Chaotic Time Series Research Papers (original) (raw)

The prediction of chaotic time series with neural networks is a traditional practical problem of dynamic systems. This paper is not intended for proposing a new model or a new methodology, but to study carefully and thoroughly several aspects of a model on which there are no enough communicated experimental data, as well as to derive conclusions that would be of interest. The recurrent neural networks (RNN) models are not only important for the forecasting of time series but also generally for the control of the dynamical system. A RNN with a sufficiently large number of neurons is a nonlinear autoregressive and moving average (NARMA) model, with "moving average" referring to the inputs. The prediction can be assimilated to identification of dynamic process. An architectural approach of RNN with embedded memory, "Nonlinear Autoregressive model process with eXogenous input" (NARX), showing promising qualities for dynamic system applications, is analyzed in this paper. The performances of the NARX model are verified for several types of chaotic or fractal time series applied as input for neural network, in relation with the number of neurons, the training algorithms and the dimensions of his embedded memory. In addition, this work has attempted to identify a way to use the classic statistical methodologies (R/S Rescaled Range analysis and Hurst exponent) to obtain new methods of improving the process efficiency of the prediction chaotic time series with NARX.

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- Time Series, Moving average, Chaotic Time Series, Dynamic System

Multi-Step ahead prediction of a chaotic time series is a difficult task that has attracted increasing interest in recent years. The interest in this work is the development of nonlinear neural network models for the purpose of building multistep ahead prediction of North and South hemisphere sunspots chaotic time series. In the literature there is a wide range of different approaches but their success depends on the predicting performance of the individual methods. Also the most popular neural models are based on the statistical and traditional feed forward neural networks. But it is seen that this kind of neural model may present some disadvantages when longterm prediction is required. In this paper focused time lagged recurrent neural network (FTLRNN) model with gamma memory is developed not only for short-term but also for long-term prediction which allows to obtain better predictions of northern and southern chaotic time series in future. The authors experimented the performance of this FTLRNN model on predicting the dynamic behavior of typical northern and southern sunspots chaotic time series. Static MLP model is also attempted and compared against the proposed model on the performance measures like mean squared error (MSE), Normalized mean squared error (NMSE) and Correlation Coefficient (r) .The standard back propagation algorithm with momentum term has been used for both the models. The various parameters like number of hidden layers, number of processing elements in the hidden layer, step size, the different learning rules, the various transfer functions like tanh, sigmoid, linear-tanh and linear sigmoid, different error norms L 1 ,L 2 (Euclidean), L 3, L 4 ,L 5 and L ∞ , and different combination of training and testing samples are exhaustively varied and experimented for obtaining the optimal values of performance measures. The obtained results indicates the superior performance of estimated dynamic FTLRNN based model with gamma memory over the static MLP NN in various performance metrics. In addition, the output of proposed FTLRNN neural network model with gamma memory closely follows the desired output for multi-step ahead prediction for all the chaotic time series considered in the study.

Artificial neural networks ͑ANN͒ are typically composed of a large number of nonlinear functions ͑neurons͒ each with several linear and nonlinear parameters that are fitted to data through a computationally intensive training process. Longer training results in a closer fit to the data, but excessive training will lead to overfitting. We propose an alternative scheme that has previously been described for radial basis functions ͑RBF͒. We show that fundamental differences between ANN and RBF make application of this scheme to ANN nontrivial. Under this scheme, the training process is replaced by an optimal fitting routine, and overfitting is avoided by controlling the number of neurons in the network. We show that for time series modeling and prediction, this procedure leads to small models ͑few neurons͒ that mimic the underlying dynamics of the system well and do not overfit the data. We apply this algorithm to several computational and real systems including chaotic differential equations, the annual sunspot count, and experimental data obtained from a chaotic laser. Our experiments indicate that the structural differences between ANN and RBF make ANN particularly well suited to modeling chaotic time series data.

- by Michael Small
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- Engineering, Algorithms, Time Series, Iterative Methods

DARWIN is an efficient evolutionary algorithm programmed to approximate the functional relation, in symbolic form, that describes the behaviour of a time series. The search procedure is based on Darwinian theories of natural selection and survival. An initial population of potential solutions is subjected to an evolutionary process described by selection, reproduction and mutation processes which are repeated over generations until an optimum individual is finally found. DARWIN is particularly useful when the dynamical model that creates the time series is nonlinear. The code, based on a previously proposed evolutionary algorithm [Phys. Rev. E 55 (1997) 2557-2568, is programmed in Fortran 77. Darwin is actually employed as a predictor in a satellite based ocean forecasting system. 

