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 α < 0.3490 and otherwise), the two distributions match well, particularly for ≤ α ≤ 21 . 1 INTRODUCTION equations to link the parameters in the Levy distribution and Tsallis distribution so that they can Researchers have conducted many studies on be approximated to each other. Various examples computational methods that are motivated by natural show that they are quite similar, but not identical. evolution [1-6]. These methods can be divided into The Levy stable process can not only be used in three main groups: genetic algorithms (GAs), simulated annealing, evolutionary algorithms, as a evolutionary programming (EP), and evolutionary model for many types of physical and economic strategies (ESs). All of these groups use various systems, it also has quite amazing applications in mutation methods to intelligently search the science and nature. In the case of animal foraging, promising region in the solution domain. Based food search patterns can be quantitatively described upon these mutation methods, researchers often use as resembling the Levy process. For example, three types of mutation variate to produce random researchers have studied reindeer, wandering mutation: Gaussian, Cauchy and Levy variates. albatrosses, and bumblebees and found that their Gaussian and Cauchy variates are special cases of random walk resembles Levy flight behavior (see the Levy process. Lee et al. (Lee and Yao, 2004) example in Viswanathan et al. (Viswanathan and introduced the Levy process, used Mantegna’s Afanasyev, etal, 2000), Edwards et al. (Edwards and algorithm (Mangetna, 1994) to produce the Levy Philips et al, 2007)). The strength of Levy flight in variate, and showed that the algorithm is useful for animal foraging is obvious, as it helps foragers find Levy’s kurtosis parameter α > 0.7 . Iwamatsu food and survive in severe environments. generated the Levy variate of the Levy-type distribution, which is just an approximation, using the algorithm proposed by Tsallis and Stariolo 2 THEORETICAL DEPLOYMENT (Iwamatsu, 2002). Iwamatsu’s contribution is the usage of Tsallis and Stariolo’s algorithm to generate In probability theory, a Lévy skew alpha-stable the Tsallis variate and apply it to the mutation in the distribution or even just a stable distribution is a four evolutionary programming. The Tsallis variate is not parameter family of continuous probability the Levy stable process, but is very similar. The distributions. The parameters are classified as paper first introduces the stable process and Tsallis location and scale parameters μ and c, and two shape distribution. Equations show that these two parameters β and α, which roughly correspond to distributions are generally different, but are identical measures of skewness and kurtosis, respectively. for two special distributions, i.e. the normal and The stable distribution has the important property of Cauchy distributions. This section also provides two 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 RELATIONSHIP BETWEEN LEVY DISTRIBUTION AND TSALLIS DISTRIBUTION ∞ parameters, a stochastic variable, which is a linear 1 α . (7) f (x;α,0,c,0) = Lα ,γ (x) = exp(-γt ) cos(tx)dt combination of several independent variables with π ∫ 0 stable distribution, has the same stable distribution. This equation is identical to in Lee’s and The Lévy skew stable probability distribution is Lα ,γ (y) defined by the Fourier transform of its characteristic Mantegan’s paper, though the current study changes function ϕ(t) (Voit, 2003) the variable y to x. 1 ∞ When α =2, the stable process in Eq. (3) becomes a f (x;α, β,c, μ) = ϕ(t)e −itx dt (1) 2π ∫ −∞ normal distribution. Using the characteristic function of a normal distribution with a zero mean and a variance of σ 2 (Papoulis, 1990), which is where ϕ(t) is defined as: 1 σ 2t 2 ϕ(t) = exp(− 1 ) , it is easy to show that the variance α 2 ϕ (t) = exp (itu − ct ()1 − iβ sgn( t)Φ ) (2) 2 2 σ 1 of Eq. (3) is 2c . As for the Cauchy distribution ( α =1), its characteristics function is where sgn(t) is just the sign of t, and Φ is given ϕ(t) = exp(− ct ) and the corresponding probability by density function is Φ = tan(πα / 2) 1 1 . (8) g2 (x) = 2 for all α except α = 1, in which case: cπ 1+ (x / c) Φ = − 2/π log t . The Tsallis distribution (Tsallis and Stariolo, 1996) () in one dimension is written as follows Note that the range of each parameter is the kurtosis ⎛ 1 ⎞ Γ⎜ ⎟ 1/ 2 ⎜ ⎟ −1/(3−q) .