Meta-Heuristic Algorithms for Hydrologic ... - Semantic Scholar

Water Resour Manage (2011) 25:1855–1879 DOI 10.1007/s11269-011-9778-1

Meta-Heuristic Algorithms for Hydrologic Frequency Analysis Yousef Hassanzadeh · Amin Abdi · Siamak Talatahari · Vijay P. Singh

Received: 28 April 2010 / Accepted: 27 January 2011 / Published online: 22 February 2011 © Springer Science+Business Media B.V. 2011

Abstract Meta-heuristic algorithms, such as the genetic algorithm and ant colony optimization, have received considerable attention in recent years due to their higher ability for solving difficult engineering optimization problems. This paper employs these techniques for estimating parameters of commonly used flood frequency distributions, and compares them with some conventional methods such as maximum likelihood, moments and probability weighted moments using annual maximum discharge data of 14 rivers from East-Azarbaijan, Iran. The results indicate that both the genetic algorithm and ant colony optimization are suitable parameter estimation alternatives. Also, the results of Monte Carlo simulation for various sample sizes, ranging from 20 to 100, demonstrate that the meta-heuristic algorithms yield accurate quantile estimates. Keywords Hydrologic frequency analysis · Conventional methods · Meta-heuristic algorithms · Parameter estimation · Monte Carlo simulation

Y. Hassanzadeh · A. Abdi (B) · S. Talatahari Faculty of Civil Engineering, University of Tabriz, Tabriz, Iran e-mail: [email protected] Y. Hassanzadeh e-mail: [email protected] S. Talatahari e-mail: [email protected] V. P. Singh Department of Biological and Agricultural Engineering, Texas A & M University, College Station, USA e-mail: [email protected]

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1 Introduction The primary objective of frequency analysis is to relate the magnitude of extreme events to their frequency of occurrence through the use of probability distributions (Chow et al. 1988). The preciseness of hydrologic frequency analysis depends on the type of statistical distributions and parameter estimation techniques. A lot of models have been developed to describe the distribution of hydrological data. However, the choice of a suitable model is still one of the major problems in frequency analysis. Likewise, there are several methods of parameter estimation, among which the most popular are the methods of maximum likelihood, moments and probability weighted moments. Using four data sets of annual maximum rainfall and annual peak flow discharge, Singh and Deng (2003) compared the entropy-based method with the methods of moments, L-moments, and the maximum likelihood estimation. The results of estimation showed that both the entropy and L-moments methods enabled the four-parameter Kappa distribution to fit the data well and a combination of the two methods can further improve parameter estimation. Mahdi and Ashkar (2004) estimated parameters of the Weibull distribution using generalized probability weighted moments, generalized moments and maximum likelihood estimation methods. Utilizing data from the Miramichi River in Canada, they showed that the maximum likelihood produced better results than the other two. Using the methods of moments, probability weighted moments, maximum likelihood, principle of maximum entropy, and least squares Öztekin (2005) estimated parameters of the three-parameter generalized Pareto distribution for observed annual maximum discharge data for 50 rivers, most of them in Turkey. It is concluded that the method of moments was superior to all other methods employed. Ashkar and Tatsambon (2007) proposed some of the principal methods such as methods of maximum likelihood, moments, probability weighted moments, and generalized probability weighted moments for fitting the generalized Pareto distribution. They used 42 lowflow events for hydrometric station on the Fish River in New Brunswick, Canada. Finally the maximum likelihood method was seen to provide the best fit for that particular data set. Hassanzadeh et al. (2008) investigated six distributions, including generalized extreme value, Pearson type 3, generalized logistic, generalized Pareto, normal and exponential. For 14 data series of annual maximum discharges they utilized the methods of moments, probability weighted moments and maximum likelihood. Using the goodness-of-fit tests they showed that the maximum likelihood method was the best parameter estimation method. The genetic algorithm (GA) has been discovered to be a useful tool to solve optimization problems in a variety of disciplines (Sivanandam and Deepa 2008). For example, Karahan et al. (2007) developed and used a GA to predict rainfall intensities for a given set of return periods. The results showed that the proposed GA can be used to develop rainfall intensity–duration–frequency relationships with lowest mean-squared error between observed and predicted intensities. They concluded that predicted intensities were in good agreement for the studied return period. Dong (2008) used the GA to optimize the parameters of instantaneous unit hydrograph (IUH). Also he utilized the approximate formula method and moment method. The results showed that the GA is more effectual than the other two methods. Reca et al. (2008) evaluated the performance of several meta-heuristic techniques containing

Meta-Heuristic Algorithms for Hydrologic Frequency Analysis

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GA, simulated annealing, tabu search, and iterated local search for optimization of looped water distribution systems. Those techniques first validated and compared by applying them to a medium-sized benchmark network. Then they applied to a large irrigation water distribution network. The results showed that the GA was more efficient when dealing with a medium-sized network, but other methods outperformed it when dealing with a real complex one. Recently, Rai et al. (2009) employed the GA to derive the unit hydrograph (UH). They used nine distributions such as Beta, Exponential, Gamma, Normal, Lognormal, Weibull, Logistic, Generalized logistic and Pearson type 3 for the determination of UH. Parameters of nine distributions functions were estimated using the real coded GA optimization technique. Those distributions were tested on 13 watersheds of different characteristics and it was observed that except for the Exponential distribution function, most other distribution functions produced UHs which were in satisfactory agreement with observed UHs. Also, Reddy and Adarsh (2010) utilized two meta-heuristic search algorithms containing GAs and particle swarm optimization to obtain optimal solutions for the design of irrigation channels. Ant Colony Optimization (ACO) is a new approach that has been applied to water distribution system optimization (Maier et al. 2003), optimal design of open channels (Nourani et al. 2009; Kaveh and Talatahari 2010), optimization of soil hydraulic parameters (Abbaspour et al. 2001), identifying optimal sampling networks that minimize the number of monitoring locations in ground water design optimization (Li and Chan Hilton 2006), deterring the optimum design of skeletal structures congaing optimum weight (Kaveh et al. 2008), among many others. The advantages of applying ACO to engineering problems are similar to those of GA. Both are multiagent randomized search techniques in which in each cycle a number of search space points are selected and tested. The random selection and the information obtained in each cycle are used to choose new points (design vectors) in subsequent cycles. ACO and GA contrary to many conventional algorithms is a global optimizer. ACO is utilized to derive operating policies for a multi-purpose reservoir system by Kumar and Reddy (2006). To formulate the ACO for reservoir operation, they modified the problem by considering a finite time series of inflows, classifying the reservoir volume into several class intervals, and determining the reservoir release for each period with respect to a predefined optimality criterion. The ACO technique is applied to a case study of Hirakud reservoir, which is a multi-purpose reservoir system located in India. The results of the two models indicate the good performance of the ACO model, in terms of higher annual power production, satisfying irrigation demands and flood control restrictions. Nourani et al. (2009) tried to optimize the overall reliability and cost effectiveness of composite channels. The models are developed to minimize the total cost, while satisfying the specified probability of the channel capacity being greater than the design flow. Also, Kaveh and Talatahari (2010) introduced an improved ACO to solve this problem and they found that the ACO algorithm performs well in finding optimal solutions. Also, Madadgar and Afshar (2009) proposed an improved continuous ACO to water resources problems. Then they examine the performance of the proposed algorithm on few well-known benchmark problems as well as a real-world water resource optimization problem. The comparison of the results emphasized the robustness of the ant colony algorithm in searching the continuous

