A new approach for tuning interval type-2 fuzzy knowledge bases using genetic algorithms
© Shukla and Tripathi; licensee Springer. 2014
Received: 15 September 2013
Accepted: 13 February 2014
Published: 27 February 2014
Fuzzy knowledge-based systems (FKBS) are significantly applicable in the area of control, classification, and modeling, having knowledge in the form of fuzzy if-then rules. Type-2 fuzzy theory is used to make these systems more capable of dealing with inherent uncertainties in real-world problems. In this paper, the authors have proposed a genetic tuning approach named lateral displacement and expansion/compression (LDEC) in which α and β parameters are calculated to adjust the parameters of interval type-2 membership functions. α tuning deals with lateral displacement, whereas β tuning carries out compression/expansion operation. The interpretability and accuracy features are considered during the development of this approach. The experimental results show the performance of the proposed approach.
Fuzzy systems, more specifically fuzzy knowledge-based systems (FKBS) or fuzzy rule-based systems (FRBS), are significantly applicable in areas like control , classification , and modeling . The essential feature of FKBS is the incorporation of human expert knowledge which is in the form of fuzzy  extended if-then rules. The major components of FKBS are fuzzification interface, inference engine, knowledge base, and defuzzification interface . Knowledge base (KB) is composed of two components: data base (DB) and rule base (RB). DB is the repository of membership functions (MFs) and scaling functions (SFs) representing linguistic values, whereas RB is the collection of knowledge related to problems in terms of fuzzy if-then rules.
The design and implementation of KB can be assumed as an optimization task. Hence, genetic algorithms (GAs) are used for learning and tuning of various parameters of KB due to their strong capacity of searching in a complicated and poorly defined search space. Such an application of GAs in developing FKBS is specifically named as genetic fuzzy systems (GFS) [5–8]. GFS have been used for handling various types of applications like predicting surface finish in ultraprecision diamond , bioaerosol detector , classification of intrusion attacks from a network traffic data , tool wear monitoring , smart base isolation system , etc.
Fuzzy systems for applications like in economics, medicine, etc. are to be developed such that the users may understand how they work by inspecting their KB and functioning. Technically, this feature is called ‘interpretability’  which is the subjective feature of a fuzzy system showing how much the system is readable/understandable to the users by observing its functionality. Accuracy  is another feature showing the closeness between the real model and the developed model. Interpretability and accuracy are contradictory with each other, i.e., one can be improved at the cost of the other, denoted by ‘interpretability-accuracy trade-off’ (I-A Trade-Off) [16–19]. For the above applications, interpretability as well as accuracy is required to be maintained at the higher level by maintaining a good I-A Trade-Off.
Interpretability and accuracy features are directly related to the approaches of developing FKBS which are domain expert method and experimental data method. In the first method, domain experts of the problem are contributing their knowledge to develop the RB of the FKBS. Such FKBS are much more interpretable. In the second method, RB is generated by using some machine learning method applied on the data set of the particular problem. The FKBS developed by the second method are less interpretable but are more generic. An idea of generating FKBS with the experimental data method guided by the domain expert method is good enough toward achieving an I-A Trade-Off with higher levels of interpretability as well as accuracy.
The special interest of this paper is the use of interval type-2 fuzzy systems (IT2FS) . The membership functions are tuned using GAs, which leads toward a new system, the ‘type-2 genetic fuzzy system’ (T2GFS).
The paper continues with the ‘Interpretability issues in FKBS’ section in which the interpretability issues of FKBS are discussed. The ‘Tuning and learning operations in FKBS’ section introduces the basics of tuning and learning approaches. The fundamentals of type-2 fuzzy systems are discussed in the ‘Type-2 fuzzy systems’ section. A new lateral displacement and expansion/compression (LDEC) tuning approach is discussed in the ‘Proposed LDEC tuning approach’ section. The genetic representation of KB and the proposed tuning approach is discussed in the ‘Genetic representation of knowledge base’ section. Experimental results are discussed in the ‘Experiments and results’ section.
Interpretability issues in FKBS
Interpretability [14, 21–23] and accuracy  are the two important features considered during the design of fuzzy systems. Basically, interpretability is identified as a feature to understand the significance of something , and it is also known with other names like comprehensibility, intelligibility, transparency, readability, understandability, etc. Also, the quantification of interpretability is a highly subjective task depending on various parameters like experience, preference, and the knowledge of the person who interprets the system functionality.