(9) α ∈(0, 2] , the skewness β ∈ −1, 1 , the scale ⎛ q −1⎞ ⎝ q −1⎠ T [] g(x;q,T) = ⎜ ⎟ ⎝ π ⎠ ⎛ 1 1 ⎞ 2 1/(q−1) c > 0 , and the location μ ∈(−∞,∞) . Assuming Γ⎜ − ⎟ ⎧ x ⎫ ⎜ ⎟ ⎨1+ (q −1) 2 /(3−q) ⎬ ⎝ q −1 2 ⎠ ⎩ T ⎭ that the distribution is symmetric ()β = 0 , the center of its location is zero ()μ = 0 , then Eq. (2) can be Note that the ranges of parameters q and T are simplified as q ∈[1,3) and T > 0 , respectively. The follow α section investigates the relationship between the ϕ (t) = exp (- ct ). (3) parameters α , c of f (x;α,0,c,0) in Eq. (1) and the q and T of g(x;q,T) in Eq. (9). Inserting Eq. (3) into (1) produces + ∞ According to Iwamatsu, when q →1 , the Tsallis 1 α −itx f (x;α,0,c,0) = exp(- ct )e dt . (4) distribution becomes a normal distribution 2π ∫ −∞ + 1 2 Let , (10) g1 (x;q →1 ,T) = exp(−(x /σ ) ) πσ 2 α γ = c (5) and when q =2, it becomes the Cauchy distribution and using the Euler formula 1 1 g (x;2,T ) = ⋅ . (11) eiθ = cosθ + isinθ (6) 2 πσ 1+ (x /σ ) 2 and considering only the real part of Eq. (6), it is 1/(3−q) easy to show that Note that σ = T is a scale parameter, and is not the usual meaning of standard deviation in a 361 ICEIS 2010 - 12th International Conference on Enterprise Information Systems normal distribution. The scale parameter σ is a function of q and T , and with different q it has which describes the probability density in Eq. (7) with scale parameter c =1（implying γ =1, through different function forms of T . For example, if Eq. (5)）at x = 0 . Recall that when x = 0 , the q =1, then σ = T , whereas if q =2, then σ = T . probability density for Tsallis distribution renders The true standard deviation of the normal 1 T ⎛ 1 ⎞ distribution in Eq. (10) is , which 1/ 2 Γ⎜ ⎟ σ 1 = σ = ⎜ ⎟ 2 2 ⎛ q −1⎞ ⎝ q −1⎠ −1/(3−q) . (17) gq (x = 0) = ⎜ ⎟ T renders the standard form of normal distribution as ⎝ π ⎠ ⎛ 1 1 ⎞ Γ⎜ − ⎟ ⎝ q −1 2 ⎠ 1 1 g (x;q →1+ ,T ) = exp(− (x /σ )2 ) .(12) Combining Eq. (16) and (17) leads to 1 σ 2π 2 1 1 ⎛ 1 ⎞ 1/ 2 Γ⎜ ⎟ As indicated above, the variance of normal Γ()1/α ⎛ q −1⎞ ⎝ q −1⎠ −1/(3−q) . (18) = ⎜ ⎟ T distribution as a special case of Levy distribution is πα ⎝ π ⎠ ⎛ 1 1 ⎞ 2 ⎜ ⎟ 2c and the variance of normal distribution as a Γ⎜ − ⎟ ⎝ q −1 2 ⎠ special case of Tsallis distribution is T . Therefore, 2 Equation (18) gives another constraint between if the two normal distributions are identical, the parameter α in Eq. (7) and parameters q and T parameters between the Levy distribution and Tsallis in Eq. (9) when γ =1. Since this equation (18) is distribution must satisfy the following constraint, derived from the special case of γ =1, this study proposes a general model between parameters α 2 T and γ in Eq. (7) and parameters q and T in Eq. 2c = . (13) 2 (9) as follows By the same token, apply the equality of the Cauchy ⎛ 1 ⎞ 1/ 2 Γ⎜ ⎟ distribution and compare Eq. (8) and (11). It is clear Γ()1/α ⎛ q −1⎞ ⎝ q −1⎠ −1/(3−q) . (19) = ⎜ ⎟ T that πα(γ 1/α ) ⎝ π ⎠ ⎛ 1 1 ⎞ Γ⎜ − ⎟ c = σ = T . (14) ⎝ q −1 2 ⎠ Note that when γ =1, Eq. (19) reduces to Eq. (18). Equations (13) and (14) establish the link between Therefore, by combining Eq. (5), (15) and (19) and parameter c of the Levy stable process in Eq. (7) making some substitution in the parameters, this and the parameters q and T of the Tsallis study obtains two equations to define the distribution in Eq. (9) for the special cases of normal relationship between (α,γ ) and ()q,T as (α =2, q =1) and Cauchy distributions (α =1, q =2). αγ 1/α = T 1/(3−q) Since this is derived only from special cases of α =1 ⎛ 1 ⎞ ⎜ ⎟ or 2, this study proposes a general model between 1/ 2 Γ⎜ ⎟ .