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Computer Engineering and Intelligent Systems www.iiste.orgISSN 2222-1719 (Paper) ISSN 2222-2863 (Online)Vol 2, No.8, 2011 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.comReceived: 2011-10-23Accepted: 2011-10-29Published:2011-11-04AbstractJeffreys (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.1. IntroductionJeffreys (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 n f (θ ; n) = + ∑ {ak cos(kθ ) + bk sin(kθ )} (1.1) 2π π k =1When n = 1 1 1 f (θ ;1) = + ( a1 cos θ + b1 sin θ ) represents the Cardioid distribution. 2π π1|Pagewww.iiste.org
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Computer Engineering and Intelligent Systems www.iiste.orgISSN 2222-1719 (Paper) ISSN 2222-2863 (Online)Vol 2, No.8, 2011The probability density and distribution functions of Cardioid distribution are respectively given by 1 f (θ ; µ , ρ ) = (1 + 2 ρ cos(θ − µ ) ) (1.2) 2π 1 1where θ , µ ∈ [−π , π ) and − <ρ< 2 2 (1.3) (θ + 2ρ sin (θ − µ ) + 2 ρ sin µ ) 1 F (θ ) = 2πIn Section 2, we make certain assumptions on arbitrary constants in the general solution of a lineardifferential equation to get Cardioid distribution and Section 3 deals with the generation of Cauchy typedistributions from Cardioid model induced by Mobius transformation/stereographic projection.2. Cardioid distribution through a Differential equationBy making use of certain assumptions on arbitrary constants in the general solution of a differentialequation we construct the pdf of Cardioid model.Theorem 2.1: d2y 1 1 + 2 ρ cos µ ρ sin µThe solution of the initial value problem +y= , y (0) = , y′(0) = admits dθ 2 2π 2π π i) the particular integral which is pdf of Uniform distribution on Unit Circle and (1 + 2 ρ cos (θ − µ ) ) 1 ii) y (θ ) = which is probability density function of Cardioid 2π distribution where −π ≤ θ , µ < π and | ρ | < 0.5.Proof: Consider a non homogeneous second order linear differential equation with constant coefficients d2y 1 +y= . (2.1) dθ 2 2πThe Particular Integral of (2.1) admits pdf of circular uniform distribution 1 yp = . 2π (2.2)General solution of the above differential equation is 1 y (θ ) = C1 cos θ + C 2 sin θ + , 2π (2.3)where C1 and C2 are arbitrary constants.Under the following initial conditions the above solution (2.3) also admits probability density function ofCardioid distribution for2|Pagewww.iiste.org
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Computer Engineering and Intelligent Systems www.iiste.orgISSN 2222-1719 (Paper) ISSN 2222-2863 (Online)Vol 2, No.8, 2011 1 + 2 ρ cos µ ρ sin µ y (0) = , y′ (0) = . 2π π (2.4)From (2.3), ρ cos µ ρ sin µ C1 = and C2 = . (2.5) π πHence 1 y (θ ) = (1 + 2 ρ cos µ cos θ + 2 ρ sin µ sin θ ) 2π (1 + 2ρ cos (θ − µ ) ) . 1 = (2.6) 2π (1 + 2ρ cos (θ − µ ) ) . 1Conveniently we write this equation as f (θ ) = 2π3. Cauchy Type Distributions Using Mobius Transformation On Cardioid DistributionAhlfors (1966) defined Mobius transformation as follows az + b“The transformation of the form w = T ( z ) = , where a, b, c and d are complex constants such that cz + dad − bc ≠ 0 is known as Bilinear transformation or Linear fractional transformation or Mobiustransformation or stereographic projection”.Minh and Farnum (2003) imposed certain restrictions on parameters a, b, c and d in T(Z) and arrived at thefollowing Cz + C with Im(C) ≠ 0. T (z) = , (3.1) z +1Where 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 sin θ θ which is real. x = T (θ ) = u + v = u + v tan , (3.2) 1 + cos θ 2 Cz + CHence the Mobius transformation defined by T ( z ) = , maps every point on the Unit circle onto z +1the real line.Form (3.2), we have θ x = u + v tan 2 x −u θ 3|Page ⇒ = tan www.iiste.org v 2 x −u ⇒ T −1 ( x ) = θ = 2 tan −1 . v
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Computer Engineering and Intelligent Systems www.iiste.orgISSN 2222-1719 (Paper) ISSN 2222-2863 (Online)Vol 2, No.8, 2011 (3.3)which maps every point on the real line onto the Unit Circle and the mapping is a bijection . f ( x ) = g (θ ( x ) ) 2 , when v > 0 (3.4) x − u 2 v 1 + v Theorem 3.1 : θ If θ follows Cardiod Distribution in [-π, π), then T (θ ) = x = u + v tan 2has a 2-parameter linear distribution on the real line given by x − u 2 1− f ( x) = 1 1 + 2 ρ v x − u 2 x − u 2 vπ 1 + 1+ v v Proof : If θ follows Cardioid Distribution with µ = 0 in [-π, π), then pdf g (θ ) is 1 g (θ ) = (1 + 2 ρ cos θ ) . 2πBy applying theorem of Minh and Farnum (2003), we have 2 f ( x) = g (θ ( x)) ,v > 0 x − u 2 v 1 + v 2 1 x − u = . 1 + 2 ρ cos 2 tan −1 x − u 2π 2 v v 1 + v x − u 2 1 − = 1 1 + 2 ρ v . (3.5) x − u 2 x − u 2 π v 1 + 1 + v v 4|Pagewww.iiste.org
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Computer Engineering and Intelligent Systems www.iiste.orgISSN 2222-1719 (Paper) ISSN 2222-2863 (Online)Vol 2, No.8, 2011and1) f ( x ) ≥ 0 ∀ − ∞ < x < ∞ ( since v > 0) ∞2) ∫ f ( x ) dx = 1. −∞is a family of distributions on the real line and are named by us as Cauchy type distributions obtainedfrom the circular model called Cardioid distribution induced by Mobious Transformation.When ρ = 0 in (3.5), we get 1 f ( x) = . x − u 2 (3.6) π v 1 + v which is the density function of the 2 – parameter Cauchy’s distribution with location parameter u and 1scale parameter v .When u = 0 and v = 1, we get f ( x ) = , which is standard Cauchy π 1 + x 2 distribution.4 GraphWe observe that the probability distribution on real line generated by using Mobius Transformation onCardioid Model is also Unimodal and symmetric for 0 < ρ < 0.5.Acknowledgements:We acknowledge Prof. (Retd.) I. Ramabhadra Sarma, Dept. of Mathematics, Acharya Nagarjuna Universityfor his suggestions which have helped in improving the presentation of this paper.ReferencesAhlfors, 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 ProbabilityDistributions, Communication in Statistics – Theory and Methods, 32, 1, pp. 1 – 9.5|Pagewww.iiste.org
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Computer Engineering and Intelligent Systems www.iiste.orgISSN 2222-1719 (Paper) ISSN 2222-2863 (Online)Vol 2, No.8, 2011 Graph of pdf of Cauchy Type model 0.12 ρ =0.05 ρ =0.1 0.1 ρ =0.2 ρ =0.3 ρ =0.4 0.08 f(x) 0.06 0.04 0.02 0 -50 -40 -30 -20 -10 0 10 20 30 40 50 x Figure – 1Graph of pdf of Cauchy Type Model6|Pagewww.iiste.org
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