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Spherical Logistic Distribution

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Abstract

Some of the spherical distributions can be constructed through proper transformation of the densities on plane. Since the logistic density on the Euclidean space has similar behavior to the normal distribution, it is of interest to extend it for spherical data. In this paper, we introduce spherical logistic distribution on the unit sphere and then study relevant statistical inferences including parameters estimation through method of moments and maximum likelihood techniques. It is shown that the spherical logistic distribution is a multimodal distribution with the marginal logistic density function. Proposed density has rotational symmetry property and this plays a key role to drive some important results related to first two moments. To investigate the proposed density in more details, some simulation studies along with analyzing real-life data are also considered.

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References

  1. Abe, T., Pewsey, A.: Sine-skewed circular distributions. Stat. Papers 52(3), 683–707 (2011)

    Article  MathSciNet  Google Scholar 

  2. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables. Dover Publications, New York (1965)

    MATH  Google Scholar 

  3. Azzalini, A.: A Class of Distributions which Includes the Normal Ones. Scand. J. Stat. 12(2), 171–178 (1985)

    MathSciNet  MATH  Google Scholar 

  4. Banerjee, A., Dhillon, I.S., Ghosh, J., Sra, S.: Clustering on the unit hypersphere using von Mises-Fisher distributions. J. Mach. Learn. Res. 6, 1345–1382 (2005)

    MathSciNet  MATH  Google Scholar 

  5. Bemmaor, A.C., Glady, N.: Modeling purchasing behavior with sudden death: a flexible customer lifetime model. Manag. Sci. 58(5), 1012–1021 (2012)

    Article  Google Scholar 

  6. Everitt, B.S., Hothorn, T.: HSAUR2: A Handbook of Statistical Analyses Using R (2nd Edition). R package version 11-6. 2013; Available from: http://CRAN.R-project.org/package=HSAUR2

  7. Gatto, R., Jammalamadaka, S.R.: The generalized von Mises distribution. Stat. Methodol. 4(3), 341–353 (2007)

    Article  MathSciNet  Google Scholar 

  8. Hornik, K., Grün, B.: movMF: An R package for fitting mixtures of von Mises-Fisher distributions. J. Stat. Softw. 58(10), 1–31 (2014)

    Article  Google Scholar 

  9. Jupp, P.E., Kent, J.T.: Fitting smooth paths to speherical data. Appl. Stat. 36(1), 34–46 (1987)

    Article  Google Scholar 

  10. Kato, S., Jones, M., et al.: An extended family of circular distributions related to wrapped Cauchy distributions via Brownian motion. Bernoulli 19(1), 154–171 (2013)

    Article  MathSciNet  Google Scholar 

  11. Kent, J.T.: The Fisher-Bingham distribution on the sphere. J. R. Stat. Soc. Ser. B (Methodological) 44(1), 71–80 (1982)

    MathSciNet  MATH  Google Scholar 

  12. Lewin, L.: Polylogarithms and Associated Functions. Elsevier, North Holland (1981)

    MATH  Google Scholar 

  13. Mardia, K.V., Jupp, P.E.: Directional Statistics. Wiley, London (2000)

    MATH  Google Scholar 

  14. Meeker, W.Q., Escobar, L.A.: Statistical Methods for Reliability Data. Wiley, New York (1998)

    MATH  Google Scholar 

  15. Rosenthal, M., Wu, W., Klassen, E., Srivastava, A.: Spherical regression models using projective linear transformations. J. Am. Stat. Assoc. 109(508), 1615–1624 (2014)

    Article  MathSciNet  Google Scholar 

  16. Ulrich, G.: Computer generation of distributions on the m-sphere. Appl. Stat. 33(2), 158–163 (1984)

    Article  MathSciNet  Google Scholar 

  17. Wood, A.T.: Simulation of the von Mises Fisher distribution. Commun. Stat.-Simul. Comput. 23(1), 157–164 (1994)

    Article  Google Scholar 

Download references

Acknowledgements

This research was in part supported by a Grant from Iran National Science Foundation [No. 95014574].

