Evolving Compact and Interpretable Takagi–Sugeno Fuzzy Models With a New Encoding Scheme

Developing Takagi–Sugeno fuzzy models by evolutionary algorithms mainly requires three factors: an encoding scheme, an evaluation method, and appropriate evolutionary operations. At the same time, these three factors should be designed so that they can consider three important aspects of fuzzy modeling: modeling accuracy, compactness, and interpretability. This paper proposes a new evolutionary algorithm that fulfills such requirements and solves fuzzy modeling problems. Two major ideas proposed in this paper lie in a new encoding scheme and a new fitness function, respectively. The proposed encoding scheme consists of three chromosomes, one of which uses unique chained possibilistic representation of rule structure. The proposed encoding scheme can achieve simultaneous optimization of parameters of antecedent membership functions and rule structures with the new fitness function developed in this paper. The proposed fitness function consists of five functions that consider three evaluation criteria in fuzzy modeling problems. The proposed fitness function guides evolutionary search direction so that the proposed algorithm can find more accurate compact fuzzy models with interpretable antecedent membership functions. Several evolutionary operators that are appropriate for the proposed encoding scheme are carefully designed. Simulation results on three modeling problems show that the proposed encoding scheme and the proposed fitness functions are effective in finding accurate, compact, and interpretable Takagi–Sugeno fuzzy models. From the simulation results, it is shown that the proposed algorithm can successfully find fuzzy models that approximate the given unknown function accurately with a compact number of fuzzy rules and membership functions. At the same time, the fuzzy models use interpretable antecedent membership functions, which are helpful in understanding the underlying behavior of the obtained fuzzy models.     

Evolving compact and interpretable Takagi-Sugeno fuzzy models with a new encoding scheme.

Developing Takagi-Sugeno fuzzy models by evolutionary algorithms mainly requires three factors: an encoding scheme, an evaluation method, and appropriate evolutionary operations. At the same time, these three factors should be designed so that they can consider three important aspects of fuzzy modeling: modeling accuracy, compactness, and interpretability. This paper proposes a new evolutionary algorithm that fulfills such requirements and solves fuzzy modeling problems. Two major ideas proposed in this paper lie in a new encoding scheme and a new fitness function, respectively. The proposed encoding scheme consists of three chromosomes, one of which uses unique chained possibilistic representation of rule structure. The proposed encoding scheme can achieve simultaneous optimization of parameters of antecedent membership functions and rule structures with the new fitness function developed in this paper. The proposed fitness function consists of five functions that consider three evaluation criteria in fuzzy modeling problems. The proposed fitness function guides evolutionary search direction so that the proposed algorithm can find more accurate compact fuzzy models with interpretable antecedent membership functions. Several evolutionary operators that are appropriate for the proposed encoding scheme are carefully designed. Simulation results on three modeling problems show that the proposed encoding scheme and the proposed fitness functions are effective in finding accurate, compact, and interpretable Takagi-Sugeno fuzzy models. From the simulation results, it is shown that the proposed algorithm can successfully find fuzzy models that approximate the given unknown function accurately with a compact number of fuzzy rules and membership functions. At the same time, the fuzzy models use interpretable antecedent membership functions, which are helpful in understanding the underlying behavior of the obtained fuzzy models.     

Interactive genetic algorithms with individual’s fuzzy fitness

Interactive genetic algorithms are effective methods to solve an optimization problem with implicit or fuzzy indices, and have been successfully applied to many real-world optimization problems in recent years. In traditional interactive genetic algorithms, many researchers adopt an accurate number to express an individual’s fitness assigned by a user. But it is difficult for this expression to reasonably reflect a user’s fuzzy and gradual cognitive to an individual. We present an interactive genetic algorithm with an individual’s fuzzy fitness in this paper. Firstly, we adopt a fuzzy number described with a Gaussian membership function to express an individual’s fitness. Then, in order to compare different individuals, we generate a fitness interval based on α-cut set, and obtain the probability of individual dominance by use of the probability of interval dominance. Finally, we determine the superior individual in tournament selection with size two based on the probability of individual dominance, and perform the subsequent evolutions. We apply the proposed algorithm to a fashion evolutionary design system, a typical optimization problem with an implicit index, and compare it with two interactive genetic algorithms, i.e., an interactive genetic algorithm with an individual’s accurate fitness and an interactive genetic algorithm with an individual’s interval fitness. The experimental results show that the proposed algorithm is advantageous in alleviating user fatigue and looking for user’s satisfactory individuals.     