Linguistic fuzzy modeling (LFM) and precise fuzzy modeling (PFM)  are two modeling approaches of fuzzy systems. In LFM, fuzzy models are developed by means of linguistic FRBS which are called Mamdani-type FKBS  mainly focusing on interpretability. On the other hand, PFM is developed considering the accuracy parameter and called Takagi-Sugeno FKBS . Accuracy improvement in LFM  and interpretability improvement in PFM  are carried out to achieve the desired I-A Trade-Off.
Interpretability in type-1 FKBS
Description of work
Interpretability improvement in high-dimensional fuzzy systems
Automatic rule generation and structure optimization for maintaining interpretability
R. Mikut et al.
Maintaining interpretability in data-based fuzzy system development along with user-controllable trade-off in between interpretability and accuracy
R. Alcala et al.
Seven hybrid techniques for developing accurate and interpretable FKBS
J. M. Alonso et al.
Highly interpretable linguistic knowledge (HILK) utilizing the features of LFM and PFM
S. M. Zhou and J. A. Gan
Identification of two interpretability levels: low level on the fuzzy set and high level on the fuzzy rule
C. Mencar and A. M. Fanelli
Introduction of semantic constraints, distinguishability, coverage, convexity, and normality
J. M. Alonso et al.
Conceptual framework for assessing the interpretability based on two issues: ‘description’ and ‘explanation’
M. J. Gacto et al.
A proposal of double-axis taxonomy: ‘complexity and semantic interpretability’ and ‘rule base and fuzzy partition’
M. Fazzolari et al.
I-A Trade-Off handling with instance selection techniques
Many other indexes and methodologies have been developed for assessing the interpretability, which are considered in this paper. These are (1) number of rules (NOR), (2) total rule length (TRL) - the sum of the number of premises in all the rules, and (3) average rule length (ARL) - calculated by TRL/NOR.
where (it measures the complexity), (it is the average normalized partition index), and cov is the average normalized coverage degree of the fuzzy partition. For strong fuzzy partition (SFP), it is equal to 1.
Similarly, a new global fuzzy index has been proposed in . In this approach, the index has been computed as the outcomes of the inference of hierarchical fuzzy system.
Tuning and learning operations in FKBS
In the literature, two types of approaches are found for tuning operations: one is related to applying SFs for handling linguistic hedges and the other is the tuning of the MF parameters. In this paper, the second approach of MF tuning is considered.
The scaling functions are responsible for adjusting the universe of discourse of input and output variables to the domain. The parameters used for tuning the scaling functions are scaling factor, upper and lower bounds (linear scaling functions), and contraction/dilation parameters (non-linear scaling function). The linguistic hedges are used and applied on the tuned MFs as discussed in [40–42]. The main linguistic hedges are as follows: very, more-or-less, extremely, very-very, positively, and negatively. Linguistic hedges are playing the role of adjectives and adverbs in the languages responsible for changing the qualitative linguistic statements.
Type-2 fuzzy systems
To implement FKBS, type-2 fuzzy sets (T2FS) [43, 44] are used having more capacity to deal with inherent uncertainties in the system to be developed. General type-2 fuzzy sets require high computational cost and type reduction complexity; hence, interval type 2 fuzzy sets [45–48] are preferred to model and implement various problems.
Here, ; when all , then A* is an interval type-2 fuzzy set.
Proposed LDEC tuning approach
α tuning operation
β tuning operation
Genetic representation of knowledge base
GAs [50, 51] are popular search techniques for ill-defined and complex search spaces. They are based on natural evolution. The initial population G(0) is generated with chromosomes representing DB and RB information and subsequently goes under evolution. During evolution, the next generation G(n + 1) is generated by applying crossover and mutation operators on the generation G(n). On each generation, each individual is evaluated by a fitness function. A termination condition is set to stop the evolution process.
In , inter-valued fuzzy sets (IVFS) have been used to implement a linguistic fuzzy rule-based classification system based on a new interval fuzzy reasoning method along with a new fuzzy rule learning process, called IVTURS-FARC.
In , the performance of a fuzzy rule-based classification system is improved using an interval-valued fuzzy set and a tuning approach using genetic algorithm. The uncertainty is modeled by the function ‘weak ignorance.’
