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Improved Chaotic Quantum-Behaved Particle Swarm Optimization Algorithm for Fuzzy Neural Network and Its Application

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Improved Chaotic Quantum-Behaved Particle Swarm Optimization Algorithm for Fuzzy Neural Network and Its Application Hindawi Mathematical Problems in Engineering Volume 2020, Article ID 9464593, 11 pages https://doi.org/10.1155/2020/9464593 Research Article Improved Chaotic Quantum-Behaved Particle Swarm Optimization Algorithm for Fuzzy Neural Network and Its Application Yuexi Peng ,1 Kejun Lei ,2 Xi Yang,2 and Jinzhang Peng 3, 1School of Physics and Electronics, Central South University, Changsha 410083, China 2College of Information Science and Engineering, Jishou University, Jishou 416000, China 3College of Physics and Mechatronics Engineering, Jishou University, Jishou 416000, China Correspondence should be addressed to Kejun Lei; [email protected] Received 7 February 2020; Accepted 5 March 2020; Published 28 March 2020 Guest Editor: Kehui Sun Copyright © 2020 Yuexi Peng et al. 0is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Traditional fuzzy neural network has certain drawbacks such as long computation time, slow convergence rate, and premature convergence. To overcome these disadvantages, an improved quantum-behaved particle swarm optimization algorithm is proposed as the learning algorithm. In this algorithm, a new chaotic search is introduced, and benchmark function experiments prove it outperforms the other five existing algorithms. Finally, the proposed algorithm is presented as the learning algorithm for Takagi–Sugeno fuzzy neural network to form a new neural network, and it is utilized in the water quality evaluation of Dongjiang Lake of Hunan province. Simulation results demonstrated the effectiveness of the new neural network. 1. Introduction is a complex high-dimensional problem, PSO algorithm has the disadvantage of premature convergence [16–19]. After Artificial neural network (ANN) [1] is an effective method to studying the results of particle convergence behavior, Sun deal with nonlinear problems. Because of the powerful fit- et al. [20] proposed a novel metaheuristic algorithm called ting ability, the ANN can nearly simulate any complex quantum-behaved particle swarm optimization (QPSO) nonlinear functional relationship without knowing the algorithm. It combines PSO algorithm with the quantum correlation between the input and the output. At present, the mechanic and has better global searching ability than that of ANN has been widely used in practical applications [2–5]. In the PSO algorithm [21, 22]. Since the QPSO algorithm was fact, fuzzy neural network (FNN) [6] can handle complex proposed, many researchers devoted to apply it for practical engineering problem as well [7–9]. It combines the ad- applications [23–26] or improve the algorithm itself [27–31]. vantages of fuzzy mathematics and ANN. Due to the in- For example, Mariani et al. [29] proposed a novel chaotic troduction of the fuzzy logic concept, it is very suitable for QPSO algorithm for the image matching, but this algorithm the classification of nonlinear or highly uncertain infor- only introduces chaos variables into the particle position mation. However, the FNN also inherits the disadvantages of initialization. In fact, the effectiveness should be better if the the ANN, such as long computation time and slow con- chaos variables are introduced both at the beginning and the vergence rate. 0erefore, how to solve these shortcomings end of the algorithm search stage. Mariani et al. [29] pro- becomes a problem to be solved urgently [10–12]. posed a QPSO algorithm combined with the Zaslavskii It is an effective way to improve the FNN performance by chaotic map and applied it to the optimization of shell and replacing the traditional learning algorithm by metaheuristic tube heat exchangers. 0en, Turgut et al. [30] proposed algorithms such as the particle swarm optimization (PSO) another chaotic QPSO algorithm for solving nonlinear algorithm [13–15]. However, when the considered problem system of equations, and the benchmark function 2 Mathematical Problems in Engineering experiments are proved that the algorithm using Logistic where Pbi(t) represents the optimal position of the particle i chaotic map has the best performance. On the basis of Ref. at the tth iteration and Pg(t) represents the global optimal [30], Turgut [31] proposed a hybrid chaotic QPSO algorithm position in the particle swarm at the tth iteration. c1 and c2 for thermal design of plate fin heat exchangers. However, are the learning factors, which provide the optimal selection this algorithm is a little complicated for setting up three function. r1 and r2 are the random numbers between (0; 1). different populations to search simultaneously, and the Li(t) is defined as convergence rate is not fast enough. � � � � 0erefore, to accelerate the convergence rate and improve Li(t + 1) � 2β(t)�mbest − Xi(t)�; (4) the optimization precision further, an algorithm called im- proved chaotic quantum-behaved particle swarm optimiza- N P � X bi; ( ) tion (ICQPSO) algorithm is proposed. Compared with the mbest 5 i� N other chaotic PSO algorithms [17, 29–31], it is simple and easy 1 to implement. 