Öz
İki
değişkenli olasılık yoğunluk fonksiyonundan, birikimli dağılım fonksiyonunu
hesaplamak için genellikle dikdörtgensel bir alanda tanımlanmış iki değişkenli
olasılık yoğunluk fonksiyonu kullanılır. Ancak uygulamada, tanım bölgesi
dikdörtgensel bir alan olmayan birçok olasılık yoğunluk fonksiyonu mevcuttur.
Bu çalışmada öncelikle dikdörtgen olmayan keyfi alanlar, çokgensel bir yaklaşım
uygulanarak tanımlanmıştır. Bu yaklaşım sonucunda elde edilen çokgensel bölge,
olasılık yoğunluk fonksiyonunun tanımlandığı sınırlarını oluşturmuştur. Böylece,
iki değişkenli parçalı olasılık yoğunluk fonksiyonu, keyfi bir alanda
tanımlanabilir. Elde edilen tanım bölgesinde birikimli dağılım fonksiyonu
hesaplamaları yapılmıştır. Bu hesaplamalarda iki tür yaklaşım kullanılmıştır.
İlk yaklaşım çokgensel alan üzerinden iki değişkenli sürekli olasılık yoğunluk
fonksiyonunun analitik integrali alınarak yapılmıştır. İkinci yaklaşım ise
seçilen olasılık yoğunluk fonksiyonun integralinin açık bir şekilde
hesaplanamaması durumunda uygulanması için geliştirilen sayısal yöntemdir.
Anahtar Kelimeler
Birikimli dağılım fonksiyonu Çokgen tabanlı olasılık yoğunluk fonksiyonu İki değişkenli dağılım fonksiyonları İki değişkenli parçalı dağılım fonksiyonları
Kaynakça
- Badiru, A. and Omitaomu, O., 2010. Handbook of Industrial Engineering Equations, Formulas, and Calculations: CRC Press, 456p.
- Boissonnat, J.D. and Teillaud, M., 2007. Effective computational geometry for curves and surfaces: Springer, 344p.
- Climate Change in Australia. (2016, 06 December). Retrieved from CSIRO and Bureau of Meteorology, http://www.climatechangeinaustralia.gov.au/.
- Douglas, D. and 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.
- Eshbach, O., Tapley, B. and Poston, T., 1990. Eshbach's handbook of engineering fundamentals: John Wiley & Sons, 2176p.
- Gudmundsson, J., Haverkort, H. and Van Kreveld, M., 2005. Constrained higher order Delaunay triangulations, Computational Geometry, 30 (3), 271-277.
- Haines, E., 1994. Point in polygon strategies: In Graphics gems IV: Academic Press, p. 24-26.
- Hormann, K. and Agathos, A., 2001. The point in polygon problem for arbitrary polygons, Computational Geometry, 20 (3), 131-144.
- Howard, W. and Musto, J., 2008. Engineering Computation: An Introduction Using MATLAB and Excel: McGraw Hill Higher Education, 330p.
- Kay, S. M., 2006. Intuitive Probability and Random Processes Using Matlab®, NY: Springer Science & Business Media, 834p.
- Kesemen, O. and Doğru, F. Z., 2011. Cumulative Distribution Functions of Two Variable in Polygonal Areas, 7. International Statistics Congress, Antalya, Turkey, p. 150-151.
- Kobayashi, H., Mark, B. and Turin, W., 2011. Probability, Random Processes, and Statistical Analysis: Applications to Communications, Signal Processing, Queueing Theory and Mathematical Finance: Cambridge University Press, 812p.
- Margalit, A. and Knott, G., 1989. An algorithm for computing the union, intersection or difference of two polygons, Computers & Graphics, 13 (2), 167-183.
- Martinez, W. L. and Martinez, A. R., 2002. Computational Statistics Handbook with MATLAB, New York: Crc Press, 731p.
- Miller, S. and Childers, D., 2012. Probability and random processes: With applications to signal processing and communications: Academic Press, 611p.
- Montgomery, D. and Runger, G., 2010. Applied statistics and probability for engineers: John Wiley & Sons, 784p.
- Preparata, F. and Shamos, M., 2012. Computational geometry: an introduction: Springer Science & Business Media, 398p.
- Roussas, G., 2003. An introduction to probability and statistical inference: Elsevier, 523p.
- Shewchuk, J., 1996. Triangle: Engineering a 2D quality mesh generator and Delaunay triangulator: In Applied computational geometry towards geometric engineering: Springer, p. 203-222.
