasets, whereas often Climate data are rare and sparse. Moreover, chaotic dynamics should not rely much on past statistics, which these models are built on. In this thesis, we explore the possibility to study dynamical systems, seen as simple proxies of Climate models, by using Neural Networks, possibly adding prior knowledge on the underlying physical processes in the spirit of Physics Informed Neural Networks, aiming to the reconstruction of the Weather (short term dynamics) and Climate (long term dynamics) of these dynamical systems as well as the estimation of unknown parameters from Data

Complexity, Emergent Systems and Complex Biological Systems:\ud Complex Systems Theory and Biodynamics. [Edited book by I.C. Baianu, with listed contributors (2011)]

An overview is presented of System dynamics, the study of the behaviour of complex systems, Dynamical system in mathematics Dynamic programming in computer science and control theory, Complex systems biology, Neurodynamics and Psychodynamics.\u

Chaotic price dynamics of agricultural commodities

Traditionally, commodity prices have been analyzed and modeled in the context of linear generating processes. The purpose of this dissertation is to address the adequacy of this work through examination of the critical assumption of independence in the residual process of linearly specified models. As an alternative, a test procedure is developed and utilized to demonstrate the appropriateness of applying generalized conditional heteroscedastic time series models (GARCH) to agricultural commodity prices. In addition, a distinction is made between testing for independence and testing for chaos in commodity prices. The price series of interest derive from the major international agricultural commodity markets, sampled monthly over the period 1960--1994. The results of the present analysis suggest that for bananas, beef, coffee, soybeans, wool and wheat seasonally adjusted growth rates, ARCH-GARCH models account for some of the non-linear dependence in these commodity price series. As an alternative to the ARCH-GARCH models, several neural network models were estimated and in some cases outperformed the ARCH family of models in terms of forecast ability. This further demonstrated the nonlinearity present in these time series. Although, further examination is needed, all prices were found to be non-linearly dependent. It was determined by use of different statistical measures for testing for deterministic chaos that wheat prices may be an example of such behavior. Therefore, their may be something to be gained in terms of short-run forecast accuracy by using semi-parametric modeling approaches as applied to wheat prices

Entropy in Dynamic Systems

In order to measure and quantify the complex behavior of real-world systems, either novel mathematical approaches or modifications of classical ones are required to precisely predict, monitor, and control complicated chaotic and stochastic processes. Though the term of entropy comes from Greek and emphasizes its analogy to energy, today, it has wandered to different branches of pure and applied sciences and is understood in a rather rough way, with emphasis placed on the transition from regular to chaotic states, stochastic and deterministic disorder, and uniform and non-uniform distribution or decay of diversity. This collection of papers addresses the notion of entropy in a very broad sense. The presented manuscripts follow from different branches of mathematical/physical sciences, natural/social sciences, and engineering-oriented sciences with emphasis placed on the complexity of dynamical systems. Topics like timing chaos and spatiotemporal chaos, bifurcation, synchronization and anti-synchronization, stability, lumped mass and continuous mechanical systems modeling, novel nonlinear phenomena, and resonances are discussed

Detection of dynamical regime transitions with lacunarity as a multiscale recurrence quantification measure

We propose lacunarity as a novel recurrence quantification measure and illustrate its efficacy to detect dynamical regime transitions which are exhibited by many complex real-world systems. We carry out a recurrence plot-based analysis for different paradigmatic systems and nonlinear empirical data in order to demonstrate the ability of our method to detect dynamical transitions ranging across different temporal scales. It succeeds to distinguish states of varying dynamical complexity in the presence of noise and non-stationarity, even when the time series is of short length. In contrast to traditional recurrence quantifiers, no specification of minimal line lengths is required and geometric features beyond linear structures in the recurrence plot can be accounted for. This makes lacunarity more broadly applicable as a recurrence quantification measure. Lacunarity is usually interpreted as a measure of heterogeneity or translational invariance of an arbitrary spatial pattern. In application to recurrence plots, it quantifies the degree of heterogeneity in the temporal recurrence patterns at all relevant time scales. We demonstrate the potential of the proposed method when applied to empirical data, namely time series of acoustic pressure fluctuations from a turbulent combustor. Recurrence lacunarity captures both the rich variability in dynamical complexity of acoustic pressure fluctuations and shifting time scales encoded in the recurrence plots. Furthermore, it contributes to a better distinction between stable operation and near blowout states of combustors