Emergence of self‐learning fuzzy systems by a new virus DNA–based evolutionary algorithm

In this article, we propose a new approach to the virus DNA–based evolutionary algorithm (VDNA‐EA) to implement self‐learning of a class of Takagi‐Sugeno (T‐S) fuzzy controllers. The fuzzy controllers use T‐S fuzzy rules with linear consequent, the generalized input fuzzy sets, Zadeh fuzzy logic and operators, and the generalized defuzzifier. The fuzzy controllers are proved to be nonlinear proportional‐integral (PI) controllers with variable gains. The fuzzy rules are discovered automatically and the design parameters in the input fuzzy sets and the linear rule consequent are optimized simultaneously by the VDNA‐EA. The VDNA‐EA uses the VDNA encoding method that stemmed from the structure of the VDNA to encode the design parameters of the fuzzy controllers. We use the frameshift decoding method of the VDNA to decode the DNA chromosome into the design parameters of the fuzzy controllers. In addition, the gene transfer operation and bacterial mutation operation inspired by a microbial evolution phenomenon are introduced into the VDNA‐EA. Moreover, frameshift mutation operations based on the DNA genetic operations are used in the VDNA‐EA to add and delete adaptively fuzzy rules. Our encoding method can significantly shorten the code length of the DNA chromosomes and improve the encoding efficiency. The length of the chromosome is variable and it is easy to insert and delete parts of the chromosome. It is suitable for complex knowledge representation and is easy for the genetic operations at gene level to be introduced into the VDNA‐EA. We show how to implement the new method to self‐learn a T‐S fuzzy controller in the control of a nonlinear system. The fuzzy controller can be constructed automatically by the VDNA‐EA. Computer simulation results indicate that the new method is effective and the designed fuzzy controller is satisfactory. © 2003 Wiley Periodicals, Inc.     

Dynamic evolution of the genetic search region through fuzzy coding

A technique for automatic exploration of the genetic search region through fuzzy coding (Sharma and Irwin, 2003) has been proposed. Fuzzy coding (FC) provides the value of a variable on the basis of the optimum number of selected fuzzy sets and their effectiveness in terms of degree-of-membership. It is an indirect encoding method and has been shown to perform better than other conventional binary, Gray and floating-point encoding methods. However, the static range of the membership functions is a major problem in fuzzy coding, resulting in longer times to arrive at an optimum solution in large or complicated search spaces.This paper proposes a new algorithm, called fuzzy coding with a dynamic range (FCDR), which dynamically allocates the range of the variables to evolve an effective search region, thereby achieving faster convergence. Results are presented for two benchmark optimisation problems, and also for a case study involving neural identification of a highly non-linear pH neutralisation process from experimental data. It is shown that dynamic exploration of the genetic search region is effective for parameter optimisation in problems where the search space is complicated.     

Fuzzy coding of genetic algorithms

A new chromosome encoding method, named fuzzy coding, is proposed for representing real number parameters in a genetic algorithm. Fuzzy coding provides the value of a parameter on the basis of the optimum number of selected fuzzy sets and their effectiveness in terms of degree of membership. Thus, it represents the knowledge associated with each parameter and is an indirect method of encoding compared with alternatives, where the parameters are directly represented in the encoding. Fuzzy coding is described and compared with conventional binary coding, gray coding, and floating-point coding. Two test examples, along with neural identification of a nonlinear pH process from experimental data, are studied. It is shown that fuzzy coding is better than the conventional methods and is effective for parameter optimization in problems where the search space is complicated.     

