Hilbert–Huang Transform Approach to Lorenz Signal Separation

2015 ◽  
Vol 07 (01n02) ◽  
pp. 1550004 ◽  
Author(s):  
Gregori J. Clarke ◽  
Samuel S. P. Shen
Keyword(s):  
Time Series ◽  
Fourier Series ◽  
Lorenz System ◽  
Initial Conditions ◽  
Fourier Series Expansion ◽  
Finite Collection ◽  
Lorenz Attractor ◽  
Lorenz Equations ◽  
Hilbert Huang Transform ◽  
Time Frequency

This study uses the Hilbert–Huang transform (HHT), a signal analysis method for nonlinear and non-stationary processes, to separate signals of varying frequencies in a nonlinear system governed by the Lorenz equations. Similar to the Fourier series expansion, HHT decomposes a data time series into a sum of intrinsic mode functions (IMFs) using empirical mode decomposition (EMD). Unlike an infinite number of Fourier series terms, the EMD always yields a finite number of IMFs, whose sum is equal to the original time series exactly. Using the HHT approach, the properties of the Lorenz attractor are interpreted in a time–frequency frame. This frame shows that: (i) the attractor is symmetric for [Formula: see text] (i.e. invariant for [Formula: see text]), even though the signs on [Formula: see text] and [Formula: see text] are changed; (ii) the attractor is sensitive to initial conditions even by a small perturbation, measured by the divergence of the trajectories over time; (iii) the Lorenz system goes through “windows” of chaos and periodicity; and (iv) at times, a system can be both chaotic and periodic for a given [Formula: see text] value. IMFs are a finite collection of decomposed quasi-periodic signals, starting from the highest to lowest frequencies, providing detection of the lower frequency signals that may have otherwise been “hidden” by their higher frequency counterparts. EMD decomposes the original signal into a family of distinct IMF signals, the Hilbert spectra are a “family portrait” of time–frequency–amplitude interplay of all IMF members. Together with viewing the IMF energy, it is easy to discern where each IMF resides in the spectra in relation to one another. In this study, the majority of high amplitude signals appear at low frequencies, approximately 0.5–1.5. Although our numerical experiments are limited to only two specific cases, our HHT analyses of time–frequency, marginal spectra, and energy and quasi-periodicity of each IMF provide a novel approach to exploring the profound and phenomena-rich Lorenz system.

NONSTATIONARY TIME SERIES CONVOLUTION: ON THE RELATION BETWEEN THE HILBERT–HUANG AND FOURIER TRANSFORM

2013 ◽  
Vol 05 (01) ◽  
pp. 1350004 ◽  
Author(s):  
MAIK NEUKIRCH ◽  
XAVIER GARCIA
Keyword(s):  
Time Series ◽  
Fourier Transform ◽  
Fourier Series ◽  
Frequency Domain ◽  
Response Function ◽  
Intrinsic Mode Functions ◽  
Hilbert Huang Transform ◽  
Time Frequency ◽  
Time Domain Response ◽  
The Imf

The Hilbert–Huang Transform (HHT) decomposes time series into intrinsic mode functions (IMF) in time-frequency domain. We show that time slices of IMFs equal time slices of Fourier series, where the instantaneous parameters of the IMF define the parameters amplitude and phase of the Fourier series. This leads to the formulation of the theorem that nonstationary convolution of an IMF with a general time domain response function translates into a multiplication of the IMF with the respective spectral domain response function which is explicitly permitted to vary over time. We conclude and show on a real world application that a de-trended signal's IMFs can be convolved independently and then be used for further time-frequency analysis. Finally, a discussion is opened on parallels in HHT and the Fourier transform with respect to the time-frequency domain.

scholarly journals ON INVESTIGATING EMD PARAMETERS TO SEARCH FOR GRAVITATIONAL WAVES

2013 ◽  
Vol 05 (02) ◽  
pp. 1350010 ◽  
Author(s):  
HIROTAKA TAKAHASHI ◽  
KEN-ICHI OOHARA ◽  
MASATO KANEYAMA ◽  
YUTA HIRANUMA ◽  
JORDAN B. CAMP
Keyword(s):  
Time Series ◽  
High Resolution ◽  
Time Series Analysis ◽  
Gravitational Waves ◽  
Basis Set ◽  
Hilbert Huang Transform ◽  
Time Frequency ◽  
Frequency Decomposition ◽  
Mode Decomposition ◽  
Optimal Values

The Hilbert–Huang transform (HHT) is a novel, adaptive approach to time series analysis. It does not impose a basis set on the data or otherwise make assumptions about the data form, and so the time-frequency decomposition is not limited by spreading due to uncertainty. Because of the high resolution of the time-frequency, we investigate the possibility of the application of the HHT to the search for gravitational waves. It is necessary to determine some parameters in the empirical mode decomposition (EMD), which is a component of the HHT, and in this paper we propose and demonstrate a method to determine the optimal values of the parameters to use in the search for gravitational waves.