Type-2 fuzzy system
D. Wu and W. W. Tan
Less computational expensive type-2 FLC is developed for real-time applications
D. Wu and W. W. Tan
GAs are used to evolve type-2 FLC
R. Sepulveda et al.
Feedback control systems for a non-linear plant using type-1 and type-2 fuzzy logic controllers
R. Martinez et al.
Type-2 fuzzy systems and GAs are used to implement track controller for unicycle mobile robot
M. H. F. Zarandi et al.
An interval type-2 fuzzy system has been developed for stock price analysis
O. Castillo et al.
An interval type-2 fuzzy logic controller has been developed using evolutionary algorithms
O. Castillo et al.
Ant colony optimization (ACO), particle swarm optimization (PSO), and GAs are used to optimize the MF parameters of a fuzzy logic controller
D. Hidalgo et al.
A footprint of uncertainty (FoU)-based type-2 fuzzy system optimization has been developed
O. Castillo and P. Mellin
A review on the optimization methods of type-2 fuzzy systems using bio-inspired computing
R. Hosseini et al.
Automatic tuning and learning approach for type-2 fuzzy systems has been proposed applied to lung CAD classification system
New proposed KB representation using GA
where CRM encodes the membership function and CRT encodes the tuning information for the membership function.
where the size of the data set is M. F(a i ) is the output obtained from FRBS for the i th example. The desired output is b i .
To perform GA operations, the following GA operators are used:
Selection: Tournament selection has been used for the selection operation.
Crossover: Crossover is the operator that generates new offspring by integrating multiple parents. A simple two-point crossover has been applied to all the chromosomes.
Mutation: This operator is used to maintain the diversity in the solutions from one generation to another generation. This operator changes the values of one or more bits in the chromosomes. In this proposed approach, a uniform mutation operator has been used in which the bits of chromosomes are altered within uniform random values at user-specified ranges.
Experiments and results
The RB generation methods used in the experiments are the decision tree (DT) method, Wang-Mendel method , and fast prototyping algorithms. The experiments are supported by the open-access free software tool ‘Guaje’ [29, 65] for type-1 fuzzy system implementation.
Description of data set
Number of attributes
Number of instances
Type-1 fuzzy system implementation
Accuracy and interpretability measures
Experiment 1 (E1)
Fuzzy partition method: hierarchical fuzzy partition (HFP) and rule generation method: Wang-Mendel method
Experiment 2 (E2)
Fuzzy partition method: strong fuzzy partition (SFP) and rule generation method: Wang-Mendel method
Experiment 3 (E3)
Fuzzy partition method: HFP and rule generation method: fuzzy decision trees
Experiment 4 (E4)
Fuzzy partition method: SFP and rule generation method: Wang-Mendel method
Type 2 fuzzy system implementation
The values of tuning parameters α and β calculated in the experiment are given in Table 5.
Experiment 5 (E5)
In this experiment (Table 6), the parameters of the genetic algorithm are as follows:
Number of generations = 2,000
Size of population = 70
Tournament size = 2
Size of population = 70
Mutation probability = 0.1
Crossover probability =0.5
Initial rules are generated by using the Wang-Mendel method.
Experiment 6 (E6)
In this experiment (Table 7), the initial rules are generated by a fuzzy decision tree with the following parameter settings:
Minimum cardinality of leaf = 1
Coverage threshold = 0.9
Minimum deviance gain = 0.001
Minimum significant level = 0.2
Pruning condition = yes
α and β parameters
Year of operation
Number of auxiliary nodes
Results of experiment 5
Results of experiment 6
The genetic algorithm parameters are the same as those in experiment 5.
Number of rules
Total number of instances
Number of wrong classification
WM + LDEC
FDT + LDEC
Type-2 fuzzy systems are strongly capable of modeling uncertainties in FKBS than type1 fuzzy systems using three-dimensional membership function representation. General type-2 fuzzy systems are deteriorating the interpretability of the systems, so IT2FS have been preferred to implement the proposed model with good interpretability.
The tuning and learning operations in the development of fuzzy systems playa vital role in improving their performance. This is considered as an optimization task and dealt properly with the application of evolutionary approaches, like GAs. The proposed tuning approach LDEC adjusts the parameters of interval type-2 fuzzy membership functions. This approach is based on the lateral displacement, expansion, and compression operations on the MFs. The proposed tuning approach is interpretable and the experimental results are found satisfactory.
