Noisy Chaotic Time Series Research Articles

Associative reasoning-based interpretable continuous decision making in industrial production process

Adaptive decision making in continuous production process is challenging for the uncertain characteristics of dynamics system state, decision-making time, as well as the correlation between the previous decision and the subsequent decision. In this paper, a data-driven continuous decision-making model with self-adaption and interpretability is proposed. For the uncertainty of industrial data, a new form of information granule with variable parameters is designed to describe the knowledge extracted from data. For the continuity of the decision-making process, a data-driven interpretable associative reasoning model is established by considering the similarity of knowledge and the attention mechanism. Aiming at the dynamic feature in the decision-making process, a data-driven associative learning method is proposed to obtain the adaptivity of the proposed model, which generates and updates routes according to the co-activation and simultaneous activities of the information granules. To reason decision results, the proposed model searches and records the reasoning routes according to the reasoning results and the objective at the decision points. Noisy chaotic time series data and a number of classification datasets are employed for experimental analyses, and the results indicate that the proposed method exhibits high reasoning accuracy, anti-noise ability, and interpretability. In addition, the experimental results based on industrial data indicate that the decision-making model is capable of giving a reasonable, effective and interpretable decision scheme and suitable for establishing a credible decision-making system.

Robust manifold broad learning system for large-scale noisy chaotic time series prediction: A perturbation perspective

Noises and outliers commonly exist in dynamical systems because of sensor disturbations or extreme dynamics. Thus, the robustness and generalization capacity are of vital importance for system modeling. In this paper, the robust manifold broad learning system(RM-BLS) is proposed for system modeling and large-scale noisy chaotic time series prediction. Manifold embedding is utilized for chaotic system evolution discovery. The manifold representation is randomly corrupted by perturbations while the features not related to low-dimensional manifold embedding are discarded by feature selection. It leads to a robust learning paradigm and achieves better generalization performance. We also develop an efficient solution for Stiefel manifold optimization, in which the orthogonal constraints are maintained by Cayley transformation and curvilinear search algorithm. Furthermore, we discuss the common thoughts between random perturbation approximation and other mainstream regularization methods. We also prove the equivalence between perturbations to manifold embedding and Tikhonov regularization. Simulation results on large-scale noisy chaotic time series prediction illustrates the robustness and generalization performance of our method.

Uncovering low dimensional macroscopic chaotic dynamics of large finite size complex systems.

In the last decade, it has been shown that a large class of phase oscillator models admit low dimensional descriptions for the macroscopic system dynamics in the limit of an infinite number N of oscillators. The question of whether the macroscopic dynamics of other similar systems also have a low dimensional description in the infinite N limit has, however, remained elusive. In this paper, we show how techniques originally designed to analyze noisy experimental chaotic time series can be used to identify effective low dimensional macroscopic descriptions from simulations with a finite number of elements. We illustrate and verify the effectiveness of our approach by applying it to the dynamics of an ensemble of globally coupled Landau-Stuart oscillators for which we demonstrate low dimensional macroscopic chaotic behavior with an effective 4-dimensional description. By using this description, we show that one can calculate dynamical invariants such as Lyapunov exponents and attractor dimensions. One could also use the reconstruction to generate short-term predictions of the macroscopic dynamics.

Noisy Chaotic time series forecast approximated by combining Reny's entropy with Energy associated to series method: application to rainfall series

This article proposes that the combination of smoothing approach considering the entropic information provided by Renyi's method, has an acceptable performance in term of forecasting errors. The methodology of the proposed scheme is examined through benchmark chaotic time series, such as Mackey Glass, Lorenz, Henon maps, the Lynx and rainfall from Santa Francisca - Cordoba, with addition of white noise by using neural networks-based energy associated (EAS) predictor filter modified by Renyi's entropy of the series. When the time series is short or long, the underlying dynamical system is nonlinear and temporal dependencies span long time intervals, in which this are also called long memory process. In such cases, the inherent nonlinearity of neural networks models and a higher robustness to noise seem to partially explain their better prediction performance when entropic information is extracted from the series. Then, to demonstrate that permutation entropy is computationally efficient, robust to outliers, and effective to measure complexity of time series, computational results are evaluated against several non-linear ANN predictors to show the predictability of noisy rainfall and chaotic time series reported in the literature.