- by JOAQUIN TINTORE
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- Time Series, Natural Selection, Fortran, Mathematical Sciences

The NARX network is a recurrent neural architecture commonly used for input-output modeling of nonlinear systems. The input of the NARX network is formed by two tapped-delay lines, one sliding over the input signal and the other one over the output signal. Currently, when applied to chaotic time series prediction, the NARX architecture is designed as a plain Focused Time Delay Neural Network (FTDNN); thus, limiting its predictive abilities. In this paper, we propose a strategy that allows the original architecture of the NARX network to fully explore its computational power to improve prediction performance. We use the well-known chaotic laser time series to evaluate the proposed approach in multi-step-ahead prediction tasks. The results show that the proposed approach consistently outperforms standard neural network based predictors, such as the FTDNN and Elman architectures.

- by Guilherme Barreto
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- Time Series, Nonlinear Time Series, Neural Network, Recurrent Neural Network

This paper investigates the prediction of a Lorenz chaotic attractor having relatively high values of Lypunov's exponents. The characteristic of this time series is its rich chaotic behavior. For such dynamic reconstruction problem, regularized radial basis function (RBF) neural network (NN) models have been widely employed in the literature. However, author recommends using a two-layer multi-layer perceptron (MLP) NN-based recurrent model. When none of the available linear models have been able to learn the dynamics of this attractor, it is shown that the proposed NN-based auto regressive (AR) and auto regressive moving average (ARMA) models with regularization have not only learned the true trajectory of this attractor, but also performed much better in multi-step-ahead predictions. However, equivalent linear models seem to fail miserably in learning the dynamics of the time series, despite the low values of Akaike's final prediction error (FPE) estimate. Author proposes to employ the recurrent NN-based ARMA model with regularization which clearly outperforms all other models and thus, it is possible to obtain good results for prediction and reconstruction of the dynamics of the chaotic time series with NN-based models.

- by Dr Sanjay Dudul
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- Information Systems, Applied Mathematics, System Identification, Time Series

The architecture and learning procedure underlying ANFIS (adaptive-network-based fuzzy inference system) is presented, which is a fuzzy inference system implemented in the framework of adaptive networks. By using a hybrid learning procedure, the proposed ANFIS can construct an input-output mapping based on both human knowledge (in the form of fuzzy if-then rules) and stipulated input-output data pairs. In the simulation, the ANFIS architecture is employed to model nonlinear functions, identify nonlinear components on-line in a control system, and predict a chaotic time series, all yielding remarkable results. Comparisons with artificial neural networks and earlier work on fuzzy modeling are listed and discussed. Other extensions of the proposed ANFIS and promising applications to automatic control and signal processing are also suggested

- by Shing How
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- Artificial Intelligence, Architecture, Signal Processing, Fuzzy Logic

In the context of time series analysis, forecasting time series is known as an important sub-study field within the associated scientific fields. At this point, especially forecasting chaotic systems has been a remarkable research approach. As being associated with the works on forecasting chaotic systems, some application areas are very interested in benefiting from advantages of forecasting time series. For instance, forecasting electroencephalogram (EEG) time series enables researchers to learn more about future status of the brain activity in terms of any physical or pathological case. In this sense, this work introduces an ANFIS–VOA hybrid system, which is based on ANFIS and a new optimization algorithm called as vortex optimization algorithm (VOA). Generally, the system provides a basic, strong enough, alternative forecasting solution way for EEG time series. The performed evaluation applications have shown that the ANFIS–VOA approach here provided effective enough solution way for forecasting EEG time series, as a result of learning–reasoning infrastructure achieved by the combination of two different artificial intelligence techniques.

- by Ahmet Arslan and +1
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- Artificial Intelligence, Fuzzy Logic, Time Series, Chaos Theory

In this paper, we review our work on a time series forecasting methodology based on the combination of unsupervised clustering and artificial neural networks. To address noise and non-stationarity, a common approach is to combine a method for the partitioning of the input space into a number of subspaces with a local approximation scheme for each subspace. Unsupervised clustering algorithms have the desirable property of deciding on the number of partitions required to accurately segment the input space during the clustering process, thus relieving the user from making this ad hoc choice. Artificial neural networks, on the other hand, are powerful computational models that have proved their capabilities on numerous hard real-world problems. The time series that we consider are all daily spot foreign exchange rates of major currencies. The experimental results reported suggest that predictability varies across different regions of the input space, irrespective of clustering algorithm. In all cases, there are regions that are associated with a particularly high forecasting performance. Evaluating the performance of the proposed methodology with respect to its profit generating capability indicates that it compares favorably with that of two other established approaches. Moving from the task of one-step-ahead to multiple-step-ahead prediction, performance deteriorates rapidly.

