Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 8s (2025) 420 https://internationalpubls.com Mixed Stereographic Circular Probability Model: A comparative study using M. L Estimation D.V. L. Prasanna1*, R. Subba Rao2 and G.V. S. R. Deekshitulu3 1Research Scholar, JNTUK, Kakinada and Assistant Professor, Department of Mathematics, Aditya University, Surampalem-533 437, A. P, India . Mail: lakshmisreee.21@gmail.com 2 Professor, Department of Mathematics, S.R.K.R Engineering College, Bhimavaram, A.P, India . Mail: rsubbarao9@gmail.com 3 Professor, Department of Mathematics, J.N.T.U.K Engineering College, Kakinada, A.P, India . Mail: dixitgvsr@gmail.com. *Corresponding author; e-mail: lakshmisreee.21@gmail.com Article History: Received: 29-10-2024 Revised:02-12-2024 Accepted:28-12-2024 Abstract: Now a days, circular statistics has a wide range of applications in the fields of Engineering, Geosciences, Environmental studies, Medicine, Psychology, Economics and Finance etc., such as analyzing angular movements in robotics, orientations in structural Engineering, Seismic wave directions, water and air flow patterns, brain waves in EEG signals, eye ball movements, modeling of economic indicators behaviors which cannot be analyzed using linear statistical methods as the data involves periodic nature. Many researchers have presented discrete and continuous circular probability models; their properties are discussed. By combining two distinct probability distributions rather than utilizing a single probability model, the T-X family of probability distributions is intended to increase the efficiency and reduce errors. We developed a mixed stereographic circular probability model with this interest. Study is done on same attributes. The maximum likelihood estimators, AIC and BIC values were determined in this study. The model’s output is compared with those of the current existing single circular probability models by using an eye data set from a glaucoma clinic at the University of Malaya Medical Centre in Malaysia. Keywords: T-X family, Inverse stereographic projection, Akaike Information Criteria, Maximum likelihood(M L) estimators, Bayesian Information Criteria. 1. Introduction Mardia and Jupp (2000)[1], Jammalamadaka S. Rao and Sen Gupta (2001)[2] discussed the methodologies like wrapping, stereographic projection, offset, and increasing sun function etc., to develop circular probability models and formulas to evaluate the various characteristics of these models. Minh and Farnum (2003)[3] developed ‘inverse stereographic methodology’ using bilinear transformation to induce new circular probability models. Motivated by this Toshihiro Abe et al., (2010)[4], A. V. Dattatreya Rao, S. V. S. Girija, and Y. Phani(2006)[8], Phani Yedlapalli et al., (2020)[14] developed various circular probability models and studied their characteristics. Rao AV Dattatreya, and S. V. S. Girija(2019)[12] published a book which describes the research work done in circular statistics in various previously mentioned methodologies. