Forecasting high waters at Venice Lagoon using chaotic time series analisys and nonlinear neural netwoks

Time series analysis using nonlinear dynamics systems theory and multilayer neural networks models have been applied to the time sequence of water level data recorded every hour at 'Punta della Salute' from Venice Lagoon during the years 1980-1994. The first method is based on the reconstruction of the state space attractor using time delay embedding vectors and on the characterisation of invariant properties which define its dynamics. The results suggest the existence of a low dimensional chaotic attractor with a Lyapunov dimension, DL, of around 6.6 and a predictability between 8 and 13 hours ahead. Furthermore, once the attractor has been reconstructed it is possible to make predictions by mapping local-neighbourhood to local-neighbourhood in the reconstructed phase space. To compare the prediction results with another nonlinear method, two nonlinear autoregressive models (NAR) based on multilayer feedforward neural networks have been developed. From the study, it can be observed that nonlinear forecasting produces adequate results for the 'normal' dynamic behaviour of the water level of Venice Lagoon, outperforming linear algorithms, however, both methods fail to forecast the 'high water' phenomenon more than 2-3 hours ahead.Publicad

The Spatio-Temporal Structure of Spiral-Defect Chaos

We present a study of the recently discovered spatially-extended chaotic state known as spiral-defect chaos, which occurs in low-Prandtl-number, large-aspect-ratio Rayleigh-Benard convection. We employ the modulus squared of the space-time Fourier transform of time series of two-dimensional shadowgraph images to construct the structure factor ${S}({\vec k},\omega )$. This analysis is used to characterize the average spatial and temporal scales of the chaotic state. We find that the correlation length and time can be described by power-law dependences on the reduced Rayleigh number ${\epsilon}$. These power laws have as yet no theoretical explanation.Comment: RevTex 38 pages with 13 figures. Due to their large size, some figures are stored as separate gif images. The paper with included hi-res eps figures (981kb compressed, 3.5Mb uncompressed) is available at ftp://mobydick.physics.utoronto.ca/pub/MBCA96.tar.gz An mpeg movie and samples of data are also available at ftp://mobydick.physics.utoronto.ca/pub/. Paper submitted to Physica

Minimal perturbations approaching the edge of chaos in a Couette flow

This paper provides an investigation of the structure of the stable manifold of the lower branch steady state for the plane Couette flow. Minimal energy perturbations to the laminar state are computed, which approach within a prescribed tolerance the lower branch steady state in a finite time. For small times, such minimal-energy perturbations maintain at least one of the symmetries characterizing the lower branch state. For a sufficiently large time horizon, such symmetries are broken and the minimal-energy perturbations on the stable manifold are formed by localized asymmetrical vortical structures. These minimal-energy perturbations could be employed to develop a control procedure aiming at stabilizing the low-dissipation lower branch state

Forecasting high waters at Venice Lagoon using chaotic time series analysis and nonlinear neural networks

Time series analysis using nonlinear dynamics systems theory and multilayer neural networks models have been applied to the time sequence of water level data recorded every hour at 'Punta della Salute' from Venice Lagoon during the years 1980–1994. The first method is based on the reconstruction of the state space attractor using time delay embedding vectors and on the characterisation of invariant properties which define its dynamics. The results suggest the existence of a low dimensional chaotic attractor with a Lyapunov dimension, DL, of around 6.6 and a predictability between 8 and 13 hours ahead. Furthermore, once the attractor has been reconstructed it is possible to make predictions by mapping local-neighbourhood to local-neighbourhood in the reconstructed phase space. To compare the prediction results with another nonlinear method, two nonlinear autoregressive models (NAR) based on multilayer feedforward neural networks have been developed. From the study, it can be observed that nonlinear forecasting produces adequate results for the 'normal' dynamic behaviour of the water level of Venice Lagoon, outperforming linear algorithms, however, both methods fail to forecast the 'high water' phenomenon more than 2–3 hours ahead

