Novel adaptive hybrid rule network based on TS fuzzy rules using an improved quantum-behaved particle swarm optimization
Introduction
Fuzzy theories have been widely applied in many fields, such as computer-aided design [1], [2], [3], [4], system identification [5], automatic control [6], [7], pattern recognition [6], data mining [8], and prediction [9], because of the ability to handle the complex problems with strong nonlinearity or high degree of uncertainty. Therefore, the fuzzy model has been proposed and proven as powerful in complex system modeling [10]. Two kinds of fuzzy models currently exist: the Mamdani [11] and the Takagi–Sugeno (TS) fuzzy models [5]. To date, the use of the TS fuzzy model to predict chaotic time series problems has gained increasing attention, and research on building the TS fuzzy rule has become a key point in practical applications.
Formulating fundamental TS rules is based on structure and parameter identification. The structure used to determine the antecedent and the consequent parts of a rule can be identified through heuristics [5], neural networks [12], fuzzy clustering methods [13], and so on. Meanwhile, the existing parameters in the rule can be identified through the least-square method [5], gradient descent [14], genetic algorithm (GA) [15], and so on. The structure and parameters, as well as the number of rules can influence prediction. An improper number of rules can neither improve the accuracy nor clear the explanation; thus, classic methods, such as increasing or merging clusters and mountain or subtractive clustering, are used. Simultaneously, new methods have also been proposed to optimize the model and improve the prediction performance, such as the TS-group method of data handling algorithm [16], the incremental smooth support vector regression algorithm [17], and the habitually linear evolving TS fuzzy model [18]. However, these methods still cannot guarantee that the accurate partition of the input samples and the correct number of rules are employed to solve chaotic time series prediction problems. Therefore, a novel rule-based fuzzy model, called adaptive hybrid rule network (AHRN) is proposed in this study to establish the solution model. AHRN is mainly established by nodes that represent fuzzy subspaces (i.e., different clusters of input data) and that are linked to others to generate rules. In the learning process, AHRN can automatically partition input samples according to their characteristics; thus, the structure can be updated adaptively. Meanwhile, the dynamic rule selection mechanism (DRSM) is used to establish the rule set according to the number of rules (RL) and rule similarity (RS). Through this mechanism, AHRN can build the most suitable set of rules. However, implementing AHRN is another key issue. Experience proves that an evolution algorithm is a better choice after several experiments that implement different kinds of algorithms have been performed in this research.
An evolutionary algorithm is a self-organizing and adaptive artificial intelligence technology that solves problems by simulating a biological evolution process and mechanism. GA [19], ant colony optimization [20], and particle swarm optimization (PSO) [21], [22] are typical examples of evolutionary algorithm. PSO has better computational efficiency, i.e., it requires less memory space and lesser CPU speed, and less number of parameters to adjust [23]. As research develops further, new algorithms based on standard PSO have been proposed, such as elite PSO with mutation [24], group decision PSO [25], multi-grouped PSO [26], optimization algorithm based on the TS fuzzy model of self-adaptive disturbed PSO and neural network [27], and quantum-behaved PSO (QPSO) [28]. QPSO has lesser parameters to control and better search capability than standard PSO [23]. QPSO has been increasingly used in chaotic prediction problems [29], [30]; hence, new algorithms based on standard QPSO have been proposed. The stochastic coefficient is one of the key factors that influences the particle movement in classic QPSO; thus, Coelho [31] generated random numbers by using Gaussian distribution sequences with zero mean and unit variance for the stochastic coefficient. This method may allow particles to move away from the current point and escape from local minima. The mean best position (mbest) is another key factor; thus, Xi [32] proposed the weighted QPSO algorithm. This algorithm modifies the calculation method for the mbest by assigning each particle with a weight coefficient that linearly decreases with a particle׳s rank; this method promises considerable influence on the movement of the particle with high fitness. To further improve performance, other valuable studies have been done. Sun [33] incorporated the improved heuristic strategies into QPSO; and the proposed method does not require using penalty functions. Moreover, it explores the optimum solution at a low computational effort. Pan [34] introduced the chaos theory into QPSO; and the proposed method uses a logistic map to generate a set of chaotic offsets and produces multiple positions around every local optimal position of the particle, and thus, convergence accuracy is better than that in typical QPSO. Sun [35] proposed modified QPSO, which substitutes the global best position (gbest) by a personal best position (pbest) of a randomly selected particle, thus exhibiting stronger global search capability than QPSO and PSO. In the present study, an improved algorithm called opinion leader-based QPSO (OLB-QPSO) is proposed with a new type of particle. This method modifies the calculation method for the mbest and uses the composed particles with multiple structures.