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space more efficiently as locating the closest, among other ant methods, to the global optimal solution. According to our knowledge, there are few studies on applying meta-heuristics to estimate parameters of flood frequency distributions. Main contribution of this study contains therefore to utilize the GA and ACO methodologies to estimate parameters of flood frequency distributions using maximum discharge data from 14 gauging stations from northwest of Iran, and compare them with the conventional methods such as moments, maximum likelihood and probability weighted moments.

2 Meta-Heuristic Algorithms Two types of algorithms were employed here: the GA and ACO. 2.1 Genetic Algorithm The GA developed by Holland (1975), is a stochastic search method that mimics natural evolution and is based on the concept of “survival of the fittest.” The GA has an advantage over traditional optimization techniques, since it does not require derivatives of the objective function and can hence be applied to solve complex and discontinuous optimization problems. The GA is inspired by biological evolution, cross-breeding, trial solutions and allows only the best solutions to survive and propagate to successive generations. It deals with a population of individual (candidate) solutions, which undergo constant changes by means of genetic operations of reproduction, crossover, and mutation. These solutions are ranked according to their fitting with respect to the objective function where the fitted individuals are more likely to reproduce and propagate to the next generation. Based on the value of the objective function, individuals (parents) are selected for reproduction of the next generation by exchanging genetic information to form children (crossover). The parents are then removed and replaced in the population by the children to keep a stable population size (Fig. 1a). The result is a new generation with (normally) better fitting. Occasionally, mutation is introduced into the population to prevent the convergence to a local optimum and help generate unexpected directions in the parameter space (Fig. 1b). The more GA iterates, the better the chance would be to generate an optimal solution. After a number of generations, the population is expected to evolve artificially, and the (near) optimal solution will be reached. The measure of success is the convergence to a population with identical members. The global optimum solution,

Fig. 1 a Crossover operator, b mutation operator

(a)

(b)


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Start NC=1; (number of iteration)

Initialize random population

Evaluate fitness for the population

Store best individual

Creating mating pool NC=NC+1; Create next generation by applying crossover

Reproduce and ignore few populations

Perform mutation

NO

The termination conditions satisfied? YES

Output values contains the best fitness

End

Fig. 2 The flowchart for the genetic algorithm

however, cannot be guaranteed, since the convexity of the objective function cannot be proven. The algorithm is repeated until a solution is found that satisfies a target, a fixed number of generations has been produced, the highest ranking solutions reach a plateau and there is no further improvement with repeated iteration, or one may run out of time or money. The GA has been widely applied in a variety of engineering optimization problems, shown to be capable of solving optimization problems with non-differentiable, nonlinear, and multimodal objective functions (Goldberg 1989; Zalzala and Fleming 1997; Gen and Cheng 2000). The GA procedure is illustrated in Fig. 2. 2.2 Ant Colony Optimization In the 1990s, the ACO was introduced as a novel nature-inspired method for solving difficult combinatorial optimization problems (Dorigo et al. 1996; Dorigo and Stützle 2004). The inspiring source of ACO is the foraging behavior of real ants. When searching for food, ants initially explore the area surrounding their nests in a random manner. As soon as an ant finds a food source, it evaluates it and carries some food back to the nest. During the return trip, the ant deposits a pheromone trail on the ground. The pheromone deposited, the amount of which may depend on the quantity and quality of the food, guides other ants to the food source. This indirect

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communication among ants via pheromone trails enables them to find shortest paths between their nest and the food source. This capability of real ant colonies has inspired the definition of artificial ant colonies that can find approximate solutions to hard combinatorial optimization problems. When ants encounter an obstacle, at first, there is an equal probability for all ants to move right or left, but after a while, the number of ants choosing the shorter path increases because of the increase in the amount of the pheromone deposited on that path. With the increase in the number of ants and pheromone on the shorter path, all of the ants will choose and move along the shorter path. The ACO can reach the performance of other general-purpose heuristics like evolutionary computation. Its main characteristic is that after each iteration, the pheromone values are updated (global updating rule). The pheromone associated with the edge joining cities i and j is updated as: τij (t + n) = (1 − ρ) .τij (t) + ρ.τij (t)

(1)

where ρ ∈ (0, 1) is the pheromone decay coefficient, and τij(t) is computed for the best ant as: τij (t) = 1/L+

(2)

where L+ is the length of navigated tour for the best ant. According to the local updating rule, after each selection, the pheromone value related to the selected path is updated by the local updating rule as: τij (t + 1) = ξ.τij (t)

(3)

where ξ is an adjustable parameter between 0 and 1, representing the persistence of the trail. In constructing a solution, ants select the following city to be visited through a stochastic mechanism. When an ant m is in a city i, the probability of going to city j is given by: ⎧ α β ⎪ τij ηij ⎪ ⎪ ⎨ α β m τij ηij pij = ⎪ ⎪ j∈Ni ⎪ ⎩ 0

if j ∈ Ni (4) otherwise

where Ni is the set of all the neighbor nodes of node i; α and β are two parameters that control the relative weight of pheromone trail versus the heuristic information; and ηij, called visibility, gives the heuristic information to ants, which is given by: ηij =

1 dij

(5)

where dij is the distance between cities i and j (Kaveh et al. 2008). The flowchart of the ACO algorithm is shown in Fig. 3.


