i-manager's Journal on Mathematics

View PDF

Volume :5 No :1 Issue :-2016 Pages :39-46

The Semicircular Reflected Gamma Distribution

Phani Yedlapalli *  S.V.S. Girija **  A.V. Dattatreya Rao ***
* Associate Professor, Department of Mathematics, Shri Vishnu Engineering College for Women, Bhimavaram, India.
** Associate Professor, Department of Mathematics, Hindu College, Guntur, India
*** Professor, Department of Statistics, Acharya Nagarjuna University, Guntur, India.

Abstract

Modeling of circular data with limited number of available circular models such as, von Mises, Wrapped Cauchy, Cardioid, etc., was done in various domains like Neuro Science, Geography, Archaeology, Remote Sensing, Spatial Analysis, Plant Phenology and Political Science. Dattatreya Rao (2007; 2011a;2011b;2011c; 2013a; 2013b; 2016) and Girija (2010; 2013a; 2013b; 2014a; 2014b), Phani (2012a; 2012b; 2013a; 2013; 2014; 2015a; 2015b; 2015c; 2015d), Radhika (2013a; 2013b; 2014;2015) and Devaraaj (2012; 2014) have introduced several new models and a few new methodologies of constructing the new circular models. These circular models are constructed by applying wrapping method, inverse stereographic projection, offsetting and the rising sun function. It is observed that, the simple projection method is not a much paid attention in constructing circular models. Glancing the literature, semicircular, arc and skewed angular data were observed in the applications and sufficient number of models for such data is not available. Motivated by these points, the authors have introduced semicircular reflected gamma distribution for modelling semicircular data by a simple projection method on reflected gamma distribution. The authors have extend it to the laxial reflected gamma distribution by a simple projection for modeling any arc of arbitrary length, and also the first four trigonometric moments has been derived for the proposed model.

Keywords :

  • Angular Data,
  • Characteristic Function,
  • Semicircular Model,
  • Stereographic Double Exponential Distribution,
  • Semicircular Laplace Distribution,
  • Stereographic Projection,
  • l-axial Data,
  • Projection Trigonometric Moments.

Introduction

Quite a lot of work was done on circular models defined on the unit circle (Jammalamadaka and Sengupta 2001); (Mardia and Jupp, 2000). In some cases, the directional/angular data do not require full circular models for modeling, this fact is noted in Jones (1968), Guardiola (2004), Byoung and Hyoung, (2008) and Phani et al. (2013a). For example, when sea turtles emerge from the ocean in search of a nesting site on dry land, a random variable having values on a semicircle is well sufficient for modeling such data. Similarly, when an aircraft is lost but its departure and its initial headings are known, a semicircular random variable is sufficient for such angular data. And few more examples of semicircular data are available in Ugai et.al, (1977).

Guardiola (2004) has obtained the semicircular normal distribution by using a simple projection and Byoung and Hyoung, (2008) have developed a family of semicircular Laplace distributions for modeling semicircular data by simple projection. Phani et al. (2013a) constructed some semicircular distributions by applying modified inverse stereographic projection. In this paper, the authors derive the Semicircular Reflected Gamma distribution by projecting Reflected Gamma distribution over a semicircular segment; this distribution generalizes semicircular Laplace distribution (Byoung and Hyoung, 2008). The authors have plot the graphs of the density function for various values of the parameters. Furthermore, the first four trigonometric moments for proposed model are derived and also they extend this model for l-axial Reflected Gamma distribution for modelling any arc of arbitrary length.

1. Methodology of Projection

Projection is defined by a one-to-one mapping given by,

(1)

where,

 

Suppose, x is randomly chosen on the interval (-∞ ,∞ ). Let F(x) and f(x) denote the cumulative distribution and probability density functions of the random variable x respectively. Then,

(2)

By Guardiola (2004), equation (2) is a random point on the semicircle. Let, G(θ) and g(θ) denote the cumulative distribution and probability density functions of this random point θ respectively. Then G(θ) and g(θ) can be written in terms of F(x) and f(x) using the following Theorem.

1.1 Theorem

For v > 0,

(3)
(4)

If a linear random variable X has a support on R, then θ has a support on and if X has support on R , then θ has a support on . These means that, after the projection is applied, we can deal semicircular data if the support of X is on R + and we can handle 4-axial circular data if the support of X is on R+ . Here v is not a parameter; therefore without loss of generality we can assume that v=1.

2. Semicircular Reflected Gamma Distribution

Here the authors recall the definition of reflected Gamma distribution.

