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Wrapped Skew Laplace Distribution on Integers:A New Probability

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Wrapped Skew Laplace Distribution on Integers:A New Probability Open Journal of Statistics, 2012, 2, 106-114 http://dx.doi.org/10.4236/ojs.2012.21011 Published Online January 2012 (http://www.SciRP.org/journal/ojs) Wrapped Skew Laplace Distribution on Integers: A New Probability Model for Circular Data K. Jayakumar1, Sophy Jacob2 1University of Calicut, Kerala, India 2MES Asmabi College, Kerala, India Email: {jkumar19, sophyjacob}@rediffmail.com Received September 19, 2011; revised October 21, 2011; accepted October 31, 2011 ABSTRACT In this paper we propose a new family of circular distributions, obtained by wrapping discrete skew Laplace distribution on Z = 0, ±1, ±2, around a unit circle. In contrast with many wrapped distributions, here closed form expressions exist for the probability density function, the distribution function and the characteristic function. The properties of this new family of distribution are studied. Keywords: Circular Distributions; Trigonometric Moments; Wrapped Laplace Distribution 1. Introduction son, wrapped Cauchy are discussed in [9]. We give a brief description of circular distribution in Section 2. In Circular data arise in various ways. Two of the most com- Section 3 we introduce and study Wrapped Discrete mon correspond to circular measuring instruments, the Skew Laplace Distribution. Section 4 deals with the es- compass and the clock. Data measured by compass usu- timation of the parameters using the method of moments. ally include wind directions, the direction and orienta- tions of birds and animals, ocean current directions, and 2. Circular Distributions orientation of geological phenomena like rock cores and fractures. Data measured by clock includes times of arri- A circular distribution is a probability distribution whose val of patients at a hospital emergency room, incidences total probability is concentrated on the circumference of of a disease throughout the year, where the calendar is a circle of unit radius. Since each point on the circum- regarded as a one-year clock. Circular or directional data ference represents a direction, it is a way of assigning also arise in many scientific fields, such as Biology, Ge- probabilities to different directions or defining a direc- ology, Meteorology, Physics, Psychology, Medicine and tional distribution. The range of a circular random vari- Astronomy [1]. able Θ measured in radians, may be taken to be 0, 2π Study on directional data can be dated back to the 18th orπ, π . century. In 1734 Daniel Bernoulli proposed to use the Circular distributions are of two types: they may be resultant length of normal vectors to test for uniformity discrete - assigning probability masses only to a count- of unit vectors on the sphere [2]. In 1918 von Mises in- able number of directions, or may be absolutely continuous. troduced a distribution on the circle by using characteri- In the latter case, the probability density function f θ zation analogous to the Gauss characterization of the exists and has the following basic properties. normal distribution on a line [2]. Later, interest was re- 1) f θ 0 newed in spherical and circular data by [3-5]. 2π Circular distributions play an important role in model- 2) f θθd1 ing directional data which arise in various fields. In re- 0 3) f θ f θ 2πk , for any integer k. That is cent years, several new unimodal circular distributions f θ is periodic with period 2π . capable of modeling symmetry as well as asymmetry have been proposed. These include, the wrapped versions Wrapped Distributions of skew normal [6], exponential [7] and Laplace [8]. Wrapped distributions provide a rich and useful class One of the common methods to analyze circular data is of models for circular data. known as wrapping approach [10]. In this approach, The special cases of the wrapped normal, wrapped Pois- given a known distribution on the real line, we wrap it Copyright © 2012 SciRes. OJS K. JAYAKUMAR ET AL. 107 around the circumference of a circle with unit radius. ||x Technically this means that if X is a random variable on 1 e , &x 0 fx (6) 2 ||x the real line with distribution function F x , the ran- 1 e , &x 0 dom variable X w of the wrapped distribution is defined by where, 0 , are inserted into Equation (5), the prob- XXw mod 2π (1) ability mass function of the resulting discrete distribution takes an explicit form in terms of the parameters