Wrapped Hb-Skewed Laplace Distribution and Its Application in Meteorology

Biometrics & Biostatistics International Journal

Research Article Open Access Wrapped hb-skewed laplace distribution and its application in meteorology

Abstract Volume 7 Issue 6 - 2018 In this paper,we introduce a new circular distribution called Wrapped Holla and Jisha Varghese, KK Jose Bhattacharya’s (HB) Skewed Laplace distribution.We obtain explicit form for the Department of Statistics, Mahatma Gandhi University, India density and derive expressions for distribution function, characteristic function and trigonometric moments. Method of maximum likelihood estimation is used for Correspondence: KK Jose, Department of Statistics, estimation of parameters. The methods are applied to a data set which consists of St.Thomas College, Mahatma Gandhi University, the wind directions of a Black Mountain ACT and compared with that of wrapped Kottayam, Kerala 686 574, India, variance gamma distribution (WvG) and generalised von Mises (GvM) distribution. Email

Keywords: circular distribution, laplace distribution, hb skewed laplace, Received: Novmber 08, 2017 | Published: Novmber 20, 2018 trigonometric moments, meteorology, wind direction

Introduction study of circular distribution because of their wide applicability. Jammalamadakka & Kozubowski3 discussed circular distributions Circular or directional data arise in different ways. The two main obtained by wrapping the classical exponential and Laplace ways correspond to the two principal circular measuring instruments, distributions on the real line around the circle. Rao et al.,4 derived the compass and the clock. Typical observations measured by the new circular models by wrapping the well known life testing models compass include wind directions and directions of migrating birds. like log normal, logistic, Weibull and extreme-value distributions. Data of a similar type arise from measurements by spirit level or Roy & Adnan5 developed a new class of circular distributions namely protractor. Typical observations measured by the clock include the wrapped weighted exponential distribution. In another work, Roy arrival times (on a 24-hour clock) of patients at a casualty unit in & Adnan6 explored wrapped generalized Gompertz distribution a hospital. Data of a similar type arise as times of year or times of and discussed its application to Ornithology. Recently, Jacob & month of appropriate events. A circular observation can be regarded Jayakumar7 derived a new family of circular distribution by wrapping as a point on a circle of unit radius, or a unit vector (i.e. a direction) geometric distribution and studied its properties. Rao et al.,8 discussed in the plane. Once an initial direction and an orientation of the circle the characteristics of wrapped Gamma distribution. Adnan & Roy9 have been chosen, each circular observation can be specified by the derived wrapped variance Gamma distribution and showed its angle from the initial direction to the point on the circle corresponding applicability to wind direction. Joshi & Jose10 introduced a wrapped to the observation. Lindley distribution and applied it for a biological data. Recently, 11 Circular data are usually measured in degrees. Sometimes it Jammalamadakka & Kozubowski introduced a general approach to is useful to measure in radians. For more details see Mardia & obtain wrapped circular distributions through mixtures. The present Jupp.1 Circular Statistics is the branch of statistics that addresses paper is arranged as follows. In section 1, we give the general the modeling and inference from circular or directional data, i.e. introduction about circular distribution. Section 2 explains the basic data with rotating values. Many interesting circular models can be theory of circular distribution. Section 3 explains about the importance generated from known probability distributions by either wrapping of Laplace distribution. In section 4, we discuss about skewed Laplace a linear distribution around the unit circle or transforming a bivariate distribution. In section 5, we consider Holla and Bhattacharya’s (HB) linear r.v. to its directional component. Two dimensional directions Skewed Laplace (HBSL) distribution. In section 6, the Wrapped Holla can be represented as angles measured with respect to some suitably and Bhattacharya’s (HB) Skewed Laplace (WHBSL) distribution is chosen ”zero direction”. Since a direction has no magnitude, it can introduced and the probability density function (pdf) is obtained. In be conveniently represented as points on the circumference of a unit section 7, we derive the cumulative distribution function (cdf) and circle centered at the origin or as unit vectors connecting the origin in section 8, the characteristic function and trigonometric moments to these points. Because of this circular representation, observations are obtained. In Section 9, maximum likelihood estimation method on such two dimensional directions are called circular data is discussed to estimate the model parameter. In section 10, we apply Jammalamadakka & Sen Guptha.2 Directional data have many unique the proposed model to a data on wind direction compared with that and novel features both in terms of modeling and in their statistical of wrapped variance gamma distribution (WvG) and generalised treatment. In meteorology, wind directions provide a natural source von Mises (GvM) distribution by using Akaike information criterion of circular data. A distribution of wind directions may arise either (AIC), Bayesian information criterion (BIC) and log-likelihood. as a marginal distribution of the wind speed and direction, or as a Finally, Section 11 summarizes the findings. conditional distribution for a given speed. Other circular data arising in meteorology include the times of day at which thunderstorms occur Theory of circular distribution and the times of year at which heavy rain occurs. A circular distribution is a probability distribution whose total probability is concentrated on the circumference of a unit circle. Recently, statisticians have shown active interest in the