Robust kernel adaptive filters based on mean p-power error for noisy chaotic time series prediction

Two robust kernel adaptive filter (KAF) algorithms, called the kernel least mean p-power (KLMP) and kernel recursive least mean p-power (KRLP), are developed by combining mean p-power error (MPE) criterion and kernel trick for noisy chaotic time series prediction (CTSP). The proposed algorithms employ the MPE to overcome the performance degradation of the CTSP when training data are corrupted by impulsive noises (especially the α-stable noises). First, the KLMP algorithm is proposed by the gradient decent method to improve the robustness of the traditional kernel least mean square (KLMS). Second, the recursion idea and the kernel method are utilized to develop a recursive KAF, namely KRLP, to improve the robustness of the traditional kernel recursive least squares (KRLS). Simulation results show that the proposed algorithms display notable robustness in CTSP when the training data contain different levels of noises, and can perform better in terms of testing MSE than other algorithms.

Symplectic geometry spectrum regression for prediction of noisy time series.

We present the symplectic geometry spectrum regression (SGSR) technique as well as a regularized method based on SGSR for prediction of nonlinear time series. The main tool of analysis is the symplectic geometry spectrum analysis, which decomposes a time series into the sum of a small number of independent and interpretable components. The key to successful regularization is to damp higher order symplectic geometry spectrum components. The effectiveness of SGSR and its superiority over local approximation using ordinary least squares are demonstrated through prediction of two noisy synthetic chaotic time series (Lorenz and Rössler series), and then tested for prediction of three real-world data sets (Mississippi River flow data and electromyographic and mechanomyographic signal recorded from human body).

Mackey-Glass noisy chaotic time series prediction by a swarm-optimized neural network

In this study, an artificial neural network (ANN) based on particle swarm optimization (PSO) was developed for the time series prediction. The hybrid ANN+PSO algorithm was applied on Mackey-Glass noiseless chaotic time series in the short-term and long-term prediction. The performance prediction is evaluated and compared with similar work in the literature, particularly for the long-term forecast. Also, we present properties of the dynamical system via the study of chaotic behaviour obtained from the time series prediction.Then, this standard hybrid ANN+PSO algorithm was complemented with a Gaussian stochastic procedure (called stochastic hybrid ANN+PSO) in order to obtain a new estimator of the predictions that also allowed us compute uncertainties of predictions for noisy Mackey-Glass chaotic time series. We study the impact of noise for three cases with a white noise level (σN) contribution of 0.01, 0.05 and 0.1.

Impact of Noise on a Dynamical System: Prediction and Uncertainties from a Swarm-Optimized Neural Network.

An artificial neural network (ANN) based on particle swarm optimization (PSO) was developed for the time series prediction. The hybrid ANN+PSO algorithm was applied on Mackey-Glass chaotic time series in the short-term x(t + 6). The performance prediction was evaluated and compared with other studies available in the literature. Also, we presented properties of the dynamical system via the study of chaotic behaviour obtained from the predicted time series. Next, the hybrid ANN+PSO algorithm was complemented with a Gaussian stochastic procedure (called stochastic hybrid ANN+PSO) in order to obtain a new estimator of the predictions, which also allowed us to compute the uncertainties of predictions for noisy Mackey-Glass chaotic time series. Thus, we studied the impact of noise for several cases with a white noise level (σ N) from 0.01 to 0.1.

LYAPUNOV-EXPONENT SPECTRUM FROM NOISY TIME SERIES

Since all kinds of noise exist in signals from real-world systems, it is very difficult to exactly estimate Lyapunov exponents from this time series. In this paper, a novel method for estimating the Lyapunov spectrum from a noisy chaotic time series is presented. We consider the higher-order mappings from neighbors into neighbors, rather than the mappings from small displacements into small displacements as usual. The influence of noise on the second-order mappings is researched, and an averaging method is proposed to cope with this noise. The mappings equations of the underlying deterministic system can be obtained from the noisy data via the method, and then the Lyapunov spectrum can be estimated. We demonstrate the performance of our algorithm for three familiar chaotic systems, Hénon map, the generalized Hénon map and Lorenz system. It is found that the proposed method provides a reasonable estimate of Lyapunov spectrum for these three systems when the noise level is less than 20%, 10% and 7%, respectively. Furthermore, our method is not sensitive to the distribution types of the noise, and the results of our method become more accurate with the increase of the length of time series.