0e ICQPSO algorithm is introduced as the where β(t) is the contraction expansion coefficient that learning algorithm in the Takagi–Sugeno fuzzy neural net- determines the convergence rate of the algorithm [22], and it work (TSFNN) [32], which can get accurate output value by is calculated by inputting most forms of information, to form the improved (T − t)β − β � neural network called ICQPSO-TSFNN. Finally, to demon- β(t) � β + max min ; (6) min T strate the effectiveness of the proposed method, we focus on the topic of the FNN for the water quality evaluation. 0e where t and T are current and maximum iteration number, ICQPSO-TSFNN is applied to evaluate the water quality of respectively. According to Ref. [22], βmax � 1.0 and the Dongjiang Lake in the Hunan Province from 2002 to 2013. βmin � 0.5. 0e position equation of the particle i is updated 0e rest of this paper is organized as follows. Section 2 by mainly introduces the proposed ICQPSO algorithm. 0e � � performance test of ICQPSO algorithm are shown in Section � � 1 Xi(t + 1) � pi(t) + β(t)�mbest − Xi(t)�ln� �; r3 ≥ 0:5; 3. 0e application of ICQPSO-TSFNN for water quality u evaluation is presented in Section 4. Finally, we summarize (7) the results and indicate future directions. � � � � 1 Xi(t + 1) � pi(t) − β(t)�mbest − Xi(t)�ln� �; r3 < 0:5; 2. ICQPSO Algorithm u (8) 0e proposed ICQPSO algorithm aims at improving the performance of dealing with high-dimensional complex where u and r3 are generated according to a uniform problems by introducing a new chaotic search method, and probability distribution at the range (0, 1). it also lays the foundation for the application of TSFNN in Assuming that the considered problem is a minimum Section 4. optimization problem, the implementation of the QPSO algorithm is summarized as follows: 2.1. QPSO Algorithm. For the QPSO algorithm [20], the (a) Initialize particles in the population with random particle movement is completely different from that of the position vectors. PSO algorithm [33]. Newtonian principles are invalid in (b) Evaluate the fitness values of each particle. quantum world as the velocity and position update cannot (c) Calculate the local attractor point as defined in be determined simultaneously. Hence, there is no velocity equation (2). vector in the particle of QPSO algorithm. 0e new state of ! (d) Calculate mbest vector according to equation (5). each particle is determined by the wave function ψ( x ; t), and there also exists jψ(t)|2 which is the probability density (e) Compare the fitness of the Pi(t) with the fitness of function of the position of each particle. Considering a one- the Pbi. If the fitness value of Pi(t) is smaller than dimensional optimization problem, the Monte Carlo sto- that of the Pbi, then replace the Pbi by the Pi(t). chastic simulation is employed to obtain the position (f) Compare the fitness of the Pbi with the fitness of the equation of the particle i: Pg. If the fitness value of the Pb is smaller than that of the P , then replace the P by the P . L (t) 1 g g bi X (t + 1) � p (t) ± i ln !; (1) i i 2 μ (g) Update the position of the particles according to equation (7) or equation (8). where pi(t) is the local attractor of the particle i (i �1, 2, ..., N, (h) Repeat Step (b) to Step (f) until the termination N is the total number of particle swarm), and it is defined by criteria are satisfied. pi(t) � φPbi(t) +(1 − φ)Pg(t); (2) Since there is no velocity limit, the QPSO algorithm can jump out of the local optimum more easily. So, it has c1r1 φ � ; (3) stronger global convergence ability than that of the PSO c1r1 + c2r2 algorithm. Mathematical Problems in Engineering 3 2.2. Chaotic Search. Chaos has inherent randomness and (d) Calculate the mbest vector according to equation (5). ergodicity [34, 35]. It allows the chaotic search to be pro- (e) Compare the fitness of the Pi(t) with the fitness of grammed and traverses every state in a certain search region, the Pbi. If the fitness of the Pi(t) is smaller than that while every state is visited only once. 0erefore, introducing of the Pbi, then replace it. chaos sequence generated by the chaotic system into the (f) Compare the fitness of the P with the fitness of the QPSO algorithm can improve the algorithm performance bi P . If the fitness of the P is smaller than that of the [28–31]. Here, the logistic map is considered, and it is de- g bi P , then replace it. fined by g (g) Update particle positions according to equation (7) x(n + ) � x(n)[ − x(n)]; ( ) 1 μ 1 9 or (8).
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