- by Michael Vrahatis
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- Information Systems, Applied Mathematics, Time Series, Neural Network

The prediction accuracy and generalization ability of neural/neurofuzzy models for chaotic time series prediction highly depends on employed network model as well as learning algorithm. In this study, several neural and neurofuzzy models with different learning algorithms are examined for prediction of several benchmark chaotic systems and time series. The prediction performance of locally linear neurofuzzy models with recently developed Locally Linear Model Tree (LoLiMoT) learning algorithm is compared with that of Radial Basis Function (RBF) neural network with Orthogonal Least Squares (OLS) learning algorithm, MultiLayer Perceptron neural network with error back-propagation learning algorithm, and Adaptive Network based Fuzzy Inference System. Particularly, cross validation techniques based on the evaluation of error indices on multiple validation sets is utilized to optimize the number of neurons and to prevent over fitting in the incremental learning algorithms. To make a fair comparison between neural and neurofuzzy models, they are compared at their best structure based on their prediction accuracy, generalization, and computational complexity. The experiments are basically designed to analyze the generalization capability and accuracy of the learning techniques when dealing with limited number of training samples from deterministic chaotic time series, but the effect of noise on the performance of the techniques is also considered. Various chaotic systems and time series including Lorenz system, Mackey-Glass chaotic equation, Henon map, AE geomagnetic activity index, and sunspot numbers are examined as case studies. The obtained results indicate the superior performance of incremental learning algorithms and their respective networks, such as, OLS for RBF network and LoLiMoT for locally linear neurofuzzy model.

- by Babak Araabi
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- Cognitive Science, Computational Complexity, Time Series, Radial Basis Function

A method for resampling time series generated by a deterministic chaotic data generating process (DGP) is proposed. Given an observed time series, this method potentially allows one to obtain an arbitrary number of time series of arbitrary length which can be considered as a product of the same unknown DGP. The notion of shadowing and brittleness of the pseudo-orbit proves to be particularly useful in characterizing the conditions for a correct resampling. A simple practical application of the method is shown. [S0031-9007 03296-1] PACS numbers: 05.45.+b

- by Silvia Golia
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- Time Series, Physical sciences, Chaotic Time Series

Two players of Rock-Paper-Scissors are modeled as adaptive agents which use a reinforcement learning algorithm and exhibit chaotic behavior in terms of trajectories of probability in mixed strategies space. This paper demonstrates that an external super-agent can exploit the behavior of the other players to predict favorable moments to play against one of the other players the symbol suggested by a sub-optimal strategy. This third agent does not affect the learning process of the other two players, whose only goal is to beat each other. The choice of the best moment to play is based on a threshold associated with the Local Lyapunov Exponent or the Entropy, each computed by using the time series of symbols played by one of the other players. A method for automatically adapting such a threshold is presented and evaluated. The results show that these techniques can be used effectively by a super-agent in a game involving adaptive agents that exhibit collective chaotic behavior. #

- by Franco Salvetti
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- Information Systems, Applied Mathematics, Game Theory, Reinforcement Learning

In this paper the multi step ahead prediction of typical Duffing Chaotic time series and the monthly sunspots real time series are carried out. These two time series are popularized due to their highly chaotic behavior. This paper compares the performance of two neural network configurations namely a Multilayer Perceptron (MLP) and proposed FTLRNN with gamma memory for the duffing time series for 1, 5,10,20,50 and 100-step ahead prediction and for monthly sunspot time series for 1, 6, 12, 18 & 24 month ahead prediction. The standard back propagation algorithm with momentum term has been used for both the models. It is seen that estimated dynamic fully recurrent model clearly outperforms the MLP NN in various performance matrices such as Mean square error (MSE), Normalized mean square error (NMSE) and correlation coefficient (r) on testing as well as training data set for multi step prediction (K=1,5,10,20,50,100) for duffing time series and for the sunspot time series for 1, 6, 12, 18 &24 month ahead prediction. In addition, the output of proposed neural network model closely follows the desired output for all the step ahead prediction. It is observed that suggested recurrent models have the remarkable capability of time series prediction. The major contribution of this paper is that Various parameters like number of processing elements, step size, momentum value in hidden layer, in output layer the various transfer functions like tanh, sigmoid, linear-tan-h and linear sigmoid, different error norms L1, L2 ,Lp to L.

- by Sanjay Dudul
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- Time Series, Neural Network, Recurrent Neural Network, Mean square error

We propose a method that allows one to estimate the parameters of model scalar time-delay differential equations from time series. The method is based on a statistical analysis of time intervals between extrema in the time series. We verify our method by using it for the reconstruction of time-delay differential equations from their chaotic solutions and for modeling experimental systems with delay-induced dynamics from their chaotic time series.