2.1 Definition1

A random variable X on the real line is said to have reflected Gamma distribution with scale parameter λ>0, shape parameter c>0, location parameter α, if the probability density and distribution functions of X are respectively given by,

(5)

and

(6)

By applying simple projection defined by a one-to-one mapping, x= ν tan(θ), ν >0, which leads to a three parameter symmetric semicircular distribution on unit semicircle.

This distribution is called as Semicircular Reflected Gamma Distribution and it is denoted by SCRG (c,σ,µ).

2.2 Definition 2

A random variable XSC on the semicircle is said to have the Semicircular reflected Gamma distribution with shape parameter c, scale parameter σ>0, and location parameter µ, denoted by SCRG ( c,σ,µ), if the probability density and the distribution functions are respectively given by,

(7)

and

(8)

where,

Clearly,

 

Hence, the proposed new model SCRG (c,σ,µ) is a semicircular model.

When c=1 in (equation 7), we get,

(9)

which is semicircular Laplace distribution (Byoung and Hyoung, (2008)).

Figure 1 shows the plots of probability density function of semicircular reflected Gamma distribution for various values of c, σ and µ=0.

Figure 1. Plots of Probability Density Function for values of c, σ and µ=0

It is observed from the plot that the proposed circular distribution is symmetric about µ=0.

3. Trigonometric Moments

The trigonometric moments of the distribution are given by,

(10)

where, φpp+iβp, with αp=E(cos pθ) and βp=E(sin pθ) being the pth order cosine and sine moments of the random angle θ, respectively. Since, the Semicircular reflected Gamma distribution is symmetric about µ=0, it follows that the sine moments are zero. Thus φpp.

3.1 Theorem

Under the density function of Semicircular reflected Gamma distribution with µ=0, the first four αp=E(cos pθ), p=1,2,3,4 are given as follows:

(11)
(12)
(13)
(14)

where,

(15)

for and is called as Meijer's G-function (Gradshteyn and Ryzhik, 2007, formula no. 3.389.2).

Proof:

The proof is the process of using some transformations. For the first cosine moment, use the transformation and then use the property of an even function. So we have the following integral,

(16)

The result α1 follows by the integral formula (Gradshteyn and Ryzhik, 2007).

To obtain second cosine moment α2, we use the transformation and the use the property of an even function. So we have the following integral,

(17)

The result follows by the same integral formula of α1. To obtain the third cosine moment α3, we use the transformation and the use the property of even function we has the following integral as,

(18)

The result follows by the same integral formula of α1.

To obtain the fourth cosine moment α4, we use the transformation and the use the property of even function. We have the following integral,

(19)

The result follows by the same integral formula of α1.

Higher-order moments can be obtained similarly.

Note that, like any other symmetric circular density, βp=E(sin pθ) are 0 as the density of semicircular reflected Gamma  distribution is symmetric about µ=0.

4. Extension to l-axial Distribution

The authors have extend the proposed model to the l-axial distribution, which is applicable to any arc of arbitrary length say 2π/l for l=1,2,... . To construct the l-axial Reflected Gamma distribution, we consider the density function of semicircular reflected Gamma distribution and use the transformation Ψ=2θ/l, l=1,2,.... The probability density function of Ψ is given by,

(20)

where,

4.1 Case 1

When, l=2, the probability density function (equation 20) is the same as the probability density function of Semicircular reflected Gamma distribution.

4.2 Case 2

When, l=1, the probability density function (equation 20) is

(21)

where -π ≤ Ψ < π, which is the probability density function of Stereographic Reflected Gamma Distribution (Phani et al.2012a).

4.3 Case 3

When, l=2 and c=1, the probability density function (equation 20) is

(22)

where , which is the probability density function of Semicircular Laplace Distribution (Byoung and Hyoung, 2008).

4.4 Case 4

When, l=1 and c=1, the probability density function (equation 20) is,

(23)

where, -π ≤ Ψ < π, which is the probability density function of Stereographic Double Exponential Distribution (Girija et al. 2014b).

5. Findings

  • The semicircular reflected Gamma distribution by projecting reflected Gamma distribution over a semicircular segment is derived, this distribution generalizes semicircular Laplace distribution (Byoung and Hyoung, 2008).
  • The graphs of the density function for various values of the parameters are plotted and observed that the proposed circular distribution is symmetric about µ=0.
  • By adopting integral formulae in terms of Meijer's G-function, the first four trigonometric moments for the proposed model are derived and also we extended this model for l-axial Reflected Gamma distribution for modeling any arc of arbitrary length.
 

