wdtoto

royal123

maha212

Vol 18 No 2 (2024): BAREKENG: Journal of Mathematics and Its Application
Articles

THE UNINFORMATIVE PRIOR OF JEFFREYS’ DISTRIBUTION IN BAYESIAN GEOGRAPHICALLY WEIGHTED REGRESSION

Pdf
  • Fachri Faisal,
  • Henny Pramoedyo,
  • Suci Astutik,
  • Achmad Efendi
Fachri Faisal
Department of Mathematics, Faculty of Mathematics and Natural Sciences, Brawijaya University, Indonesia
Henny Pramoedyo
Department of Statistics, Faculty of Mathematics and Natural Sciences, Brawijaya University, Indonesia
Suci Astutik
Department of Statistics, Faculty of Mathematics and Natural Sciences, Brawijaya University, Indonesia
Achmad Efendi
Department of Statistics, Faculty of Mathematics and Natural Sciences, Brawijaya University, Indonesia
DOI: https://doi.org/10.30598/barekengvol18iss2pp1229-1236
Published May 25, 2024
Keywords
  • prior and posterior,
  • uninformative Jeffreys prior,
  • marginal posterior distributions
How to Cite
[1]
F. Faisal, H. Pramoedyo, S. Astutik, and A. Efendi, “THE UNINFORMATIVE PRIOR OF JEFFREYS’ DISTRIBUTION IN BAYESIAN GEOGRAPHICALLY WEIGHTED REGRESSION”, BAREKENG: J. Math. & App., vol. 18, no. 2, pp. 1229-1236, May 2024.

Copyright (c) 2024 Fachri Faisal, Henny Pramoedyo, Suci Astutik, Achmad Efendi

Creative Commons License

This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Abstract

When using the Bayesian method for estimating parameters in a geographically weighted regression model, the choice of the prior distribution directly impacts the posterior distribution.

The distribution known as the Jeffreys prior is an uninformative type of prior distribution and is invariant to reparameterization. In cases where information about the parameter is not available, the Jeffreys' prior is utilized. The data was fitted with an uninformative Jeffreys' prior distribution, which yielded a posterior distribution that was utilized for estimating parameters. This study aims to derive the prior and marginal posterior distributions of the Jeffreys'  and  in Bayesian geographically weighted regression (BGWR). The marginal posterior distributions of  and  can be obtained by integrating the other parameters of a common posterior distribution. Based on the results and discussion, the Jeffreys prior in BGWR with the likelihood function  is . On the other hand, the marginal posterior distribution of  follows a normal multivariate distribution, that is, , while the marginal posterior distribution of  follows an inverse gamma distribution, that is,  . As further research, it is necessary to follow up on several limitations of the results of this research, namely numerical simulations and application to a particular case that related to the results of the analytical studies that we have carried out.

Pdf

Downloads

Download data is not yet available.

