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

Year 2022, Volume: 4 Issue: 2, 96 - 119, 31.12.2022
https://doi.org/10.51541/nicel.1093030

Abstract

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.

Supporting Institution

Selcuk University (TR) (Faculty Development Program (FDP))

Project Number

2016-OYP-063

References

  • Abbasi, B., Jahromi, A. H. E., Arkat, J. and Hosseinkouchack, M. (2006), Estimating the parameters of Weibull distribution using simulated annealing algorithm, Applied Mathematics and Computation, 183(1), 85-93.
  • Abbasi, B., Niaki, S. T. A., Khalife, M. A. and Faize, Y. (2011), A hybrid variable neighborhood search and simulated annealing algorithm to estimate the three parameters of the Weibull distribution, Expert Systems with Applications, 38(1), 700-708.
  • Acıtaş, Ş., Aladağ, Ç. H. and Şenoğlu, B. (2019), A new approach for estimating the parameters of Weibull distribution via particle swarm optimization: An application to the strengths of glass fibre data, Reliability Engineering System Safety, 183, 116-127.
  • Akaike, H., Petrov, B. N. and Csaki, F. (1973), Second international symposium on information theory. In: Akadémiai Kiadó, Budapest.
  • Akay, B. and Karaboğa, D. (2012), A modified artificial bee colony algorithm for real-parameter optimization, Information Sciences, 192, 120-142.
  • Balakrishnan, N. and Wang, J. (2000), Simple efficient estimation for the three-parameter gamma distribution, Journal of Statistical Planning Inference, 85(1-2), 115-126.
  • Basak, I. and Balakrishnan, N. (2012), Estimation for the three-parameter gamma distribution based on progressively censored data, Statistical Methodology, 9(3), 305-319.
  • Bowman, K., Shenton, L. and Karlof, C. (1995), Estimation problems associated with the three parameter gamma distribution, Communications in Statistics-Theory Methods, 24(5), 1355-1376.
  • Chen, C.-T. (2000), Extensions of the TOPSIS for group decision-making under fuzzy environment, Fuzzy sets systems, 114(1), 1-9.
  • Clifford, C. and Jones, W. (1982), Modified moment and maximum likelihood estimators for parameters of the three-parameter gamma distribution, Communications in Statistics-Simulation, 11(2), 197-216.
  • Cohen, A. C. and Whitten, B. J. (1986), Modified moment estimation for the three-parameter gamma distribution, Journal of Quality Technology, 18(1), 53-62.
  • Dumonceaux, R. and Antle, C. E. (1973), Discrimination between the log-normal and the Weibull distributions, Technometrics, 15(4), 923-926.
  • Eberhart, R. and Kennedy, J. (1995), A new optimizer using particle swarm theory. Paper presented at the Proceedings of the Sixth International Symposium on Micro Machine and Human Science.
  • Goldberg, D. E. and Holland, J. H. (1988), Genetic algorithms and machine learning. Machine learning, 3(2), 95-99.
  • Gui, L., Xia, X., Yu, F., Wu, H., Wu, R., Wei, B., . . . He, G. (2019), A multi-role based differential evolution. Swarm Evolutionary Computation.
  • Hirose, H. (1995), Maximum likelihood parameter estimation in the three-parameter gamma distribution, Computational Statistics & Data Analysis, 20(4), 343-354.
  • Holland, J. H. (1975), Adaptation in natural and artificial systems Ann Arbor, The University of Michigan Press, 1.
  • Hwang, C.-L. and Yoon, K. (1981). Multiple attribute decision making: methods and applications a state-of-the-art survey (Vol. 186). New York: Springer Science & Business Media.
  • Johnson, N., Kotz, S., & Balakrishnan, N. (1994). Lognormal distributions. Continuous Univariate Distributions (Vol. 1), John Wiley & Sons, New York, US.
  • Johnson, N. L., Kotz, S. and Balakrishnan, N. (1994), Univariate continuous distributions, John Wiley & Sons, New York, US.
  • Karaboğa, D. and Öztürk, C. (2011), A novel clustering approach: Artificial Bee Colony (ABC) algorithm, Applied Soft Computing, 11(1), 652-657.
  • Kirkpatrick, S., Gelatt, C. D. and Vecchi, M. P. (1983), Optimization by simulated annealing. science, 220(4598), 671-680.
  • Lakshmi, V. and Vaidyanathan, V. (2016), Three-parameter gamma distribution: Estimation using likelihood, spacings and least squares approach, Journal of Statistics Management Systems, 19(1), 37-53.
  • Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H. and Teller, E. (1953), Equation of state calculations by fast computing machines. The Journal of Chemical Physics, 21(6), 1087-1092.