fuzzy knowledge-based system
genetic fuzzy systems
- Palm R, Drainkov D, Hellendorn H: Model Based Fuzzy Control. Springer, Berlin; 1997.View ArticleGoogle Scholar
- Kuncheva LI: Fuzzy Classifier Design. Studies in Fuzziness and Soft Computing. Springer, Berlin; 2000.View ArticleMATHGoogle Scholar
- Pedrycz W: Fuzzy Modelling: Paradigms and Practices. Kluwer, Boston; 1996.View ArticleMATHGoogle Scholar
- Ross TJ: Fuzzy Logic with Engineering Applications. Wiley, Chichester; 2009.Google Scholar
- Cordon O, Herrera F, Hoffmann F, Magdalena L: Genetic Fuzzy Systems: Evolutionary Tuning and Learning of Fuzzy Knowledge Bases. World Scientific, Singapore; 2001.View ArticleMATHGoogle Scholar
- Herrera F: Genetic fuzzy systems: taxonomy, current research trends and prospects. Evol. Intel. 2008, 1: 27–46. 10.1007/s12065-007-0001-5View ArticleGoogle Scholar
- Herrera F: Genetic fuzzy systems: status, critical considerations and future directions. Int. J. Comput. Intell. Res. 2005, 1(1):59–67.View ArticleGoogle Scholar
- Cordon O, Gomide F, Herrera F, Hoffmann F, Magdalena L: Ten years of genetic fuzzy systems: current framework and new trends. Fuzzy Set. Syst. 2005, 141: 5–31.MathSciNetView ArticleMATHGoogle Scholar
- Roy SS: Design of genetic fuzzy expert system for predicting surface finish in ultra-precision diamond tuning of metal matrix composite. J. Mater. Process. Technol. 2006, 173: 337–344. 10.1016/j.jmatprotec.2005.12.003View ArticleGoogle Scholar
- Pulkkinen P, Hytonen J, Koivisto H: Developing a bioaerosol detector using hybrid genetic fuzzy systems. Eng. Appl. Artif. Intel. 2008, 21: 1330–1346. 10.1016/j.engappai.2008.01.006View ArticleGoogle Scholar
- Tseng CH, Kwong S, Wang H: Genetic fuzzy rule mining approach and evaluation of feature selection techniques for anomaly intrusion detection. Pattern Recogn. 2007, 40: 2373–2391. 10.1016/j.patcog.2006.12.009View ArticleMATHGoogle Scholar
- Achiche S, Balazinski M, Baron L, Jemielniak K: Tool wear monitoring using genetically-generated fuzzy knowledge bases. Eng. Appl. Artif. Intel. 2002, 15: 303–314. 10.1016/S0952-1976(02)00071-4View ArticleGoogle Scholar
- Kim HS, Roschke PN: Design of fuzzy logic controller for smart base isolation system using genetic algorithms. Eng. Struct. 2006, 28: 84–96. 10.1016/j.engstruct.2005.07.006View ArticleGoogle Scholar
- Cassilas J, Cordon O, Herrera F, Magdalena L: Interpretability Issues in Fuzzy Modeling. Studies in Fuzziness and Soft Computing. Springer, Berlin; 2003.View ArticleGoogle Scholar
- Cassilas J, Cordon O, Herrera F, Magdalena L: Accuracy Improvements in Linguistic Fuzzy Modeling. Studies in Fuzziness and Soft Computing. Springer, Berlin; 2003.View ArticleGoogle Scholar
- Alcala R, A-Fdez J, Cassilas J, Cordon O, Herrera F: Hybrid learning models to get the interpretability-accuracy trade-off in fuzzy modeling. Soft. Comput. 2006, 10: 717–734. 10.1007/s00500-005-0002-1View ArticleGoogle Scholar
- Shukla PK, Tripathi SP: A survey on interpretability-accuracy (I-A) trade-off in evolutionary fuzzy systems. Proceedings of 5th International Conference on Genetic and Evolutionary Computing (ICGEC 2011), Kitakyushu, 29 Aug–1 Sept 2011
- Shukla PK, Tripathi SP: A review on the interpretability-accuracy trade-off in evolutionary multi-objective fuzzy systems (EMOFS). Information 2012, 3(3):256–277.View ArticleGoogle Scholar
- Shukla PK, Tripathi SP: Interpretability issues in evolutionary multi-objective fuzzy knowledge base systems. In Proceedings of 7th International Conference on Bio-inspired Computing: Theories and Applications (BIC-TA 2012) Advances in Intelligent Systems and Computing, vol. 201. Edited by: Bansal JC. Springer, New Delhi; 2012:473–484.Google Scholar
- Liang Q, Mendel JM: Interval type-2 fuzzy logic systems: theory and design. IEEE Trans. Fuzzy Syst. 2000, 8(5):535–550. 10.1109/91.873577View ArticleGoogle Scholar
- Alonso JM, Magdalena L: Special issue on interpretable fuzzy systems. Inform. Sci. 2011, 181: 4331–4339. 10.1016/j.ins.2011.07.001MathSciNetView ArticleGoogle Scholar
