Non-Uniform Random Number Generation from Arbitrary Bivariate Distribution in Polygonal Area

Buğra Kaan Tiryaki; Orhan Kesemen

Non-Uniform Random Number Generation from Arbitrary Bivariate Distribution in Polygonal Area

Öz

Bivariate non-uniform random numbers are usually generated in a rectangular area. However, this is generally not useful in practice because the arbitrary area in real-life is not always a rectangular area. Therefore, the arbitrary area in real-life can be defined as a polygonal approach. Non-uniform random numbers are generated from an arbitrary bivariate distribution within a polygonal area by using the rejection and the inversion methods. Three examples are given for non-uniform random number generation from an arbitrary bivariate distribution function in polygonal areas. In these examples, the non-uniform random number generation is discussed in the triangular area, the Korea mainland and the Australia mainland. The non-uniform random numbers are generated in these areas from the arbitrary probability density function. The observed frequency values are calculated with using both methods in the simulation study and the generated random numbers are tested with the chi-square goodness of fit test to determine whether or not they come from the given distribution. Also, both methods are compared each other with a simulation study.

Anahtar Kelimeler

Bivariate distribution function,Piecewise probability density function; Non-uniform random number generation; Rejection method; Inversion method

Kaynakça

[1] Tausworthe, R. 1965. Random numbers generated by linear recurrence modulo two. Mathematics of Computation, 19(90), 201-209.
[2] Hayashi, S., Tanaka, Y., Kodama, E. 2001. A new manufacturing control system using Mahalanobis distance for maximising productivity. Semiconductor Manufacturing Symposium, 2001 IEEE International, 59-62.
[3] Dougherty, E. 1999. Random processes for image and signal processing. SPIE Optical Engineering Press, 592s.
[4] Breiman, L. 2001. Random forests. Machine learning, 45(1), 5-32.
[5] Gottselig, J., Adam, M., Retey, J., Khatami, R., Achermann, P., Landolt, H. 2006. Random number generation during sleep deprivation: effects of caffeine on response maintenance and stereotypy. Journal of sleep research, 15(1), 31-40.
[6] Gonzalo, J. 1994. Five alternative methods of estimating long-run equilibrium relationships. Journal of econometrics, 60(1), 203-233.
[7] Kobayashi, H., Mark, B., Turin, W. 2011. Probability, Random Processes, and Statistical Analysis. Cambridge University Press Textbooks, New York, 812s.
[8] Berg, H. 1993. Random walks in biology. Princeton University Press, 152s.

[9] Beard, R. 2013. Risk theory: the stochastic basis of insurance. Springer Science & Business Media, 195s.
[10] Rukhin, A., Soto, J., Nechvatal, J., Smid, M., Barker, E. 2001. A statistical test suite for random and pseudorandom number generators for cryptographic applications. Booz-Allen and Hamilton Inc McLean Va.
[11] Ross, S. 2006. Simulation. Elsevier, 285s.
[12] Yannakakis, G., Togelius, J. 2011. Experience-driven procedural content generation. Affective Computing, IEEE Transactions on, 2(3), 147-161.
[13] Fuller, W. 2011. Sampling statistics (Vol. 560). John Wiley & Sons, 472s.
[14] Devroye, L. 1986. Non-uniform Random Variate Generation. Springer, New York, 843s.
[15] Rubinstein, R., Kroese, D. P. 2008. Simulation and the Monte Carlo method. John Wiley & Sons, 304s.
[16] Walck, C. 2007. Handbook on statistical distributions for experimentalists, 190s.
[17] Boes, D., Graybill, F., Mood, A. 1974. Introduction to the Theory of Statistics. McGraw-Hill, 564s.
[18] Chu, D., Fotouhi, A. 2009. Distance between bivariate beta random points in two rectangular cities. Communications in Statistics—Simulation and Computation, 38(2), 257-268.
[19] Haight, F. 1964. Some probability distributions associated with commuter travel in a homogeneous circular city. Operations Research, 12(6), 964-975.
[20] Wang, M., Kennedy, W. 1990. Comparison of algorithms for bivariate normal probability over a rectangle based on self-validated results from interval analysis. Journal of Statistical Computation and Simulation, 37(1-2), 13-25.
[21] Robert, C., Casella, G. 2013. Monte Carlo Statistical Methods. Springer Science & Business Media, New York, 649s.
[22] Gentle, J. E. 2006. Random number generation and Monte Carlo methods. Springer Science & Business Media, New York, 247s.
[23] Johnson, M. E. 2013. Multivariate statistical simulation: A guide to selecting and generating continuous multivariate distributions. John Wiley & Sons, Canada, 240s.
[24] Kesemen, O., Doğru, F. 2011. Cumulative Distribution Functions of Two Variable in Polygonal Areas. 7. International Statistics Congress, 28 April – 1 May, Antalya, 150-151.
[25] Kesemen, O., Uluyurt, T. 2013. Bivariate Chi-Square Goodness of Fit Test in Polygonal Areas. 8. International Statistics Congress, 28-30 October, Antalya, 280-281.
[26] L'Ecuyer, P. 2012. Random Number Generation. ss 35-71. Genttle J.E., Härdle W. K., Mori Y., ed. 2012. Handbooks of Computational Statistics Springer Berlin Heidelberg, 1192s.
[27] Burden, R., Faires, J. 2011. Numerical Analysis. Brooks/Cole 7, 912s.
[28] Thomas, D., Luk, W. 2007. Non-uniform random number generation through piecewise linear approximations. IET Computers A Digital Techniques, 1(4), 312-321.
[29] Douglas, D., Peucker, T. 1973. Algorithms for the reduction of the number of points required to represent a digitized line or its caricature. Cartographica: The International Journal for Geographic Information and Geovisualization, 10(2), 112-122.
[30] Haines, E. 1994. Point in polygon strategies. pp 24-26. Heckbert P. S., ed. 1994. Graphics gems IV, Academic Press Professional, Cambridge, 575s.
[31] Hormann, K., Agathos, A. 2001. The point in polygon problem for arbitrary polygons. Computational Geometry, 20(3), 131-144.
[32] Gudmundsson, J., Haverkort, H., Van Kreveld, M. 2005. Constrained higher order Delaunay triangulations. Computational Geometry, 30(3), 271-277.
[33] Shewchuk, J. 1996. Triangle: Engineering a 2D quality mesh generator and Delaunay triangulator. ss 203-222. Lin M.C., Manocha D., eds. 1996. Applied Computational Geometry Towards Geometric Engineering. Lecture Notes in Computer Science, Springer Berlin Heidelberg, 222s.
[34] Forbes, C., Evans, M., Hastings, N., Peacock, B. 2011. Statistical distributions. New Jersey: John Wiley & Sons, 212s.
[35] Earth Observation Group (EOG). https://www.ngdc.noaa.gov/eog/. (Date of Access: 27.02.2018).
[36] Climate Change in Australia. Retrieved from CSIRO and Bureau of Meteorology. http://www.climatechangeinaustralia.gov.au/ (Date of Access: 12.06.2016).