Determining membership functions and minimum fuzzy support in finding fuzzy association rules for classification problems

This paper propose a new method, that employs the genetic algorithm, to find fuzzy association rules for classification problems based on an effective method for discovering the fuzzy association rules, namely the fuzzy grids based rules mining algorithm (FGBRMA). It is considered that some important parameters, including the number and shapes of membership functions in each quantitative attribute and the minimum fuzzy support, are not easily user-specified. Thus, the above-mentioned parameters are automatically determined by a binary string or chromosome is composed of two substrings: one for each quantitative attribute by the coding method proposed by Ishibuchi and Murata, and the other for the minimum fuzzy support. In each generation, the fitness value, which maximizes the classification accuracy rate and minimizes the number of fuzzy rules, of each chromosome can be obtained. When reaching the termination condition, a chromosome with maximum fitness value is then used to test its performance. For classification generalization ability, the simulation results from the iris data and the appendicitis data demonstrate that proposed method performs well in comparison with other classification methods.     

求解非线性规划问题的改进直觉模糊遗传算法

提出一种改进的直觉模糊遗传算法用于求解带有多维约束的非线性规划问题。以遗传算法在迭代寻优中的个体适应度大小构造相应可行解的隶属度和非隶属度函数,将非线性规划问题直觉模糊化转化为直觉模糊非线性规划问题,通过建立直觉模糊推理系统,自适应地调节遗传算法的交叉率和变异率;并采用一种改进的选择策略,将个体按适应度值大小排序、等量分组,对适应度低的个体组随机选择复制,保留不可行解中可能隐含的有利寻优信息,增强种群个体的多样性和竞争性。仿真实验结果表明,该算法求解非线性规划问题时是可行和有效的。     

Solving Japanese nonograms by Taguchi-based genetic algorithm

A Taguchi-based genetic algorithm (TBGA) is proposed to solve Japanese nonogram puzzles. The TBGA exploits the power of global exploration inherent in the traditional genetic algorithm (GA) and the abilities of the Taguchi method in efficiently generating offspring. In past researches, the GA with binary encoding and inappropriate fitness functions makes a huge search space size and inaccurate direction for searching the solution of a nonogram. Consequently, the GA does not easily converge to the solution. The proposed TBGA includes the effective condensed encoding, the improved fitness function, the modified crossover, the modified mutation, and the Taguchi method for solving Japanese nonograms. The systematic reasoning ability of the Taguchi method is incorporated in the modified crossover operation to select the better genes to achieve crossover, and eventually enhance the GA. In this study, the condensed encoding can make sure that the chromosome is a feasible solution in all rows for Japanese nonograms. In the reconstruction process of a Japanese nonogram, the numbers in the left column are used as encoding conditions, and the numbers in the top row with the improved fitness function are employed to evaluate the reconstruction result. From the computational experiments, the proposed TBGA approach is effectively applied to solve nonograms and better than a GA does.     

采用新的DNA进化算法自动设计Takagi-Sugeno模糊控制器

提出一种新颖的基于DNA的进化算法(DNA-EA)来自动设计一类Trakagi-Sugeno (TS)模糊控制器.TS模糊控制器采用带有线性规则后项的TS模糊规则,连续输 入模糊集,Zadeh模糊逻辑和常用的重心反模糊器.TS模糊控制器被证明是带有可变增 益的非线性PI控制器.DNA-EA被用于自动获取TS模糊规则,并同时优化模糊规则前 项和后项中的设计参数.DNA-EA采用由生物DNA结构启发得到的DNA编码方法来编 码模糊控制器的设计参数.在DNA-EA中,引入了受微生物进化现象启发的基因转移和细 菌变异操作.另外,也引入了基于DNA遗传操作的框构变异操作.DNA编码方法非常适 合于复杂知识的表达,基于基因水平的遗传操作也很容易引入到DNA-EA中.染色体的长 度是可变的,且可插入或删除部分碱基序列.作为示例,给出了采用DNA-EA来自动设计 TS模糊控制器用于控制一类非线性系统的方法.DNA-EA能自动地构造模糊控制器.计 算机仿真结果表明,DNA-EA是有效的,且优化得到的模糊控制器是满意的.     

