A Nonlinear Generalization of Singular Value Decomposition and Its Applications to Mathematical Modeling and Chaotic Cryptanalysis (original) (raw)

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.

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.

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.

On the Calculation of Chaotic Features for Nonlinear Time Series

Recent developments in the theory of nonlinear dynamics have paved the way for analyzing signals generated from nonlinear biological systems. This study is aimed at investigating the application of nonlinear analysis in differentiating between patients and healthy persons, as well as in investigating the relation between ergodicity and stationarity in the dynamics of the heart. The nonlinear analysis in this work includes attractor reconstruction, estimation of the correlation dimension, calculation of the largest Lyapunov exponent, the approximate entropy, the sample entropy, and a Poincaré plot. Four groups of electrical cardiograph (ECG) signals have been investigated. Our results, obtained from clinical data, confirm the previous studies; this allows one to distinguish between a healthy group and a group of patients with more confidence than the standard methods for heart rate time series. Furthermore we extended our understanding of heart dynamics using entropies and a Poincaré plot along with the correlation dimension and the largest Lyapunov exponent. We have also obtained the results that stationarity and ergodicity are related to each other in heart dynamics.

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.

Recognizing hidden frequencies in a chaotic time series

Physics Letters A, 1998

Using the method of phase space reconstruction (PSR), it is possible to detect dynamically connected hidden frequency components in a chaotic time series. Here, we suggest to use the method of singular value decomposition (SVD) in tandem with the PSR. With this new scheme, the procedure to search for the hidden frequencies can be simplified substantially and the search efficiency can be improved greatly. We use the Lorenz model, the Riissler model, and the hyperchaotic Rijssler model to illustrate how this PSR/SVD scheme works. 0 1998 Elsevier Science B.V.

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.

Identification of Variable Relations from Multivariate Chaotic Time Series Data

2007

Abstract| Identifying relations among di erent variables becomes a very practical issue as it could determine values of the variables, which are hardly or expensively measured, from values of the variables, which can be measured easily. One of the key problems related is to determine embedding dimensions from multivariate data. A simple and e ective method is proposed to nd embedding dimensions in this paper. Several examples are provided.

Some Aspects of Chaotic Time Series Analysis

2001

We address two aspects in chaotic time series analysis, namely the definition of embedding parameters and the largest Lyapunov exponent. It is necessary for performing state space reconstruction and identification of chaotic behavior. For the first aspect, we examine the mutual information for determination of time delay and false nearest neighbors method for choosing appropriate embedding dimension. For the second aspect we suggest neural network approach, which is characterized by simplicity and accuracy.

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.