- by Vladimir I.Ponomarenko
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- Engineering, Mathematical Sciences, Physical sciences, Chaotic Time Series

This study presents a novel application and comparison of higher order neural networks (HONNs) to forecast benchmark chaotic time series. Two models of HONNs were implemented, namely functional link neural network (FLNN) and pi-sigma neural network (PSNN). These models were tested on two benchmark time series; the monthly smoothed sunspot numbers and the Mackey-Glass time-delay differential equation time series. The forecasting performance of the HONNs is compared against the performance of different models previously used in the literature such as fuzzy and neural networks models. Simulation results showed that FLNN and PSNN offer good performance compared to many previously used hybrid models. Keywords— Chaotic time series; Sunspot time series; Mackey-Glass time series; higher order neural network; pi-sigma neural network; functional link neural network.

- by WADDAH WAHEEB and +1
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- Artificial Neural Networks, Time Series Forecasting, Chaotic Time Series, Higher order neural network

We construct complex networks from pseudoperiodic time series, with each cycle represented by a single node in the network. We investigate the statistical properties of these networks for various time series and find that time series with different dynamics exhibit distinct topological structures. Specifically, noisy periodic signals correspond to random networks, and chaotic time series generate networks that exhibit small world and scale free features. We show that this distinction in topological structure results from the hierarchy of unstable periodic orbits embedded in the chaotic attractor. Standard measures of structure in complex networks can therefore be applied to distinguish different dynamic regimes in time series. Application to human electrocardiograms shows that such statistical properties are able to differentiate between the sinus rhythm cardiograms of healthy volunteers and those of coronary care patients.

- by Michael Small
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- Time Series, Kinetics, Time series analysis, Signal Transduction

We study the dynamics of on-line learning for a large class of neural networks and learning rules, including backpropagation for multilayer perceptrons. In this paper, we focus on the case where successive examples are dependent, and we analyze how these dependencies affect the learning process. We define the representation error and the prediction error. The representation error measures how well the environment is represented by the network after learning. The prediction error is the average error that a continually learning network makes on the next example. In the neighborhood of a local minimum of the error surface, we calculate these errors. We find that the more predictable the example presentation, the higher the representation error, i.e, the less accurate the asymptotic representation of the whole environment. Furthermore we study the learning process in the presence of a plateau. Plateaus are flat spots on the error surface, which can severely slow down the learning process. In particular, they are notorious in applications with multilayer perceptrons. Our results, which are confirmed by simulations of a multilayer perceptron learning a chaotic time series using backpropagation, explain how dependencies between examples can help the learning process to escape from a plateau.

- by Wim Wiegerinck
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- Learning Networks, Neural Network, Multidisciplinary, Probability

In this paper, division algebras are proposed as an elegant basis upon which to extend support vector regression (SVR) to multidimensional targets. Using this framework, a multitarget SVR called X -SVR is proposed based on an -insensitive loss function that is independent of the coordinate system or basis used. This is developed to dual form in a manner that is analogous to the standard -SVR. The H -SVR is compared and contrasted with the least-square SVR (LS-SVR), the Clifford SVR (C-SVR), and the multidimensional SVR (M-SVR). Three practical applications are considered: namely, 1) approximation of a complex-valued function; 2) chaotic time-series prediction in 3-D; and 3) communication channel equalization. Results show that the H -SVR performs significantly better than the C-SVR, the LS-SVR, and the M-SVR in terms of mean-squared error, outlier sensitivity, and support vector sparsity.

- by Daniel Lai
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- Algebra, Support Vector Machines, Multidisciplinary, Mean square error

Evolving Takagi-Sugeno (eTS) fuzzy models and the method for their on-line identification has been recently introduced for both MISO and MIMO case. In this paper, the mechanism for rule-base evolution, one of the central points of the algorithm together with the recursive clustering and modified recursive least squares (RLS) estimation, is studied in detail. Different scenarios are considered for the rule base upgrade and modification. The radius of influence of each fuzzy rule is considered to be a vector instead of a scalar as in the original eTS approach, allowing different areas of the data space to be covered by each input variable. Simulation results using a well-known benchmark (Mackey-Glass chaotic time-series prediction) are presented.

- by Antonio Dourado and +2
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- Chaotic Time Series, Fuzzy Rules, Rule Based, Recursive Least Square

Abstract-A general method is developed to generate fuzzy rules from numerical data. This new method consists of five steps: Step 1 divides the input and output spaces of the given numerical data into fuzzy regions; Step 2 generates fuzzy... more

- by li wang
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- Time Series Prediction, Associative Memory, Chaotic Time Series, Fuzzy Rules

This paper presents a new numerical approach to the study of nonperiodicity in signals, which can complement the maximal Lyapunov exponent method for determining chaos transitions of a given dynamical system. The proposed technique is based on the continuous wavelet transform and the wavelet multiresolution analysis. A new parameter, the scale index, is introduced and interpreted as a measure of the degree of the signal's non-periodicity. This methodology is successfully applied to three classical dynamical systems: the Bonhoeffer-van der Pol oscillator, the logistic map, and the Henon map.

