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Computer Engineering and Intelligent Systems www.iiste.org
ISSN 2222-1719 (Paper) ISSN 2222-2863 (Online)
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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 n
f (θ ; n) = + ∑ {ak cos(kθ ) + bk sin(kθ )} (1.1)
2π π k =1
When n = 1
1 1
f (θ ;1) = + ( a1 cos θ + b1 sin θ ) represents the Cardioid distribution.
2π π
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The 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 1
where θ , µ ∈ [−π , π ) and − <ρ<
2 2
(1.3)
(θ + 2ρ sin (θ − µ ) + 2 ρ sin µ )
1
F (θ ) =
2π
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:
d2y 1 1 + 2 ρ cos µ ρ sin µ
The 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 2π
The 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 of
Cardioid distribution for
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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 π
Hence
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 ) ) .
1
Conveniently we write this equation as f (θ ) =
2π
3. Cauchy Type Distributions Using Mobius Transformation On Cardioid Distribution
Ahlfors (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 + d
ad − bc ≠ 0 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
Cz + C with Im(C) ≠ 0.
T (z) = , (3.1)
z +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
 sin θ   θ  which is real.
x = T (θ ) = u + v   = u + v tan   ,
(3.2)
 1 + cos θ  2
Cz + C
Hence the Mobius transformation defined by T ( z ) = , maps every point on the Unit circle onto
z +1
the 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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(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  
2
has 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 ) .
2π
By 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  
  
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ISSN 2222-1719 (Paper) ISSN 2222-2863 (Online)
Vol 2, No.8, 2011
and
1) 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 obtained
from 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
1
scale parameter v .When u = 0 and v = 1, we get f ( x ) = , which is standard Cauchy
π 1 + x 2 
 
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 < ρ < 0.5.
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.
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Computer Engineering and Intelligent Systems www.iiste.org
ISSN 2222-1719 (Paper) ISSN 2222-2863 (Online)
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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 Model
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