Title Science and Technology Indonesia e-ISSN:2580-4391 p-ISSN:2580-4405 Vol. 7, No. 1, January 2022 Research Paper A Simulation Study on The Simulated Annealing Algorithm in Estimating The Parameters of Generalized Gamma Distribution Amani Idris A. Sayed1,2* and Sabri Shamsul Rijal Muhammad1 1School of Mathematical Sciences, University Sains Malaysia, Pulau Pinang, 11800, Malaysia 2Department of Mathematics, Jazan University, Jazan, 45142, Saudi Arabia *Corresponding Author e-mail: amani.sayed1983@yahoo.com Abstract ThisstudyproposedahybridcomputationalmodelbyincorporatingSimulatedAnnealingalgorithm(SA) inthemaximumlikelihood in estimation the parameters of Generalized Gamma Distribution (GGD). The purpose is to improve the searching capacity of max- imum likelihood estimator for Generalized Gamma Distribution (GGD). A simulated Annealing algorithm (SA) is one of the global search heuristics computational approach inspired by the metallurgical process, in which metal is rapidly heated to a high temper- ature, then cooled slowly until it reaches the lowest-energy state, increasing its strength and making it easier to work with. It has been applied in approximating global optimization in a large search space for various optimization problems. This study employs a Simulated Annealing algorithm (SA) to improve the global search capacity of maximum likelihood methods (MLE) in estimating the parameters of the Generalized Gamma Distribution (GGD). The Generalized Gamma Distribution (GGD) constitutes an extensive family that contains nearly all of the most commonly used distributions including the exponential, Weibull, and lognormal distribu- tion. The performances of the proposed estimation method are computed based on their biases and mean square errors through a simulation study. The study reveals that the Simulated Annealing algorithm (SA) performs better than the classical estimators in estimating the parameters of Generalized Gamma Distribution (GGD). Keywords Simulated Annealing Algorithm, Generalized Gamma Distribution, Parameters Estimation Methods, Log-Likelihood Function. Received: 28 October 2021, Accepted: 08 January 2022 https://doi.org/10.26554/sti.2022.7.1.84-90 1. INTRODUCTION The Gamma Distribution (GD) is one of the most important statistical distributions for its widely used in many �elds of data modelling and forecasting of various phenomena. It is useful in handling large scales of data due to its �exibility to be transformed to other distributions such as exponential and normal distributions. The important features of the Gener- alized Gamma Distribution (GGD) made it useful in various areas of research which include business, engineering and other lifetime analysis (Lawless, 2003; Tang and Cheong, 2004; Eric et al., 2020). The probability density function of the gamma distribution is given as. fXK (x, U, \) = { 1 Γ(U) [ 1 \ ]U xU−1e x \ , x > 0 0, otherwise (1) where a and q are the shape and scale parameters of the gamma distribution model respectively. We employ a Gen- eralized Gamma Distribution (GGD) in this work, which was established by Amoroso (1925). The gamma distribution is among the most widely statistical distribution employed in modelling a di�erent kind of lifetime data set with application in modelling di�erent phenomena characterized with mono- tone failure rates. It requires a little measure of parameters for learning. The �rst generalization of the gamma distribu- tion can be found in Amoroso (1925), which analyzed and applied a GGD to �tting the income rates based on the GGD with four parameters has been studied by Johnson et al. (1995). The study on the four gamma distribution was later reduced to the GGD according to Stacy (1962) by setting the location parameter to zero. Mudholkar and Srivastava (1993) used the exponentiated method to derive the gamma distribution model for data anlysis. Stacy (1962) developed the generalized version of the gamma distribution based three-parameters. Agarwal and Al-Saleh (2001) modelled the distribution of hazard rates using the gamma distribution model. Similar distribution was emplyed by Balakrishnan and Peng (2006) in creating a gen- eralized version of gamma distribution to modelling frailty https://crossmark.crossref.org/dialog/?doi=10.26554/sti.2022.7.1.84-90&domain=pdf https://doi.org/10.26554/sti.2022.7.1.84-90 Sayed and Muhammad Science and Technology Indonesia, 7 (2022) 84-90 data. Another