- by Rafael Benítez Suárez and +1
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- Signal Processing, Chaos Theory, Wavelets, Chaotic Time Series

In the field of chaotic time series analysis, there is a lack of a distributional theory for the main quantities used to characterize the underlying data generating process (DGP). In this paper a method for resampling time series generated by a chaotic dynamical system is proposed. The basic idea is to develop an algorithm for building trajectories which lie on the same attractor of the true DGP, that is with the same dynamical and geometrical properties of the original data. We performed some numerical experiments on some short noise-free and high-noise series confirming that we are able to correctly reproduce the distribution of the largest finite-time Lyapunov exponent and of the correlation dimension.

- by Silvia Golia
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- Statistics, Time Series, Neural Network, Distribution Theory
- by Ljiljana Kolar-anić
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- Time Series, Numerical Simulation, Multifractal Analysis, Spectrum

In this paper, we investigate the accurate prediction ability of the chaotic time series. The considered chaotic system is Mackey Glass time series. After inspection of Mackey Glass equations, we investigate sensitivity and effect of parameters. The next step is to find unknown parameters using Genetic Algorithm. The result of GA shows that it is impossible to estimate accurate value of parameters. Hence, the new method has been proposed to improve the prediction by finding similar structures. At last by some examples it is shown that prediction with 100 steps ahead is possible.

- by Hamid Khaloozadeh and +1
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- Time Series, Genetic Algorithm, Chaotic System, Chaotic Time Series

In this paper, we propose a time delay dynamic neural network (TDDNN) to track and predict a chaotic time series systems. The application of artificial neural networks to dynamical systems has been constrained by the non-dynamical nature of popular network architectures. Many of the drawbacks caused by the algebraic structures can be overcome with TDDNNs. TDDNNs have time delay elements in their states. This approach provides the natural properties of physical systems. The minimization of a quadratic performance index is considered for trajectory tracking applications. Gradient computations are presented based on adjoint sensitivity analysis. The computational complexity is significantly less than direct method, but it requires a backward integration capability. We used Levenberg-Marquardt parameter updating method.

Chaos theory refers to the behaviour of certain deterministic nonlinear dynamical systems whose solutions, although globally stable, are locally unstable. These chaotic systems describe aperiodic, irregular, apparently random and erratic trajectories, i.e., deterministic complex dynamics. One of the properties that derive from this local instability and that allow characterizing these deterministic chaotic systems is their high sensitivity to small changes in the initial conditions, which can be measured by using the so-called Lyapunov exponents. The detection of chaotic behaviour in the underlying generating process of a time series has important methodological implications. When chaotic behaviour is detected, then it can be concluded that the irregularity of the series is not necessarily random, but the result of some deterministic dynamic process. Then, even if such process is unknown, it will be possible to improve the predictability of the time series and even to control or stabilize the evolution of the time series. This article provides a summary of the main current concepts and methods for the detection of chaotic behaviour from time series.

- by Loren Escot
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- Chaos Theory, Dynamical systems and Chaos, Complexity Economics, Chaos Control

A trial is made to explore the applicability of chaos analysis outside the commonly reported analysis of a single chaotic time series. Two cross-correlated streamflows, the Little River and the Reed Creek, Virginia, USA, are analysed with regard to the chaotic behaviour. Segments of missing data are assumed in one of the time series and estimated using the other complete time series. Linear regression and artificial neural network models are employed. Two experiments are conducted in the analysis: (a) fitting one global model and (b) fitting multiple local models. Each local model is in the direct vicinity of the missing data. A nonlinear noise reduction method is used to reduce the noise in both time series and the two experiments are repeated. It is found that using multiple local models to estimate the missing data is superior to fitting one global model with regard to the mean squared error and the mean relative error of the estimated values. This result is attributed to the chaotic behaviour of the streamflows under consideration.

- by S. Simonovic and +1
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- Environmental Engineering, Civil Engineering, Time Series, Cross Correlation

The accuracy of a model to forecast a time series diminishes as the prediction horizon increases, in particular when the prediction is carried out recursively. Such decay is faster when the model is built using data generated by highly dynamic or chaotic systems. This paper presents a topology and training scheme for a novel artificial neural network, named "Hybrid-connected Complex Neural Network" (HCNN), which is able to capture the dynamics embedded in chaotic time series and to predict long horizons of such series. HCNN is composed of small recurrent neural networks, inserted in a structure made of feed-forward and recurrent connections and trained in several stages using the algorithm back-propagation through time (BPTT). In experiments using a Mackey-Glass time series and an electrocardiogram (ECG) as training signals, HCNN was able to output stable chaotic signals, oscillating for periods as long as four times the size of the training signals. The largest local Lyapunov Exponent (LE) of predicted signals was positive (an evidence of chaos), and similar to the LE calculated over the training signals. The magnitudes of peaks in the ECG signal were not accurately predicted, but the predicted signal was similar to the ECG in the rest of its structure.