Submit Manuscript | http://medcraveonline.com Biom Biostat Int J. 2018;7(6):525‒530. 525 © 2018 Varghese et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and build upon your work non-commercially. Copyright: Wrapped hb-skewed laplace distribution and its application in meteorology ©2018 Varghese et al. 526

Since each point on the circumference represents a direction, such a Skewed version of Laplace distribution distribution is a way of assigning probabilities to different directions or defining a directional distribution. The range of a circular random Skewed - Laplace distributions are considered as a generalization variable (rv) θ , measured in radians, may be taken to be [0, 2 π ) or of the symmetric Laplace distribution. The discrete analogue of the [- π , π ). Circular distributions are essentially of two types: discrete symmetric Laplace distribution has been derived recently Kozubowski 11 and continuous. In continuous case, a probability density function et al., as in the case of normal distribution. The traditional models (pdf) f(θ )exists and has the following basic properties: based on normal distribution are very often not supported by real life data in many studies because of long tails and asymmetry present (if ) (θ )≥ 0; in these data. The symmetric Laplace distribution(being heavier θ tailed than the normal) and the asymmetric Laplace distribution can account for leptokurtic and skewed data. Thus Laplace distribution is 2π (ii ) fθθ d = 1. a natural, often superior alternative to the normal law. The Laplace ∫0 distributions find applications in diversified fields. The classical (iii ) f (θ ) = f ( θπ+ k .2 ) symmetric Laplace distribution is regarded as an appropriate model in areas such as ocean engineering and fracture problems. It has also for any integer k. been found to be a suitable model for the formation of sand dunes. Distribution function The asymmetric Laplace distributions are being used in modeling of financial data. There is a growing interest in stochastic modeling One way of specifying a distribution on the unit circle is by with Laplace distribution and its asymmetric extensions. The means of its distribution function. Let f(θ ) be the probability Laplace distribution is commonly encountered in image and speech density function of a continuous random variable Θ i.e.f(θ ) is a compression applications. This distribution is also applicable for wind non-negative 2 π periodic function such that f(θπ+ 2 ) = f(θ ) and shear data, error distributions in navigation, encoding and decoding of 2π ∫ fd(θθ ) =1. The distribution function F(θ ) can be defined over analog signals, inventory management and quality control, astronomy 0 θ and the biological and environment sciences. any interval (θθ, ) by F(θ )-F(θ )= 2 fd()θθ. Suppose that an 12 2 1 ∫θ 1 In the last several decades various forms of skewed Laplace initial direction and orientation of the unit circle have been chosen 0 distributions have appeared in the literature. Kozubowski & (generally 0 direction and anticlockwise orientation). Then F(θ ) is Nadarajah13 studied different forms of skewed Laplace distributions. θ defined as F(θ )=∫ fd( φφ ) obviously it holds that, F(2 π ) = 1. Mc Gill,14 Holla & Bhattacharya,15 Thomas Agnan & Poiraud- 0 Casanoa,16 Fernandez & Steel17 and Yu & Zhang18 proposed Laplace distribution different skewed Laplace distributions. Here we consider Holla and Bhattacharya’s Skewed Laplace distribution and derived its circular The Laplace distribution was first introduced by Pierre Simon density function, distribution function, characteristic function and Laplace in 1774 and is often called the ‘first law of errors’. A continuous trigonometric moments. random variable X is said to follow classical Laplace distribution with the parameters θ and λ , if its pdf is given by Johnson et al.12 Holla and Bhattacharya’s skewed Laplace 1|−−x θ | fx( ) = e ; −∞ 0. (1) distribution X 2λλ Mc Gill14 considered distributions with the pdf given by The random variable X following Laplace distribution (1) is denoted by L(θλ , ) , where the constantsθ and λ are known as the  1 x − β  exp  ,if x ≤ β , location and scale parameter respectively. The corresponding cdf is 2ψψ  obtained as fx( )= (5)  1 β − x exp ,if x > β . 1 −−|x θλ |/    ex;,≤ θ 2φφ 2 FxX ( )= (2) where −∞ 0 and ψ >0. These  −−|x θλ |/ 1− 1 / 2 ex; >θ . distributions are also known as the two-piece double exponential.  Holla and Bhattacharya’s skewed Laplace distribution is a variation of Mc Gill’s skewed Laplace distribution studied by Holla & A standard form of the pdf (1) is obtained by putting θλ= 0, = 1 Bhattacharya.15 The pdf of this distribution is given by and is given by  üüü exp{ (ββ−≥ )}; , 1  fy( ) = e−||y ; −∞ < y < ∞ . (3) fx( )= (6) Y 2  (1−−üüü ) exp{ (ββ )}; < . This form is sometimes called ‘Poisson’s first law of error’. Alternatively, the standard density function (3) is obtained by where −∞<0. and 0