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 8s (2025) 421 https://internationalpubls.com Transformed-transformer families of linear probability distributions are developed to increase the efficiency and reduce the errors rather than utilizing single probability models. Alzaatreh, Lee et al.,(2013)[5] developed a method for inducing this type of models using quantile functions. Later so many researchers like Rambli, Adzhar et al., (2015)[6], Okasha, Mahmoud K et al., (2015)[7], R. Subba Rao, R. R. L. Kantam and G. Prasad, (2016)[9] and Korkmaz, Mustafa Ç et al.,(2017)[10] worked on mixed linear probability models. New half circular Burr type models were induced by Rambli, Adzhar, et al.,(1029) [13],Abuzaid, Ali H(2018)[11] using inverse stereographic projection and the AIC,BIC and parameter values are estimated. Ayesha Iftikhar et al.,(2022)[15] generated a modified hc burr III distribution and conducted a comparative study with the existing half circular models. Section 2 describes the fundamental definitions needed for this research work. In section 3 parameters of the Stereographic Weibull Rayleigh Distribution (SWRD) are estimated using M L method of estimation. Section 4 shows the comparative study of SWRD with current existing circular probability models and follows section 5 discusses the conclusion and scope of the work. 2. Fundamental definitions T-X family: For a random variable X, the T-X class of distributions’ probability distribution function is J(x) = ∫ m(t)dt = M{W(L(x))} W(L(x)) a (2.1) and the associated density function is j(x) = d dx W(L(x))m{W(L(x))} (2.2) When a continuous random variable T>0 specified on [0, ∞) has a cumulative probability distribution M(t) and a probability density function m(t). L(x) is the cumulative distribution function of a random variable X. Maximum likelihood estimators: The most common method for estimating the parameters of the probability model that best fits a given data set is maximum likelihood estimation. Consider a random sample of size n, 𝑧𝑇 = (𝑧1 , 𝑧2, . . . , 𝑧𝑛), taken from a population with distribution𝑓 (. ∅⁄ ), with ∅ , a vector of parameters. ∅ has the likelihood function 𝐿 (∅ 𝑧⁄ ) = ∏ 𝑓 (𝑧𝑖 ∅⁄ )𝑛 𝑖=1 . The log-likelihood function 𝑙𝑜𝑔𝐿 (∅ 𝑧⁄ ) can be maximized to estimate the parameters. Akaike Information Criteria: A statistical measure called the Akaike Information Criterion (AIC) is used to assess and contrast various models according to their complexity and goodness of fit. A better trade-off between model fit and complexity is indicated by lower AIC values. In most cases, the model with the lowest AIC is chosen when evaluation different models. The formula for calculating AIC is Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 8s (2025) 422 https://internationalpubls.com 𝐴𝐼𝐶 = 2𝑘 − 2𝑙𝑛 �̂� Bayesian Information Criteria: A statistical measure called the Bayesian Information Criterion (BIC) is used to assess a model’s goodness of fit while penalizing for model complexity to prevent over fitting. 