  • Nagatsuka, H., Balakrishnan, N. and Kamakura, T. (2014), A consistent method of estimation for the three-parameter gamma distribution, Communications in Statistics-Theory Methods, 43(18), 3905-3926.
  • Opricovic, S. and Tzeng, G.-H. (2004), Compromise solution by MCDM methods: A comparative analysis of VIKOR and TOPSIS, European journal of operational research, 156(2), 445-455.
  • Örkcü, H., Aksoy, E. and Dogan, M. İ. (2015), Estimating the parameters of 3-p Weibull distribution through differential evolution, Applied Mathematics and Computation, 251, 211-224.
  • Örkcü, H., Özsoy, V. S., Aksoy, E. and Dogan, M. I. (2015), Estimating the parameters of 3-p Weibull distribution using particle swarm optimization: A comprehensive experimental comparison, Applied Mathematics and Computation, 268, 201-226.
  • Özsoy, V. S., Örkcü, H. H. and Bal, H. (2017), Particle Swarm Optimization applied to parameter estimation of thefour-parameter Burr III distribution, Iranian Journal of Science and Technology, Transactions A: Science, 42, 1-15.
  • Price, K., Storn, R. M. and Lampinen, J. A. (2006), Differential evolution: a practical approach to global optimization, Springer Science & Business Media, Germany
  • Rajasekhar, A., Lynn, N., Das, S. and Suganthan, P. N. (2017), Computing with the collective intelligence of honey bees–a survey. Swarm Evolutionary Computation, 32, 25-48.
  • Ranneby, B. (1984), The Maximum Spacing Method: An Estimation Method Related to the Maximum Likelihood Method, Scandinavian Journal of Statistics, 11(2), 93-112.
  • RStudio, (2021), https://www.rstudio.com/products/rstudio/download/, Access Date: 06.10.2021.
  • Shin, J.-Y., Heo, J.-H., Jeong, C. and Lee, T. (2014), Meta-heuristic maximum likelihood parameter estimation of the mixture normal distribution for hydro-meteorological variables, Stochastic environmental research and risk assessment, 28(2), 347-358.
  • Stone, M. (1979), Comments on model selection criteria of Akaike and Schwarz, Journal of the Royal Statistical Society. Series B, 41(2), 276-278.
  • Storn, R. (1996), On the usage of differential evolution for function optimization, Paper presented at the Fuzzy Information Processing Society, Biennial Conference of the North American.
  • Storn, R. and Price, K. (1997), Differential evolution–a simple and efficient heuristic for global optimization over continuous spaces, Journal of global optimization, 11(4), 341-359.
  • Şahin, A. and Pehlivan, N. Y. (2017), Evaluation of life quality by integrated method of AHP and TOPSIS based on interval type-2 fuzzy sets, Hacettepe Journal of Mathematics and Statistics, 46(3), 519-531.
  • Talbi, E.-G. (2009), Metaheuristics: from design to implementation, John Wiley & Sons, Canada.
  • Vaidyanathan, V. and Lakshmi, R. V. (2015), Parameter Estimation in Multivariate Gamma Distribution, Statistics, Optimization Information Computing, 3(2), 147-159.
  • Vera, J. F. and Díaz-García, J. A. (2008), A global simulated annealing heuristic for the three-parameter lognormal maximum likelihood estimation, Computational Statistics & Data Analysis, 52(12), 5055-5065.
  • Yalçınkaya, A., Şenoğlu, B. and Yolcu, U. (2018), Maximum likelihood estimation for the parameters of skew normal distribution using genetic algorithm, Swarm and Evolutionary Computation, 38, 127-138.
  • Yang, X.-S. (2010), Engineering optimization: An introduction with metaheuristic applications. John Wiley & Sons,Cambridge.
  • Yonar, A. (2020), Metaheuristic approaches for estimating parameters of univariate and multivariate distributions. PhD thesis (Unpublished), Selçuk University, Konya,Turkey.
  • Yonar, A. and Pehlivan, N. Y. (2020a), Artificial Bee Colony with Levy Flights for Parameter Estimation of 3-p Weibull Distribution, Iranian Journal of Science and Technology, Transactions of Electrical Engineering, 44, 851-864.
  • Yonar, A. and Pehlivan, N. Y. (2020b), A novel differential evolution algorithm approach for estimating the parameters of Gamma distribution: An application to the failure stresses of single carbon fibres, Hacettepe Journal of Mathematics and Statistics, 49(4), 1493-1514.
  • Zoraghi, N., Abbasi, B., Niaki, S. T. A. and Abdi, M. (2012), Estimating the four parameters of the Burr III distribution using a hybrid method of variable neighborhood search and iterated local search algorithms, Applied Mathematics and Computation, 218(19), 9664-9675.

