Regularized local linear prediction of chaotic time series (original) (raw)
Related papers
Iterative SVD method for noise reduction of low-dimensional chaotic time series
Mechanical Systems and Signal Processing, 1999
A new simple method using singular value decomposition (SVD) is presented for reducing noise from a sampled signal where the deterministic signal is from a low-dimensional chaotic dynamical system. The technique is concerned particularly with improving the reconstruction of the phase portrait. This method is based on time delay embedding theory to form a trajectory matrix. SVD is then used iteratively to distinguish the deterministic signal from the noise. Under certain conditions, the method can be used almost blindly, even in the case of a very noisy signal (e.g. a signal to noise ratio of 6 dB). The algorithm is evaluated for a chaotic signal generated by the Duffing system, to which white noise is added.
Singular-spectrum analysis: A toolkit for short, noisy chaotic signals
Physica D: Nonlinear Phenomena, 1992
Singular-spectrum analysis (SSA) is developed further, based on experience with applications to geophysical time series. It is shown that SSA provides a crude but robust approximation of strange attractors by tori, in the presence ol noise. The method works well for short, noisy time series. The lagged-covariance matrix of the processes studied is the basis of SSA. We select subsets of eigenelements and associated principal components (PCs) in order to provide (i) a noise-reduction algorithm, (ii) a detrending algorithm, and (iii) an algorithm for the identification of oscillatory components. Reconstructed components (RCs) are developed to provide optimal reconstruction of a dynamic process at precise epochs, rather than averaged over the window length of the analysis. SSA is combined with advanced spectral-analysis methods-the maximum entropy method (MEM) and the multi-taper method (MTM)-to refine the interpretation of oscillatory behavior. A combined SSA-MEM method is also used for tile prediction of selected subsets of RCs. The entire toolkit is validated against a set of four prescribed time series generated by known processes, quasi-periodic or chaotic. It is also applied to a time series of global surface air temperatures, 130 years long, which has attracted considerable attention in the context of the global warming issue and provides a severe test for noise reduction and prediction.
Nonlinear Forecasting, Chaos and Statistics
Woodward Conference, 1992
Many natural and experimental time series are generated by a combination of coherent, low-dimensional dynamics and stochastic, high-dimensional dynamics. A famous example is the sunspot time series. In the first part of this paper a nonlinear forecasting algorithm is used to attempt to identify how much of the irregularity in an aperiodic time series is due to low-dimensional chaos, as opposed to high-dimensional noise. The algorithm is applied to experimentally generated time series from coupled diodes, fluid turbulence and flame dynamics, and compared to dimension calculations. Theoretical results concerning the limitations of such forecasting algorithms in the presence of noise are reviewed in the second part of the paper. These results use a combination of ideas from dynamical systems and statistics. In particular, the state space reconstruction problem is investigated using properties of likelihood functions at low noise levels.
Forecasting chaotic time series: Global vs. local methods
Introduction Predicting the continuation of a time series is an interesting problem, with important applications in almost all fields of human activity. The standard theory views the series as a realization of a random process[1], which is appropiate for systems with many irreducible degrees of freedom. However, for deterministic time series associated to systems with complex chaotic dynamics, only a few degrees of freedom are relevant. Furthermore, even if chaos prevents long-term predictions, the intrinsic determinism in the series offers new possibilities for short-term forecasting[2]. On this basis, many algorithms have been recently devised to reconstruct the underlying dynamics and allow accurate predictions of the next few values in the future[3]. Given the observations of a system x i N 0 , the problem is then the reconstruction of the time-series dynamics x t = F (X t<F14.
Notes on forecasting a chaotic series using regression
Technological Forecasting and Social Change, 1991
It is well known that a nonlinear recursive equation can produce a chaotic sequence at certain values of the parameter. Furthermore, in the chaotic regime, extremely small changes in the initial value or in the value of the parameter produce very large changes in the sequence. It is surmising therefore that a short segment of a chaotic sequence can be used to reconstruct large portions of the sequence and to forecast future values of the sequence over short ranges. Furthermore, while the accuracy of the forecasts is dependent on the precision of the data, the relationship is much less sensitive than might have been expected. This is demonstrated by fitting a two-parameter model to two different types of chaotic equations: one a polynomial and the other a trigonometric function. This leads to the expectation that under certain circumstances, it may be possible to forecast values in a chaotic series over limited ranges, even if initial data are somewhat degraded.
Chaotic signal processing by use of second order statistical methods and surrogate data analysis
The cleaning of signals contaminated by noise is a major concern in real world systems, where short noisy signals are frequently encountered. In linear analysis the problem can be dealt with by extracting sharp narrowband linear signals from broadband noise in the Fourier domain, but this cannot be used for nonlinear signals, since nonlinear structures can be difficult to distinguish from broadband noise. Under these circumstances, it is better to attempt to differentiate between the signal and the noise in the time domain, by assuming that the observed signal s(t), is the sum of the desired signal s 1 (t) and some other signals s 2 (t), s 3 (t) … s m (t). In this paper, the use of singular spectrum analysis and related methods to this end is investigated. The signal s(t) is decomposed and the constituent signals are characterized by Monte Carlo simulations in which surrogate signals are generated which can serve as a benchmark for the detection and removal of noise from the original signal.
Short term chaotic time series prediction using symmetric LS-SVM regression
2005
In this article, we illustrate the effect of imposing symmetry as prior knowledge into the modelling stage, within the context of chaotic time series predictions. It is illustrated that using Least-Squares Support Vector Machines with symmetry constraints improves the simulation performance, for the cases of time series generated from the Lorenz attractor, and multi-scroll attractors. Not only accurate forecasts are obtained, but also the forecast horizon for which these predictions are obtained is expanded.
Chaotic time series. Part I. Estimation of some invariant properties in state-space
Modeling, Identification and Control: A Norwegian Research Bulletin, 1994
Certain deterministic non-linear systems may show chaotic behaviour. Time series derived from such systems seem stochastic when analyzed with linear techniques. However, uncovering the deterministic structure is important because it allows for construction of more realistic and better models and thus improved predictive capabilities. This paper describes key features of chaotic systems including strange attractors and Lyapunov exponents. The emphasis is on state space reconstruction techniques that are used to estimate these properties, given scalar observations. Data generated from equations known to display chaotic behaviour are used for illustration. A compilation of applications to real data from widely di erent elds is given. If chaos is found to be present, one may proceed to build non-linear models, which is the topic of the second paper in this series.
Acta applicandae mathematicae, 2010
Singular Value Decomposition (SVD) is a powerful tool in linear algebra and has been extensively applied to Signal Processing, Statistical Analysis and Mathematical Modeling. We propose an extension of SVD for both the qualitative detection and quantitative determination of nonlinearity in a time series. The method is to augment the embedding matrix with additional nonlinear columns derived from the initial embedding vectors and extract the nonlinear relationship using SVD. The paper demonstrates an application of nonlinear SVD to identify parameters when the signal is generated by a nonlinear transformation. Examples of maps (Logistic map and Henon map) and flows (Van der Pol oscillator and Duffing oscillator) are used to illustrate the method of nonlinear SVD to identify parameters. The paper presents the recovery of parameters in the following scenarios: (i) data generated by maps and flows, (ii) comparison of the method for both noisy and noise-free data, (iii) surrogate data analysis for both the noisy and noise-free cases. The paper includes two applications of the method: (i) Mathematical Modeling and (ii) Chaotic Cryptanalysis.