Analysis of noisy chaotic time series prediction error

For the noisy chaotic series prediction problem, traditional methods are quite empirical, and are lacking in the analysis of the composition of the prediction error, thereby ignoring the the difference between chaotic dynamics reconstruction and prediction model. Based on the composition of actual prediction error, the predictor bias error and input disturbance error are defined in this paper and two kinds of global forecasts, ensemble least-square method and regularization method are analysed. It is shown that the ensemble least-square method is suitable for the reconstruction of chaotic dynamics, but has a greater influence on the predictor error. On the other hand, the regularization method can improve the sensitivity of the predictor, but it can be influenced by the input perturbation error. Two simulation examples are used to demonstrate the difference between the chaotic dynamical reconstruction and the establishment of prediction model, and to compare the ensamble least-square method and the regularization method, and at the same time indicate that the actual prediction error is influenced both by the input disturbance error and by the predictor error. In practice, a balance should be stricken between the two, in order to optimize the model prediction accuracy.

Estimating the largest Lyapunov exponent and noise level from chaotic time series

A novel method for estimating simultaneously the largest Lyapunov exponent (LLE) and noise level (NL) from a noisy chaotic time series is presented in this paper. We research the influence of noise on the average distance of different pairs of points in an embedding phase space and provide a rescaled formula for calculating the LLE when the time series is contaminated with noise. Our algorithm is proposed based on this formula and the invariant of the LLE in different dimensional embedding phase spaces. With numerical simulation, we find that the proposed method provides a reasonable estimate of the LLE and NL when the NL is less than 10% of the signal content. The comparison with Kantz algorithm shows that our method gives more accurate results of the LLE for the noisy time series. Furthermore, our method is not sensitive to the distribution of the noise.

Noise-level estimation of noisy chaotic time series based on the invariant of the largest Lyapunov exponent

A novel method of estimating the noise level from a noisy chaotic time series based on the invariant of the largest Lyapunov exponent is presented in this paper. The influence of noise on the distance between two points in an embedding phase space is considered, and then based on the invariant of the largest Lyapunov exponent in a different dimensional embedding phase space, the algorithm is proposed to estimate the noise level. Simulation results show that the estimated values of noise level agree well with the true values when the noise level is less than 10%. And this method is not sensitive to the distribution of noise. Therefore, the method is useful for estimating the noise level of noisy chaotic time series.

Recursive spectral analysis of natural time series based on eigenvector matrix perturbation for online applications

Singular spectrum analysis (SSA) is a well-studied approach in signal processing. SSA has originally been designed to extract information from short noisy chaotic time series and to enhance the signal-to-noise ratio. SSA is good for offline applications; however, many applications, such as modelling, analysis, and prediction of time-varying and non-stationary time series, demand for online analysis. This study introduces a recursive algorithm called recursive SSA as a modification to regular SSA for dynamic and online applications. The proposed method is based on eigenvector matrix perturbation approach. After recursively calculating the covariance matrix of the trajectory matrix, R-SSA updates the eigenvalues and eigenvectors for new samples by considering the effect of the new sample as perturbation in the covariance matrix and its singular value decomposition. The eigenvalues and eigenvectors adapt simultaneously to track their true values as would be calculated from the current covariance matrix. Analysis of two well-known chaotic time series: Mackey-Glass and Lorenz chaotic time series and two natural time series: Darwin sea-level pressure and Sunspot number as non-stationary processes are considered in this study to examine the performance of the proposed recursive method. The obtained results depict the power of the proposed method in online spectral analysis of non-linear time-varying systems.

Modified correlation entropy estimation for a noisy chaotic time series

A method of estimating the Kolmogorov-Sinai (KS) entropy, herein referred to as the modified correlation entropy, is presented. The method can be applied to both noise-free and noisy chaotic time series. It has been applied to some clean and noisy data sets and the numerical results show that the modified correlation entropy is closer to the KS entropy of the nonlinear system calculated by the Lyapunov spectrum than the general correlation entropy. Moreover, the modified correlation entropy is more robust to noise than the correlation entropy.

Markov chain Monte Carlo method in Bayesian reconstruction of dynamical systems from noisy chaotic time series

The impossibility to use the MCMC (Markov chain Monte Carlo) methods for long noisy chaotic time series (TS) (due to high computational complexity) is a serious limitation for reconstruction of dynamical systems (DSs). In particular, it does not allow one to use the universal Bayesian approach for reconstruction of a DS in the most interesting case of the unknown evolution operator of the system. We propose a technique that makes it possible to use the MCMC methods for Bayesian reconstruction of a DS from noisy chaotic TS of arbitrary long duration.