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Authors and Affiliations

  1. Department of Statistics, Tarbiat Modares University, Tehran, Iran

    M. Moghimbeygi & M. Golalizadeh

Authors
  1. M. Moghimbeygi
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  2. M. Golalizadeh
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Correspondence to M. Golalizadeh.

Appendix

Appendix

To calculate the normalized constant of SL distribution represented in (2.1), we need to work with the spherical coordinate. To do so, we end up with

$$\begin{aligned} C^{-1}_d(b, \kappa )= & {} \int _{{\mathbb {S}}^{d-2}} \int _0^\pi \frac{e^{\kappa \cos \theta _1} \sin ^{d-2} \theta _1}{(b-1+e^{\kappa \cos \theta _1})^{2}} d\theta _1 d_{{\mathbb {S}}^{d-2}}V, \end{aligned}$$

where

$$\begin{aligned} d_{{\mathbb {S}}^{d-2}}V=\sin ^{d-2}\theta _1 \sin ^{d-3}\theta _2\cdots \sin \theta _{d-2}d\theta _1 \cdots d\theta _{d-1}, \end{aligned}$$

is the volume element of \({\mathbb {S}}^{d-2}\) for the spherical coordinate systems, identified by \(0\le \theta _j \le \pi \) for \(j=1,..,d-2\) and \(0\le \theta _{d-1} <2\pi .\) Note that the integral involved with all of \(\theta \)’s excepts \(\theta _1\) is equal to the volume of \({\mathbb {S}}^{d-2}, \) (see, for example, [13], for more details). Now, substituting \(\cos \theta _1\) with t we have

$$\begin{aligned} C^{-1}_d(b, \kappa )= & {} \frac{2\pi ^{(d-1)/2}}{\varGamma (d/2-1/2)}\int _{-1}^1\frac{e^{kt}(1-t^2)^{(d-3)/2}}{(b-1+e^{kt})^2}dt \\= & {} \frac{2\pi ^{(d-1)/2}}{\varGamma (d/2-1/2)}\sum _{i=0}^\infty (-1)^i {(d-3)/2\atopwithdelims ()i} \int _{-1}^1\frac{t^{2 i}e^{kt}}{(b-1+e^{kt})^2}dt \\= & {} \frac{2\pi ^{(d-1)/2}}{\varGamma (d/2-1/2)} \sum _{i=2}^\infty \sum _{j=1}^i {(d-3)/2\atopwithdelims ()i-1} \bigg [ \frac{(-1)^{i+j} t^{i-j} i ! }{(b-1)\kappa ^{j+1}(i-j)!} {\text {Li}}_j (-e^{\kappa t}/(b-1))\bigg ]_{-1}^1 \\- & {} \frac{2\pi ^{(d-1)/2}}{\varGamma (d/2-1/2)}\sum _{i=0}^\infty (-1)^i {(d-3)/2\atopwithdelims ()i} \bigg [ \frac{t^{2i}}{e^{\kappa t}+b-1} \bigg ]_{-1}^1 \\= & {} \frac{2\pi ^{(d-1)/2}}{\varGamma (d/2-1/2)} \sum _{i=2}^\infty \sum _{j=1}^i {(d-3)/2\atopwithdelims ()i-1} \frac{i!}{(i-j)!} \frac{(-1)^{i+j} {\text {Li}}_j ( \frac{-e^{\kappa t}}{b-1}) -{\text {Li}}_j ( \frac{-e^{-\kappa t}}{b-1}) }{(b-1)\kappa ^{j+1}}, \end{aligned}$$

where \((1-t^2)^{(d-3)/2}=\sum _{i=0}^\infty (-1)^i {(d-3)/2\atopwithdelims ()i} t^{2i}\) and \({\text {Li}}_j\) is polylogarithmic function. This will complete our proof.

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Moghimbeygi, M., Golalizadeh, M. Spherical Logistic Distribution. Commun. Math. Stat. 8, 151–166 (2020). https://doi.org/10.1007/s40304-018-00171-2

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