基于实数编码遗传算法的PID控制系统设计

提出一种使用遗传算法确定PID控制系统参数的方法。它以使用工程中常用的频域指标作为目标函数,使方法更具有实效。根据这一思想,使用C语言设计遗传算法的各个部分,如编码方式、适应度函数、遗传操作算子等。最后,对实际的算例进行计算并将结果在Matlab环境下进行了仿真。仿真结果表明:本方法快速、有效。     

Generating fuzzy rules for target tracking using a steady-stategenetic algorithm

Radar target tracking involves predicting the future trajectory of a target based on its past positions. This problem has been dealt with using trackers developed under various assumptions about statistical models of process and measurement noise and about target dynamics. Due to these assumptions, existing trackers are not very effective when executed in a stressful environment in which a target may maneuver, accelerate, or decelerate and its positions be inaccurately detected or missing completely from successive scans. To deal with target tracking in such an environment, recent efforts have developed fuzzy logic-based trackers. These have been shown to perform better as compared to traditional trackers. Unfortunately, however, their design may not be easier. For these trackers to perform effectively, a set of carefully chosen fuzzy rules are required. These rules are currently obtained from human experts through a time-consuming knowledge acquisition process of iterative interviewing, verifying, validating, and revalidating. To facilitate the knowledge acquisition process and ensure that the best possible set of rules be found, we propose to use an automatic rule generator that was developed based on the use of a genetic algorithm (GA). This genetic algorithm adopts a steady-state reproductive scheme and is referred to as the steady-state genetic algorithm (SSGA) in this paper. To generate fuzzy rules, we encode different rule sets in different chromosomes. Chromosome fitness is then determined according to a fitness function defined in terms of the number of track losses and the prediction accuracy when the set of rules it encodes is tested against training data. The rules encoded in the fittest chromosome at the end of the evolutionary process are taken to be the best possible set of fuzzy rules     

MULTITUNING FUZZY CONTROL SYSTEMS DESIGN

A multituning fuzzy control system structure that involves two simple, but effective tuning mechanisms, is proposed: one is called fuzzy control rule tuning mechanism (FCRTM); the other is called dynamic scalar tuning mechanism (DSTM). In FCRTM, it is used to generate the necessary control rules with a center extension method. In DSTM, it contains three fuzzy IF-THEN rules for determining the appropriate scaling factors for the fuzzy control system. In this paper, a method based on the genetic algorithm (GA) is proposed to simultaneously choose the appropriate parameters in FCRTM and DSTM. That is, the proposed GA-based method can automatically generate the required rule base of fuzzy controller and efficiently determine the appropriate map for building the dynamic scalars of fuzzy controller. A multiobjective fitness function is proposed to determine an appropriate parameter set such that not only the selected fuzzy control structure has fewer fuzzy rules, but also the controlled system has a good control performance. Finally, an inverted pendulum control problem is given to illustrate the effectiveness of the proposed control scheme.     

Surrogate model-assisted interactive genetic algorithms with individual’s fuzzy and stochastic fitness

We propose a surrogate model-assisted algorithm by using a directed fuzzy graph to extract a user’s cognition on evaluated individuals in order to alleviate user fatigue in interactive genetic algorithms with an individual’s fuzzy and stochastic fitness. We firstly present an approach to construct a directed fuzzy graph of an evolutionary population according to individuals’ dominance relations, cut-set levels and interval dominance probabilities, and then calculate an individual’s crisp fitness based on the out-degree and in-degree of the fuzzy graph. The approach to obtain training data is achieved using the fuzzy entropy of the evolutionary system to guarantee the credibilities of the samples which are used to train the surrogate model. We adopt a support vector regression machine as the surrogate model and train it using the sampled individuals and their crisp fitness. Then the surrogate model is optimized using the traditional genetic algorithm for some generations, and some good individuals are submitted to the user for the subsequent evolutions so as to guide and accelerate the evolution. Finally, we quantitatively analyze the performance of the presented algorithm in alleviating user fatigue and increasing more opportunities to find the satisfactory individuals, and also apply our algorithm to a fashion evolutionary design system to demonstrate its efficiency.     

