Jeffreys (1961) introduced Cardioid distribution and used it to modeling directional spectra of ocean waves.Here an attempt is made to derive pdf of cardioid model as a solution of a second order non homogeneouslinear differential equation having constant coefficients with certain initial conditions. We also arrive atnew unimodal and symmetric distribution on real line from Cardioid model induced by Mobiustransformation called Cauchy type models.Keywords: Circular model, Mobius transformation, Cardioid and Uniform distributions, Cauchy typemodels

Rao AV, Dattatreya

SVS, Girija

Y, Phani

English

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Computer Engineering and Intelligent Systems    www.iiste.org ISSN 2222-1719 (Paper) ISSN 2222-2863 (Online) Vol 2, No.8, 2011   1 | P a g e  www.iiste.org  Differential Approach to Cardioid Distribution Dattatreya Rao AV Department of Statistics, Acharya Nagarjuna University Guntur, India E-mail: avdrao@gmail.com    Girija SVS (Corresponding Author)  Department of Mathematics, Hindu College Guntur, India E-mail: svs.girija@gmail.com    Phani Y  Department of Mathematics, Swarnandhra College of Engineering and Technology Seetharampuram – 534 280, Narasapur, India   E-mail: phaniyedlapalli23@gmail.com   Received: 2011-10-23  Accepted: 2011-10-29 Published:2011-11-04  Abstract Jeffreys (1961) introduced Cardioid distribution and used it to modeling directional spectra of ocean waves. Here an attempt is made to derive pdf of cardioid model as a solution of a second order non homogeneous linear differential equation having constant coefficients with certain initial conditions. We also arrive at new unimodal and symmetric distribution on real line from Cardioid model induced by Mobius transformation called “Cauchy type models”. Keywords: Circular model, Mobius transformation, Cardioid and Uniform distributions, Cauchy type models.  1. Introduction Jeffreys (1961) introduced Cardioid distribution and used it to modeling directional spectra of ocean waves. Following  Fejer’s theorem in Fernandez (2006) we may define a family of circular distributions by                     (1.1)    When n = 1  ( )1 11 1(  ;1) cos sin2f a bθ θ θpi pi= + +  represents the Cardioid distribution. { }11 1( ; ) cos( ) sin( )  2nk kkf n a k b kθ θ θpi pi== + +∑Computer Engineering and Intelligent Systems    www.iiste.org ISSN 2222-1719 (Paper) ISSN 2222-2863 (Online) Vol 2, No.8, 2011   2 | P a g e  www.iiste.org  The probability density and distribution functions of Cardioid distribution are respectively given by                     (1.2)  where              and                      (1.3)  In Section 2, we make certain assumptions on arbitrary constants in the general solution of a linear differential equation to get Cardioid distribution and Section 3 deals with the generation of Cauchy type distributions from Cardioid model induced by Mobius transformation/stereographic projection.  2. Cardioid distribution through a Differential equation By making use of certain assumptions on arbitrary constants in the general solution of a differential equation we construct the pdf of Cardioid model. Theorem 2.1:  The solution of the initial value problem ( ) ( )221 1 2 cos sin, 0 , 02 2d y y y ydρ µ ρ µpi pi piθ+′+ = = =  admits  i) the particular integral which is pdf of Uniform distribution on Unit Circle and ii) ( ) ( )( )1 1 2 cos2y θ ρ θ µpi= + −  which is probability density function of Cardioid distribution  where              and  Proof: Consider a non homogeneous second order linear differential equation with constant coefficients                     (2.1)  The Particular Integral of (2.1) admits pdf of circular uniform distribution                     (2.2)  General solution of the above differential equation  is                       (2.3) where C1 and C2 are arbitrary constants. Under the following initial conditions the above solution (2.3) also admits probability density function of Cardioid distribution for ( )1(  ; , ) 1 2 cos( )2f θ µ ρ ρ θ µpi= + −,  [ , )θ µ pi pi∈ −1 12 2ρ− < <( ) ( )( )1 2 sin 2 sin2F θ θ ρ θ µ ρ µpi= + − +,pi θ µ pi− ≤ < | | 0.5.ρ <221.2d y yd piθ+ =1.2pypi=( ) 1 2 1cos sin ,2y C Cθ θ θ pi= + +Computer Engineering and Intelligent Systems    www.iiste.org ISSN 2222-1719 (Paper) ISSN 2222-2863 (Online) Vol 2, No.8, 2011   3 | P a g e  www.iiste.org                      (2.4)    From (2.3),             and         (2.5)   Hence   (2.6)  Conveniently we write this equation as   3. Cauchy Type Distributions Using Mobius Transformation On Cardioid Distribution   Ahlfors (1966) defined Mobius transformation as follows “The transformation of the form ( ) ,  az bw T zcz d+= =+where a, b, c and d are complex constants such that 0ad bc− ≠ is known as Bilinear transformation or Linear fractional transformation or Mobius transformation or stereographic projection”.  