Ensembles of the Lorenz attractor

1993 ◽  
Vol 441 (1912) ◽  
pp. 291-312 ◽  
Keyword(s):  
Closed System ◽  
Temporal Evolution ◽  
Initial Conditions ◽  
Lorenz Attractor ◽  
Present Formulation ◽  
Lorenz Equations ◽  
Averaged Equations ◽  
Sensitive Dependence ◽  
Large Ensembles ◽  
The Individual

The temporal evolution of small and large ensembles is examined for the Lorenz equations. It is shown that a general closed system of ensemble averaged equations can be developed for the Lorenz equations, independent of the size of the ensemble. The closure is based on the method of Rothmayer & Black (1993). The present formulation gives a deterministic set of equations for the description of ensembles of the Lorenz attractor. Large ensemble solutions of the strange attractor computed in this study are found to be regular, without the sensitive dependence on initial conditions which characterizes the individual chaotic members of the ensemble.

scholarly journals Numerical Counterexamples of Lorenz System in Implicit Time Scheme

2018 ◽  
Author(s):  
Xuan Chen
Keyword(s):  
Lorenz System ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Explicit Scheme ◽  
Implicit Time ◽  
Lorenz Equations ◽  
Consistent System ◽  
Runge Kutta Method ◽  
Self Consistent ◽  
The One

In nonlinear self-consistent system, Lorenz system (Lorenz equations) is a classic case with chaos solutions which are sensitively dependent on the initial conditions. As it is difficult to get the analytical solution, the numerical methods and qualitative analytical methods are widely used in many studies. In these papers, Runge-Kutta method is the one most often used to solve these differential equations. However, this method is still a method based on explicit time scheme, which would be the main reason for the chaotic solutions to Lorenz system. In this work, numerical experiments based on implicit time scheme and explicit scheme are setup for comparison, the results show that: in implicit time scheme, the numerical solutions (counterexamples) are without chaos; for an original volume, the volume shrinks exponentially fast to 0 in common.

scholarly journals Instantaneous Granger Causality with the Hilbert-Huang Transform

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
João Rodrigues ◽  
Alexandre Andrade
Keyword(s):  
Time Series ◽  
Granger Causality ◽  
Fourier Transforms ◽  
Theoretical Background ◽  
Hilbert Huang Transform ◽  
Time Frequency ◽  
Error Covariance ◽  
Mode Decomposition ◽  
Frequency Representation ◽  
Limited Frequency

Current measures of causality and temporal precedence have limited frequency and time resolution and therefore may not be viable in the detection of short periods of causality in specific frequencies. In addition, the presence of nonstationarities hinders the causality estimation of current techniques as they are based on Fourier transforms or autoregressive model estimation. In this work we present a combination of techniques to measure causality and temporal precedence between stationary and nonstationary time series, that is sensitive to frequency-specific short episodes of causality. This methodology provides a highly informative time-frequency representation of causality with existing causality measures. This is done by decomposing each time series into intrinsic oscillatory modes with an empirical mode decomposition algorithm and, subsequently, calculating their complex Hilbert spectrum. At each time point the cross-spectrum is calculated between time series and used to measure coherency and compute the transfer function and error covariance matrices using the Wilson-Burg method for spectral factorization. The imaginary part of coherency can then be computed as well as several Granger causality measures in the previous matrices. This work covers the most important theoretical background of these techniques and tries to prove the usefulness of this new approach while pointing out some of its qualities and drawbacks.

scholarly journals Fourier Series Approximations toJ2-Bounded Equatorial Orbits

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Wei Wang ◽  
Jianping Yuan ◽  
Yanbin Zhao ◽  
Zheng Chen ◽  
Changchun Chen
Keyword(s):  
Fourier Series ◽  
Initial Conditions ◽  
Polar Angle ◽  
Elliptic Functions ◽  
Dynamical Analysis ◽  
Fourier Series Expansion ◽  
Elementary Functions ◽  
Mathematical Functions ◽  
Jacobian Elliptic Functions ◽  
Closed Form Solutions

The current paper offers a comprehensive dynamical analysis and Fourier series approximations ofJ2-bounded equatorial orbits. The initial conditions of heterogeneous families ofJ2-perturbed equatorial orbits are determined first. Then the characteristics of two types ofJ2-bounded orbits, namely, pseudo-elliptic orbit and critical circular orbit, are studied. Due to the ambiguity of the closed-form solutions which comprise the elliptic integrals and Jacobian elliptic functions, showing little physical insight into the problem, a new scheme, termed Fourier series expansion, is adopted for approximation herein. Based on least-squares fitting to the coefficients, the solutions are expressed with arbitrary high-order Fourier series, since the radius and the flight time vary periodically as a function of the polar angle. As a consequence, the solutions can be written in terms of elementary functions such as cosines, rather than complex mathematical functions. Simulations enhance the proposed approximation method, showing bounded and negligible deviations. The approximation results show a promising prospect in preliminary orbits design, determination, and transfers for low-altitude spacecrafts.

scholarly journals Time–frequency time–space LSTM for robust classification of physiological signals

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Tuan D. Pham
Keyword(s):  
Time Series ◽  
Network Architecture ◽  
Time Series Data ◽  
Short Term Memory ◽  
Physiological Signals ◽  
Series Data ◽  
Time Frequency ◽  
Time Space ◽  
Deep Recurrent Neural Network

AbstractAutomated analysis of physiological time series is utilized for many clinical applications in medicine and life sciences. Long short-term memory (LSTM) is a deep recurrent neural network architecture used for classification of time-series data. Here time–frequency and time–space properties of time series are introduced as a robust tool for LSTM processing of long sequential data in physiology. Based on classification results obtained from two databases of sensor-induced physiological signals, the proposed approach has the potential for (1) achieving very high classification accuracy, (2) saving tremendous time for data learning, and (3) being cost-effective and user-comfortable for clinical trials by reducing multiple wearable sensors for data recording.

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