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Microstate and Omega Complexity Analyses of the Resting-state Electroencephalography

Microstate and Omega Complexity Analyses of the Resting-state Electroencephalography

Published on: June 15, 2018

Constructing 1/omegaalpha noise from reversible Markov chains.

Sveinung Erland¹, Priscilla E Greenwood

¹Department of Mathematics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|October 13, 2007

Summary

This summary is machine-generated.

This study identifies conditions for generating 1/omega^alpha noise from reversible Markov chains. The findings extend to hidden Markov chains and continuous state spaces, demonstrating long memory properties.

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Area of Science:

Probability Theory
Stochastic Processes
Signal Processing

Background:

1/omega^alpha noise, also known as pink noise, is prevalent in various natural and artificial systems.
Understanding its generation mechanisms is crucial for modeling and analysis in diverse fields.
Previous work has explored point process models for 1/omega noise.

Purpose of the Study:

To establish sufficient conditions for generating 1/omega^alpha noise from reversible Markov chains.
To extend these findings to hidden Markov chains and continuous state spaces.
To explore the relationship between Markov chain properties and noise characteristics.

Main Methods:

Analysis of reversible Markov chains on finite state spaces.
Application of eigendecomposition of the probability transition matrix.
Representation of covariance functions and spectral densities.
Generalization to hidden Markov chains and aggregations of AR1-processes.
Construction of noise in continuous state spaces.

Main Results:

Sufficient conditions for 1/omega^alpha noise output from reversible Markov chains are provided.
Examples exhibiting 1/omega^alpha noise in specific frequency ranges are constructed.
The results are shown to extend to hidden Markov chains and continuous state spaces, exhibiting long memory.
A random walk on a specific state space is shown to be complementary to a known 1/omega noise model.

Conclusions:

Reversible Markov chains provide a framework for generating 1/omega^alpha noise.
The spectral properties of Markov chains are directly linked to the generated noise characteristics.
The methodology allows for the construction of long-memory noise processes in both discrete and continuous settings.