Delay Coordinate Space Research Articles

Estimating Kolmogorov–Sinai entropy from time series of high-dimensional complex systems

A method is presented to estimate KS entropy from time series of a high-dimensional complex system. We focus on partitioned entropy, which measures the complexity of data points localized in a delay-coordinate space. By quantifying the time evolution of the partitioned entropy, we show that its growth rate gives a good estimate of the KS entropy. Our method is computationally efficient, since a time series is discretized into a coarse-grained symbolic sequence by a vector quantization technique. Using numerical data sets generated from prototypical models of chaotic systems, we showed that our approach works quite well for high-dimensional complex systems. Application to experimental data measured from physical models of the vocal membranes confirmed that our method is effective for characterizing the complexity of real-world systems.

Color Recurrence Plots for Bearing Fault Diagnosis.

This paper presents bearing fault diagnosis using the image classification of different fault patterns. Feature extraction for image classification is carried out using a novel approach of Color recurrence plots, which is presented for the first time. Color recurrence plots are created using non-linear embedding of the vibration signals into delay coordinate space with variable time lags. Deep learning-based image classification is then performed by building the database of the extracted features of the bearing vibration signals in the form of Color recurrence plots. A Series of computational experiments are performed to compare the accuracy of bearing fault classification using Color recurrence plots. The standard bearing vibration dataset of Case Western Reserve University is used for those purposes. The paper demonstrates the efficacy and the accuracy of a new and unique approach of scalar time series extraction into two-dimensional Color recurrence plots for bearing fault diagnosis.

Permutation entropy-based 2D feature extraction for bearing fault diagnosis

Bearing fault diagnosis based on the classification of patterns of permutation entropy is presented in this paper. Patterns of permutation entropy are constructed by using non-uniform embedding of the vibration signal into a delay coordinate space with variable time lags. These patterns are interpreted, processed and classified by employing deep learning techniques based on convolutional neural networks. Computational experiments are used to compare the accuracy of classification with other methods and to demonstrate the efficacy of the presented early defect detection and classification method.

Permutation Entropy Based on Non-Uniform Embedding

A novel visualization scheme for permutation entropy is presented in this paper. The proposed scheme is based on non-uniform attractor embedding of the investigated time series. A single digital image of permutation entropy is produced by averaging all possible plain projections of the permutation entropy measure in the multi-dimensional delay coordinate space. Computational experiments with artificially-generated and real-world time series are used to demonstrate the advantages of the proposed visualization scheme.

Data-driven spectral decomposition and forecasting of ergodic dynamical systems

Abstract We develop a framework for dimension reduction, mode decomposition, and nonparametric forecasting of data generated by ergodic dynamical systems. This framework is based on a representation of the Koopman and Perron–Frobenius groups of unitary operators in a smooth orthonormal basis of the L 2 space of the dynamical system, acquired from time-ordered data through the diffusion maps algorithm. Using this representation, we compute Koopman eigenfunctions through a regularized advection–diffusion operator, and employ these eigenfunctions in dimension reduction maps with projectible dynamics and high smoothness for the given observation modality. In systems with pure point spectra, we construct a decomposition of the generator of the Koopman group into mutually commuting vector fields that transform naturally under changes of observation modality, which we reconstruct in data space through a representation of the pushforward map in the Koopman eigenfunction basis. We also establish a correspondence between Koopman operators and Laplace–Beltrami operators constructed from data in Takens delay-coordinate space, and use this correspondence to provide an interpretation of diffusion-mapped delay coordinates for this class of systems. Moreover, we take advantage of a special property of the Koopman eigenfunction basis, namely that the basis elements evolve as simple harmonic oscillators, to build nonparametric forecast models for probability densities and observables. In systems with more complex spectral behavior, including mixing systems, we develop a method inspired from time change in dynamical systems to transform the generator to a new operator with potentially improved spectral properties, and use that operator for vector field decomposition and nonparametric forecasting.

Evolutionary algorithms for the selection of time lags for time series forecasting by fuzzy inference systems

Time series forecasting by fuzzy inference systems based on optimal non-uniform attractor embedding in the multidimensional delay phase space is analyzed in this paper. A near-optimal set of time lags is identified by evolutionary algorithms after the optimal dimension of the reconstructed phase space is determined by the FNN (false nearest neighbors) algorithm. The fitness function is constructed in such a way that it represents the spreading of the attractor in the delay coordinate space but does not contain any information on prediction error metrics. The weighted one-point crossover rule enables an effective identification of near-optimal sets of non-uniform time lags which are better than the globally optimal set of uniform time lags. Thus the reconstructed information on the properties of the underlying dynamical system is directly elaborated in the fuzzy prediction system. A number of numerical experiments are used to test the functionality of this method.