sort of GGD suggested by Nadarajah and Gupta (2007) was used to �t drought data. Another generalization of distribution by Stacys used an exponentiated approach to GGD model and used it to lifetime and survival analysis (Ortega et al., 2012). A study by Chen et al. (2017) employed a generalized version of the gamma distribution (GGD) with three param- eters in modelling the �ood frequency data analysis. Zhao et al. (2018) employed the three-parameter GGD model in calculating the statistical characteristics of high-resolution SAR pictures. Similarly, Mead et al. (2018) propose the modi�ed version of GGD to evaluate greater �exibility and appropria- teess in representing data from a practical standpoint, and they derived a variety of identities and properties of this distribution, which include explicit representations for lifetime expression. Researchers in the �elds of statistical distribution have tried a variety of methods in estimating the parameters of the GGD. This include, moment method (Khodabina and Ahmadabadib, 2010), quantile methods (Nagatsuka and Balakrishnan, 2012), modi�ed moment methods (Dey et al., 2021), evolutionary algorithm (Yonar and Yapici, 2020), maximum likelihood (Hi- rose, 1995), Bayesian approach (Khodabina and Ahmadabadib, 2010; Pradhan and Kundu, 2011). However, because parame- ters involve leading to the same density function, it is di�cult and challenging face by researchers in estimating the parame- ters of the generalized version of the gamma distribution us- ing methods like moment methods and maximum likelihood. The likelihood equations must be solved simultaneously in the maximum likelihood technique. To explore other approach in estimating the parameter of the generalized version of the gamma distribution, conventional methods were developed. It also showcases the incapability of the conventional/traditional method such as moment method and maximum likelihood estimator to estimate the parameters of the proposed GGD distribution. However, the results are unappealing due to their intricacy of the traditional method. As a result, it appears that estimating the parameters of the GGD is quite challenging and not straightforward. There is the need to incorporating other search approach such as heuristics method to overcome such challenges and limitation associated with the conventional methods. In this study, a stochastic computational methods based on the Simulated Annealing algorithm (SA) has been incorporated to reduce the complexity involve in the searching for the optimal parameters of the generalized version of the gamma distribution model. In this study, a simulation study has been carried out according to the proposed model. The remainder of this paper is organized as follows: The simulated annealing algorithm and Generalize gamma distribution and its parameters have been presented in Section 2. A simulated annealing algorithm for optimization of gamma distribution pa- rameters is studied in Section 3. The experimental results and with discussion have been presented in Section 4. Conclusions and recommendation the of the study in Section 5. 2. SIMULATED ANNEALING ALGORITHM The Simulated Annealing algorithm (SA) aims to generate new solutions using a random process through a series of probability distributions. This random operation does not necessarily improve the objective function, but even so, it may be accepted (Franzin and Stützle, 2019). The algorithm was originally used as an optimization process in Metallurgical engineering to achieve the minimum energy by gradually reducing the atomic motion that reduces the uniformity of lattice defects, thereby reducing the metal temperature (Du and Swamy, 2016). The Simulated Annealing algorithm (SA) cannot be a�ected by any restraint at any local minimum and has an indulgent acceptance of any changes that arise in the objective function, thus its e�ciency makes it useful in many �elds, including �nance and the mathematical and statistical sciences (Crama and Schyns, 2003; Orús et al., 2019; Abubakar and Sabri, 2021b; Abubakar and Sabri, 2021a). The acceptance probability in the SA algorithm can be written as, P = e −(ΔZ) H (2) Where ΔZ denote the increase in the objective function and H represent the controlling parameter. Some studies includ- ing (Abbasi et al., 2006) and (Bertsimas and Tsitsiklis, 1993), presented the most general form of the Simulated Annealing algorithm (SA) which can be written in the following steps: i. Start with initial solutions X=X0. ii. Generate a random chance of the solutions X. iii. Evaluate the objective function at the randomly generated solutions. iv. Setting an initial controlling parameter H=H0 and an ap- proach to reduce it as the process goes. The general form of the SA algorithm, which has been ex- plained above, works toward minimizing the control parameter H, in adjusting the procedure which can be implemented in other problems. The purpose is to explore the feasibility of Simulated Annealing algorithm (SA) in estimating the param- eters of proposed model in relation to conventional method. And to �nd out how the estimated parameters will maximize the log-likelihood function. To maximize the log-likelihood function, the controlling parameter H in the algorithm must be maximized too, so the objective function should be multiplied by (-1) according to (Sabri and Sarsour, 2019). The deter- mination of the control parameter and generating initial and neighbouring solutions based on the SA procedure should be carefully observed. Also, the data sample should be big enough to obtain an accurate result. The performance of the Simu- lated Annealing algorithm (SA) has been assessed according to Mean Absolute Error (MAE) based on the simulated data set for di�erent sample size. The e�ectiveness of Mean Absolute Error (MAE) has been demonstrated in several studies, includ- ing (Peng and Yao, 2003; Franses, 2016; Frías-Paredes et al., 2018; Sayed and Sabri, 2022). The following section presents © 2022 The Authors. Page 85 of 90 Sayed and Muhammad Science and Technology Indonesia, 7 (2022) 84-90 the generalised gamma distribution along with its parameters and statistical properties. 2.1 Generalize Gamma Distribution and its Parameters The most common pdf form of the generalized version of the gamma distribution (GGD) can be written as, f (t; U, V , c) = V Γ(U)c ( t c )U V−1 e−( t c) U (3) Where U>0 and V>0 are the shape parameters, c>0 is the scale parameter and Γ(U) is the gamma function that de�nes as, Γ(U) = ∞∫ 0 xU−1e−xdx (4) Because such a distribution is associated with convergent di�culties of the MLE, some reparameterization with the pa- rameters was used to alleviate these di�culties as follow, ` = ln(c) + 1 V ln ( 1 _2 ) (5) f = 1 V √ U (6) _ = 1 √ U (7) where -∞< ` <∞, f > 0, 0 < _. Based on this reparameter- ization and the values of the parameters, the GGD can include special cases of some distributions such as The Gamma Dis- tribution (GD), The Exponential Distribution (ED), Weibull Distribution (WD), and The Lognormal Distribution (LD) as shown in Figure 1. In this case, a continuous random variable Xk taking a non-negative real value is said to follow the gamma distribu- tion for K-years period. Xk distributed using two parameters Gamma distribution U and \ with mean=U\ and variance=U\2. Then, K=2,...,4, we impose a new growth rate, W by letting Xk=(1+W)k−1X1 . Because the growth rate W is associated with the scale of the data, this results from the development of the Gamma distribution to become a generalized gamma distribu- tion as follows, fXK (X , U, (1+W) K−1 \) =  1 Γ(U) [ 1 (1+W)K−1\ ]U xU−1e x (1+W)K−1\ , x > 0 0, otherwise (8) Figure 1. Generalized Gamma Distribution Based on The Di�erent Parameters With the parameters U and (1 + W)K−1\ and having the mean=U(1+ W)K−1\ and variance=U [ (1+ W)K−1\ ]2 . Estimat- ing the parameters along with the growth parameter W of Equa- tion (8) using the method of the moment is not an easy task because it’s di�cult to construct the values of them from the mean and variance equations unless we �x the value of one of them and solve for the other two. From Equation (8), the likelihood function (L) of XK is L(U, (1+ W)K1\ |xK) =− ln [Γ(UK)] − UK ln((1+ W)K−1\K) + ln(xUK−1K ) − XK (1+ W)K−1\K (9) and, the log-likelihood function is then be simpli�ed as, l(U, (1+ W)K1\ |xK) =− K∗∑ K−1 ln [Γ(UK)] − K∗∑ K=1 UK ln((1+ W)K−1\K) + K∗∑ K=1 ln(xUK−1 K ) − K∗∑ K−1 XK (1+ W)K−1\K (10) It is observed by various studies include (Abbasi et al., 2006) and Abubakar and Sabri (2021b) that the maximum likelihood approach used to estimate the three parameters of our gen- eralized gamma model is a di�cult process, this is because is hard to get a straightforward di�erentiation of the resulting log-likelihood function in Equation (10). Therefore, it is math- ematically