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Linear Approximation in Time Domain01:21

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Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
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The frequency-domain technique, commonly used in analyzing and designing feedback control systems, is effective for linear, time-invariant systems. However, it falls short when dealing with nonlinear, time-varying, and multiple-input multiple-output systems. The time-domain or state-space approach addresses these limitations by utilizing state variables to construct simultaneous, first-order differential equations, known as state equations, for an nth-order system.
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The Discrete-Time Fourier Series (DTFS) is a fundamental concept in signal processing, serving as the discrete-time counterpart to the continuous-time Fourier series. It allows for the representation and analysis of discrete-time periodic signals in terms of their frequency components. Unlike its continuous counterpart, which utilizes integrals, the calculation of DTFS expansion coefficients involves summations due to the discrete nature of the signal.
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Complex network approach to fractional time series.

Pouya Manshour1

  • 1Physics Department, Persian Gulf University, Bushehr 75169, Iran.

Chaos (Woodbury, N.Y.)
|November 2, 2015
PubMed
Summary

The horizontal visibility algorithm effectively maps stochastic time series onto complex networks, revealing correlations. Topological network properties, including node degree and assortativity, can predict the Hurst exponent for fractional processes.

Area of Science:

  • Complex networks
  • Time series analysis
  • Statistical physics

Background:

  • Visibility graph algorithms map time series to complex networks for correlation analysis.
  • The standard visibility algorithm has limitations in capturing correlation aspects.

Purpose of the Study:

  • To evaluate the suitability of visibility graph algorithms for time series correlation analysis.
  • To introduce and apply the horizontal visibility algorithm for mapping fractional processes onto complex networks.
  • To investigate the relationship between network topology and the Hurst exponent of time series.

Main Methods:

  • Mapping fractional processes onto complex networks using the horizontal visibility algorithm.
  • Analyzing degree distributions and their parabolic exponential forms.

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  • Calculating network topological properties like maximum eigenvalue and degree assortativity.
  • Employing Spearman correlation coefficient to link network properties with time series data.

    • Main Results:

    • Degree distributions of mapped fractional processes exhibit Hurst-dependent parabolic exponential forms.
    • Network topological quantities, including maximum eigenvalue and degree assortativity, can predict the Hurst exponent.
    • An exception was noted for anti-persistent fractional Gaussian noises, addressed by incorporating Spearman correlation.

    Conclusions:

    • The horizontal visibility algorithm is a viable tool for analyzing correlations in stochastic time series.
    • Network topological properties offer predictive power for the Hurst exponent in fractional processes.
    • Spearman correlation refines the analysis for specific cases like anti-persistent fractional Gaussian noises.