Gamma Dağılımının Parametrelerinin Tahmini için Metasezgisel Yöntemlerin Değerlendirilmesi ve Karşılaştırılması

Year 2022, Volume: 4 Issue: 2, 96 - 119, 31.12.2022
https://doi.org/10.51541/nicel.1093030

Abstract

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.

Project Number

2016-OYP-063

References

  • Abbasi, B., Jahromi, A. H. E., Arkat, J. and Hosseinkouchack, M. (2006), Estimating the parameters of Weibull distribution using simulated annealing algorithm, Applied Mathematics and Computation, 183(1), 85-93.
  • Abbasi, B., Niaki, S. T. A., Khalife, M. A. and Faize, Y. (2011), A hybrid variable neighborhood search and simulated annealing algorithm to estimate the three parameters of the Weibull distribution, Expert Systems with Applications, 38(1), 700-708.
  • Acıtaş, Ş., Aladağ, Ç. H. and Şenoğlu, B. (2019), A new approach for estimating the parameters of Weibull distribution via particle swarm optimization: An application to the strengths of glass fibre data, Reliability Engineering System Safety, 183, 116-127.
  • Akaike, H., Petrov, B. N. and Csaki, F. (1973), Second international symposium on information theory. In: Akadémiai Kiadó, Budapest.
  • Akay, B. and Karaboğa, D. (2012), A modified artificial bee colony algorithm for real-parameter optimization, Information Sciences, 192, 120-142.
  • Balakrishnan, N. and Wang, J. (2000), Simple efficient estimation for the three-parameter gamma distribution, Journal of Statistical Planning Inference, 85(1-2), 115-126.
  • Basak, I. and Balakrishnan, N. (2012), Estimation for the three-parameter gamma distribution based on progressively censored data, Statistical Methodology, 9(3), 305-319.
  • Bowman, K., Shenton, L. and Karlof, C. (1995), Estimation problems associated with the three parameter gamma distribution, Communications in Statistics-Theory Methods, 24(5), 1355-1376.
  • Chen, C.-T. (2000), Extensions of the TOPSIS for group decision-making under fuzzy environment, Fuzzy sets systems, 114(1), 1-9.
  • Clifford, C. and Jones, W. (1982), Modified moment and maximum likelihood estimators for parameters of the three-parameter gamma distribution, Communications in Statistics-Simulation, 11(2), 197-216.
  • Cohen, A. C. and Whitten, B. J. (1986), Modified moment estimation for the three-parameter gamma distribution, Journal of Quality Technology, 18(1), 53-62.
  • Dumonceaux, R. and Antle, C. E. (1973), Discrimination between the log-normal and the Weibull distributions, Technometrics, 15(4), 923-926.
  • Eberhart, R. and Kennedy, J. (1995), A new optimizer using particle swarm theory. Paper presented at the Proceedings of the Sixth International Symposium on Micro Machine and Human Science.
  • Goldberg, D. E. and Holland, J. H. (1988), Genetic algorithms and machine learning. Machine learning, 3(2), 95-99.
  • Gui, L., Xia, X., Yu, F., Wu, H., Wu, R., Wei, B., . . . He, G. (2019), A multi-role based differential evolution. Swarm Evolutionary Computation.
  • Hirose, H. (1995), Maximum likelihood parameter estimation in the three-parameter gamma distribution, Computational Statistics & Data Analysis, 20(4), 343-354.