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Start NC=1; (number of iteration)

Initialize the pheromone trail Locate The initial position of ants Set t=0; Each ant selects a path the variable

t=t+1 Perform local updating rule Is all ants select a value for all variables?

NO

YES

Count the fitness function for each ant

NC=NC+1;

Perform global updating rule

NO

The termination conditions satisfied? YES

Output values contains the maximum pheromones

End

Fig. 3 The flowchart for the ant colony optimization

3 Application In order to utilize GA and ACO algorithms for estimating parameters of flood frequency distributions, a region of East-Azarbaijan in Iran was considered. The information related to the selected region, description of goodness-of-fit tests, the parameter estimation methods, and quantile estimation functions are presented in what follows.

3.1 Study Area and Data Availability The present study was carried out for the catchments of East-Azarbaijan which lie between latitude 36◦ to 39◦ North and longitude 45◦ to 48◦ East. The total geographical area spans over 45,491 km2 . Annual maximum discharge data from 14 streamflow gauging sites lying in East-Azarbaijan, northwest of Iran (Fig. 4) and varying over 27–50 years in record length were obtained from East-Azarbaijan regional water corporation of the Iran ministry of energy (see Table 1).

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East-Azarbaijan #

#

Daryan

Pole Sanikh Hervi Zinjanab #

#

#

Saeed Abad

#

Lighvan

#

Sahzab #

Iran

Bostan Abad

#

Ghirmizi Gol Shishavan Tazekand # #

Maghanjigh #

#

Gheshlaghe Amir

N

Shirin Kandi W

#

E S

70

0

70

140 Kilometers

Fig. 4 Study area showing location of streamflow gauging stations

3.2 Goodness-of-fit Tests Various criteria can be employed to evaluate the suitability of a probability distribution for describing a set of data. Statistical goodness-of-fit tests are used to determine whether selected distributions are consistent with the given set of observations (Stedinger et al. 1993). Two popular goodness-of-fit tests were utilized in this paper.

Table 1 Characteristics of stations selected for the study No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Station Bostan Abad Daryan Ghirmizi Ghol Gheshlaghe Amir Hervi Lighvan Maghanjigh Pole Sanikh Sahzab Saeed Abad Shishavan Shirin Kandi Tazekand Zinjanab

Sample size (Year) 33 33 40 29 32 50 27 39 29 29 38 38 30 38

Geographical characteristics Height (m) Latitude 1725 1616 1800 1520 1920 2150 1650 1352 1855 1850 1270 1365 1610 2100

37◦ –50

38◦ –14 37◦ –43 37◦ –19 37◦ –55 37◦ –50 37◦ –20 38◦ –11 37◦ –59 37◦ –59 37◦ –28 37◦ –01 37◦ –29 37◦ –51

Longitude 46◦ –50 45◦ –36 46◦ –06 46◦ –17 46◦ –29 46◦ –26 46◦ –25 46◦ –09 47◦ –39 46◦ –35 45◦ –53 46◦ –16 46◦ –16 46◦ –19


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3.2.1 Graphical Display Graphical techniques can be used to visually assess the adequacy of a fitted distribution. Probability plots or Quantile–Quantile (Q–Q) plots are a means of comparing observed data to a theoretical distribution. 3.2.2 Statistical Test Criteria Though the Q–Q plots are useful, they could be subjective and cannot indicate the statistical significance of the fit. Three test criteria, such as coefficient of determination (CD), coefficient of efficiency (CE), and root mean square error (RMSE), were used (Nash and Sutcliffe 1970; Kite 1977; Wang et al. 2009): n

CD =

Qoi − Qo

i=1

n

Qoi − Qo

Qci − Qc

2 n

i=1

Qci − Qc

2 1/2 , −1 ≤ CD ≤ 1

(6)

i=1

n

CE = 1 −

(Qoi − Qci )2

i=1 n

Qoi − Qo

2 , −∞ < CE ≤ 1

(7)

i=1

RMSE =

1 (Qoi − Qci )2 n i=1 n

1/2 (8)

Where Qo is the average of observed discharges; Qc represents the average of computed discharges; Qoi is the i-th observed discharge; and Qci is the discharge computed from the selected distribution corresponding to the same probability as that for the observed discharge obtained using a plotting position formula.

Table 2 Quantile functions of various distributions studied No.

Distribution

I

Generalized Extreme Value (GEV)

II

Pearson Type 3 (PE3)

III

Lognormal Type 3 (LN3)

IV

Generalized Logistic (GLOG)

V

Generalized Pareto (GPAR)

VI

Kappa (KAP)

VII

Wakeby (WAK)

aK

T

bu

standardized normal variable

frequency factor

Equations for quantile estimation

x (F) = u + α/k 1 − (− ln F)k a x (F) = α β + γ + KT α 2 β x (F) = α + exp (μ + σ u)b

x (F) = ξ + α/k 1 − {(1 − F) /F}k

x (F) = ξ + α/k 1 − {1/ (1 − F)}−k

k x (F) = ξ + α/k 1 − 1 − F h / h x (F) = ξ + α 1 − (1 − F)β /β − γ 1 − (1 − F)−δ /δ

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3.3 Parameter Estimation Matlab software was employed for estimating statistical distribution parameters using the GA and ACO algorithms as well as MML, MOM and PWM. The target function (objective function) which is selected for attaining the most optimum result with the two meta-heuristic algorithms was calculated as: ⎛ n

(Qoi − Qci )2

⎜ i=1 Minimize ⎜ n ⎝

Qoi − Qo

⎞

⎟ 2 ⎟ ⎠

(9)

i=1

3.4 Quantile Estimation After estimation of distribution parameters, quantiles (x(F)) were estimated by using the equations (Rao and Hamed 2000) given in Table 2, wherein F represents the cumulative probability of non-exceedence, and u, k, h, α, β, ξ , γ and δ are parameters of the statistical distributions.