- by Saul E. Pomares Hernandez
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- Cognitive Science, Time Series, Back Propagation, Neural Network

We have applied topological methods to analyze chaotic time series data from the Belousov-Zhabotinskii reaction. First, the periodic orbits shadowed by the data set were identified. Next, a three-dimensional embedding without self-intersections was constructed from the data set. The topological structure of that flow was visualized by constructing a branched manifold such that every periodic orbit in the flow could be held by the branched manifold. The branched manifold, or induced template, was computed using the three lowest-period orbits. The organization of the higher-period orbits predicted by this induced template was compared with the organization of the orbits reconstructed from the data set with excellent results. The consequences of the presence of certain knots found in the data are discussed.

- by Hernan Solari
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- Applied Mathematics, Nonlinear, Time Series Data, Template

Chaos-synchronization-based multiparameter estimation of a multiply delayed feedback system is investigated. We propose an adaptive method that can estimate all the parameters of the response system using the driving signal only. In the past few years, various methods have been developed for estimation of multiparameters of a chaotic system but most of them require more than one time series to estimate all the parameters of a chaotic or hyperchaotic system. The proposed method requires only a single chaotic time series to estimate all the parameters. A sufficient condition for synchronization is derived and it is shown that the numerical results well support the analytic calculations. The synchronized system has applications in cryptographic encoding for digital and analog signals, which is shown with an example.

- by Dibakar Ghosh
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- Engineering, Nonlinear Optics, Time Series, Temporal dynamics

In this paper we describe the winning entry of the time series prediction competition which was part of the International Workshop on Advanced Black-Box Techniques for Nonlinear Modeling ,h eld at K.U. Leuven, Belgium on July 8–10, 1998. We also describe the source of the data set, a nonlinear transform of a 5-scroll generalized Chua’s circuit. Participants were given

Bitcoin açık kaynaklı bir kod olarak yayınlanan ve blok zinciri (blockchain) teknolojisine dayanan ilk kriptopara birimidir. Kriptopara birimlerinin avantajı, merkezi olmayan yapılar olması ve bu sayede merkez bankalarına ihtiyaç duymayıp işlem maliyetlerinin az olmasıdır. Bu çalışmanın amacı, son zamanlarda popülerliği artan ve en köklü kriptopara birimi olan Bitcoin getirilerinin kaotik yapıya sahip olup olmadığını tespit etmektir. Başlangıç koşullarına aşırı duyarlı olan seriler kaotik dinamiklere sahiptir. Eğer seriler kaotik özelliklere sahipse, geleneksel yöntemlerle incelenmeleri yanıltıcı sonuçlar verebilmektedir. Bu amaçla, 19.12.2011-29.01.2018 dönemine ait Bitcoin getiri serisi kullanılarak ilk olarak BDS (Brock, Dechert ve Scheinkman) testi ile doğrusal olmayan bağımlılık test edilmiş, ardından serideki uzun dönemli bellek yapısını belirlemek için dönüştürülmüş genişlik (rescaled range-R/S) yöntemi uygulanarak Hurst üsteli elde edilmiştir. Ardından, yanlış en yakın komşular yöntemi ile uygun gömme boyutu belirlenmiştir. Serideki kaotik davranışı tespit etmek için korelasyon boyutu hesaplanmış ve Lyapunov üsteli değeri pozitif bulunmuştur. Sonuç olarak, serinin doğrusal olmayan dinamikler içerdiği, uzun belleğe sahip olduğu ve serinin kaotik özellikler taşıdığı bulgusu elde edilmiştir.

- by Eda Yalçın Kayacan
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- Bitcoin, Kaos Teorisi, Chaotic Time Series

This paper and its companion are devoted to the evaluation of the impact of chaos-based techniques on communications systems with asynchronous code division multiple access. Sequences obtained by repeating a truncated and quantized chaotic time series are compared with classical m-sequences and Gold sequences by means of a performance index taken from communication theory which is here defined and thoroughly discussed. This analysis reveals that, unlike conventional sequences, chaotic spreading codes can be generated for any number of users and allocated bandwidth. Numerical simulations are reported, showing that systems based on chaotic spreading sequences perform generally better than the conventional ones. Some analytical tools easing the comprehension of these advantages are here summarized and proved in Part II where formal arguments are developed and discussed to ensure general applicability of chaotic spreading codes.