  • Holland, J. H. (1975), Adaptation in natural and artificial systems Ann Arbor, The University of Michigan Press, 1.
  • Hwang, C.-L. and Yoon, K. (1981). Multiple attribute decision making: methods and applications a state-of-the-art survey (Vol. 186). New York: Springer Science & Business Media.
  • Johnson, N., Kotz, S., & Balakrishnan, N. (1994). Lognormal distributions. Continuous Univariate Distributions (Vol. 1), John Wiley & Sons, New York, US.
  • Johnson, N. L., Kotz, S. and Balakrishnan, N. (1994), Univariate continuous distributions, John Wiley & Sons, New York, US.
  • Karaboğa, D. and Öztürk, C. (2011), A novel clustering approach: Artificial Bee Colony (ABC) algorithm, Applied Soft Computing, 11(1), 652-657.
  • Kirkpatrick, S., Gelatt, C. D. and Vecchi, M. P. (1983), Optimization by simulated annealing. science, 220(4598), 671-680.
  • Lakshmi, V. and Vaidyanathan, V. (2016), Three-parameter gamma distribution: Estimation using likelihood, spacings and least squares approach, Journal of Statistics Management Systems, 19(1), 37-53.
  • Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H. and Teller, E. (1953), Equation of state calculations by fast computing machines. The Journal of Chemical Physics, 21(6), 1087-1092.
  • Nagatsuka, H., Balakrishnan, N. and Kamakura, T. (2014), A consistent method of estimation for the three-parameter gamma distribution, Communications in Statistics-Theory Methods, 43(18), 3905-3926.
  • Opricovic, S. and Tzeng, G.-H. (2004), Compromise solution by MCDM methods: A comparative analysis of VIKOR and TOPSIS, European journal of operational research, 156(2), 445-455.
  • Örkcü, H., Aksoy, E. and Dogan, M. İ. (2015), Estimating the parameters of 3-p Weibull distribution through differential evolution, Applied Mathematics and Computation, 251, 211-224.
  • Örkcü, H., Özsoy, V. S., Aksoy, E. and Dogan, M. I. (2015), Estimating the parameters of 3-p Weibull distribution using particle swarm optimization: A comprehensive experimental comparison, Applied Mathematics and Computation, 268, 201-226.
  • Özsoy, V. S., Örkcü, H. H. and Bal, H. (2017), Particle Swarm Optimization applied to parameter estimation of thefour-parameter Burr III distribution, Iranian Journal of Science and Technology, Transactions A: Science, 42, 1-15.
  • Price, K., Storn, R. M. and Lampinen, J. A. (2006), Differential evolution: a practical approach to global optimization, Springer Science & Business Media, Germany
  • Rajasekhar, A., Lynn, N., Das, S. and Suganthan, P. N. (2017), Computing with the collective intelligence of honey bees–a survey. Swarm Evolutionary Computation, 32, 25-48.
  • Ranneby, B. (1984), The Maximum Spacing Method: An Estimation Method Related to the Maximum Likelihood Method, Scandinavian Journal of Statistics, 11(2), 93-112.
  • RStudio, (2021), https://www.rstudio.com/products/rstudio/download/, Access Date: 06.10.2021.
  • Shin, J.-Y., Heo, J.-H., Jeong, C. and Lee, T. (2014), Meta-heuristic maximum likelihood parameter estimation of the mixture normal distribution for hydro-meteorological variables, Stochastic environmental research and risk assessment, 28(2), 347-358.