Citation: Varghese J, Jose KK. Wrapped hb-skewed laplace distribution and its application in meteorology. Biom Biostat Int J. 2018;7(6):525‒530. DOI: 10.15406/bbij.2018.07.00255 Copyright: Wrapped hb-skewed laplace distribution and its application in meteorology ©2018 Varghese et al. 527

1 EXüüü β +− p c The variance is 3−+ 4pp ( 1) VX( )= c2 The n th moment is n n k! µβnk− × +−k − n' =∑  k pp( 1) (1 ) (7) k =0 k c Particular case 1 When p = , Holla and Bhattacharya’s skewed Laplace 2 distribution reduces to classical Laplace distribution. The pdf of Holla and Bhattacharya’s skewed Laplace distribution is given by (7) 1 For p = , (7) becomes, 2 1  cexp{ c (ββ−≥ x )}; if , 2 fx( )=  1 (1−− )c exp{ c ( xββ )}; if x < .  2 Figure 1 Density for p=0.5, c =1, β =− 0.25 ; Density for p=0.5, c =1, β It can be simplified as = 0.25 . 1 fx( ) = c exp{ cx |− β |} . 2 where −∞<<;x ∞ −∞<β <; ∞ c >0, which is the classical Laplace distribution. Wrapped Holla and Bhattacharya’s(HB) skewed Laplace distribution Let X follows Holla and Bhattacharya’s skewed Laplace distribution with pdf (7).Then the wrapped Holla and Bhattacharya’s skewed Laplace random variable is defined as θπ=X ( mod 2), such that θ ∈ [0, 2 π ).The pdf of the corresponding wrapped distribution is given by

∞ f(θ )=∑ fm ( θπ+ 2 ) m=−∞  ∞  pc exp{ c (βθ− )}∑ exp( − 2m π c ); if θ+≥ 2 π m β ,  m=β  =  (8)  β −1 (1−−p ) c exp{ c ( x β )} exp(2m π c ); if θπ+2 m < β .  ∑  m=−∞ According to Jammalamadakka and Kozubowski (2004, 2017), we can obtain the pdf of WHB skewed Laplace distribution as pccexp[ (βℵℵℵ θ 2 π 2 βπ )] (1pcc ) exp[ (θ β 2 βπ )] f (θ )= [exp(2πc )− 1] (9) where θ ∈ [0, 2 π ), −∞<<β ∞ , 0

0. Figure 2 Density for p=0.5, c = 1.5 , β = 0.2 ; Density for p=0.5, c = 1.5 The density plot for different values of the parameters are given in β − the following Figure 1–4. , = 0.2 .