On the predictability of time series by metric entropy

Thesis (Master)--Izmir Institute of Technology, Mechanical Engineering, Izmir, 2006Includes bibliographical references (leaves: 48-49)Text in English; Abstract: Turkish and Englishxi, 55 leavesThe computation of the metric entropy, a measure of the loss of information along the attractor, from experimental time series is the main objective of this study. In this study, replacing the current warning systems (simple threshold based, on/off circuits), a new and promising prognosis system is tried to be achieved by the metric entropy, i.e. Kolmogorov . Sinai entropy, from chaotic time series. Additional to metric entropy, correlation dimension and time series statistical parameters were investigated.Condition monitoring of ball bearings and drill bits was achieved in the light of practical considerations of time series applications. Two different accelerated bearing run-to-failure test rigs were constructed and the prediction tests were performed.However, as a reason of shaft failure in both structures during the experiments, none of them is completed. Finally, drill bit breakage experiments were carried out. In the experiments, 10 small drill bits (1 mm ) were tested until they broke down, while vibration data were consecutively taken in equal time intervals. After the analysis, a consistent decrement in variation of metric entropy just before the breakage was observed. As a result of the experiment results, metric entropy variation could be proposed as an early warning system

Experiments with a Malkus-Lorenz water wheel: Chaos and Synchronization

We describe a simple experimental implementation of the Malkus-Lorenz water wheel. We demonstrate that both chaotic and periodic behavior is found as wheel parameters are changed in agreement with predictions from the Lorenz model. We furthermore show that when the measured angular velocity of our water wheel is used as an input signal to a computer model implementing the Lorenz equations, high quality chaos synchronization of the model and the water wheel is achieved. This indicates that the Lorenz equations provide a good description of the water wheel dynamics.Comment: 12 pages, 7 figures. The following article has been accepted by the American Journal of Physics. After it is published, it will be found at http://scitation.aip.org/ajp

Dislocated negative feedback control with partial replacement between chaotic Lorenz systems

In order to obtain asymptotical synchronization, we combine negative feedback control and dislocated negative feedback control with partial replacement to the nonlinear terms of the response system, a coupling version that was less explored. All these unidirectional coupling schemes are applied between Lorenz systems where we consider some values for the control parameters that lead to chaotic behavior. The sufficient conditions for global stable synchronization are obtained from a different approach of the Lyapunov direct method for the transversal system. In one of the coupling we apply a result based on the classification of the symmetric matrix AT +A as negative definite,where A is characterizing the transversal system. In the other couplings presented here, the sufficient conditions are based on the derivative increase of an appropriate Lyapunov function. In fact, the effectiveness of the coupling between systems with equal dimension follows from the analysis of the synchronization error, e(t), and, if the system variables can be bounded by positive constants, then the derivative of an appropriate Lyapunov function can be increased as required by the Lyapunov direct method