References

  1. S. Astutik, A. B. Astuti, A. Efendi, Darmanto, D. Irsandy, and F. Y. D. A. S. Saniyawati, Analisis Bayesian: Teori dan Aplikasi dengan R. Malang: UB Press, 2023.
  2. S. T. Rachev, J. S. J. Hsu, B. S. Bagasheva, and F. J. Fabozzi, Bayesian Methods in Finance, 1st editio. John Wiley & Sons, Inc., Hoboken, New Jersey., 2008.
  3. A. S Fotheringham; C. Brunsdon; M. Charlton, Geographically Weighted Regression: The Analysis of Spatially Varying Relationships. West Sussex: John Wiley & Sons Ltd, 2002.
  4. A. Kashki, M. Karami, R. Zandi, and Z. Roki, “Evaluation of the effect of geographical parameters on the formation of the land surface temperature by applying OLS and GWR, A case study Shiraz City, Iran,” Urban Clim., vol. 37, p. 100832, May 2021, doi: 10.1016/j.uclim.2021.100832.
  5. A. Nazarpour, G. Rostami Paydar, F. Mehregan, S. J. Hejazi, and M. A. Jafari, “Application of geographically weighted regression (GWR) and singularity analysis to identify stream sediment geochemical anomalies, case study, Takab Area, NW Iran,” J. Geochemical Explor., vol. 235, p. 106953, Apr. 2022, doi: 10.1016/j.gexplo.2022.106953.
  6. A. Ristea, O. Kounadi, and M. Leitner, “Geosocial media data as predictors in a GWR application to forecast crime hotspots,” Leibniz Int. Proc. Informatics, LIPIcs, vol. 114, no. 56, pp. 1–7, 2018, doi: 10.4230/LIPIcs.GIScience.2018.56.
  7. Y.-C. Chiou, R.-C. Jou, and C.-H. Yang, “Factors affecting public transportation usage rate: Geographically weighted regression,” Transp. Res. Part A Policy Pract., vol. 78, pp. 161–177, Aug. 2015, doi: 10.1016/j.tra.2015.05.016.
  8. A. S. Fotheringham, R. Crespo, and J. Yao, “Exploring, modelling and predicting spatiotemporal variations in house prices,” Ann. Reg. Sci., vol. 54, no. 2, pp. 417–436, 2015, doi: 10.1007/s00168-015-0660-6.
  9. N. Subedi, L. Zhang, and Z. Zhen, “Bayesian geographically weighted regression and its application for local modeling of relationships between tree variables,” iForest - Biogeosciences For., vol. 11, no. 5, pp. 542–552, Oct. 2018, doi: 10.3832/ifor2574-011.
  10. J. P. LeSage, “A Family of Geographically Weighted Regression Models,” 2004, pp. 241–264. doi: 10.1007/978-3-662-05617-2_11.
  11. I. G. N. M. Jaya and N. Sunengsih, “Bayesian Geographically Weighted Regression Dalam Pemodelan Angka Incidence Rate,” Euclid, vol. 5, no. 1, p. 33, 2018, doi: 10.33603/e.v5i1.707.
  12. I. Sodikin, H. Pramoedyo, and S. Astutik, “Geographically weighted regression and bayesian geographically weighted regression modelling with adaptive gaussian kernel weight function on the poverty level in west Java Province,” Int. J. Humanit. Relig. Soc. Sci., vol. 2, no. 1, pp. 21–30, 2017.
  13. J. Geweke, “Bayesian Treatment of The Independent Student-t Linear Model,” J. Appl. Econom., vol. 8, no. September 1992, pp. S19–S40, 1993.
  14. Z. Ma, Y. Xue, and G. Hu, “Geographically Weighted Regression Analysis for Spatial Economics Data: A Bayesian Recourse,” Int. Reg. Sci. Rev., vol. 44, no. 5, pp. 582–604, Sep. 2021, doi: 10.1177/0160017620959823.
  15. R. van de Schoot et al., “Bayesian statistics and modelling,” Nat. Rev. Methods Prim., vol. 1, no. 1, p. 1, Jan. 2021, doi: 10.1038/s43586-020-00001-2.
  16. F. A. Moala and G. Moraes, “Objective Prior Distributions to Estimate the Parameters of the Poisson-Exponential Distribution,” Rev. Colomb. Estadística, vol. 46, no. 1, pp. 93–110, Jan. 2023, doi: 10.15446/rce.v46n1.95989.
  17. Y.-Y. Zhang, T.-Z. Rong, and M.-M. Li, “The Bayes Estimators of the Variance and Scale Parameters of the Normal Model With a Known Mean for the Conjugate and Noninformative Priors Under Stein’s Loss,” Front. Big Data, vol. 4, Jan. 2022, doi: 10.3389/fdata.2021.763925.
  18. M. Teimouri, “Fast Bayesian Inference for Birnbaum-Saunders Distribution,” Comput. Stat., vol. 38, no. 2, pp. 569–601, Jun. 2023, doi: 10.1007/s00180-022-01234-3.
  19. A. Ly, J. Verhagen, and E.-J. Wagenmakers, “Harold Jeffreys’s default Bayes factor hypothesis tests: Explanation, extension, and application in psychology,” J. Math. Psychol., vol. 72, pp. 19–32, Jun. 2016, doi: 10.1016/j.jmp.2015.06.004.
  20. Gelman.A;Carlin;J.B;Stern.H.S;Dunson.D;Vehtari.A; Rubin.D, Bayesian Data Analaysis, 3rd Editio. 2014.
  21. B. Puza, Bayesian methods for statistical analysis. Acton ACT 2601, Australia: ANU eView, 2015.
  22. D. R. S. Saputro, F. Amalia, P. Widyaningsih, and R. C. Affan, “Parameter estimation of multivariate multiple regression model using bayesian with non-informative Jeffreys’ prior distribution,” J. Phys. Conf. Ser., vol. 1022, no. 1, 2018, doi: 10.1088/1742-6596/1022/1/012002.