Reconstruction of chaotic dynamics by on-line EM algorithm

In this paper, we discuss the reconstruction of chaotic dynamics by using a normalized Gaussian network (NGnet), which is a network of local linear regression units. The NGnet is trained by an on-line EM algorithm in order to learn the vector field of the chaotic dynamics. We investigate the robustness of our approach to two kinds of noise processes: system noise and observation noise. System noise disturbs the system dynamics itself. Observation noise is added when the state variables are observed. It is shown that the trained NGnet is able to reproduce a chaotic attractor, which approximates the complexity and instability of the original chaotic attractor well, even under various noise conditions. The trained NGnet also shows good prediction performance. When only part of the dynamical variables are observed, the delay embedding method is used. The NGnet is then trained to learn the vector field in the delay coordinate space. Experiments show that the chaotic dynamics embedded in the delay coordinate space is able to be learned under the two types of noise.

Symbolic dynamics analysis of nonstationary data from a model of a magnetic system with solitons.

The probability distribution of a complexity measure is used to characterize chaotic states: an estimator of the algorithmic complexity of a time series of symbolic words is calculated within a fixed length time window, which sweeps through the time series analyzed. The words are derived through a symbolic dynamics scheme applied in an m-dimensional delay coordinate space. Time intervals instead of the variables of the system are used. The chaotic states of a model of a magnetic domain wall are characterized better by the methods presented than with the use of fractal dimensions and new intermittent states of the system were easily identified. Using an artificial nonstationary signal composed of different chaotic states of the Bloch wall as a test for chaos-chaos intermittency we demonstrate that the method developed is suitable for the detection and characterization of intermittency. It is also shown that nonstationarity in the form of a slow monotonic drift in the control parameter may extend the stability range of periodic states of the spatially extended system studied-a trackinglike phenomenon.

A SIMPLE GEOMETRICAL STRUCTURE UNDERLYING SPEECH SIGNALS OF THE JAPANESE VOWEL /a/

We provide several pieces of evidence for possible chaotic dynamics in the irregular behavior of normal speech signals of the Japanese vowel /a/. First, principal component analysis demonstrates that a simple geometric structure underlying the complex speech signal is well reconstructed in a three-dimensional delay-coordinate space. Observations of the reconstructed speech trajectory at multiple cross sections also display speech dynamics with stretching, folding and compressing. Second, Lyapunov spectrum analysis indicates sensitive dependence on initial conditions with a positive Lyapunov exponent for the speech signals of several different speakers. Third, nonlinear modeling analysis with an artificial neural network shows that the nonlinear dynamics of the vowel sound is well reproduced by a deterministic dynamical model.

Characteristics of Dimension Functions Associated With Cutting Measurements

The information dimension, D(0), of attractors associated with orthogonal turning is determined from experimental tool-workpiece relative acceleration data. Let E≡dimension of a delay coordinate space, n≡number of generic data points and m≡ number of reference points on the attractor. It is shown that properties of D(0) as a function of E, denoted by D(0):E, are unchanging, invariant, over large intervals of n and m. The qualitative properties of D(0):E discriminate between various cutting cases. This discrimination can be based on relatively small data sets. The computation of D(0) is shown to be robust in the sense that estimated values of D(0) are invariant or slowly varying over intervals of n and m.

Plateau onset for correlation dimension: When does it occur?

Chaotic experimental systems are often investigated using delay coordinates. Estimated values of the correlation dimension in delay coordinate space typically increase with the number of delays and eventually reach a plateau (on which the dimension estimate is relatively constant) whose value is commonly taken as an estimate of the correlation dimension ${\mathit{D}}_{2}$ of the underlying chaotic attractor. We report a rigorous result which implies that, for long enough data sets, the plateau begins when the number of delay coordinates first exceeds ${\mathit{D}}_{2}$. Numerical experiments are presented. We also discuss how lack of sufficient data can produce results that seem to be inconsistent with the theoretical prediction.