Prediction and control of nonlinear dynamical systems using machine learning

Künstliche Intelligenz und Machine Learning erfreuen sich in Folge der rapide gestiegenen Rechenleistung immer größerer Popularität. Sei es autonomes Fahren, Gesichtserkennung, bildgebende Diagnostik in der Medizin oder Robotik – die Anwendungsvielfalt scheint keine Grenzen zu kennen. Um jedoch systematischen Bias und irreführende Ergebnisse zu vermeiden, ist ein tiefes Verständnis der Methoden und ihrer Sensitivitäten von Nöten. Anhand der Vorhersage chaotischer Systeme mit Reservoir Computing – einem künstlichen rekurrenten neuronalem Netzwerk – wird im Rahmen dieser Dissertation beleuchtet, wie sich verschiedene Eigenschaften des Netzwerks auf die Vorhersagekraft und Robustheit auswirken. Es wird gezeigt, wie sich die Variabilität der Vorhersagen – sowohl was die exakte zukünftige Trajektorie betrifft als auch das statistische Langzeitverhalten (das "Klima") des Systems – mit geeigneter Parameterwahl signifikant reduzieren lässt. Die Nichtlinearität der Aktivierungsfunktion spielt hierbei eine besondere Rolle, weshalb ein Skalierungsparameter eingeführt wird, um diese zu kontrollieren. Des Weiteren werden differenzielle Eigenschaften des Netzwerkes untersucht und gezeigt, wie ein kontrolliertes Entfernen der "richtigen" Knoten im Netzwerk zu besseren Vorhersagen führt und die Größe des Netzwerkes stark reduziert werden kann bei gleichzeitig nur moderater Verschlechterung der Ergebnisse. Dies ist für Hardware Realisierungen von Reservoir Computing wie zum Beispiel Neuromorphic Computing relevant, wo möglichst kleine Netzwerke von Vorteil sind. Zusätzlich werden unterschiedliche Netzwerktopologien wie Small World Netzwerke und skalenfreie Netzwerke beleuchtet. Mit den daraus gewonnenen Erkenntnissen für bessere Vorhersagen von nichtlinearen dynamischen Systemen wird eine neue Kontrollmethode entworfen, die es ermöglicht, dynamische Systeme flexibel in verschiedenste Zielzustände zu lenken. Hierfür wird – anders als bei vielen bisherigen Ansätzen – keine Kenntnis der zugrundeliegenden Gleichungen des Systems erfordert. Ebenso wird nur eine begrenzte Datenmenge verlangt, um Reservoir Computing hinreichend zu trainieren. Zudem ist es nicht nur möglich, chaotisches Verhalten in einen periodischen Zustand zu zwingen, sondern auch eine Kontrolle auf komplexere Zielzustände wie intermittentes Verhalten oder ein spezifischer anderer chaotischer Zustand. Dies ermöglicht eine Vielzahl neuer potenzieller realer Anwendungen, von personalisierten Herzschrittmachern bis hin zu Kontrollvorrichtungen für Raketentriebwerke zur Unterbindung von kritischen Verbrennungsinstabilitäten. Als Schritt zur Weiterentwicklung von Reservoir Computing zu einem verbesserten hybriden System, das nicht nur rein datenbasiert arbeitet, sondern auch physikalische Zusammenhänge berücksichtigt, wird ein Ansatz vorgestellt, um lineare und nichtlinearen Kausalitätsstrukturen zu separieren. Dies kann verwendet werden, um Systemgleichungen oder Restriktionen für ein hybrides System zur Vorhersage oder Kontrolle abzuleiten.Artificial intelligence and machine learning are becoming increasingly popular as a result of the rapid increase in computing power. Be it autonomous driving, facial recognition, medical imaging diagnostics or robotics – the variety of applications seems to have no limits. However, to avoid systematic bias and misleading results, a deep understanding of the methods and their sensitivities is needed. Based on the prediction of chaotic systems with reservoir computing – an artificial recurrent neural network – this dissertation sheds light on how different properties of the network affect the predictive power and robustness. It is shown how the variability of the predictions – both in terms of the exact short-term predictions and the long-term statistical behaviour (the "climate") of the system – can be significantly reduced with appropriate parameter choices. The nonlinearity of the activation function plays a special role here, thus a scaling parameter is introduced to control it. Furthermore, differential properties of the network are investigated and it is shown how a controlled removal of the right nodes in the network leads to better predictions, whereas the size of the network can be greatly reduced while only moderately degrading the results. This is relevant for hardware realizations of reservoir computing such as neuromorphic computing, where networks that are as small as possible are advantageous. Additionally, different network topologies such as small world networks and scale-free networks are investigated. With the insights gained for better predictions of nonlinear dynamical systems, a new control method is designed that allows dynamical systems to be flexibly forced into a wide variety of dynamical target states. For this – unlike many previous approaches – no knowledge of the underlying equations of the system is required. Further, only a limited amount of data is needed to sufficiently train reservoir computing. Moreover, it is possible not only to force chaotic behavior to a periodic state, but also to control for more complex target states such as intermittent behavior or a specific different chaotic state. This enables a variety of new potential real-world applications, from personalized cardiac pacemakers to control devices for rocket engines to suppress critical combustion instabilities. As a step toward advancing reservoir computing to an improved hybrid system that is not only purely data-based but also takes into account physical relationships, an approach is presented to separate linear and nonlinear causality structures. This can be used to derive system equations or constraints for a hybrid prediction or control system