I'd like to know if anyone has a link/article/book on relationship between Levy,Cauchy and Gaussian Distributions. In my research on an evolutionary algorithm I used these distributions for candidate solution generation and on some benchmarks Cauchy Distribution led to much better results. I want to know what characteristics of Gaussian, Cauchy and Levy Distributions I can use to justify the results.
Thanks in advance.
Hossein,
The papers below are good starting points for literature on the subject:
X. Yao, Y. Liu, G. Lin, "Evolutionary programming made faster," IEEE Trans. on Evolutionary Computation, vol. 3, no.2, pp. 82-102, 1999.
http://dx.doi.org/10.1109/4235.771163
X. Yao, Y. Liu, G. Lin, "Evolutionary programming using mutations based on Levi probability distribution," IEEE Trans. on Evolutionary Computation, vol. 8, no.1, pp. 1-24, 2004.
http://dx.doi.org/10.1109/TEVC.2003.816583
Richer, T.J.; Blackwell, T.M., "The Lévy Particle Swarm," Evolutionary Computation, 2006. CEC 2006. IEEE Congress on , vol., no., pp.808,815, 16-21 July 2006
http://dx.doi.org/10.1109/CEC.2006.1688394
http://dx.doi.org/10.1109/4235.771163
http://dx.doi.org/10.1109/TEVC.2003.816583
http://dx.doi.org/10.1109/CEC.2006.1688394
Try "the distribution in nature and entropy principle" and my book "entropy God's dice game.
http://arxiv.org/abs/0907.4852
Dear Hossein,
If you want t use the distributions in meta heuristic algorithms, I suggest that you look into section 10.3 of following book:
Yang, X. S. (2010). Engineering optimization: an introduction with metaheuristic applications. John Wiley & Sons.
Levy flights are a random walk whose step length is drawn from the Levy distribution, Levy distribution is a distribution of the sum of N identically and independently distribution random variables whose Fourier transform takes the following form.
The normal and Cauchy distributions are special cases of Levy distribution with beta (the parameter of Levy distribution) equal to 2 and 1 respectively.
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