  • Stone, M. (1979), Comments on model selection criteria of Akaike and Schwarz, Journal of the Royal Statistical Society. Series B, 41(2), 276-278.
  • Storn, R. (1996), On the usage of differential evolution for function optimization, Paper presented at the Fuzzy Information Processing Society, Biennial Conference of the North American.
  • Storn, R. and Price, K. (1997), Differential evolution–a simple and efficient heuristic for global optimization over continuous spaces, Journal of global optimization, 11(4), 341-359.
  • Şahin, A. and Pehlivan, N. Y. (2017), Evaluation of life quality by integrated method of AHP and TOPSIS based on interval type-2 fuzzy sets, Hacettepe Journal of Mathematics and Statistics, 46(3), 519-531.
  • Talbi, E.-G. (2009), Metaheuristics: from design to implementation, John Wiley & Sons, Canada.
  • Vaidyanathan, V. and Lakshmi, R. V. (2015), Parameter Estimation in Multivariate Gamma Distribution, Statistics, Optimization Information Computing, 3(2), 147-159.
  • Vera, J. F. and Díaz-García, J. A. (2008), A global simulated annealing heuristic for the three-parameter lognormal maximum likelihood estimation, Computational Statistics & Data Analysis, 52(12), 5055-5065.
  • Yalçınkaya, A., Şenoğlu, B. and Yolcu, U. (2018), Maximum likelihood estimation for the parameters of skew normal distribution using genetic algorithm, Swarm and Evolutionary Computation, 38, 127-138.
  • Yang, X.-S. (2010), Engineering optimization: An introduction with metaheuristic applications. John Wiley & Sons,Cambridge.
  • Yonar, A. (2020), Metaheuristic approaches for estimating parameters of univariate and multivariate distributions. PhD thesis (Unpublished), Selçuk University, Konya,Turkey.
  • Yonar, A. and Pehlivan, N. Y. (2020a), Artificial Bee Colony with Levy Flights for Parameter Estimation of 3-p Weibull Distribution, Iranian Journal of Science and Technology, Transactions of Electrical Engineering, 44, 851-864.
  • Yonar, A. and Pehlivan, N. Y. (2020b), A novel differential evolution algorithm approach for estimating the parameters of Gamma distribution: An application to the failure stresses of single carbon fibres, Hacettepe Journal of Mathematics and Statistics, 49(4), 1493-1514.
  • Zoraghi, N., Abbasi, B., Niaki, S. T. A. and Abdi, M. (2012), Estimating the four parameters of the Burr III distribution using a hybrid method of variable neighborhood search and iterated local search algorithms, Applied Mathematics and Computation, 218(19), 9664-9675.
There are 47 citations in total.

Details

Primary Language English
Subjects Statistics
Journal Section Articles
Authors

Aynur Yonar 0000-0003-1681-9398

Nimet Yapıcı Pehlivan 0000-0002-7094-8097

Project Number 2016-OYP-063
Publication Date December 31, 2022
Published in Issue Year 2022 Volume: 4 Issue: 2

Cite

APA Yonar, A., & Yapıcı Pehlivan, N. (2022). Evaluation and Comparison of Metaheuristic Methods to Estimate the Parameters of Gamma Distribution. Nicel Bilimler Dergisi, 4(2), 96-119. https://doi.org/10.51541/nicel.1093030