- by Riccardo Rovatti
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- Privacy, Time Series, Modeling, Communication systems

The NARX network is a dynamical neural architecture commonly used for inputoutput modeling of nonlinear dynamical systems. When applied to time series prediction, the NARX network is designed as a feedforward Time Delay Neural Network (TDNN), i.e. without the feedback loop of delayed outputs, reducing substantially its predictive performance. In this paper, we show that the original architecture of the NARX network can be easily and efficiently applied to long-term (multi-stepahead) prediction of univariate time series. We evaluate the proposed approach using two real-world data sets, namely the well-known chaotic laser time series and a variable bit rate (VBR) video traffic time series. All the results show that the proposed approach consistently outperforms standard neural network based predictors, such as the TDNN and Elman architectures.

- by José Maria and +1
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- Engineering, Time Series, Neural Network, Time Series Prediction

Sparse learning machines provide a viable framework for modeling chaotic time-series systems. A powerful state-space reconstruction methodology using both support vector machines (SVM) and relevance vector machines (RVM) within a multiobjective optimization framework is presented in this paper. The utility and practicality of the proposed approaches have been demonstrated on the time series of the Great Salt Lake (GSL) biweekly volumes from 1848 to 2004. A comparison of the two methods is made based on their predictive power and robustness. The reconstruction of the dynamics of the Great Salt Lake volume time series is attained using the most relevant feature subset of the training data. In this paper, efforts are also made to assess the uncertainty and robustness of the machines in learning and forecasting as a function of model structure, model parameters, and bootstrapping samples. The resulting model will normally have a structure, including parameterization, that suits the information content of the available data, and can be used to develop time series forecasts for multiple lead times ranging from two weeks to several months.

- by Mac McKee
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- Environmental Engineering, Civil Engineering, Water, Time Series

Time-frequency analysis is performed for chaotic flow with a power spectrum estimator based on the phase-space neighborhood. The relation between the reference phase point and its nearest neighbors is demonstrated. The nearest neighbors, representing the state recurrences in the phase space reconstructed by time delay embedding, actually cover data segments with similar wave forms and thus possess redundant information, but recur with no obvious temporal regularity. To utilize this redundant recurrence information, a neighborhood-based spectrum estimator is devised. Then time-frequency analysis with this estimator is performed for the Lorenz time series, the Rössler time series, experimental laser data, and colored noise. Features revealed by the spectrogram can be used to distinguish noisy chaotic flow from colored noise.

- by Michael Small
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- Engineering, Time Series, Nonlinear dynamics, Statistical Physics

This paper investigates error-entropy-minimization in adaptive systems training. We prove the equivalence between minimization of error's Renyi entropy of order and minimization of a Csiszar distance measure between the densities of desired and system outputs. A nonparametric estimator for Renyi's entropy is presented, and it is shown that the global minimum of this estimator is the same as the actual entropy. The performance of the error-entropy-minimization criterion is compared with mean-square-error-minimization in the short-term prediction of a chaotic time series and in nonlinear system identification.

An increasingly popular method of encoding chaotic time-series from physical experiments is the so-called threshold crossings technique, where one simply replaces the real valued data with symbolic data of relative positions to an arbitrary partition at discrete times. The implication has been that this symbolic encoding describes the original dynamical system. On the other hand, the literature on generating partitions of non-hyperbolic dynamical systems has shown that a good partition is non-trivial to find. It is believed that the generating partition of non-uniformly hyperbolic dynamical system connects "primary tangencies", which are generally not simple lines as used by a threshold crossings. Therefore, we investigate consequences of using itineraries generated by a non-generating partition. We do most of our rigorous analysis using the tent map as a benchmark example, but show numerically that our results likely generalize. In summary, we find the misrepresentation of the dynamical system by "sample-path" symbolic dynamics of an arbitrary partition can be severe, including (sometimes extremely) diminished topological entropy, and a high degree of non-uniqueness. Interestingly, we find topological entropy as a function of misplacement to be devil's staircase-like, but surprisingly non-monotone.

- by Erik Bollt
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- Applied Mathematics, Symbolic Dynamics, Chaotic Time Series, Dynamic System

Software-reliability models (SRMs) are used for the assessment and improvement of reliability in software systems. These models are normally based on stochastic processes, with the nonhomogeneous Poisson process being one of the most prominent model forms. An underlying assumption of these models is that software failures occur randomly in time. This assumption has never been quantitatively tested. Our contribution in this paper is to conduct an experimental investigation that contrasts random processes with nonlinear deterministic processes as a model for software failures. We study two sets of real-world softwarereliability data using the techniques of chaotic time-series analysis. We have found that both appear to arise from a deterministic process, rather than a stochastic process, and that both show some evidence of chaotic dynamics. In addition, we have conducted a series of k-steps-ahead forecasting experiments in the datasets, pitting a number of well-known stochastic SRMs against radial basis function networks (RBFNs), which are deterministic in nature. The out-of-sample prediction results from the RBFNs showed an improvement of roughly 25% over the best of the stochastic models, for both of our datasets. Finally, we propose a causal model to explain these results, which hypothesizes that faults in a program are distributed over a fractal subset of the program's input space.