- Straszewicz, S., 2014. Mathematical Problems and Puzzles: from the Polish Mathematical Olympiads: Elsevier, 376p.
- Walck, C., 2007. Handbook on statistical distributions for experimentalists: University of Stockholm, 202p.
Öz
Generally
bivariate probability density function defined in a rectangular area is used to
calculate the cumulative distribution function from the bivariate probability
density function. However, definition limits of the probability density
functions being non-rectangular are in existence in practice. In this paper,
primarily arbitrary non-rectangular areas are defined by applying a polygonal
approach. The polygonal area obtained as a result of this approach constitutes
boundaries of the probability density function. Thus, the bivariate piecewise
probability density function can be defined in an arbitrary area. Then the
cumulative distribution function is calculated in the obtained area. Two types
of approaches are used for these calculations. The first approach is applied to
take integral analytically of bivariate continuous probability density function
in the polygonal area. The second approach is developed a numerical method
since the explicit integral of the selected probability density function cannot
be found.
Anahtar Kelimeler
Cumulative distribution function Probability density function based on polygon Bivariate distribution functions Bivariate piecewise distribution functions
Kaynakça
- Badiru, A. and Omitaomu, O., 2010. Handbook of Industrial Engineering Equations, Formulas, and Calculations: CRC Press, 456p.
- Boissonnat, J.D. and Teillaud, M., 2007. Effective computational geometry for curves and surfaces: Springer, 344p.
- Climate Change in Australia. (2016, 06 December). Retrieved from CSIRO and Bureau of Meteorology, http://www.climatechangeinaustralia.gov.au/.
- Douglas, D. and 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.
- Eshbach, O., Tapley, B. and Poston, T., 1990. Eshbach's handbook of engineering fundamentals: John Wiley & Sons, 2176p.
- Gudmundsson, J., Haverkort, H. and Van Kreveld, M., 2005. Constrained higher order Delaunay triangulations, Computational Geometry, 30 (3), 271-277.
- Haines, E., 1994. Point in polygon strategies: In Graphics gems IV: Academic Press, p. 24-26.
- Hormann, K. and Agathos, A., 2001. The point in polygon problem for arbitrary polygons, Computational Geometry, 20 (3), 131-144.
- Howard, W. and Musto, J., 2008. Engineering Computation: An Introduction Using MATLAB and Excel: McGraw Hill Higher Education, 330p.
- Kay, S. M., 2006. Intuitive Probability and Random Processes Using Matlab®, NY: Springer Science & Business Media, 834p.
- Kesemen, O. and Doğru, F. Z., 2011. Cumulative Distribution Functions of Two Variable in Polygonal Areas, 7. International Statistics Congress, Antalya, Turkey, p. 150-151.
- Kobayashi, H., Mark, B. and Turin, W., 2011. Probability, Random Processes, and Statistical Analysis: Applications to Communications, Signal Processing, Queueing Theory and Mathematical Finance: Cambridge University Press, 812p.
- Margalit, A. and Knott, G., 1989. An algorithm for computing the union, intersection or difference of two polygons, Computers & Graphics, 13 (2), 167-183.
- Martinez, W. L. and Martinez, A. R., 2002. Computational Statistics Handbook with MATLAB, New York: Crc Press, 731p.
- Miller, S. and Childers, D., 2012. Probability and random processes: With applications to signal processing and communications: Academic Press, 611p.
- Montgomery, D. and Runger, G., 2010. Applied statistics and probability for engineers: John Wiley & Sons, 784p.
- Preparata, F. and Shamos, M., 2012. Computational geometry: an introduction: Springer Science & Business Media, 398p.
- Roussas, G., 2003. An introduction to probability and statistical inference: Elsevier, 523p.
- Shewchuk, J., 1996. Triangle: Engineering a 2D quality mesh generator and Delaunay triangulator: In Applied computational geometry towards geometric engineering: Springer, p. 203-222.
- Straszewicz, S., 2014. Mathematical Problems and Puzzles: from the Polish Mathematical Olympiads: Elsevier, 376p.
- Walck, C., 2007. Handbook on statistical distributions for experimentalists: University of Stockholm, 202p.
Ayrıntılar
| Birincil Dil | İngilizce |
|---|---|
| Konular | Mühendislik |
| Bölüm | Makaleler |
| Yazarlar | |
| Yayımlanma Tarihi | 15 Ocak 2019 |
| Gönderilme Tarihi | 9 Nisan 2018 |
| Kabul Tarihi | 22 Haziran 2018 |
| Yayımlandığı Sayı | Yıl 2019 Cilt: 9 Sayı: 1 |