Table 3 Values of estimated parameters of different distributions by MML, MOM, PWM, GA and ACO methods for Shirin Kandi station Distribution

Parameter

GEV

μ α k γ α β α μ σ ξ α k ξ α k ξ α k h ξ α β γ δ

PE3

LN3

GLOG

GPAR

KAP

WAK

Method MML

MOM

PWM

GA

ACO

4.9267 2.6854 −0.0445 0.2763 2.0918 3.0232 −2.0420 2.0744 0.4055 5.9257 1.8088 −0.2357 1.2400 7.0083 0.3278 –

4.9160 2.7822 −0.0277 0.9658 2.4370 2.3120 −2.3358 2.1108 0.3983 6.1753 1.9150 −0.1322 2.3541 4.9109 0.1565 –

–

–

4.8831 2.6685 −0.0631 0.8502 2.3398 2.4576 −1.5168 1.9987 0.4364 5.9319 1.8263 −0.2111 2.0698 5.9019 0.3027 5.9145 1.8190 −0.2169 −0.9850 0.2302 3.8183 7.4009 19.7117 0.1323

4.8637 2.6011 −0.1029 1.4381 2.7318 1.9079 0.0886 1.7332 0.5463 5.9501 1.8599 −0.2240 2.6405 4.2996 0.0739 5.7422 1.9623 −0.2085 −0.7564 0.7118 3.4371 5.9256 18.5787 0.1398

4.8517 2.5409 −0.1149 1.2333 2.5511 2.1368 −0.3572 1.8175 0.5102 5.9862 1.8479 −0.2360 2.5803 4.4562 0.0859 5.8816 1.9527 −0.2104 −0.7769 1.1335 2.9431 5.5039 21.0004 0.1278


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3.5 Monte Carlo Simulation A Monte Carlo simulation was conducted to evaluate the performance of a parameter estimation method, by generating and analysing samples from various distributions with known parameters. For various methods, this approach allows the investigator to establish bench marks in terms of relative bias (RBIAS) and relative root mean square error (RRMSE) for selected quantiles. RBIAS and RRMSE are given as: 1 RBI AS = m i=1 m

ˆ T,i − QT Q QT

(10)

Table 4 Values of the goodness-of-fit tests for various distributions at Shirin Kandi station Distribution

Method

Goodness-of-fit test CD

CE

RMSE

GEV

MML MOM PWM GA ACO MML MOM PWM GA ACO MML MOM PWM GA ACO MML MOM PWM GA ACO MML MOM PWM GA ACO PWM GA ACO PWM GA ACO

0.9853 0.9844 0.9860 0.9868 0.9867 0.9821 0.9834 0.9832 0.9837 0.9836 0.9843 0.9841 0.9851 0.9855 0.9858 0.9870 0.9809 0.9869 0.9875 0.9877 0.9731 0.9777 0.9715 0.9788 0.9786 0.9874 0.9878 0.9881 0.9876 0.9891 0.9886

0.9691 0.9685 0.9714 0.9735 0.9734 0.9637 0.9667 0.9660 0.9676 0.9672 0.9675 0.9678 0.9697 0.9713 0.9717 0.9749 0.9607 0.9736 0.9750 0.9754 0.9512 0.9558 0.9436 0.9577 0.9574 0.9739 0.9756 0.9760 0.9742 0.9773 0.9765

0.6423 0.6491 0.6182 0.5950 0.5960 0.6965 0.6672 0.6745 0.6578 0.6622 0.6593 0.6558 0.6361 0.6200 0.6149 0.5792 0.7253 0.5942 0.5760 0.5755 0.7727 0.7688 0.8681 0.7516 0.7550 0.5910 0.5753 0.5559 0.5636 0.5530 0.5541

PE3

LN3

GLOG

GPAR

KAP

WAK

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⎡

1 RRMSE = ⎣ m i=1 m

ˆ T,i − QT Q QT

2 ⎤1/2 ⎦

(11)

Where m is the number of Monte Carlo samples for sample sizes of 20, 50 and ˆ T,i is 100 values, QT is the quantile estimated using the known parameters, and Q the quantile computed by using the parameters estimated from sample i of each simulation for the selected sample (Singh and Guo 1995; Yu and Naghavi 1994; Kim and Heo 2002).

GEV (GA)

20 15

15

10

10

5

5

0

0

5

10

15

20

Computed Discharge (m3/s)

PE3 (GA)

0

20

20

15

15

10

10

5

5

0

0

5

10

15

20

LN3 (GA)

0

20

20

15

15

10

10

5

5

0

0

5

10 GLOG (GA)

15

20

0

20

20

15

15

10

10

5

5

0

0

5

10

GEV (ACO)

20

15

0

5

10

15

20

15

20

10 GLOG (ACO)

15

20

10

15

20

PE3 (ACO)

0

5

10 LN3 (ACO)

0

5

0 20 0 5 Observed Discharge (m3/s)

Fig. 5 Quantile–Quantile plots for various distributions fitted with the GA and ACO methods to the annual maximum flows observed at Shirin Kandi station


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4 Results and Discussion A ranking scheme was developed to evaluate the overall goodness of fit of each distribution by comparing the three categories of test criteria described earlier. Ranking was assigned to each distribution for every test category according to the relative magnitude of the test statistic. A distribution with the highest CD, highest CE and lowest RMSE was given the best rank. Tables 3 and 4 summarize values of estimated parameters of different distributions and the results of three statistical tests for Shirin Kandi station, respectively. Also, Fig. 5 illustrates the probability plots for the same station. Generally, the results of goodness-of-fit tests for selection of the most appropriate distribution function show that meta-heuristic methods are best methods for parameter estimation. Also there was very little difference between GA and ACO for each distribution. Further, the goodness-of-fit assessment for all 14 stations reveals that Wakeby distribution and then Kappa distribution produced better fits. A possible reason could be that these distributions have more parameters compared to others that have fewer parameters. On the other hand, the use of statistical distributions with four or five parameters has been advocated on two grounds: versatility and the