6. Recommendations

For analyzing semicircular data, in various applications like aircraft lost problem, sea turtles emerging from the ocean in search of a nesting site on dry land, etc., the proposed semicircular model is worth trying.

Conclusion

In this paper, the authors have investigated the semicircular distribution which follows by projecting the reflected Gamma distribution onto a semicircular segment. The density and distribution functions of a semicircular reflected Gamma distribution admit explicit forms, as do trigonometric moments. Meijer's G-function played an important role in the derivation of trigonometric moments. Furthermore, the first four trigonometric moments for the proposed model are derived and also the authors have extended this model for l-axial Reflected Gamma distribution for modeling any arc of arbitrary length. As this distribution is symmetric, promising for modeling symmetrical semicircular as well as l-axial data. The Semicircular Reflected Gamma distribution generalizes semicircular Laplace distribution (Byoung and Hyoung, 2008). The authors have plotted the graphs of the density function for various values of the parameters.

Acknowledgement

The authors thank the referee for their valuable suggestions which have helped in improving the presentation of the paper.

References

[1]. Byoung J.A and Hyoung M.K, (2008). “A New Family of Semicircular Models: The Semicircular Laplace Distributions”. Communications of the Korean Statistical Society, Vol.15 (5), pp.775- 781.
[2]. Dattatreya Rao A.V., Girija S.V.S. and Phani, (2016). “Stereographic Logistic Model – Application to Noisy Scrub Birds Data”. Chilean Journal of Statistics.
[3]. Dattatreya Rao A.V., Ramabhadra Sarma I. and Girija S.V.S, (2007). “On Wrapped Version of Some Life Testing Models”. Communication Statistics. - Theories and Methods, Vol.36(11), pp.2027- 2035.
[4]. Dattatreya Rao A.V., Girija S.V.S. and Phani Y, (2011a). “On Stereographic Logistic Model”. Proceedings of NCAMES, pp.139 -141.
[5]. Dattatreya Rao A.V., Girija S.V.S. and Phani Y, (2011b). “Differential Approach to Cardioid Distribution”. Computer Engineering and Intelligent Systems, Vol.2(8), pp.1-6.
[6]. Dattatreya Rao A.V, S.V.S.Girija and A.J.V. Radhika, (2011c). “A Note on Offset Cauchy Distribution”. Proceedings of the 5 International Conference of IMBIC on Mathematical Sciences for Advancement of Science and Technology. pp.133-139.
[7]. Dattatreya Rao A.V, S.V.S. Girija, and V.J. Devaraaj, (2013a). “On the Rising Sun Wrapped Lognormal and the Rising Sun Wrapped Exponential Models”. International Journal of Statistics and Systems, Vol.3(1), pp.1-10.
[8]. Devaraaj V. J, (2012). “Some Contributions to Circular Statistics”. Unpublished PhD. Thesis, Acharya Nagarjuna University.
[9]. Devaraaj, S.V.S. Girija, and A.V.Dattatreya Rao, (2014). “Estimation of Parameters in Cardioid Distribution from Censored Samples”. International Journal Innovative Research in Science & Engineering (IJIRSE), pp.1-8.
[10]. Girija S.V.S, (2010). Construction of New Circular Models. Project number 36254, and ISBN 978-3-639-27939-9 VDM Verlag Dr. Müller GmbH & Co. KG.
[11]. Girija S.V.S, A.V. Dattatreya Rao, and Y. Phani, (2013a). “On Stereographic Lognormal Distribution”. International Journal of Advances in Applied Sciences (IJAAS), Vol.2(3), pp.125- 132.
[12]. Girija S.V.S, A J V Radhika and A.V. Dattatreya Rao, (2013b). “On Bimodal Offset Cauchy Distribution”. Journal of the Applied Mathematics, Statistics and Informatics (JAMSI), Vol.9(1).
[13]. Girija S.V.S, A.J.V. Radhika and A.V.Dattatreya Rao, (2014a). “On Offset l-Arc Models”. Mathematics and Statistics, Vol.2(3), pp.127-136.