Enhancement of Chinese speech based on nonlinear dynamics

Based on recently observed nonlinear dynamic features of human speech, the local projection (LP) method, originally developed for noisy chaotic time series, is generalized and adapted to the enhancement of Chinese speech. The analysis of minimum embedding dimensions estimated by the false nearest neighbor algorithm shows that all the basic phonemes and syllables in Chinese can be faithfully embedded in some low-dimensional phase space. Over-embedding is applied to reconstruct the dynamics of continuous speech in some extended phase space of higher dimension, thus solving the problem of nonstationarity in continuous speech. A generalization of the LP method, named the local subspace method, is presented for speech enhancement in the phase space. It is demonstrated that, the local subspace method is essentially an extension of the well-known linear subspace technique in the local phase space, and the LP method is the least square case of this generalization. Noise reduction is then carried out in the local phase space. Results show that the LP method, with 2 or 3 iterations, achieves better performances than the local subspace method. For both isolated and continuous speech with additive white noise, experiments show the superiority of the LP method over two other popular algorithms.

Noise Smoothing for Nonlinear Time Series Using Wavelet Soft Threshold

In this letter, a new threshold algorithm based on wavelet analysis is applied to smooth noise for a nonlinear time series. By detailing the signals decomposed onto different scales, we smooth the details by using the updated thresholds to different characters of a noisy nonlinear signal. This method is an improvement of Donoho's wavelet methods to nonlinear signals. The approach has been successfully applied to smoothing the noisy chaotic time series generated by the Lorenz system as well as the observed annual runoff of Yellow River. For the nonlinear dynamical system, an attempt is made to analyze the noise reduced data by using multiresolution analysis, i.e., the false nearest neighbors, correlation integral, and autocorrelation function, to verify the proposed noise smoothing algorithm

Modified Bayesian approach for the reconstruction of dynamical systems from time series

Some recent papers were concerned with applicability of the Bayesian (statistical) approach to reconstruction of dynamic systems (DS) from experimental data. A significant merit of the approach is its universality. But, being correct in terms of meeting conditions of the underlying theorem, the Bayesian approach to reconstruction of DS is hard to realize in the most interesting case of noisy chaotic time series (TS). In this work we consider a modification of the Bayesian approach that can be used for reconstruction of DS from noisy TS. We demonstrate efficiency of the modified approach for solution of two types of problems: (1) finding values of parameters of a known DS by noisy TS; (2) classification of modes of behavior of such a DS by short TS with pronounced noise.

Sturdy cycles in the chaotic Tribolium castaneum data series

Because of the inherent discreteness of individuals, population dynamical models must be discrete variable systems. In case of strong nonlinearity, such systems interacting with noise can generate a great variety of patterns from nearly periodic behavior through complex combination of nearly periodic and chaotic patterns to noisy chaotic time series. The interaction of a population consisting of discrete individuals and demographic noise has been analyzed in laboratory population data Henson et al. (Science 294 (2001) 602; Proc. Roy. Soc. Ser. B 270 (2003) 1549). In this paper we point out that some of the cycles are fragile, i.e. they are sensitive to the discretization algorithm and to small variation of the model parameters, while others remain “sturdy” against the perturbations. We introduce a statistical algorithm to detect disjoint, nearly-periodic patterns in data series. We show that only the sturdy cycles of the discrete variable models appear in the data series significantly. Our analysis identified the quasiperiodic 11-cycle (emerging in the continuous model) to be present significantly only in one of the three experimental data series. Numerical simulations confirm that cycles can be detected only if noise is smaller than a certain critical level and population dynamics display the largest variety of nearly-periodic patterns if they are on the border of “grey” and “noisy” regions, defined in Domokos and Scheuring (J. Theor. Biol. 227 (2004) 535).

Estimation of parameters in one-dimensional maps from noisy chaotic time series

The problem of parameter estimation in model maps from noisy time series is addressed. We suggest a new technique for a special case of one-dimensional maps and chaotic signals. It is based on the maximum likelihood (ML) principle and evaluation of the cost function via backward iterations of a model map. We demonstrate in numerical experiments and, in part, justify theoretically that this “backward ML technique” gives more accurate estimates than previously known techniques for low and moderate noise levels. In particular, global optimisation of the cost function becomes much easier; biases in the estimates vanish as the time series length N increases; variances of the estimates decrease as fast as N − α where α depends on the original system, typical values being about α = 2.0 under mild conditions on the original systems.