challenging in searching for the optimal parameters of the generalized gamma distribution simultaneously from Equation (10) as di�erentiating the log-likelihood function to- wards U couldn’t be derived directly. Therefore, it is easier to incorporate with another useful method such as the Simulated Annealing algorithm (SA) to search for the gamma distribution parameters, which works with the log-likelihood function as an objective function and aims to maximize it. The SA procedure © 2022 The Authors. Page 86 of 90 Sayed and Muhammad Science and Technology Indonesia, 7 (2022) 84-90 use in estimating the parameters of the generalized gamma distribution has been presented in the following section. 3. SIMULATED ANNEALING ALGORITHM FOR GAMMA PARAMETERS ESTIMATION In this work, a simulated annealing algorithm is used to estimate the parameters of the generalized gamma distribution and detect how close the estimated parameters are to the true ones. Furthermore, we observe how the estimated parameters are going to maximize the log-likelihood function. This procedure involves some steps, as it is shown in Figure 2. Figure 2. Block Diagram of The Proposed Framework Step1: Variables Designing Start with generating the data sample randomly by simu- lation method using Microsoft Excel. From Equation (1) the cumulative density function of X when K=1 is, FX1(X ∗) = X∗∫ 0 1 Γ(U) ( 1 \ )U xU−1e− x \ dx = Gamma(U, \ , x∗) (11) If the value of Fx1(x ∗)=p, then we �nd x as, Gamma(U, \ , x∗) = p → x∗ = Gamma−1(U, \ , p) (12) In Excel, the value of p is computed as the function of p=randa(). The cumulative density function of X when K>1 is, FXK (X ∗) = X∗∫ 0 1 Γ(U) [ 1 (1+ W)K−1\ ]U xU−1e − x (1+W)K−1\ dx (13) and Gamma(U, (1+ W)K−1\ , x∗) = p (14) Thus, x∗ = Gamma−1(U, (1+ W)K−1\ , p) (15) So, in Microsoft Excel x∗ is generated and distributed using CDF of the gamma distribution to calculate the log-likelihood function in the following few steps: 1. Generate p using the function =rand(). 2. Select initial values for U, \ , W. 3. Generate x∗ using the inverse gamma function Gamma−1 (U,(1+ W)K−1\ , p). 4. Find the cumulative gamma distribution FX1(x ∗) using the initial parameters. 5. Find the inverse log of FX1(x ∗) and calculate the log- likelihood function. 6. Estimate a set of 100 gamma parameters of U, \ , W using the simulated annealing method. 7. Find the cumulative gamma distribution FX1(x ∗) using the mean of the 100 estimated parameters. 8. Find the inverse log of FX1(x ∗) and calculate the log- likelihood function. Step 2: Simulated Annealing Optimization As mentioned earlier, it is only helpful to incorporate a use- ful method such as simulated annealing to estimate Gamma parameters that maximize the log-likelihood function. The process of SA requires advanced programming, hence, in this study, MATLAB coding is used to estimate 100 values of the Gamma parameters. The SA algorithm is explained as follows: 1. Get a big enough sample of data x∗. 2. Determine the objective function f = (l, x∗) where l is the maximum likelihood. 3. Select controlling parameters for the SA, for example, Ho, H, A, B such that: while H > Ho, H = AH. 4. Select lower and upper bounds of random variables lb = [., ., .] , ub = [., ., .]. 5. Generate random values a, b, and c within the lower and upper bounds. 6. Compute the likelihood function l at random values. 7. Generate neighbour values a1, b1, and c1 within the lower and upper bounds. 8. Compute the likelihood function l1 at the neighbour values. 9. If l1 > l, then l1 = l, and a = a1, b = b1, c = c1. 10. Else generate a random value rn(0,1). 11. If e− (l1−l) H >r then a = a1, b = b1, c = c1. 12. Print 100 sets of a, b, and c which are Û,\̂, and Ŵ respec- tively. 4. EXPERIMENT RESULT AND DISCUSSION In this study, 4 sets with di�erent samples sizes of 50, 100, 200, and 500 were generated using Microsoft Excel for the values of K=2,3,4,5 accordingly. The CDF of the gamma distribution with three initial parameters (U=4, \=0.3, W=-0.05) was distributed across all data sets generated. Then the SA algorithm was used to further estimate 100 sets of the gamma parameters for each data sample size. The performance of the © 2022 The Authors. Page 87 of 90 Sayed and Muhammad Science and Technology Indonesia, 7 (2022) 84-90 SA algorithm was evaluated using the Mean Absolute Error (MAE) of the estimated parameters Û,\̂, and Ŵ for each sample size as follows, 1 3 ( |Û − U | + |\̂ − \ | + |Ŵ − W | ) (16) The CDF of gamma distribution was calculated for each sample size with the initial parameters and the mean of the estimated parameters to see how close the distributions will be for the di�erent data sets. Finally, the inverse of the maximum likelihood is calculated based on the initial parameters, then the average of the estimated parameters for each sample size to observe on how the means of our estimated parameters sets will maximize the maximum likelihood. Figure 3. Box Plots of The MAE for Each Sample Size Sample 50, 100, 200 and 500 Figure 4. CDF of The GGD for 50 Sample Sizes Table 1. Estimated Parameters of The GGD Via SA Sample size (n) Mean of the estimate Likelihood at the initial estimate Likelihood at the Mean 500 4, 0.31, -0.052 504.7 524.27 200 4.2, 0.29, -0.051 194.03 202.09 100 4.2, 0.27, -0.052 93.25 125.57 50 4.1, 0.33, -0.051 45.65 55.79 According to box plots in Figure 3, the larger the sample size, the closer the average absolute error of the estimated pa- rameters to zero. The sample of size 50 has a larger spread Figure 5. CDF of The GGD for 100 Sample Sizes Figure 6. CDF of The GGD for 200 Sample Sizes while the spread of the data gets smaller when the size of the data sample gets larger, which means that the 100 sets of esti- mated parameters Û, \̂, and Ŵ fall closer to the initial parame- ters the larger the data sample gets, which stresses the e�ciency of the algorithm. Figures 4 to 7, displayed the CDF of the GGD function based on the data set generated using both the initial parameters and the mean of the estimated parameters for each sample size. It can be observed that the estimated parameters of the generalized gamma distribution are getting closer to the true parameters as the sample size increases. Table 1 reports the mean of the estimate, the likelihood value of the mean and the initial estimate. It is seen that the Gamma accu- racy of the estimated parameters is getting closer to the true parameters when the sample size becomes large. Moreover, the Figure 7. CDF of The GGD for 500 Sample Sizes © 2022 The Authors. Page 88 of 90 Sayed and Muhammad Science and Technology Indonesia, 7 (2022) 84-90 mean estimated parameters using the SA algorithm maximize the likelihood function optimally. When it comes to parameter estimation for a Generalized Gamma Distribution (GGD), the classical method produces a lot of inaccuracies. Although the amount of inaccuracies in SA is low, the o�ered settings are occasionally surprising and unrealistic. This reveals that the simulated annealing algorithm can work optimally with maxi- mum likelihood in estimating the parameters of the generalized gamma distribution model. 5. CONCLUSION The goal of this study was to explore a simple approach to �t a Generalized Gamma Distribution (GGD) using simulated data, not to propose a new theoretical estimation of the parameters of the generalized gamma distribution. This study shows the e�ciency of using the simulated annealing algorithm based on to estimate the parameters of the generalized Gamma dis- tribution for a randomly generated data sample based on the simulation method explained above. The results presented in the tables and �gures demonstrated that the parameters of gen- eralized gamma distribution can be optimally estimated using a simulated annealing algorithm incorporated in the maximum likelihood method and the ability to simplify the use of such a generalized distribution to model important phenomena that other distributions cannot. Implementing the simulated an- nealing algorithm on other distributions from the same family will be our future direction. The study presented in this pa- per can also contribute to modelling di�erent real-life data set such as internal rate of return and option pricing in the �nan- cial market, wind speed and reliability problems in reliability engineering. 6. ACKNOWLEDGEMENT I would like to thank Jazan University, Saudi Arabia for sponsor- ship opportunity given to me to pursue PhD at the Universiti Sains Malaysia. I am thankful to my family for their uncounted support and encouraged during my study period. 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Page 90 of 90 INTRODUCTION SIMULATED ANNEALING ALGORITHM Generalize Gamma Distribution and its Parameters SIMULATED ANNEALING ALGORITHM FOR GAMMA PARAMETERS ESTIMATION EXPERIMENT RESULT AND DISCUSSION CONCLUSION ACKNOWLEDGEMENT