GPAR (GA)

GPAR (ACO)

20

20

15

15

10

10

5

5

0

0

5

10

15

20

0

0

5

Computed Discharge (m3/s)

KAP (GA) 20

20

15

15

10

10

5

5

0

0

5

10

15

20

0

0

5

WAK (GA) 20

15

15

10

10

5

5

0

5

10

15

20

0

0

Observed Discharge (m3/s)

Fig. 5 (continued)

15

20

10

15

20

15

20

WAK (ACO)

20

0

10 KAP (ACO)

5

10

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ability to represent data from mixed populations. As an example, when using the GA and ACO, the estimated RMSE is obtained to be 0.55 for the Kappa and Wakeby distributions, while it is found to be 0.6 for the method of PWM for the Shirin Kandi station (Table 4). The value of h for the Kappa distribution with the PWM method was equal to −1 and for other methods, this can be considered approximately equal to −1. From Section B.6 (Appendix B), it can be concluded that when h = −1, the Kappa distribution is similar to generalized logistic distribution. Comparison of the generalized logistic and Kappa shows that other parameters (ξ , α and k) have the same values. Therefore, it can be concluded that the generalized logistic distribution after the Wakeby and Kappa distributions provides better fits. This is also shown in Table 4 for Shirin Kandi station. For the Wakeby distribution, the γ parameter varies considerably compared to other parameters for some stations. Therefore, if one can define a good domain for

Table 5 Values of quantiles for various return periods at Shirin Kandi station Distribution GEV

PE3

LN3

GLOG

GPAR

KAP

WAK

Method MML MOM PWM GA ACO MML MOM PWM GA ACO MML MOM PWM GA ACO MML MOM PWM GA ACO MML MOM PWM GA ACO PWM GA ACO PWM GA ACO

Return period (years) 10

25

50

100

200

11.28 11.38 11.34 11.45 11.38 11.45 11.53 11.48 11.64 11.63 11.34 11.42 11.39 11.49 11.48 11.13 11.06 11.04 11.23 11.31 11.57 11.85 11.86 11.74 11.89 11.04 11.25 11.20 11.07 11.20 11.26

14.16 14.22 14.34 14.72 14.67 14.14 14.40 14.29 14.67 14.57 14.15 14.25 14.33 14.82 14.69 14.48 14.34 14.20 14.57 14.73 14.18 14.77 14.21 14.86 14.95 14.24 14.61 14.69 14.52 14.71 14.76

16.37 16.38 16.69 17.35 17.36 16.08 16.50 16.35 16.92 16.72 16.27 16.37 16.57 17.47 17.20 17.46 15.92 16.95 17.50 17.78 16.69 16.72 15.60 17.25 17.39 17.03 17.53 17.68 17.42 17.67 17.69

18.64 18.57 19.13 20.17 20.26 17.98 18.57 18.36 19.14 18.85 18.41 18.52 18.86 20.26 19.82 20.92 18.28 20.10 20.89 21.32 17.90 18.47 16.73 19.42 19.53 20.25 20.87 21.05 20.59 20.94 20.90

20.96 20.79 21.66 23.18 23.38 19.84 20.62 20.36 21.36 21.97 20.59 20.70 21.20 23.21 22.56 24.97 20.85 23.73 24.82 25.47 18.86 20.04 17.65 21.49 21.55 23.96 24.71 24.84 24.08 24.54 24.40


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this parameter, a better result can be expected. Also, in order to have a valid Wakeby distribution, it is necessary that δ < 1 (Landwehr et al. 1979). The PWM method for Daryan station finds δ = 4.5785 which does not lead to a valid solution, while GA and ACO find −1.4492 and −2.0203, respectively.

Table 6 Values of RBIAS at Shirin Kandi station Distribution GEV

Sample size 20 50 100

PE3

20 50 100

LN3

20 50 100

GLOG

20 50 100

GPAR

20 50 100

KAP

20 50 100

WAK

20 50 100

Method

Return period (years) 10 25

50

100

200

GA ACO GA ACO GA ACO GA ACO GA ACO GA ACO GA ACO GA ACO GA ACO GA ACO GA ACO GA ACO GA ACO GA ACO GA ACO GA ACO GA ACO GA ACO GA ACO GA ACO GA ACO

−0.0282 0.0013 0.0255 −0.0037 0.0024 0.0051 0.0013 0.0188 0.0018 0.0003 −0.0110 −0.0014 0.0058 0.0481 −0.0064 0.0042 −0.0064 −0.0023 0.0076 0.0047 0.0023 −0.0021 0.0018 −0.0004 0.0370 0.0392 0.0302 0.0321 0.0198 0.0212 −0.0159 −0.0125 −0.0194 0.0008 −0.0204 −0.0087 −0.0173 −0.0199 −0.0202 −0.0214 −0.0233 −0.0254

0.0565 0.0854 0.0490 0.0283 0.0186 0.0179 0.0607 0.1422 0.0200 0.0410 0.0062 0.0244 0.0808 0.1122 0.0185 0.0388 0.0151 0.0162 0.0210 0.0186 0.0184 0.0125 0.0137 0.0093 0.0715 0.0842 0.0335 0.0352 0.0239 0.0305 0.0052 0.0042 −0.0124 0.0049 −0.0142 −0.0005 0.0076 0.0032 0.0001 −0.0116 −0.0195 −0.0212

0.0898 0.1777 0.0672 0.0418 0.0318 0.0290 0.1231 0.2714 0.0373 0.0735 0.0262 0.0484 0.1306 0.1469 0.0348 0.0566 0.0271 0.0277 0.0340 0.0247 0.0192 0.0148 0.0164 0.0119 0.0970 0.1017 0.0486 0.0569 0.0296 0.0388 0.0473 0.0362 −0.0085 0.0130 −0.0133 0.0040 0.0257 0.0124 0.0106 −0.0099 −0.0018 −0.0183