[14]. Girija S.V.S., A.V. Dattatreya Rao and Phani Yedlapalli, (2014b). “New Circular Model Induced by Inverse Stereographic projection on Double Exponential Model-Application to Birds Migration Data”. Journal of Applied Mathematics, Statistics and Informatics (JAMSI), Vol.10(1), pp.5-17.
[15]. Gradshteyn and Ryzhik, (2007). Table of Integrals, Series and Products. 7 edition, Academic Press.
[16]. Guardiola J.H, (2004). “The Semicircular Normal Distribution”. Ph.D. Dissertation, Institute of Statistics, Baylor University.
[17]. Jammalamadaka S. Rao and Sen Gupta A, (2001). Topics in Circular Statistics. World Scientific Press, Singapore.
[18]. Jones T.A, (1968). “Statistical Analysis of Orientation Data”. Journal of Sedimentary Petrology, Vol.38, pp.61 -67.
[19]. Mardia K.V and Jupp P.E, (2000). Directional Statistics. John Wiley, Chichester.
[20]. Phani Y, Radhika A.J.V, Girija S.V.S. and Dattatreya Rao A.V, (2012b). “Modeling Ants Data Using Stereographic Reflected Gamma Distribution”. ANU Journal of Physical Science, Vol.4(1), pp.15-38.
[21]. Phani Y, S.V.S Girijaand A.V and Dattatreya Rao, (2012a). “Circular Model Induced by Inverse Stereographic Projection on Extreme-Value Distribution”. International Journal on Engineering Science and Technology, Vol.2(5), pp.881-888.
[22]. Phani Y, (2013). “On Stereographic Circular and Semicircular Models”. Unpublished PhD. Thesis, Acharya Nagarjuna University.
[23]. Phani Yedlapalli, Girija S.V.S., and Dattatreya Rao A.V., (2013a). “On Construction of Stereographic Semi Circular Models”. Journal of Applied Probability and Statistics, Vol.8(1), pp.75-90.
[24]. Phani Yedlapalli, A.J.V. Radhika, S.V.S. Girija and A.V. Dattatreya Rao, (2014). “New Circular Models induced by Modified Inverse Stereographic Projection on Arc Tan Exponential – Type Distribution”. International Journal of Mathematical Archive, Vol.5(4), pp.1-6.
[25]. Phani Y, Girija, S.V.S. and Dattatreya Rao A.V, (2015a). “ l -Axial Wrapped Exponential Distribution”. International Journal of Scientific and Innovative Mathematical Research (IJSIMR) , Special Issue, Vol.3(2), pp.463-467.
[26]. Phani Y., V. Sastry, Girija, S.V.S. and Dattatreya Rao A.V, (2015b). “A Note on Trigonometric Moments of Stereographic Circular / Semicircular Generalized Gamma Model”. International Journal of Advanced Research in Computer Science and Software Engineering (IJARCSSE), Vol.5(5).
[27]. Phani, Y, V. Sastry, S.V.S.Girija and A.V. Dattatreya Rao, (2015c). “A Note on Trigonometric moments of Marshall – Olkin Stereographic Circular Logistic Distribution”. International Research Journal of Engineering and Technology (IRJET), Vol.2(3).
[28]. Phani Yedlapalli, R.V. Babu, S.V.S.Girija and A.V.D. Rao, (2015d). “Stereographic –l-axial Exponential and Stereographic Circular Exponential Distribution”. International Journal of Scientific and Innovative Mathematical Research (IJSIMR). Special Issue, Vol.3(5), pp.108-114.
[29]. Radhika A.J.V, S.V.S.Girija and A.V. Dattatreya Rao, (2013a). “On Univariate Offset Pearson Type II Model – Application To Live Data”. International Journal of Mathematics and Statistics Studies, Vol.1(1), pp.1-9.
[30]. Radhika A.J.V, S.V.S. Girija and A.V. Dattatreya Rao, (2103b). “On Rising Sun von Mises and Rising Sun Wrapped Cauchy Circular Models”. Journal of the Applied Mathematics, Statistics and Informatics (JAMSI), Vol.9(2), pp.61- 67.
[31]. Radhika A.J.V, (2014). “Mathematical Tools in the Construction of New Circular Models”. Unpublished PhD. Thesis, Acharya Nagarjuna University,
[32]. Radhika A.J.V, S.V.S. Girija, and A.V. Dattatreya Rao, (2015). “On Characteristics of the Arc Offset Pareto Model”. International Journal Scientific and Innovative Mathematical Research (IJSIMR), Vol.3, pp.858-862.
[33]. Ugai. S.K., Nishijima, M. and Kan T, (1977). “Characteristics of Raindrop Size and Raindrop Shape”. Open Symposium URSI Commission, pp.225-230.