𝐵𝐼𝐶 = 𝑘𝑙𝑛(𝑛) − 2𝑙𝑛 �̂� Where 𝑘 is the parameters count, n is the number of data points, and �̂� is the model’s M L function value Inverse Stereographic Projection: Inverse Stereographic Projection is defined as a one-to-one mapping that is given by, 𝑇(Ф) = 𝑦 = 𝑘 + 𝑙 𝑠𝑖𝑛Ф 1+𝑐𝑜𝑠Ф = 𝑘 + 𝑙𝑡𝑎𝑛 ( Ф 2 ) where y∈(-∞, ∞), Ф∈[-π, π), k∈R, and l>0. (2.3) Then by Minh and Farnum(2003)[4] 𝑇−1(𝑦) = Ф = 2𝑡𝑎𝑛−1 ( 𝑦−𝑘 𝑙 ) (2.4) is a random point on the unit circle. Stereographic Weibull Rayleigh Distribution: If the probability density function and distribution function of a random variable 𝜃 on a unit circle are, respectively, defined by, ℎ(𝜃; 𝛼, 𝜆) = 𝑙2𝛼 2𝜆𝑝2 𝑡𝑎𝑛 ( 𝜃 2 ) 𝑠𝑒𝑐2 𝜃 2 ( 𝑙2 2𝜆𝑝2 𝑡𝑎𝑛2 ( 𝜃 2 )) 𝛼−1 𝒆 −( 𝒍𝟐 𝟐𝝀𝒑𝟐 𝒕𝒂𝒏𝟐( 𝜽 𝟐 )) 𝜶 , (2.5) 𝛼 > 0, 𝜆 > 0, 𝑝 > 0,0 ≤ 𝜃 ≤ 𝜋 H(𝜃; 𝛼, 𝜆) = 1 − 𝑒 −( 𝒍𝟐 𝟐𝝀𝒑𝟐 𝒕𝒂𝒏𝟐( 𝜽 𝟐 )) 𝜶 then the variable is said to have a Stereographic Weibull Rayleigh Distribution with shape parameter 𝛼 > 0 and scale parameter 𝜆 > 0. 3. Parameter Estimation of SWRD: The likelihood function of SWRD is given by 𝐿 = ∏ℎ(𝜃) 𝑛 𝑖=1 𝐿 = ∏ ( 𝑙2𝛼 2𝜆𝑝2 𝑡𝑎𝑛 ( 𝜃 2 ) 𝑠𝑒𝑐2 𝜃 2 ( 𝑙2 2𝜆𝑝2 𝑡𝑎𝑛2 ( 𝜃 2 )) 𝛼−1 𝑒 −(( 𝑙2 2𝜆𝑝2 𝑡𝑎𝑛2( 𝜃 2 )) 𝛼 ) ) 𝑛 𝑖=1 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 8s (2025) 423 https://internationalpubls.com 𝑙𝑜𝑔𝐿 = ∑ 𝑙𝑜𝑔 𝑙2 2𝑝2 𝑛 𝑖=1 + ∑ 𝑙𝑜𝑔𝛼 𝑛 𝑖=1 + ∑ 𝑙𝑜𝑔 1 𝜆 𝑛 𝑖=1 + ∑ 𝑙𝑜𝑔 (𝑡𝑎𝑛 𝜃𝑖 2 𝑠𝑒𝑐2 𝜃𝑖 2 ) 𝑛 𝑖=1 + (𝛼 − 1) ∑ 𝑙𝑜𝑔 𝑙2 2𝜆𝑝2 𝑛 𝑖=1 𝑡𝑎𝑛2 𝜃𝑖 2 − ∑ ( 𝑙2 2𝜆𝑝2 𝑡𝑎𝑛2 𝜃𝑖 2 ) 𝛼 𝑛 𝑖=1 (3.1) Partially differentiate (6) with respect to α and equate this expressions to zero, 𝜕𝑙𝑜𝑔𝐿 𝜕𝛼 = 0 𝑛 𝛼 + ∑ 𝑙𝑜𝑔 ( 𝑙2 2𝜆𝑝2 𝑡𝑎𝑛2 𝜃𝑖 2 ) − ∑ ( 𝑙2 2𝜆𝑝2 𝑡𝑎𝑛2 𝜃𝑖 2 ) 𝛼 𝑙𝑜𝑔 ( 𝑙2 2𝜆𝑝2 𝑡𝑎𝑛2 𝜃𝑖 2 ) = 0 𝑛 𝑖=1 𝑛 𝑖=1 Let 𝑙2 2𝜆𝑝2 𝑡𝑎𝑛2 𝜃𝑖 2 = 𝐴 𝑛 𝛼 + ∑ 𝑙𝑜𝑔𝐴 − ∑ 𝐴𝛼 𝑙𝑜𝑔𝐴 = 0𝑛 𝑖=1 𝑛 𝑖=1 (3.2) Partially differentiate (6) with respect to λ and equate this expressions to zero, 𝜕𝑙𝑜𝑔𝐿 𝜕𝜆 = 0 𝑛𝛼 𝜆 − ∑ 𝛼𝐴𝛼−1 ( 𝑙2 2𝑝2 𝑡𝑎𝑛2 𝜃𝑖 2 ) = 0𝑛 𝑖=1 (3.3) The maximum likelihood estimators (MLEs) of the unknown parameters can be obtained by solving equations (7) and (8) concurrently, although these problems are not analytically solvable. fitdistrplus, AdequacyModel and pracma packages in R are used to analyze this model at MLE. 4. Comparative Study Using the above mentioned R packages the AIC,BIC and parameter values are estimated for hc-MBIII, hc-Log logistic, hc-Burr III, , hc-Gamma, hc-GIW and hc-Burr-XII distribution(Iftikar Ayesha et al.,[15] ) and the results are tabulated below. Table 1: Comparative study Distribution AIC CAIC BIC A W K-S (p-value) SWRD 13.3696 14.03627 15.45864 0.9184 0.15577 0.16388(0.6255) hc-MBIII 25.23826 26.50142 28.64474 0.44517 0.06778 0.127358 (0.84968) hc-Burr III 26.74478 7.34478 29.01577 0.815283 0.11913 0.183918(0.41799) hc-GIW 43.54607 44.80923 6.95255 2.4295 0.41253 0.27321(0.06453) hc-Log logistic 26.14425 26.74425 28.41524 0.74076 0.10753 0.116541(0.913587) Hc-Gamma 26.17461 26.77461 28.4456 0.8166877 0.127228 0.16989(0.520301) Hc-Burr-XII 26.69013 27.29013 28.96112 0.818624 0.119508 0.16551(0.55440) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 8s (2025) 424 https://internationalpubls.com The proposed stereographic circular distribution, SWRD fits the eye data better than the models previously mentioned since smaller AIC values signify a better fit which are graphically depicted as follows.