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 orπ, 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,4θπ, fk   PYk  So if g θ represents a circular density and f x the    *** 11pq 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 02θ 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 variable X.   2πX (mod2πm) 2πr 3. Wrapped Discrete Skew Laplace is a random variable on the lattice , rm0,1, , 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 PYk , PprkmrmΘ   , 0,1, , 1  m k f j  . (5)  rkm j 11pqq   k 0, 1, 2, k 1 pq

11pq 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

qq * mod 2π m rmpq0,1, , 1 and ,  0,1 and we denote it by 1 1(1)pqrkm rr()km WDSL  pqm, ,  . qpp 1 pq kk 1 Following are the graphs of wrapped discrete skew 11pq Laplace distribution for various values of κ, σ and m. In 11ppmrpm1 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)pqqp11pqskew Laplace distribution for κ = 0.5, σ = 1 and for m = 5,   1 pq 11pqmm10, 20, 30, 40, 50 and 100 are given. The graph of the pdf of wrapped discrete skew Laplace distribution for κ = for rm0,1 , , 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.

m1  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 1 ppr function is given by 2πr   (11) PrmΘ  m , 0,1, , 1. mr m r m m 1 p 11pqqp11pq 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”.

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”.

r 2πr 1 qq mr m r m k qppq11 PΘ m ,0r ,1,, m 1. (12) 1(1)pq   mq1 F θ      mm r0 1 pq  11pq  when p = q, we have   (14) mr r mk m m k1 2πr 1(pp p ) qppqqp1 1 (1 )(1 )(1 )  PΘ  m , mp1(1)p (13) 11(1)pq pmm q rm0,1, , 1 rm 0,1, , 1 . which is the probability mass function of wrapped discrete Laplace distribution. 3.4. Probability Generating Function and Characteristic Function of WDSL pqm, , 3.3. Distribution Function of WDSL pqm, ,  The probability generating function of WDSL  pqm,,  The distribution function, F θ is given by is given by 11pq m1  s  Psqmmr1( p )11p m q mmmr p (1)() q sp  mm     11pq p 1q r0  q  (15) mmm m m 11pq 1pqsq11sp q  11pq pmm1q qs1 sp  Also, we have P11 . where nm 0 (mod ) when s  einm2π we have If  t is the characteristic function of a linear random variable X, then the characteristic function of the in2π m 1(1pq) wrapped random variable, X , is n , for Pe  w 1 pe1in2π mn qei2π m n  0, 1, 2, [2]. Thus for the wrapped discrete skew Laplace distribution, we have

in nEe , for n  0, 1, 2, in2πr Er em , for 0,1, ,m 1 (16)  m1 11pq in2π r qpppqmr1 m r 1 m 1 m pqmr(1 m ) e m  mm r0 111pq p q On simplification it reduces to 11pq n , n 0, 1, 2， ,n  0 (mod m ) (17) 1epqinm2π  1einm2π

Again, we have nn2π 2π 111pq pqpq   cos  ipq sin  11pq mm  n in2π  in2π 22 nni mm nn2π 2π 1epq e  pq 1cossinpq p q  p  q  mm  where and n2π n2π 11pq1 pqpq cos 1(1)pqpq    sin    m m      n 22 n 2 2  nn2π  2π  nn2π 2π  1()cos()sinpq p q  p  q 1pq (p q )cos (p  q )sin( )    mm   mm  (18) (19)

Proposition 3.1 If Θ ~WDSL(,,)pqm then Θd ΘΘ1  2 11 pq   n where Θ and Θ are two independently distributed n j inm2π inm2π 1 2 1epq 1e wrapped g eometric random variables with probability 11 mass functions  inm2π inm2π r r 1ep 1eq 2πr pp1 2πr qq1   ,  PΘ1 =  m PΘ2 =  m 11pp11qq m 11p m 11q  11 Proof.  We have, ABeeinm2π  CDinm2π 11pq   n 1 1epqinm2π 1einm2π where A, B, C, D > 0, A – B = C – D = 1 and A  ,  1 p

nm0, 1, 2, , n 0 (mod ) p 1 q B ,,C D and 1, pqq 11 1p1qTherefore, nn ,  121epqinm2π 1einm2π  n nn12 nk Therefore, k  11  1(1)pq   inm2π  inm2π  nn() j1 (eAB )(e CD ) 12 inm2π i2πnm    1epq 1e k (eABinm2π )(e CD  inm2π )  3.5. Infinite Divisibility 3.6. Stability with Respect to Geometric We know that the geometric distribution with probability Summation mass function, PX k(1 u )k uu ,  0,1, is infinitely divisible, so wrapped geometric distribution is Proposition 3.2. Let 12,,  be identically and in- infinitely divisible. Hence Θ ~(,,)WDSLpqm is infi- dependently distributed WDSL (,,p q m ) angular ran- nitely divisible. By the well known factorisation proper- dom variables, and let Nu be a geometric random va- ties of geometric law [14], Θ ~WDSL (,,) p q m admits a riable with probability mass function (1 uu )k1 , representation involving wrapped negative binomial dis- k 1, 2, , independent of j s . Then the angular tribution, d,  k 1,2, where  Nu nn12 nk n j random variable  d (mod2π) has the WDSL d WW .  j n1n2 j1 Consider s, rm,  distribution with 2 p sq srand 2 p pq11  pqu  pq   11  p  qu  4 pq

