Radon-Nikodym derivatives in case of rational spectral densities

2006 ◽  
pp. 1-15
Author(s):  
Mátyás Arató
Keyword(s):  
Spectral Densities

Transformation of non-Gaussian random processes, signals and noise in the nonlinear differential systems by the method of cumulative functions

2018 ◽  
Vol 15 (1) ◽  
pp. 84-93
Author(s):  
V. I. Volovach ◽  
V. M. Artyushenko
Keyword(s):  
Spectral Characteristics ◽  
Random Processes ◽  
Spectral Densities ◽  
Differential Systems ◽  
Non Gaussian ◽  
Nonlinear Differential Systems

Reviewed and analyzed the issues linked with the torque and naguszewski cumulant description of random processes. It is shown that if non-Gaussian random processes are given by both instantaneous and cumulative functions, it is assumed that such processes are fully specified. Spectral characteristics of non-Gaussian random processes are considered. It is shown that higher spectral densities exist only for non-Gaussian random processes.

scholarly journals Gaussian processes with skewed Laplace spectral mixture kernels for long-term forecasting

2021 ◽  
Author(s):  
Kai Chen ◽  
Twan van Laarhoven ◽  
Elena Marchiori
Keyword(s):  
Gaussian Processes ◽  
Multivariate Time Series ◽  
Heavy Tail ◽  
Lottery Ticket ◽  
Spectral Densities ◽  
Heavy Tailed ◽  
Long Term Forecasting ◽  
The Time Domain ◽  
Spectral Mixture

AbstractLong-term forecasting involves predicting a horizon that is far ahead of the last observation. It is a problem of high practical relevance, for instance for companies in order to decide upon expensive long-term investments. Despite the recent progress and success of Gaussian processes (GPs) based on spectral mixture kernels, long-term forecasting remains a challenging problem for these kernels because they decay exponentially at large horizons. This is mainly due to their use of a mixture of Gaussians to model spectral densities. Characteristics of the signal important for long-term forecasting can be unravelled by investigating the distribution of the Fourier coefficients of (the training part of) the signal, which is non-smooth, heavy-tailed, sparse, and skewed. The heavy tail and skewness characteristics of such distributions in the spectral domain allow to capture long-range covariance of the signal in the time domain. Motivated by these observations, we propose to model spectral densities using a skewed Laplace spectral mixture (SLSM) due to the skewness of its peaks, sparsity, non-smoothness, and heavy tail characteristics. By applying the inverse Fourier Transform to this spectral density we obtain a new GP kernel for long-term forecasting. In addition, we adapt the lottery ticket method, originally developed to prune weights of a neural network, to GPs in order to automatically select the number of kernel components. Results of extensive experiments, including a multivariate time series, show the beneficial effect of the proposed SLSM kernel for long-term extrapolation and robustness to the choice of the number of mixture components.

scholarly journals Rational spectral densities and strong mixing

1992 ◽  
Vol 42 (2) ◽  
pp. 267-283 ◽  
Author(s):  
R Cheng
Keyword(s):  
Strong Mixing ◽  
Spectral Densities

An imaging system detectivity metric using energy and power spectral densities

2016 ◽  
Author(s):  
Bradley L. Preece ◽  
David Haefner ◽  
Georges Nehmetallah
Keyword(s):  
Imaging System ◽  
Spectral Densities ◽  
Power Spectral ◽  
Power Spectral Densities

Experimental determination of spectral densities in quasi-one-dimensional electron systems

1998 ◽  
Vol 249-251 ◽  
pp. 175-179
Author(s):  
B. Kardynal ◽  
C.H.W. Barnes ◽  
E.H. Linfield ◽  
D.A. Ritchie ◽  
J.T. Nicholls ◽  
...  
Keyword(s):  
Experimental Determination ◽  
Electron Systems ◽  
Dimensional Electron ◽  
Spectral Densities ◽  
One Dimensional

Power Spectral Density of Nonlinear System Response: The Recursion Method

1993 ◽  
Vol 60 (2) ◽  
pp. 358-365 ◽  
Author(s):  
R. Vale´ry Roy ◽  
P. D. Spanos
Keyword(s):  
Broad Class ◽  
Formal Solution ◽  
Power Series Expansion ◽  
Kolmogorov Equation ◽  
System Response ◽  
Lanczos Algorithm ◽  
Spectral Densities ◽  
Recursion Method ◽  
Approximate Form ◽  
Single Well

Spectral densities of the response of nonlinear systems to white noise excitation are considered. By using a formal solution of the associated Fokker-Planck-Kolmogorov equation, response spectral densities are represented by formal power series expansion for large frequencies. The coefficients of the series, known as the spectral moments, are determined in terms of first-order response statistics. Alternatively, a J-fraction representation of spectral densities can be achieved by using a generalization of the Lanczos algorithm for matrix tridiagonalization, known as the “recursion method.” Sequences of rational approximations of increasing order are obtained. They are used for numerical calculations regarding the single-well and double-well Duffing oscillators, and Van der Pol type oscillators. Digital simulations demonstrate that the proposed approach can be quite reliable over large variations of the system parameters. Further, it is quite versatile as it can be used for the determination of the spectrum of the response of a broad class of randomly excited nonlinear oscillators, with the sole prerequisite being the availability, in exact or approximate form, of the stationary probability density of the response.

Sign in / Sign up

Export Citation Format

Share Document