(In graph SSCWR shows the curve of SWRD) 5. Conclusion In this research work, we estimated the parameters of the mixed circular probability model ‘SWRD’ using Maximum Likelihood estimation technique. Also AIC, BIC values for this model are evaluated and compared these results with the existing circular models like hc-MBIII, hc-Log logistic, hc-Burr III, hc-Gamma, hc-GIW and hc-Burr-XII. In comparison to these models, the proposed model is best fit to the data representing the posterior corneal curvature of 23 eye patients. Thus researches and those who work with angular data can considerably benefit from include this distribution in the theoretical framework. Acknowledgement: The authors thank their respective college managements for their continuous support and constant encouragement. We are also thankful to all the reviewers and editor for their valuable suggestions to improve the quality of the research paper. References: [1] Abuzaid, Ali H. "A half circular distribution for modeling the posterior corneal curvature." Communications in Statistics-Theory and Methods 47.13 (2018): 3118-3124. [2] Alzaatreh, Ayman, Carl Lee, and Felix Famoye. "A new method for generating families of continuous distributions." Metron 71.1 (2013): 63-79. [3] Dattatreya Rao, A. V., S. V. S. Girija, and Y. Phani. "Stereographic logistic model-application to noisy scrub birds data." Chilean Journal of Statistics 7.2 (2016): 69-79. [4] Iftikhar Ayesha, Azeem Ali, and Muhammad Hanif. "Half circular modified burr− III distribution, application with different estimation methods." Plos one 17.5 (2022): e0261901. [5] Jammalamadaka S. Rao and Sen Gupta A. Topics in Circular Statistics, World Scientific Press, Singapore, 2001. [6] Korkmaz, Mustafa Ç., et al. "The Burr X Pareto distribution: properties, applications and VaR estimation." Journal of Risk and Financial Management 11.1 (2017): 1. [7] Mardia K.V. and Jupp P.E. Directional Statistics, John Wiley, Chichester, 2000. [8] Minh Do Le, Farnum Nicholas R. “Using Bilinear Transformations to Induce Probability Distributions”, Communication in Statistics-Theory and Methods, Vol/Issue: 32(1). Pp. 1-9, 2003. [9] Okasha, Mahmoud K., and Mariam Y. Matter. "On the three-parameter Burr type XII distribution and its application to heavy tailed lifetime data." Journal: Journal of Advances in Mathematics 10.4 (2015): 3429 -3442. [10] Phani Yedlapalli, S.V.S. Girija, A.V. D. Rao, and Sastry, K.L.N. A new family of semicircular and Circular Arc tan - exponential type distribution. Thai Journal of Mathematics. 18(2): 775-781(2020). [11] Rambli, Adzhar, et al. "Outlier detection in a new half -circular distribution." AIP Conference Proceedings. Vol. 1682. No. 1. AIP Publishing, 2015. [12] Rambli, Adzhar, et al. "A half-circular distribution on a circle." Sains Malaysiana 48.4 (2019): 887-892. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 8s (2025) 425 https://internationalpubls.com [13] Rao AV Dattatreya, and S. V. S. Girija. Angular statistics. Chapman and Hall/CRC, 2019. [14] R. Subba Rao, G. Prasad, R. R. L. Kantam,’Acceptance sampling based on life tests: Pareto -Rayleigh model’, International Journal of Advanced Research in Science and Engineering, Vol. No 8, Issue No.08, August 2016, ISSN 2319-8354 [15] Toshihiro Abe, Kunio Shimizu and Arthur Pewsey, Symmetric Unimodal Models for Directional Data Motivated by Inverse Stereographic Projection, J. Japan Statist. Soc., Vol. 40 (No. 1), (2010), pp. 45-61.