Predicting equation from Chaotic data by Nonlinear Singular Value Decomposition (original) (raw)
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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.
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 value decomposition to determine the dynamics of a chaotic regime oscillator
2019
Targeting the hybrid analog-digital private communication field, this paper aims to estimate the parameters of an analog circuit model. An oscilloscope stores the samples of a voltage in a.csv file. The data series is processed using a digital signal technique the singular value decomposition. Singular values and corresponding right-eigenvectors are used to estimated the values of the parameters of the model characterizing the circuit that produced the measured output. The decomposition is performed using small windows of samples of the output of a jerk-type circuit from the literature and an averaging operation improves the estimation.
Regularized local linear prediction of chaotic time series
Physica D: Nonlinear Phenomena, 1998
Local linear prediction, based on the ordinary least squares (OLS) approach, is one of several methods that have been applied to prediction of chaotic time series. Apart from potential linearization errors, a drawback of this approach is the high variance of the predictions under certain conditions. Here, a different set of so-called linear regularization techniques, originally derived to solve ill-posed regression problems, are compared to OLS for chaotic time series corrupted by additive measurement noise. These methods reduce the variance compared to OLS, but introduce more bias. A main tool of analysis is the singular value decomposition (SVD), and a key to successful regularization is to damp the higher order SVD components. Several of the methods achieve improved prediction compared to OLS for synthetic noise-corrupted data from well-known chaotic systems. Similar results were found for real-world data from the R-R intervals of ECG signals. Good results are also obtained for real sunspot data, compared to published predictions using nonlinear techniques.
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.
Modelling and prediction of time series using singular value decomposition and neural networks
Computers & Electrical Engineering, 1995
A new approach for modelling and prediction of a time series with nearly periodic or quasiperiodic structure has been proposed; one of the main objectives is to produce one-period ahead prediction. The proposed modelling involves two stages: (i) the singular value decomposition (SVD) based orthogonahzation with due consideration of the prime periodicity; and (ii) neural network modelling of the orthogonalized components. Orthogonalization causes compaction of information, while the neural network models the non-linear relationship. The proposed approach yields good prediction performance and at the same time, is computationally efficient and numerically robust.
Application of chaotic noise reduction techniques to chaotic data trained by ANN
Sadhana, 2001
We propose a novel method of combining artificial neural networks (ANNs) with chaotic noise reduction techniques that captures the metric and dynamic invariants of a chaotic time series, e.g. a time series obtained by iterating the logistic map in chaotic regimes. Our results indicate that while the feedforward neural network is capable of capturing the dynamical and metric invariants of chaotic time series within an error of about 25%, ANNs along with chaotic noise reduction techniques, such as Hammel's method or the local projective method, can significantly improve these results. This further suggests that the effort on the ANN to train data corresponding to complex structures can be significantly reduced. This technique can be applied in areas like signal processing, data communication, image processing etc.
Singular-value decomposition approach to time series modelling
IEE Proceedings F Communications, Radar and Signal Processing, 1983
In various signal processing applications, as exemplified by spectral analysis, deconvolution and adaptive filtering, the parameters of a linear recursive model are to be selected so that the model is 'most' representative of a given set of time series observations. For many of these applications, the parameters are known to satisfy a theoretical recursive relationship involving the time series' autocorrelation lags. Conceptually, one may then use this recursive relationship, with appropriate autocorrelation lag estimates substituted, to effect estimates for the operator's parameters. A procedure for carrying out this parameter estimation is given which makes use of the singular-value decomposition (SVD) of an extended-order autocorrelation matrix associated with the given time series. Unlike other SVD modelling methods, however, the approach developed does not require a full-order SVD determination. Only a small subset of the matrix's singular values and associated characteristic vectors need be computed. This feature can significantly alleviate an otherwise overwhelming computational burden that is necessitated when generating a full-order SVD. Furthermore, the modelling performance of this new method has been found empirically to excel that of a near maximum-likelihood SVD method as well as several other more traditional modelling methods. 'The symbol [«,, n 2 ] denotes the set of integers satisfying n x < n < n2 while [ n,, °°) specifies the set of integers satisfying n > n,.
A Non-linear Generalization of Singular Value Decomposition and its Application to Cryptanalysis
2007
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 paper illustrates nonlinear SVD with the help of data generated from nonlinear maps and flows (differential equations). 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 also demonstrates an application of nonlinear SVD to cryptanalysis where the encrypted signal is generated by a nonlinear transformation. A comparison of the method for both noise-free and noisy data along with their surrogate counterparts is included.