Distribution function One way of specifying a distribution on the unit circle is by means of its cumulative distribution function (cdf). Let f(θ ) be the probability density function of a continuous random variable Θ . Hence f(θ ) is a non-negative periodic function with period 2 π such 2π that f(θπ+ 2 ) = f(θ ) and fd(θθ ) =1. The distribution function ∫0

F(θ ) can be defined over any interval(θθ12, ) by F(θ2 ) - F(θ1 ) = θ 2 fd()θθ. Suppose that an initial direction and orientation of the ∫θ1 0 unit circle have been chosen (generally 0 direction and anticlockwise orientation). Then F(θ ) is defined as

θ F()=θ fd() θθ ∫0 Obviously it follows that, F(2π ) = 1 The cdf of (10) is given by

θ F()=θ fd() θθ ∫0

θ pccexp[ (βℵℵℵ θ 2 π 2 βπ )] (1pcc ) exp[ (θ β 2 βπ )] = dθ ∫0 [exp(2πc )− 1] Figure 3 Density for p=0.75, c =1, β =0; Density for p=0.1, c = 0.5 , β =1− . On simplification, we get

pcexp[ (β+ 2 π − 2 βπ )][1 − exp( −cθ )] +− (1 p )exp[ c (2βπ − β )][exp( c θ ) − 1] F(θ )= [exp(2πc )− 1] (10) The survival function and hazard rate function can be obtained from the corresponding density function and distribution function. Characteristic function and trigonometric moments The characteristic function of a random angleθ is the doubly-

infinite sequence of complex numbers {φp :p = 0,± 1,...} given by itθ φt =(Ee )

2π = eitθ dF()θ ∫0

2π pccexp[ (βℵℵℵ θ 2 π 2 βπ )] (1pcc ) exp[ (θ β 2 βπ )] = editθ θ ∫0 [exp(2πc )− 1]

On simplification, we get pcexp[ c (β+− 2 π 2 βπ )][exp[(it − c )2π ] − 1] φ = t (it−− c )[exp(2π c ) 1]

(1−p ) c exp[ c ( −+β 2 βπ )][exp[(it + c )2π ] − 1] + (11) (it+− c )[exp(2π c ) 1] By the definition of trigonometric moments, we have

φαpp= + i β p; p =±± 1, 2,...

αθp =Ep (cos )

Figure 4 Density for p=0.5, c = 1.5 , β =0; Density for p=0.5, c = 0.2 , 2π üüü∫ pfθθθ d β = 0.25 . 0

∞ 2π pccexp[ (βℵℵℵ θ 2 π 2 βπ )] (1pcc ) exp[ (θ β 2 βπ )] 1 = cos pdθθf üüüüüüüℵℵℵ +∑ ppcos p +∈ sin p (14) ∫0 [exp(2πc )− 1] 2π  p=1 On simplification, we get Substituting the values of α p and β p given in (13) and (14) in c2 (15), we get an alternative expression for the PDF of the wrapped HB αp = {p exp[(1− 2πβ ) cp ] +− (1 )exp[(2π − 1) β c ]} (12) pc22+ skewed Laplace distribution.

