Symplectic geometry spectrum analysis of nonlinear time series

Xie, Huimin; Guo, Tianruo; Sivakumar, Bellie; Liew, Alan Wee‐Chung; Dokos, Socrates

doi:10.1098/rspa.2014.0409

Cited by 34 publications

(19 citation statements)

References 49 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Once such critical points are identified, one could study the properties of the local flow equations of the symplectic gradient of the Hamiltonian function numerically to gain insight into the nature of the stability of these critical points. Already in other contexts, for example, in applications of the theory of chaotic kinematics to oceanographic and atmospheric sciences, condensed matter, particle, accelerator and plasma physics, and also in string theory, symplectic geometry has proven to be a useful tool [139][140][141][142][143][144].…”

Section: Discussionmentioning

confidence: 99%

An approach to stability analyses in general relativity via symplectic geometry

Kocherlakota

Joshi

2019

Arab. J. Math.

View full text Add to dashboard Cite

We begin with a review of the statements of non-linear, linear and mode stability of autonomous dynamical systems in classical mechanics, using symplectic geometry. We then discuss what the Arnowitt-Deser-Misner (ADM) phase space and the ADM Hamiltonian of general relativity are, what constitutes a dynamical system, and subsequently present a nascent attempt to draw a formal analogy between the notions of stability in these two theories. We wish to note that we have not discussed here the construction of the reduced phase space by forming the quotient space of the constrained phase space with the gauge orbits. Our approach here is pedagogical and geometric, and the motivation is to unify and simplify a formal understanding of the statements regarding the stability of stationary solutions of general relativity. That is, typically the governing equations of motion of a Hamiltonian dynamical system are simply the flow equations of the associated symplectic Hamiltonian vector field, defined on phase space, and the non-linear stability analysis of its critical points have simply to do with the divergence of its flow there. Further, the linear stability of a critical point is related to the properties of the tangent flow of the Hamiltonian vector field.In the second half of this work, we posit that a study of the genericity of a particular black hole or naked singularity spacetime forming as an endstate of gravitational collapse is equivalent to an inquiry of how sensitive the orbits of the symplectic Hamiltonian vector field of general relativity are to changes in initial data. We demonstrate this by conducting a restricted non-linear stability analysis of the formation of a Schwarzschild black hole, working in the usual initial value formulation of general relativity. * k.prashant@tifr.res.in † psjprovost@charusat.ac.in arXiv:1902.08219v2 [gr-qc]

show abstract

Section: Discussionmentioning

confidence: 99%

An approach to stability analyses in general relativity via symplectic geometry

Kocherlakota

Joshi

2019

Arab. J. Math.

View full text Add to dashboard Cite

show abstract

“…The components corresponding to smaller eigenvalues are regarded to relate primarily to the less important components or noise in the time series. The analysis of eigenvalues are also called as the symplectic geometry spectrums analysis (SGSA) [6,18,19]. The corresponding components are also regarded as symplectic geometry mode decomposition (SGMD) [7,8,20,21].…”

Section: Symplectic Principal Component Analysis (Spca) Of a Time Seriesmentioning

confidence: 99%

Symplectic Geometry and Its Applications on Time Series Analysis

Lei¹

2021

Structure Topology and Symplectic Geometry

View full text Add to dashboard Cite

This chapter serves to introduce the symplectic geometry theory in time series analysis and its applications in various fields. The basic concepts and basic elements of mathematics relevant to the symplectic geometry are introduced in the second section. It includes the symplectic space, symplectic transformation, Hamiltonian matrix, symplectic principal component analysis (SPCA), symplectic geometry spectrum analysis (SGSA), symplectic geometry mode decomposition (SGMD), and symplectic entropy (SymEn), etc. In addition, it also briefly reviews the applications of symplectic geometry on time series analysis, such as the embedding dimension estimation, nonlinear testing, noise reduction, as well as fault diagnosis. Readers who are familiar with the mathematical preliminaries may omit the second section, i.e. the theory part, and go directly to the third section, i.e. the application part.

show abstract

“…Note that the wavelet transform in step (1) can be replaced by other time-frequency representation methods, such as short time Fourier transform, continuous wavelet transform, wavelet packet transform, S transform, multivariate empirical mode decomposition, and sympelctic geometry spectrum analysis, among others [1,[26][27][28][29]. However, the manner to construct the multi-scale matrices is the same, arranging those coefficients at the same scale or sub-band into one matrix.…”

Section: Ms2d 2 Pcamentioning

confidence: 99%

Multiscale Two-Directional Two-Dimensional Principal Component Analysis and Its Application to High-Dimensional Biomedical Signal Classification

Xie

Zhou

Guo

et al. 2016

IEEE Trans. Biomed. Eng.

Self Cite

View full text Add to dashboard Cite

Goal: Time-frequency analysis incorporating the wavelet transform followed by principal component analysis (WT-PCA) has been a powerful approach for the analysis of biomedical signals such as electromyography (EMG), electroencephalography (EEG), electrocardiography (ECG), and Doppler ultrasound. Time-frequency coefficients at various scales were usually transformed into a one-dimensional array using only a single or a few signal channels. The steady improvement of biomedical recording techniques has increasingly permitted the registration of a high number of channels. However, WT-PCA is not applicable to high-dimensional recordings due to the curse of dimensionality and small sample size problem. In this study, we present a multi-scale two-directional two-dimensional principal component analysis (MS2D 2 PCA) method for the efficient and effective extraction of essential feature information from high-dimensional signals. Multi-scale matrices constructed in the first step incorporate the spatial correlation and physiological characteristics of sub-band signals among channels. In the second step, the two-directional two-dimensional principal component analysis operates on the multi-scale matrices to reduce the dimension, rather than vectors in conventional PCA. Results are presented from an experiment to classify 20 hand movements using 89-channel EMG signals recorded in stroke survivors, which illustrates the efficiency and effectiveness of the proposed method for high-dimensional biomedical signal analysis.

show abstract

Symplectic geometry spectrum analysis of nonlinear time series

Cited by 34 publications

References 49 publications

An approach to stability analyses in general relativity via symplectic geometry

An approach to stability analyses in general relativity via symplectic geometry

Symplectic Geometry and Its Applications on Time Series Analysis

Multiscale Two-Directional Two-Dimensional Principal Component Analysis and Its Application to High-Dimensional Biomedical Signal Classification

Contact Info

Product

Resources

About