0.1938 0.2393 0.0913 0.0819 0.0426 0.0442 0.2219 0.5063 0.0612 0.1184 0.0576 0.0888 0.1944 0.1851 0.0543 0.0759 0.0402 0.0400 0.0462 0.0407 0.0201 0.0272 0.0188 0.0144 0.1391 0.1422 0.0809 0.0894 0.0451 0.0519 0.0519 0.0930 −0.0019 0.0252 −0.0108 0.0103 0.0289 0.0179 0.0182 −0.0027 0.0005 −0.0142

−0.0094 0.0331 0.0160 0.0012 0.0094 0.0104 0.0236 0.0687 0.0086 0.0184 −0.0054 0.0095 0.0421 0.0815 0.0054 0.0225 0.0045 0.0066 0.0171 0.0126 0.0127 0.0113 0.0109 0.0064 0.0552 0.0571 0.0314 0.0332 0.0228 0.0287 −0.0139 −0.0107 −0.0146 0.0018 −0.0185 −0.0031 −0.0112 −0.0129 −0.0135 −0.0158 −0.0205 −0.0236

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Values of quantiles for various return periods for Shirin Kandi station are shown in Table 5. The differences between the GA and ACO methods compared to the others could be due to presence of local optima. Also to assess the performance of GA and ACO methods, Monte Carlo sampling experiments were conducted. For each distribution, three sample sizes (20, 50 and

Table 7 Values of RRMSE at Shirin Kandi station Distribution

Sample

Method

size GEV

20 50 100

PE3

20 50 100

LN3

20 50 100

GLOG

20 50 100

GPAR

20 50 100

KAP

20 50 100

WAK

20 50 100

GA ACO GA ACO GA ACO GA ACO GA ACO GA ACO GA ACO GA ACO GA ACO GA ACO GA ACO GA ACO GA ACO GA ACO GA ACO GA ACO GA ACO GA ACO GA ACO GA ACO GA ACO

Return period (years) 10

25

50

100

200

0.1804 0.1667 0.1067 0.1008 0.0700 0.0759 0.1926 0.1897 0.1251 0.1296 0.0794 0.0701 0.1946 0.1739 0.1092 0.1069 0.0798 0.0772 0.1689 0.1809 0.1150 0.1100 0.0751 0.0772 0.2349 0.2236 0.1133 0.1259 0.0875 0.0815 0.1556 0.1606 0.0978 0.1015 0.0756 0.0809 0.1458 0.1427 0.1029 0.0954 0.0729 0.0715

0.3003 0.2644 0.1477 0.1392 0.1019 0.1099 0.3051 0.3115 0.1856 0.1932 0.1131 0.1182 0.2714 0.2491 0.1474 0.1436 0.1126 0.1038 0.2871 0.3211 0.1676 0.1735 0.1201 0.1167 0.2806 0.2719 0.1677 0.1777 0.1173 0.1190 0.2036 0.2246 0.1172 0.1256 0.0919 0.0980 0.2002 0.1937 0.1253 0.1164 0.0894 0.0882

0.4642 0.4484 0.1973 0.1886 0.1432 0.1503 0.4309 0.4198 0.2755 0.2586 0.1760 0.1835 0.3430 0.3476 0.1851 0.1879 0.1469 0.1316 0.4472 0.5801 0.2258 0.2521 0.1827 0.1714 0.4643 0.4513 0.2625 0.2613 0.1428 0.1512 0.2337 0.2554 0.1329 0.1450 0.1104 0.1166 0.2496 0.2391 0.1410 0.1395 0.1085 0.1058

0.7366 0.8398 0.2622 0.2566 0.1989 0.2021 0.5981 0.5906 0.4213 0.4365 0.2816 0.2913 0.4237 0.5026 0.2288 0.2510 0.1872 0.1641 0.7015 0.7789 0.3003 0.3636 0.2797 0.2830 0.6730 0.6413 0.4645 0.4312 0.1917 0.2078 0.3127 0.3339 0.1481 0.1639 0.1256 0.1287 0.3082 0.2899 0.1677 0.1592 0.1229 0.1197

1.7079 1.1995 0.3433 0.3468 0.2660 0.2705 0.8187 0.8216 0.6474 0.6519 0.4490 0.4549 0.5124 0.7563 0.2775 0.3399 0.2328 0.2005 1.1023 1.6600 0.3931 0.5187 0.3278 0.5921 1.2346 1.2135 0.8531 0.8613 0.3314 0.3407 0.4010 0.4598 0.1624 0.1817 0.1328 0.1437 0.3889 0.3552 0.1764 0.1602 0.1316 0.1288


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100) with 500 random samples for selected quantiles corresponding to return periods of 10, 25, 50, 100 and 200 years were considered. Values of RBIAS and RRMSE for Shirin Kandi station are shown in Tables 6 and 7, respectively. The results indicate that the performance of the GA and ACO methods is comparable and adequate for various sample sizes. Also the GA and ACO yield the minimum RBIAS and RRMSE for the Wakeby and Kappa distributions. Appropriate GA parameters identified during this study are summarized below: The number of generations = 1000; population size = 200; stall generation = 5000; stall time limit = 2000; selection = stochastic uniform; crossover function = scattered; mutation function = Gaussian. The number of ants in the ACO algorithm was selected as 100. The value of the other parameters for the ACO algorithm is summarized as: the number of generations = 1000; ρ the pheromone decay coefficient = 0.4; ξ the persistence of the trail in local updating rule = 0.6; and α is set to 1.0 but β is set to 0.4 (Kaveh and Talatahari 2010).

