Gamma Dağılımının Parametrelerinin Tahmini için Metasezgisel Yöntemlerin Değerlendirilmesi ve Karşılaştırılması

Öz

Üç parametreli (3-p) Gamma dağılımı çarpık verilerin modellenmesinde kullanılan en popular dağılımlardan biri olduğundan bu dağılımın parametrelerinin tahmini çok önemlidir. Olabilirlik fonksiyonunu maksimize eden parametreleri bulan En Çok Olabilirlik (ML) yöntemi yaygın olarak kullanılan bir parametre tahmini yöntemidir.3-p Gamma dağılımının parametrelerinin tahmini için olabilirlik fonksiyonunu maksimize etmek çok zordur. Bu çalışmada, 3-p Gamma dağılımının parametrelerinin ML tahminlerini elde etmek için beş tane iyi bilinen metasezgisel yöntem: Tavlama Benzetimi (SA), Genetik Algoritma (GA), Parçacık Sürüsü Optimizasyonu (PSO), Diferansiyel Gelişim (DE) ve Yapay Arı Kolonisi (ABC) önerilmektedir. 3-p Gamma dağılımının tahmini probleminde metasezgisel yöntemlerin etkinliğinin araştırılması için Monte-Carlo simülasyon çalışmaları yapılmaktadır. Algoritmaların çözüm kalitesi ve hesaplama zamanı arasındaki farklar istatistiksel testler ile araştırılmaktadır. Ayrıca, metasezgisel algoritmaların parametre tahminindeki performanslarına göre sıralanması için çok kriterli karar verme yöntemlerinden biri olan TOPSIS yöntemi önerilmektedir. Sonuçlar, metasezgisel algoritmaların çözüm kalitesi, hesaplama zamanı, basitlik ve sağlamlılık kriterleri göz önüne alındığında DE’nin diğerlerinden daha iyi olduğunu göstermektedir. Ayrıca, önerilen metasezgisel yöntemlerin uygulanabilirliğini göstermek için gerçek bir yaşam verisi analizi sunulmaktadır.

Anahtar Kelimeler

• Gamma Dağılımı
• En Çok Olabilirlik Tahmini
• Metasezgisel Yöntemler
• Monte-Carlo Simülasyonu
• TOPSIS

Evaluation and Comparison of Metaheuristic Methods to Estimate the Parameters of Gamma Distribution

Öz

Parameter estimation of three parameter (3-p) Gamma distribution is very important as it is one of the most popular distributions used to model skewed data. Maximum Likelihood (ML) method based on finding estimators that maximize the likelihood function, is a well-known parameter estimation method. It is rather difficult to maximize the likelihood function formed for the parameter estimation of the 3-p Gamma distribution. In this study, five well known metaheuristic methods, Simulated Annealing (SA), Genetic Algorithm (GA), Particle Swarm Optimization (PSO), Differential Evolution (DE), and Artificial Bee Colony (ABC), are suggested to obtain ML estimates of the parameters for the 3-p Gamma distribution. Monte-Carlo simulations are performed to examine efficiencies of the metaheuristic methods for the parameter estimation problem of the 3-p Gamma distribution. Also, differences between solution qualities and computation time of the algorithms are investigated by statistical tests. Moreover, one of the multi-criteria decision-making methods, Technique for Order Performance by Similarity to Ideal Solution (TOPSIS), is preferred for ranking the metaheuristic algorithms according to their performance in parameter estimation. Results show that Differential Evolution is superior to the others for this problem in consideration of all the criteria of solution quality, computation time, simplicity, and robustness of the metaheuristic algorithms. In addition, an analysis of real-life data is presented to demonstrate the implementation of the suggested metaheuristic methods.

Anahtar Kelimeler

• Gamma Distribution
• Maximum Likelihood Estimation
• Metaheuristic Methods
• Monte-Carlo Simulation
• TOPSIS
• Gamma Dağılımı
• En Çok Olabilirlik Tahmini
• Metasezgisel Yöntemler
• Monte-Carlo Simülasyonu
• TOPSIS

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