This study investigates (1) how the fuzzy map of data can be identified based on the correct data partition, (2) how the explanation of fuzzy rules can be improved to enhance robustness, and (3) how the local optimal solution can be avoided.
This paper is organized as follows. In Section 2, the construction of AHRN, including structure and DRSM analyses, is described. In Section 3, several key factors required to implement AHRN are discussed in detail. These factors include QPSO, OLB-QPSO, and the compose particle. In Section 4, three experiments are presented. Box–Jenkins gas furnace data and the Mackey–Glass chaotic time series are used to validate prediction capability, whereas exhausted gas temperature (EGT) samples are employed to demonstrate the performance of AHRN that uses OLB-QPSO with composed particles in actual engineering. Finally, the conclusions are provided in Section 5.
Section snippets
Construction of AHRN
The samples used in AHRN are as follows:where n is the dimension of the input vector of a sample, m is the capacity of the samples, is the ith variable in the input vector of the jth sample, and is the output vector of the jth sample.
The output of a sample in AHRN can be obtained by the weighted sum of different TS fuzzy rules because the TS fuzzy rules model is suitable to for solving nonlinear
Learning process of AHRN
AHRN provides an approximation model by using several algorithms for model learning. For example, evolutionary algorithms can be used, but swarm intelligence algorithms may be better choices. Among swarm intelligence algorithms, the OLB-QPSO with composed particles is selected for AHRN in this study. This algorithm is an improved QPSO.
Experiments and analyses
AHRN can be employed to predict a chaotic time series. This study investigated the prediction capability of AHRN by using Box–Jenkins gas furnace data and the Mackey–Glass chaotic time series, and obtained excellent results. To demonstrate the practicability of AHRN in the engineering field, the EGT of a particular aero engine was used. To simplify experiment manipulation, the interface of an application programmed for the experiments is shown in Fig. 12. All experiments were run by MATLAB
Conclusion
AHRN has been introduced in this study. The structure of AHRN can be updated adaptively according to actual samples, with DRSM providing the most suitable rule set to obtain the optimal solution. Several algorithms can be applied to implement the learning process of AHRN; however, OLB-QPSO with composed particles is selected in this research. This algorithm obtains the mbest based on the opinion leader, and the structure of the composed particle is composed of SSP, RSP, and AISP. The
Acknowledgments
The authors are grateful to the anonymous reviewers for their very helpful comments and constructive suggestions with regard to this paper. This paper is supported by National Natural Science Foundation of China (Grant no. 51075083), the major project of National Defense Foundation of China, and also by the advance research project of General Armament Department.
Lin Lin received the Ph.D degree in Mechanical Design from Harbin Institute of Technology, Harbin, China, in 2003. She is currently a professor in the School of Mechatronics Engineering in Harbin Institute of Technology. Her research interests include time series prediction, neural networks, fuzzy modeling, and mechanical product sheme intelligent design.
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Lin Lin received the Ph.D degree in Mechanical Design from Harbin Institute of Technology, Harbin, China, in 2003. She is currently a professor in the School of Mechatronics Engineering in Harbin Institute of Technology. Her research interests include time series prediction, neural networks, fuzzy modeling, and mechanical product sheme intelligent design.
Feng Guo received the B.A. and M.A degrees from the Newmedia technology and Art Department in Harbin Institute of Technology, Harbin, China, in the year 2006 and 2008 respectively. He is currently working towards the Ph.D. degree in School of Mechatronics Engineering in Harbin Institute of Technology. His research interests include fuzzy modeling and control, particle swarm optimization, and mechanical product sheme intelligent design.
Xiaolong Xie received the M.E. degrees in Mechanical Design & Theory from Harbin Institute of Technology, Harbin, China, in 2011. He is currently working towards the Ph.D. degree in School of Mechatronics Engineering in Harbin Institute of Technology. His research interests include computer-aided design, and knowledge-based engineering.
Bin Luo received the B.A degrees from the School of Mechatronics Engineering in Harbin Engineering University. He is currently working towards the M.A degree in School of Mechatronics Engineering in Harbin Institute of Technology. His research interests include Multi-objective optimization and intelligent design.