5 Conclusions Various statistical distributions and parameter estimation methods are used to find the most suitable distribution for annual maximum discharge. This procedure is based on a number of graphical and numerical criteria to test the descriptive and predictive abilities of each distribution. The assessment procedure includes the application of meta-heuristic methods, such as the GA and ACO, for estimating parameters of statistical distributions, and evaluating the performance of these algorithms by comparison with the methods of maximum likelihood, moments and probability weighted moments. It is found that these meta-heuristic algorithms can identify parameters of the statistical distributions which in turn can provide reasonable estimates of various flood quantiles at 14 sites located in East-Azarbaijan, Iran. Comparing the results of the meta-heuristic algorithms (i.e. GA and ACO) with conventional methods the following conclusions are drawn: • • •

• •

The GA and ACO are sufficiently effective optimization tools compared to other methods for parameter estimation of statistical distributions and can be used for more complex systems involving nonlinear optimization problems. When an approximate search domain for each parameter is determined, parameter estimation using the GA and ACO becomes easier. In the case of GA and ACO, the number of parameters does not affect the results. Also, an optimum result can always be found using these algorithms, while in the case of conventional methods the increase in the number of parameters not only would make calculation complex but also might hinder reaching the solutions. In addition, with the GA and ACO algorithms, the danger of local optima cannot be ruled out. Contrary to conventional methods of optimization, the GA and ACO can find the optimum solutions without employing differentiation of the objective function and without solving nonlinear complex equations. The GA and ACO are multi-agent approaches. It means that in each iteration a number of solutions is generated containing the values of all parameters. The other methods are a point to point search approach. This means that these

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•


methods need less time for computation than GA and ACO. Although large computational time is an obvious disadvantage of the GA and ACO methods, they are reasonable alternatives to conventional optimization algorithms for solving hydrological problems involving complex nonlinear equations. For the case of flood-frequency analysis, Monte Carlo simulated data are used to evaluate and compare the performance of the GA and ACO. The results indicate that the both algorithms are comparable for various sample sizes studied.

Acknowledgements The authors are grateful to Dr. A.M. Khorshiddoust, Dr. S. Darbandi from University of Tabriz for their helpful comments that substantially improved the manuscript.

Appendix A: Three Other methods of Parameter Estimation A.1 Probability Weighted Moments (PWM) The probability weighted moments for a random variable x and its cumulative distribution function (CDF) F(x) can be defined as (Greenwood et al. 1979):

M p,r,s = E x [F (x)] [1 − F (x)] p

r

s

#1 x p [F (x)]r [1 − F (x)]s dF

=

(A1)

0

where p, r and s are real numbers. One of its special cases relates to s = 0, p = 1: βr = M1,r,0 = E x [F (x)]r =

#1 x [F (x)]r dF,

r = 0, 1, 2, ...

(A2)

0

where βr is linear in x and of sufficient generality for parameter estimation (Hosking 1986). Hosking (1990) introduced L-moments, which are linear functions of PWM’s. For any distribution the rth L-moment λr is calculated as follows: λr+1 =

r

Pr,∗ j β j,

r = 0, 1, 2, ...

(A3)

j=0

Pr,∗ j = (−1)r− j

% $ %$ (−1)r− j (r + j)! r+ j r = j j j ! (r − j)!

(A4)

For example, the first four moments expressed as linear combinations of probability weighted moments are (Hosking 1990): λ1 = β0

(A5)

λ2 = 2β1 − β0

(A6)

λ3 = 6β2 − 6β1 + β0

(A7)

λ4 = 20β3 − 30β2 + 12β1 − β0

(A8)

Also, ratios of L-moments are expressed as τr = λr /λ2 for r = 3, 4, ....


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In the above mentioned relations, λ1 is the mean, λ2 is the L-standard deviation, λ2 /λ1 is the L-coefficient of variation (L-Cv ), τ 3 is the L-coefficient of skewness (L-Cs ) and τ 4 is the L-coefficient of kurtosis (L-Ck ). The relationships for evaluating sample L-moments are defined as: 1 (i − 1) (i − 2) · · · (i − r) xi , x1 ≤ x2 ≤ · · · ≤ xn , r = 0, 1, ..., n − 1 n i=1 (n − 1) (n − 2) · · · (n − r) (A9) n

br =

lr+1 =

r

Pr,∗ j b j, r = 0, 1, ..., n − 1

(A10)

j=0

Also, the ratio of L-moments is introduced by tr = lr /l2 estimators considering r = 3, 4, .... A.2 Method of moments (MOM) Estimates of the parameters of a probability distribution function are obtained in the MOM by equating the moments of the sample with the moments of the probability distribution function. For a distribution with k parameters, α1 , α2 , ..., αk which are to be estimated, the first k sample moments are set equal to the corresponding population moments that are given in terms of unknown parameters. These k equations are then solved simultaneously for the unknown parameters α1 , α2 , ..., αk (Rao and Hamed 2000). For a distribution with probability density function (PDF) f (x), moment of r rank about the origin is: μr

#+∞ = xr f (x) dx

(A11)

−∞

And the corresponding central moments will be: μr =

#+∞

x − μ1

r

f (x) dx

(A12)

−∞

in which

μ1

is defined in Eq. A11.

A.3 Method of Maximum Likelihood (MML) Estimation by the ML method involves the choice of parameter estimates that produce a maximum probability of occurrence of the observations. For a distribution with a PDF given by f (x) and parameters α1 , α2 , ..., αk , the likelihood function is defined as the joint PDF of the observations conditional on given values of the parameters α1 , α2 , ..., αk in the form (Rao and Hamed 2000): L (α1 , α2 , . . . , αk ) =

n & i=1

f (xi ; α1 , α2 , . . . , αk )

(A13)

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The values of α1 , α2 , ..., αk that maximize the likelihood function are computed by partial differentiation with respect to α1 , α2 , ..., αk and setting these partial derivatives equal to zero as in Eq. A14. The resulting set of equations are then solved simultaneously to obtain the values of α1 , α2 , ..., αk , ∂ L (α1 , α2 , . . . , αk ) = 0, i = 1, 2, ..., k ∂αi

(A14)

In many cases it is easier to maximize the natural logarithm of the likelihood function by using ∂ ln L (α1 , α2 , . . . , αk ) = 0, i = 1, 2, . . . , k ∂αi

(A15)

Appendix B: Frequency Distributions B.1 Generalized Extreme Value (GEV) The PDF of the GEV distribution can be expressed as: $ % 1 x − u 1/k−1 −[1−k( x−u )]1/k α f (x) = 1−k e α α

(B1)

The range of the variable x depends on the sign of the parameter k. When k is negative the variable x can take on values in the range u + α/k < x < ∞ which make it suitable for flood frequency analysis. However, when k is positive variable x becomes upper bounded and takes on values in the range −∞ < x < u + α/k which may not be acceptable for analyzing floods unless there is sufficient evidence that such an upper bound does exist. When k = 0 the GEV distribution reduces to the type I extreme value distribution (EV1). The GEV CDF is of the form: ' $ % ( x − u 1/k F (x) = exp − 1 − k (B2) α B.2 Pearson Type 3 (PE3) The PDF of the PE3 distribution is given as: $ % 1 x − γ β−1 −( x−γ ) f (x) = e α α (β) α