Proof. Now we show that the above function coincides with

Let 1,2, be identically and independently distrib- the characteristic function of WDSL  s, r, m distri- uted an gular ran dom variables following WDSL (,,pqm ) bution with s and r as given above. Setting Equation (20) and Nu is a geometric random variable with mean 1 u . equal to ns, r , which is the characteristic function Conditio ning on the distribution of Nu we can write the of WDSL s, r, m distribution, produces the equation characteristic function of the right hand side of the above 1(1)sr equation as (Equation (20)) n (21) um, in2π  in2π i   mod 2π mm 1 Nu  i1 kk1 EEee1uu1esr (1e ) k 1   upqm,, (20) up11q    11u pqm,, in2π in2π 1e pmm1eq 1 upq 1 1 up11 q    inm2π in2π m 1epq 1e  1 upq 1 1 (22)

That is, Dividing Equation (23) by Equation (24), we get in2π in2π  sq mm r  . Substituting the value of r in Equation (23) we get  1es 1erpqu1 1 p   in22in 11pqusqps     11 sqpq (25) mm 11sr  1e p 1eq  On simplification, Equation (25) reduces to  qs2  s q p11 p  q u  p 0 (26)  1 upq1  1        2 That is, f sqssqp 11 p qup. which shou ld hold for each p, q. This will happen when the following two equations hold simultaneously. Since pp, q 0, f 00 , and fupq1110     . 11pqur11srq (23) Therefore, f s admits a unique solution in the in- 11pqus11srp (24) terval 0, 1 and is given by

2 qp11 p  qu  qp  11 p  qu  4 qp  2q (27) 2 p  2 qp11 p  qu  qp  11 p  qu  4 qp

Remark 3.1. Wrapped discrete skew Laplace distribu- is given by tion is infinitely divisible since a circular random vari- 11pq  able obtained by wrapping an infinitely divisible random ()n  in2π in2π  variable is infinitely divisible by [1,11] . mm 1epq1e  3.7. Trigonometric Moments The above expression can also be expressed in the th The n trigonometric moment of the WDSL  pqm, ,  form

n2π ()sinpq  m 1 i tan1  222 n2π 1cpq p q os n2π n2π m  np 11 q  1  pqp  qcospqsin e mm

in  ne where n 0,1 is the p th mean resultant length and and  n 0, 2π is the p th mean direction, for n  1,2, , n2π pq sin 2   11pq  1  pqpqnm   cos2π 1 m n    n  tan (29)  n2π 1 (28) 1cpq p q os 2 2 m pq sinn 2 m  The length of the mean resultant vector, 11 pq    22   111 22 nn2π  2π 1cossinpqp q  p  q  mm  and the mean direction, 2π pq sin  111 m 1 tan  tan  1 2π 1cpq p q os m

The circular variance, 11pq  V 11  0 22 nn2π 2π 1cpq p q os p  q sin mm The circular standard deviation,

22 nn2π 2π 1cpq p q os p  q sin mm 2ln  ln 0 2 11pq  The measure of skewness, Using Equations (33)-(34) and for a fixed value of “m” we can find estimates for “p” and “q”. 032 where E sin sin 2 12 V0 2  We have, The measure of kurtosis, 2π 1cospq p q 4  a1 m 0 2 (1V0 )    where 2 E cos 2 2 2  2π V b1  0 pq sin  m 4. Estimation That is Method of Moments 2π  2π  apq11sin b 1 pqpq cos mm  Let 1,, k be a random sample of size n taken from   the WDSL (,,pqm ) distribution with parameters p, q, which gives and m. Then the nth sample trigonometric moment about the zero direction, 2π 2π bq111cos aq sin  b mm  mann ib n (30) p  (35) 2π 2π where bq11 bcos a 1 sin  1 k mm  anj cosn (31) k j1 Substituting the value of “p” in terms of “q” in Equa- and tion (33) or in Equation (34) we will get an equation in 1 k “q” and solving that we can find the estimate of “q” and bnnj sin  (32) thus “p”. k j1 Corresponding population moment is ni   nn REFERENCES Equating the sample moments to the corresponding popula- [1] K. V. Mardia and P. E. Jupp, “Directional Statistics,” 2nd tion moments, we get nn a and nn b for n  1, 2, . Thus, we have Edition, Wiley, New York, 2001. [2] K. V. Mardia, “Statistics of Directional Data,” Academic 2π Press, London, 1972. 11pq1 pqpq   cos m [3] R. A. Fisher, “Dispersion on a Sphere,” Proceedings of a1  22(33) 2π 2π the Royal Society of London A, Vol. 217, No. 1130, 1953, 1cpq p q os p q si n  pp. 295-305. doi:10.1098/rspa.1953.0064 m m [4] J. A. Greenwood and D. Durand, “The Distribution of and Length and Components of the Sum of n Random Unit Vectors,” Annals of Mathematical Statistics, Vol. 26, No. 2π 2, 1955, pp. 233-246. doi:10.1214/aoms/1177728540 11pqpq  sin m [5] G. S. Watson and E. J. William, “On the Construction of b1  22 nn2π  2π Significance Tests on the Circle and the Sphere,” Biometrika, 1cpq p q os p  q sin  Vol. 43, 1956, pp. 344-352. mm  [6] A. Pewsey, “A Wrapped Skew—Normal Distribution on (34) the Circle,” Communications in Statistics: Theory and

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