A fuzzy self-tuning parallel genetic algorithm for optimization

The genetic algorithm (GA) is now a very popular tool for solving optimization problems. Each operator has its special approach route to a solution. For example, a GA using crossover as its major operator arrives at solutions depending on its initial conditions. In other words, a GA with multiple operators should be more robust in global search. However, a multiple operator GA needs a large population size thus taking a huge time for evaluation. We therefore apply fuzzy reasoning to give effective operators more opportunity to search while keeping the overall population size constant. We propose a fuzzy self-tuning parallel genetic algorithm (FPGA) for optimization problems. In our test case FPGA there are four operators—crossover, mutation, sub-exchange, and sub-copy. These operators are modified using the eugenic concept under the assumption that the individuals with higher fitness values have a higher probability of breeding new better individuals. All operators are executed in each generation through parallel processing, but the populations of these operators are decided by fuzzy reasoning. The fuzzy reasoning senses the contributions of these operators, and then decides their population sizes. The contribution of each operator is defined as an accumulative increment of fitness value due to each operator's success in searching. We make the assumption that the operators that give higher contribution are more suitable for the typical optimization problem. The fuzzy reasoning is built under this concept and adjusts the population sizes in each generation. As a test case, a FPGA is applied to the optimization of the fuzzy rule set for a model reference adaptive control system. The simulation results show that the FPGA is better at finding optimal solutions than a traditional GA.     

基于VISIT算法设计模糊分类器的进化方法

介绍了在没有数据分布先验知识的情况下,用进化方法直接从训练数据中建立紧致模糊分类系统的方法。使用VISIT算法获取每个个体模糊系统,再用遗传算法从中搜索最优的模糊系统。规则和隶属函数是在进化过程中自动建立和优化的。为了同时有效地评价系统的精度和紧致性,用一个模糊专家系统作适应度函数。在2个基准分类问题上的实验结果表明了新方法的有效性。     

基于模糊遗传算法的模糊调度研究

就模糊调度问题的研究现状、研究方法进行了评述 ,探讨了加工时间与交货期的模糊化操作以及基于模糊遗传算法的模糊调度系统 ,归纳了模糊遗传算法中模糊数的操作和遗传算法的编码 ,并指出了存在的不足 .指出模糊生产环境下模糊加工时间与模糊交货期综合问题的研究是当今全球性市场竞争机制所带来的必然趋势.     

改进的区间值模糊集交互式遗传算法

区间值模糊集的交互式遗传算法,能有效缓解用户的疲劳,同时避免用户因一时无法给出确定值而浪费掉的时间,大大加快了收敛速度。首先采用区间值模糊集的方法表示对个体进行评价的适应度值,即为区间适应度值,然后对其进行排序,按照排序结果采用交互式遗传算法进行全局搜索。整个过程符合人的思维过程,能有效搜索到用户满意的个体。将该方法应用于图像检索系统中,结果表明该方法有效地提高了检索速度,并且取得了较好的检索结果。     

Integrating fuzzy knowledge by genetic algorithms

We propose a genetic algorithm-based fuzzy knowledge integration framework that can simultaneously integrate multiple fuzzy rule sets and their membership function sets. The proposed approach consists of two phases: fuzzy knowledge encoding and fuzzy knowledge integration. In the encoding phase, each fuzzy rule set with its associated membership functions is first transformed into an intermediary representation and then further encoded as a string. The combined strings form an initial knowledge population, which is then ready for integration. In the knowledge-integration phase, a genetic algorithm is used to generate an optimal or nearly optimal set of fuzzy rules and membership functions from the initial knowledge population. Two application domains, the hepatitis diagnosis and the sugarcane breeding prediction, were used to show the performance of the proposed knowledge-integration approach. Results show that the fuzzy knowledge base derived using our approach performs better than every individual knowledge base     

基于类神经网络模型的电路演化实现方法

为解决目前数字型演化硬件研究中存在的电路编码困难问题,提出一个可用矩阵形式描述组合电路的类神经网络门级电路模型,讨论在此模型上进行电路编码的具体方法.根据编码矩阵特点,对标准遗传算法进行改进,设计遗传操作算子、适应度评估方法等.通过无刷直流电动机电子换相电路的成功演化实例,验证了采用矩阵编码和改进遗传算法实现数字电路演...