Ayrıntılar

Birincil Dil

Türkçe

Konular

-

Bölüm

-

Yazarlar

Buğra Kaan Tiryaki

Orhan Kesemen

Yayımlanma Tarihi

15 Ağustos 2018

Gönderilme Tarihi

10 Temmuz 2017

Kabul Tarihi

-

Yayımlandığı Sayı

Yıl 2018 Cilt: 22 Sayı: 2

IZ

https://izlik.org/JA82YD92ZG

Kaynak Göster

RIS / Bibtex

APA

Tiryaki, B. K., & Kesemen, O. (2018). Non-Uniform Random Number Generation from Arbitrary Bivariate Distribution in Polygonal Area. Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 22(2), 443-457. https://izlik.org/JA82YD92ZG

AMA

1.Tiryaki BK, Kesemen O. Non-Uniform Random Number Generation from Arbitrary Bivariate Distribution in Polygonal Area. Süleyman Demirel Üniv. Fen Bilim. Enst. Derg. 2018;22(2):443-457. https://izlik.org/JA82YD92ZG

Chicago

Tiryaki, Buğra Kaan, ve Orhan Kesemen. 2018. “Non-Uniform Random Number Generation from Arbitrary Bivariate Distribution in Polygonal Area”. Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Dergisi 22 (2): 443-57. https://izlik.org/JA82YD92ZG.

EndNote

Tiryaki BK, Kesemen O (01 Ağustos 2018) Non-Uniform Random Number Generation from Arbitrary Bivariate Distribution in Polygonal Area. Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Dergisi 22 2 443–457.

IEEE

[1]B. K. Tiryaki ve O. Kesemen, “Non-Uniform Random Number Generation from Arbitrary Bivariate Distribution in Polygonal Area”, Süleyman Demirel Üniv. Fen Bilim. Enst. Derg., c. 22, sy 2, ss. 443–457, Ağu. 2018, [çevrimiçi]. Erişim adresi: https://izlik.org/JA82YD92ZG

ISNAD

Tiryaki, Buğra Kaan - Kesemen, Orhan. “Non-Uniform Random Number Generation from Arbitrary Bivariate Distribution in Polygonal Area”. Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Dergisi 22/2 (01 Ağustos 2018): 443-457. https://izlik.org/JA82YD92ZG.

JAMA

1.Tiryaki BK, Kesemen O. Non-Uniform Random Number Generation from Arbitrary Bivariate Distribution in Polygonal Area. Süleyman Demirel Üniv. Fen Bilim. Enst. Derg. 2018;22:443–457.

MLA

Tiryaki, Buğra Kaan, ve Orhan Kesemen. “Non-Uniform Random Number Generation from Arbitrary Bivariate Distribution in Polygonal Area”. Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Dergisi, c. 22, sy 2, Ağustos 2018, ss. 443-57, https://izlik.org/JA82YD92ZG.

Vancouver

1.Buğra Kaan Tiryaki, Orhan Kesemen. Non-Uniform Random Number Generation from Arbitrary Bivariate Distribution in Polygonal Area. Süleyman Demirel Üniv. Fen Bilim. Enst. Derg. [Internet]. 01 Ağustos 2018;22(2):443-57. Erişim adresi: https://izlik.org/JA82YD92ZG