Probability density of the empirical wavelet coefficients of a noisy chaos

This study presents a stochastic approach for the analysis of nonchaotic, chaotic, random and nonchaotic, random and chaotic, and random dynamics of a nonlinear system. The analysis utilizes a Markov process approximation, direct numerical simulations, and a generalized stochastic Melnikov process. The Fokker-Planck equation along with a path integral solution procedure are developed and implemented to illustrate the evolution of probability density functions. Numerical integration is employed to simulate the noise effects on nonlinear responses. In regard to the presence of additive ideal white noise, the generalized stochastic Melnikov process is developed to identify the boundary for noisy chaos. A mathematical representation encompassing all possible dynamical responses is provided. Numerical results indicate that noisy chaos is a possible intermediate state between deterministic and random dynamics. A global picture of the system behavior is demonstrated via the transition of probability density function over its entire evolution. It is observed that the presence of external noise has significant effects over the transition between different response states and between co-existing attractors.

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On the probability density function of the derotated phase of complex wavelet coefficients

2008 IEEE International Symposium on Circuits and Systems ◽

10.1109/iscas.2008.4542061 ◽

2008 ◽

Cited By ~ 1

Author(s):

Yothin Rakvongthai ◽

Soontorn Oraintara

Keyword(s):

Probability Density Function ◽

Probability Density ◽

Density Function ◽

Wavelet Coefficients ◽

Complex Wavelet

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Nonparametric estimation of the quadratic functional of multimodal probability density

Metrologiya ◽

10.32446/0132-4713.2019-3-17-2910.32446/ ◽

2019 ◽

pp. 17-29

Author(s):

А. V. Lapko ◽

V. A. Lapko

Keyword(s):

Probability Density ◽

Nonparametric Estimation ◽

Quadratic Functional

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Modified algorithm for fast bandwidth selection for kernel estimates of multidimensional probability densities

Izmeritel`naya Tekhnika ◽

10.32446/0368-1025it.2020-11-9-13 ◽

2020 ◽

pp. 9-13

Author(s):

A. V. Lapko ◽

V. A. Lapko

Keyword(s):

Probability Density ◽

Optimal Parameter ◽

Random Variables ◽

Kernel Functions ◽

Independent Random Variables ◽

Functional Dependencies ◽

Approximation Properties ◽

Probability Density Estimation ◽

Multidimensional Probability ◽

Selection Of

An original technique has been justified for the fast bandwidths selection of kernel functions in a nonparametric estimate of the multidimensional probability density of the Rosenblatt–Parzen type. The proposed method makes it possible to significantly increase the computational efficiency of the optimization procedure for kernel probability density estimates in the conditions of large-volume statistical data in comparison with traditional approaches. The basis of the proposed approach is the analysis of the optimal parameter formula for the bandwidths of a multidimensional kernel probability density estimate. Dependencies between the nonlinear functional on the probability density and its derivatives up to the second order inclusive of the antikurtosis coefficients of random variables are found. The bandwidths for each random variable are represented as the product of an undefined parameter and their mean square deviation. The influence of the error in restoring the established functional dependencies on the approximation properties of the kernel probability density estimation is determined. The obtained results are implemented as a method of synthesis and analysis of a fast bandwidths selection of the kernel estimation of the two-dimensional probability density of independent random variables. This method uses data on the quantitative characteristics of a family of lognormal distribution laws.

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