(B3)

The variable x can take on values in the range γ < x < ∞. Generally α can be positive or negative, but for negative values of α the distribution becomes upper bounded and therefore not suitable for analyzing flood maxima. The PE3 CDF is of the form: % #x $ 1 x − γ β−1 −( x−γ ) F (x) = (B4) e α dx α (β) α γ

Quantile estimation for this distribution is given by Eq. II in Table 2, where KT is the frequency factor corresponding to return period of T year, and can be evaluated


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by Wilson–Hilferty transformation (Rao and Hamed 2000). The Wilson–Hilferty approximation is quite accurate for Cs ≤ 1 and maybe sufficiently accurate for Cs as high as 2. ( ' $ % 3 2 CS CS KT = (B5) u− + 1 − 1 , Cs > 0 CS 6 6 where Cs is the skewness coefficient of the data and u is the standardized normal variable. log (x − a) − μ y u= (B6) σy B.3 Lognormal Type 3 (LN3) The PDF of the LN3 distribution is given as: ' ( 2 1 1 exp − 2 log (x − a) − μ y f (x) = √ 2σ y (x − a) σ y 2π

(B7)

where μ y and σ y2 are the location and scale parameters, which correspond to the mean and variance of the logarithm of the shifted variable (x − a). B.4 Generalized Logistic (GLOG) The PDF of the GLOG distribution is given as: ' $ $ % % (−2 1 x − ξ 1/k x − ξ 1/k−1 1+ 1−k 1−k f (x) = α α α

(B8)

The variable x takes on values in the range: ξ + α/k ≤ x < ∞ for k ≤ 0 and −∞ < x ≤ ξ + α/k for k > 0. The GLOG CDF is of the form: ' % (−1 $ x − ξ 1/k (B9) F (x) = 1 + 1 − k α B.5 Generalized Pareto (GPAR) The PDF of the GPAR distribution is given as: $ % x − ξ 1/k−1 1 1−k f (x) = α α The CDF is written as:

% $ x − ξ 1/k F (x) = 1 − 1 − k α

(B10)

(B11)

The variable x takes on values in the range: ξ ≤ x < ∞ for k ≤ 0 and ξ ≤ x ≤ ξ + α/k for k > 0. The special case of k being 0 yields the exponential distribution, whereas the special case of k = 1 yields the uniform distribution on [ξ , ξ + α].

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B.6 Four-Parameter Kappa Distribution (KAP) The Kappa distribution has received only limited attention from the hydrologic community. Hosking and Wallis (1993) were probably the first to employ this distribution to generate artificial data for assessing the goodness of fit of different frequency distributions. Hosking (1994) described the properties of the Kappa distribution, derived using the L-moments method of parameter estimation, and discussed an application for modeling maximum precipitation data (Singh and Deng 2003). The probability density function of the KAP distribution is given as: ' $ % % (1/ h−1 $ 1 x − ξ 1/k−1 x − ξ 1/k f (x) = 1−k 1−h 1−k α α α

(B12)

The distribution function is given as: % (1/ h $ x − ξ 1/k F (x) = 1 − h 1 − k α '

(B13)

The lower and upper bounds of random variable x will also depend on the parameter values as follows: ξ + α 1 − 1/ hk /k ≤ x ≤ ξ + α/k if h > 0, k > 0 ξ + α log h ≤ x < ∞ if h > 0, k = 0 if h > 0, k < 0 ξ + α 1 − 1/ hk /k ≤ x < ∞ (B14) −∞ < x ≤ ξ + α/k if h ≤ 0, k > 0 −∞ < x < ∞ if h ≤ 0, k = 0 ξ + α/k ≤ x < ∞ if h ≤ 0, k < 0 Of the four-parameters in Eq. B13, ξ is a location parameter α is a scale parameter, and k and h are shape parameters. Apart from the restriction α > 0, all parameter values yield valid distribution functions (Hosking 1994). Equation B13 specializes into the following distribution functions, depending on the values of the shape parameters h and k, introduced in Table 8 (Parida 1999).

Table 8 Family of distributions generated by the kappa distribution with different values of h and k h

k

Distribution

1 0 −1 1 0 −1 1 0

= 0 = 0 = 0 0 0 0 1 1

Generalized Pareto distribution Generalized Extreme Value distribution Generalized Logistic distribution Exponential distribution Gumbel distribution Logistic distribution Uniform distribution Reverse Exponential distribution


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B.7 Five-Parameter Wakeby Distribution (WAK) The quantile function of the Wakeby distribution is given by: x (F) = ξ + α 1 − (1 − F)β /β − γ 1 − (1 − F)−δ /δ

(B15)

The Wakeby distribution is analytically defined only in the inverse form in Eq. B15. Therefore, explicit expressions cannot be obtained either for the probability density function or the distribution function. Although moments of x can be obtained as functions of the parameters, the inverse relationship cannot be readily derived. Consequently, moment estimates of the parameters are not feasible. Similarly, maximum likelihood estimates of the parameters are not easily obtained. Only the PWM method is presently considered for this distribution. The Wakeby distribution was proposed for flood frequency analysis by Houghton (1978). It was considered to be a superior distribution for flood frequency analysis. The probability weighted moments method was used to estimate the parameters of the Wakeby distribution by Landwehr et al. (1979). The Wakeby distribution is potentially useful in flood frequency analysis for several reasons discussed by Greenwood et al. (1979). One of these is the large number of parameters in the Wakeby distribution which permits better fitting of data than by distributions characterized by fewer parameters. Another reason is that it can accommodate a variety of flows ranging from low flows to floods. In Eq. B15, ξ is a location parameter, α and γ are scale parameters, and β and δ are shape parameters. The range of x is such that ξ ≤ x < ∞ if δ ≥ 0 and γ > 0; ξ ≤ x ≤ ξ + α/β − γ /δ if δ < 0 and γ = 0 (Tao et al. 2002).

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