Maximum likelihood estimation βθp =Ep (sin ) In this section, the maximum likelihood estimators of the unknown 2π parameters β of the WHB skewed Laplace distribution are üüü pfθθθ d (,,pc ) ∫0 derived. Let θθθ123, , ,... θn be a random sample of size n from WHB skewed Laplace distribution, then the the likelihood function is 2π pccexp[ (βℵℵℵ θ 2 π 2 βπ )] (1pcc ) exp[ (θ β 2 βπ )] = sin pdθθn ∫0 [exp(2πc )− 1] Lüüü∏ fbθi σβ On simplification, we get i=1 n pc pccexp[ (βℵℵℵ θii 2 π 2 βπ )] (1pcc ) exp[ (θ β 2 βπ )] β = {(p− 1)exp[(2π −− 1) βcp ] exp[(1 − 2πβ ) c ]} (13) =∏  p 22+ [exp(2πc )− 1] pc i=1 n 1 n According to Jammalamadaka and SenGupta (2001), an βℵℵℵ θ π βπ θ β βπ = 2π c ∑[ pccexp[ (ii 2 2 )] (1pcc ) exp[ ( 2 )]] alternative expression for the PDF of the wrapped distribution using e −1 i=1 the trigonometric moments is given by The log likelihood function is given by

n 2π c  logL=− n log[ e −+ 1] log∑{pc exp[ c (β−+ θii 2 π − 2 βπ )] +− (1p ) c exp[ c (θ −+ β 2 βπ )]} (15) i=1 The partial derivatives of the log likelihood with respect to p, c and Application in meteorology β are obtained as n The following application shows the effectiveness of Wrapped ∑eeccüüüβ− θii + π − βπ− θ−+ β βπ Holla and Bhattacharya’s Skewed Laplace (WHBSL) distribution, the ∂log L  = i=1 (16) data wind directions comprising hourly measurements of three days ∂p n ccüüüβ− θii + π − βπ +− θ−+ β βπ at a site on Black Mountain, ACT, Australia, was reported in Dr. M.A. ∑pe (1p ) e 19 i=1 Cameron and discussed in Fisher. The main goal was to provide a n regular monitoring of climate change into ACT. Climate change is c(β−+ θi 22) π − βπ 2π c ∑pe üüü+c β −+ θi π − βπ the greatest threat facing the world today, wind generated electricity ∂−logL 2π ne = + i=1 is one of a number of ways that we can reduce our reliance on fossil ∂ 2π cn− c (e 1) ccüüüβ− θii + π − βπ θ−+ β βπ fuel-generated electricity and therefore reduce our greenhouse gas ∑pce +−(1p ) ce i=1 production and limit climate change. The geography and dynamics n wind direction this area are important elements of climate system. This c(θi −+ β 2) βπ ∑(1−pe ) [1+c (θi −+ β 2 βπ )] study is important given the growing evidence of the ACT climate + i=1 change and biosphere to global change. The data (n= 22, degree) are n (17) ccüüüβ− θii + π − βπ θ−+ β βπ as given below (Table 1). ∑pce +−(1p ) ce i=1 Table 1 Wind direction data n ∑cüüü−π  peccüüüβ− θii + π − βπ −−p e θ−+ β βπ ∂log L  = i=1 0 15 90 150 182 220 235 ∂β n (18) ccüüüβ− θii + π − βπ θ−+ β βπ ∑pe +−(1p ) e 240 245 250 255 265 270 280 285 i=1 300 315 330 335 340 345 Inorder to estimate the parameters,we have to solve the normal Here we compute the estimates of the unknown parameters with equations respect to the three distributions, namely WHBSL (,cp ,β ), WvG ∂∂∂logLLL log log (,,,µλα βγ ,) and GvM (µµκκδ , , , ,) and obtain the values üü§üü§ü (19) 1212 ∂pc ∂∂β for loglikelihood, Akaike information criterion (AIC) and Bayesian information criterion (BIC). Table 2 provides the relevant numerical Since (20) cannot be solved analytically, numerical iteration summaries for the three fits with the goodness of fit. Based on these technique is used to get a solution for p,c and β . One may use the values, we can conclude that the WHBSL (,cp ,β ) distribution is nlm package in R software to get the maximum likelihood estimators comparatively better than the other two distributions in modeling the (MLE) for p,c and β . present data.20

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Citation: Varghese J, Jose KK. Wrapped hb-skewed laplace distribution and its application in meteorology. Biom Biostat Int J. 2018;7(6):525‒530. DOI: 10.15406/bbij.2018.07.00255