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RELATIONSHIP BETWEEN LEVY DISTRIBUTION
AND TSALLIS DISTRIBUTION
Jyhjeng Deng
Industrial Engineering & Technology Management Department, DaYeh University, Chuang-Hua, Taiwan
Keywords: Mutator, Stable process.
Abstract: This paper describes the relationship between a stable process, the Levy distribution, and the Tsallis
distribution. These two distributions are often confused as different versions of each other, and are
commonly used as mutators in evolutionary algorithms. This study shows that they are usually different, but
are identical in special cases for both normal and Cauchy distributions. These two distributions can also be
related to each other. With proper equations for two different settings (with Levy’s kurtosis parameter
<
0.3490 and otherwise), the two distributions match well, particularly for
21
.
1 INTRODUCTION
Researchers have conducted many studies on
computational methods that are motivated by natural
evolution [1-6]. These methods can be divided into
three main groups: genetic algorithms (GAs),
evolutionary programming (EP), and evolutionary
strategies (ESs). All of these groups use various
mutation methods to intelligently search the
promising region in the solution domain. Based
upon these mutation methods, researchers often use
three types of mutation variate to produce random
mutation: Gaussian, Cauchy and Levy variates.
Gaussian and Cauchy variates are special cases of
the Levy process. Lee et al. (Lee and Yao, 2004)
introduced the Levy process, used Mantegna’s
algorithm (Mangetna, 1994) to produce the Levy
variate, and showed that the algorithm is useful for
Levy’s kurtosis parameter
0.7>
. Iwamatsu
generated the Levy variate of the Levy-type
distribution, which is just an approximation, using
the algorithm proposed by Tsallis and Stariolo
(Iwamatsu, 2002). Iwamatsu’s contribution is the
usage of Tsallis and Stariolo’s algorithm to generate
the Tsallis variate and apply it to the mutation in the
evolutionary programming. The Tsallis variate is not
the Levy stable process, but is very similar. The
paper first introduces the stable process and Tsallis
distribution. Equations show that these two
distributions are generally different, but are identical
for two special distributions, i.e. the normal and
Cauchy distributions. This section also provides two
equations to link the parameters in the Levy
distribution and Tsallis distribution so that they can
be approximated to each other. Various examples
show that they are quite similar, but not identical.
The Levy stable process can not only be used in
simulated annealing, evolutionary algorithms, as a
model for many types of physical and economic
systems, it also has quite amazing applications in
science and nature. In the case of animal foraging,
food search patterns can be quantitatively described
as resembling the Levy process. For example,
researchers have studied reindeer, wandering
albatrosses, and bumblebees and found that their
random walk resembles Levy flight behavior (see
example in Viswanathan et al. (Viswanathan and
Afanasyev, etal, 2000), Edwards et al. (Edwards and
Philips et al, 2007)). The strength of Levy flight in
animal foraging is obvious, as it helps foragers find
food and survive in severe environments.
2 THEORETICAL DEPLOYMENT
In probability theory, a Lévy skew alpha-stable
distribution or even just a stable distribution is a four
parameter family of continuous probability
distributions. The parameters are classified as
location and scale parameters μ and c, and two shape
parameters β and α, which roughly correspond to
measures of skewness and kurtosis, respectively.
The stable distribution has the important property of
stability. Except for possibly different shift and scale
360
Deng J. (2010).
RELATIONSHIP BETWEEN LEVY DISTRIBUTION AND TSALLIS DISTRIBUTION.
In Proceedings of the 12th International Conference on Enterprise Information Systems - Artificial Intelligence and Decision Support Systems, pages
360-367
DOI: 10.5220/0003002103600367
Copyright
c
SciTePress