Minh and Farnum (2003) imposed certain restrictions on parameters a, b, c and d in T(Z) and arrived at the following            with Im(C) 0.≠       (3.1)  Where  C = u –i v and C = u + i v.  The Mobius transformation defined by (3.1) is a real-valued for any z on the Unit Circle               which is real.   (3.2)  Hence the Mobius transformation defined by ( ) ,1Cz CT zz+=+ maps every point on the Unit circle onto the real line. Form (3.2), we have   ( ) ( )1 2 cos sin0 , 0 .2y yρ µ ρ µpi pi+′= =1cosC ρ µpi= 2sin.C ρ µpi=( ) ( )( )( )1 1 2 cos cos 2 sin sin21 1 2 cos .2y θ ρ µ θ ρ µ θpiρ θ µpi= + += + −( ) ( )( )1 1 2 cos .2f θ ρ θ µpi= + −( ) ,1Cz CT zz+=+( ) sin tan ,1 cos 2x T u v u vθ θθθ   = = + = +   +   ( )1 1tan2tan22 tan .x u vx uvx uT xvθθθ− − = +   −  ⇒ =   − ⇒ = =   Computer Engineering and Intelligent Systems    www.iiste.org ISSN 2222-1719 (Paper) ISSN 2222-2863 (Online) Vol 2, No.8, 2011   4 | P a g e  www.iiste.org                         (3.3) which maps every point on the real line onto the Unit Circle and the mapping is a bijection .             when v > 0       (3.4)  Theorem 3.1 :    If θ follows Cardiod Distribution in [-π, π), then ( ) tan2T x u v θθ  = = +    has a 2-parameter linear distribution on the real line given by ( )22211 1 211x uvf xx ux uvvvρpi  − −      = +   −−     + +               Proof :  If θ follows Cardioid Distribution with 0µ =  in [-π, π), then pdf ( )g θ is                      By applying theorem of Minh and Farnum (2003), we have                            (3.5)   ( ) ( )( ) 22 ,1f x g xx uvvθ= − +     ( ) ( )1 1 2 cos .2g θ ρ θpi= +2122222( ) ( ( )) , 012 1. 1 2 cos 2 tan2111 1 2 .11f x g x vx uvvx uvx uvvx uvx ux uvvvθρpiρpi−= > − +       −  = +       −   +       − −      = +   −−     + +             Computer Engineering and Intelligent Systems    www.iiste.org ISSN 2222-1719 (Paper) ISSN 2222-2863 (Online) Vol 2, No.8, 2011   5 | P a g e  www.iiste.org  ( )( )and1) 0 ( since 0)2) 1.f x x vf x dx∞−∞≥ ∀ − ∞ < < ∞ >=∫ is a family of distributions on the real line and are named by us as Cauchy type distributions obtained from the circular model called Cardioid distribution induced by Mobious Transformation. When 0ρ =  in (3.5), we get                  (3.6)   which is the density function of the 2 – parameter  Cauchy’s distribution with location parameteru and  scale parameter v .When u = 0 and v = 1, we get ( ) 211f x xpi=  + , which is standard Cauchy distribution. 4 Graph   We observe that the probability distribution on real line generated by using Mobius Transformation on Cardioid Model is also Unimodal and symmetric for 0 0.5.ρ< <  Acknowledgements:  We acknowledge Prof. (Retd.) I. Ramabhadra Sarma, Dept. of Mathematics, Acharya Nagarjuna University for his suggestions which have helped in improving the presentation of this paper. References Ahlfors, L.V. (1966), Complex Analysis, 2nd ed. New York, McGraw-Hill, 76 – 89. Fernandez-Duran,J.J.(2006), Models for Circular-Linear and Circular -Circular Data, Biometrics, 63, 2, pp. 579-585. Girija, S.V.S., (2010), New Circular Models, VDM – VERLAG, Germany. Jeffreys, H.,(1961), Theory of Probability, 3rd edition, Oxford University Press. Minh, Do Le & Farnum, Nicholas R. (2003), Using Bilinear Transformations to Induce Probability Distributions, Communication in Statistics – Theory and Methods, 32, 1, pp. 1 – 9. ( ) 21 .1f xx uvvpi= − +     Computer Engineering and Intelligent Systems    www.iiste.org ISSN 2222-1719 (Paper) ISSN 2222-2863 (Online) Vol 2, No.8, 2011   6 | P a g e  www.iiste.org     Figure – 1Graph of pdf of Cauchy Type Model   -50 -40 -30 -20 -10 0 10 20 30 40 5000.020.040.060.080.10.12xf(x)Graph of pdf of Cauchy Type model  ρ=0.05ρ=0.1ρ=0.2ρ=0.3ρ=0.4This academic article was published by The International Institute for Science, Technology and Education (IISTE).  The IISTE is a pioneer in the Open Access Publishing service based in the U.S. and Europe.  The aim of the institute is Accelerating Global Knowledge Sharing.  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Differential Approach to Cardioid Distribution

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Differential Approach to Cardioid Distribution

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