- by Scott Dick
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- Engineering, Stochastic Process, Soft Computing, Chaos Theory

We address the detection of unstable periodic orbits from experimentally measured transient chaotic time series. In particular, we examine recurrence times of trajectories in the vector space reconstructed from an ensemble of such time series. Numerical experiments demonstrate that this strategy can yield periodic orbits of low periods even when noise is present. We analyze the probability of finding periodic orbits from transient chaotic time series and derive a scaling law for this probability. The scaling law implies that unstable periodic orbits of high periods are practically undetectable from transient chaos.

- by Mukesh Dhamala
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- Time Series, Time series analysis, Vector Space, Chaotic Time Series

Topological methods have recently been developed for the analysis of dissipative dynamical systems that operate in the chaotic regime. They were originally developed for three-dimensional dissipative dynamical systems, but they are applicable to all ''low-dimensional'' dynamical systems. These are systems for which the flow rapidly relaxes to a three-dimensional subspace of phase space. Equivalently, the associated attractor has Lyapunov dimension

- by Robert Gilmore
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- Time Series, Modern physics, Physical sciences, Time Series Data

- by Eugen Diaconescu
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- Time Series, Prediction, Neural Network, Recurrent Neural Network

Functional networks are a general framework useful for solving a wide range of problems in probability, statistics, and engineering applications. In this article, we demonstrate that functional networks can be used for many general purposes including (a) solving nonlinear regression problems without the rather strong assumption of a known functional form, (b) modeling chaotic time series data, (c) nding conjugate families of distribution functions needed for the applications of Bayesian statistical techniques, (d) analyzing the problem of stability with respect to maxima operations, which are useful in the theory and applications of extreme values, and (e) modeling the reproductivity and associativity laws that have many applications in applied probability. We also give two speci c engineering applications-analyzing the Ikeda map with parameters leading to chaotic behavior and modeling beam stress subject to a given load. The main purpose of this article is to introduce functional networks and to show their power and usefulness in engineering and statistical applications. We describe the steps involved in working with functional networks including structural learning (speci cation and simpli cation of the initial topology), parametric learning, and model-selection procedures. The concepts and methodologie s are illustrated using several examples of applications.

- by Enrique Castillo
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- Statistics, Extreme Value Theory, Model Selection, Nonlinear Regression

This report was prepared as an account of work sponsored by an agency of the United States Government Neither the United States Government nor any agency themf, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or proctss disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or servia by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, m mmendotion, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. DISCLAIMER Portions of this document may be illegible in electronic image products. Images are produced from the best available original document.

- by James M McDonough
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- Applied Mathematics, Stochastic Process, Turbulence, Time Series

We propose the original methods for reconstructing model delay-differential equations from chaotic time series for various classes of time-delayed feedback systems including: i) scalar time-delay systems with arbitrary nonlinear function, ii) high-order time-delay systems, iii) systems with several coexisting delays, and iv) coupled time-delay systems. These methods are based on the statistical analysis of time intervals between extrema in the time series of time-delay systems and the projection of infinite-dimensional phase space of these systems to suitably chosen low-dimensional subspaces. The methods allow one to recover the delay times, the nonlinear functions, and the parameters characterizing the inertial properties of the systems and to define the a priori unknown order of a time-delay system. In the case of coupled time-delay systems the methods are able to define also the type, strength, and direction of coupling and can be used for the analysis of unidirectional and mutual coupling of time-delay systems for a wide range of the coupling coefficients variation. The proposed methods are efficient for the analysis of short time series under sufficiently high levels of noise. The methods are successfully applied to recovery of standard time-delay systems from their simulated time series corrupted with noise and to modeling various electronic oscillators with delayed feedback from their experimental time series. The proposed methods are applied to the problem of hidden message extraction in the communication systems with nonlinear mixing of information signal and chaotic signal of a time-delay system. Different ways for encryption and decryption of information in these communication schemes are investigated. Using both numerical and experimental data we obtained a high quality of the information signal extraction from the transmitted signal for different message signals and different configurations of the chaotic transmitter with a priori unknown parameters.

- by Vladimir I.Ponomarenko
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- Time Series, Statistical Analysis, Phase Space, Chaotic Time Series

In view of several limitations of gradient search techniques (e.g. backpropagation), global search techniques, including evolutionary programming and genetic algorithms (GAs), have been proposed for training neural networks (NNs). However, the eectiveness, ease-of-use, and eciency of these global search techniques have not been compared extensively with gradient search techniques. Using ®ve chaotic time series functions, this paper empirically compares a genetic algorithm with backpropagation for training NNs. The chaotic series are interesting because of their similarity to economic and ®nancial series found in ®nancial markets. Ó

- by Jatinder Gupta
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- Engineering, Genetic Algorithms, Neural Network, Genetic Algorithm