p* = and the distribution function of X w is given by 1 e and q* e . Fw θ F θ 2πkF 2πk , k (2) Definition 3.1 A random variable Y has a discrete * k 0, 1, 2, skew Laplace distribution with parameters p (0,1) and q* (0,1) denoted by DSL pq**, , if By this approach, we are accumulating probability over all the overlapping points x θθ,2 π,4θπ, fk PYk So if g θ represents a circular density and f x the *** 11pq pk,0,1,2, (7) density of the real valued random variable, we have ** 1 pq qk* , 0,1,2, g θ f θ 2πk , (3) k The characteristic function of Y is given by 02θ π 1(1)pq** By this technique, both discrete and continuous ttR, (8) wrapped distributions may be constructed. In particular, 1(1)pe**it qeit if X has a distribution concentrated on the points k In this paper, we study the probability distribution ob- x , k 0, 1, 2,and m is an integer, the prob- 2πm tained by wrapping discrete skew Laplace distribution on Z 0,,1, 2 around a unit circle. ability function of X w is given by As we know, reduction modulo 2π wraps the line onto 2πr Pr X pr km the circle, reduction modulo 2πm (if m is a positive in- w th m k (4) teger) wraps the integers onto the group of m root of 1, r 0,1, , m 1 regarded as a subgroup of the circle. That is, if X is a random variable on the integers, then Θ, defined by where “p” is the probability function of the random vari- able X. 2 πX (mod2πm) 2πr 3. Wrapped Discrete Skew Laplace is a random variable on the lattice , rm0,1, , 1 Distribution m on the circle. The probability function of Θ is given by 3.1. Discrete Skew Laplace Distribution Equation (4). In particular, if X has a discrete skew Laplace distribu- Discrete Laplace distribution was introduced by [11] tion with parameters p* and q* , then the probability following [12], who defined a discrete analogue of the 2πr normal distribution. The probability mass function of a distribution of the wrapped random variable is general Discrete Normal random variable Y can be writ- m ten in the form given by fk 2πr PYk , PprkmrmΘ , 0,1, , 1 m k f j . (5) rkm j 11pqq k 0, 1, 2, k 1 pq 11pq where “f” is the probability density function of a normal pr distribution with mean µ and variance 2 [13]. 1 pq Using Equation (5), for any continuous random vari- 11)pqp( rkm able X on R, we can define a discrete random variable Y 1 pq that has integer support on Z. When the skew Laplace k density where pp * mod 2π m and Copyright © 2012 SciRes. OJS 108 K. JAYAKUMAR ET AL. qq * mod 2π m rmpq0,1, , 1 and , 0,1 and we denote it by 1 1(1)pqrkm rr()km WDSL pqm, , . qpp 1 pq kk 1 Following are the graphs of wrapped discrete skew 11pq Laplace distribution for various values of κ, σ and m. In 11ppmrpm1 q m (9) Figure 1, the graph of the pdf of wrapped discrete skew 1 pq Laplace distribution for κ = 0.25, σ = 1 and for m = 5, 10, pqmr 1 m ,r0,1, , m 1 20, 30, 40, 50 and 100 are given. In Figure 2, the graph of the pdf of wrapped discrete mr m r m 1(1)pqqp11pqskew Laplace distribution for κ = 0.5, σ = 1 and for m = 5, 1 pq 11pqmm10, 20, 30, 40, 50 and 100 are given. The graph of the pdf of wrapped discrete skew Laplace distribution for κ = for rm0,1 , , 1 and pq,0,1 0.25, σ = 1 and for m = 5, 10, 20, 30, 40, 50 and 100 are Again, we have given in Figure 3. m1 pw θ 1 3.2. Special Cases r 0 When either “p” or “q” converges to zero, we obtain the Hence P represents a probability distribution. w following two special cases: ~WDSL (,0,) p m with Definition 3.2 An angular random variable “Θ” is p 0,1 is a wrapped geometric distribution with said to follow wrapped skew Laplace distribution on in- probability mass function tegers with parameters p, q and m if its probability mass r 2πr 1 pp function is given by (11) PrmΘ m , 0,1, , 1. mr m r m m 1 p 11pqqp11pq pw θ (10) while ~(0,,)WDSL q m with q 0,1 is a wrapped 1 pq 11pqmm geometric distribution with probability mass function Figure 1. Wrapped discrete skewed Laplace distribution for σ = 1 , κ = 0.25 and for different values of “m”. Copyright © 2012 SciRes. OJS K. JAYAKUMAR ET AL. 109 Figure 2. Wrapped discrete skewed Laplace distribution for σ = 1 , κ = 0.25 and for different values of “m”. Figure 3. Wrapped discrete skewed Laplace distribution for σ = 1 , κ = 0.25 and for different values of “m”. Copyright © 2012 SciRes. OJS 110 K. JAYAKUMAR ET AL. r 2πr 1 qq mr m r m k qppq11 PΘ m ,0r ,1,, m 1. (12) 1(1)pq mq1 F θ mm r0 1 pq 11pq when p = q, we have (14) mr r mk m m k1 2πr 1(pp p ) qppqqp1 1 (1 )(1 )(1 ) PΘ m , mp1(1)p (13) 11(1)pq pmm q rm0,1, , 1 rm 0,1, , 1 .
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