- Alonso JM, Magdalena L, Gonzalez-Rodriguez G: Looking for a good fuzzy system interpretability index: an experimental approach. Int. J. Approx. Reason. 2009, 51: 115–134. 10.1016/j.ijar.2009.09.004MathSciNetView ArticleGoogle Scholar
- Jin Y: Fuzzy modeling of high dimensional systems: complexity reduction and interpretability improvement. IEEE Trans. Fuzzy Syst. 2000, 8(2):212–221. 10.1109/91.842154View ArticleGoogle Scholar
- Cassilas J, Cordon O, Herrera F, Magdalena L: Interpretability improvements to find the balance interpretability-accuracy in fuzzy modeling: an overview. In Interpretability Issues in Fuzzy Modeling, Studies in Fuzziness and Soft Computing. Edited by: Cassilas J, Cordon O, Herrera F, Magdalena L. Springer, Heidelberg; 2003:3–22.View ArticleGoogle Scholar
- Mamdani EH: Applications of fuzzy algorithms for controlling a simple dynamic plant. Proceedings of Institution of Electrical Engineers 1974, 121(12):1585–1588. 10.1049/piee.1974.0328View ArticleGoogle Scholar
- Takagi T, Sugeno M: Fuzzy identification of systems and its application to modeling and control. IEEE Trans. Syst. Man Cybern. 1985, 15: 116–132.View ArticleMATHGoogle Scholar
- Guillaume S: Designing fuzzy inference system from data: an interpretability oriented review. IEEE Trans. Fuzzy Syst. 2001, 9(3):426–443. 10.1109/91.928739MathSciNetView ArticleGoogle Scholar
- Mikut R, Jakel J, Groll L: Interpretability issues in data based learning of fuzzy systems. Fuzzy Set. Syst. 2005, 150: 179–197. 10.1016/j.fss.2004.06.006MathSciNetView ArticleMATHGoogle Scholar
- Alonso JM, Magdalena L: HILK++: an interpretability guided fuzzy modeling methodology for learning readable and comprehensible fuzzy rule based classifiers. Soft. Comput. 2011, 15(10):1959–1980. 10.1007/s00500-010-0628-5View ArticleGoogle Scholar
- Alonso JM, Magdalena L, Guillaume S: HILK: a new methodology for designing highly interpretable linguistic knowledge bases using fuzzy logic formalism. Int. J. Intell. Syst. 2008, 23(7):761–794. 10.1002/int.20288View ArticleMATHGoogle Scholar
- Zhou SM, Gan JQ: Low level interpretability and high level interpretability: a unified view of data-driven interpretable fuzzy system modeling. Fuzzy Set. Syst. 2008, 159: 3091–3131. 10.1016/j.fss.2008.05.016MathSciNetView ArticleGoogle Scholar
- Mencar C, Fanelli AM: Interpretability constraints for fuzzy information granulation. Inform. Sci. 2008, 178: 4585–4618. 10.1016/j.ins.2008.08.015MathSciNetView ArticleGoogle Scholar
- Gacto MJ, Alcala R, Herrera F: Interpretability of linguistic fuzzy rule based systems: an overview of interpretability measures. Inform. Sci. 2011, 181: 4340–4360. 10.1016/j.ins.2011.02.021View ArticleMATHGoogle Scholar
- Fazzolari M, Giglio B, Alcala R, Marcelloni F, Herrera F: A study on the application of instance selection techniques in genetic fuzzy rule based classification systems: accuracy-complexity trade-off. Knowledge Based Syst 2013, 54: 32–41.View ArticleGoogle Scholar
- Nauck DD: Measuring interpretability in rule based classification systems. In Proceedings of FUZZ-IEEE. Missouri; 25–28 May 2003
- Alonso JM, Guillaume S, Magdalena L: A hierarchical fuzzy system for assessing interpretability of linguistic knowledge bases in classification problems. Proceedings of IPMU 2006, Information Processing and Management of Uncertainty in Knowledge Based Systems, Paris, 2–7 July 2006 348–355.
- Smith SF Dissertation. In A learning system based on genetic adaptive algorithms. Department of Computer Science, University of Pittsburgh; 1980.Google Scholar
- Booker LB Dissertation. In Intelligent behavior as an adaptation to the task environment. Department of Computer and Communication Sciences, University of Michigan; 1982.Google Scholar
- Venturini G: SIA: A supervised inductive algorithm with genetic search for learning attribute based concepts. Proceedings of European Conference on Machine Learning, Vienna 5–7 Apr 1993
- Shi H, Ward R, Kharma N: Expanding the definitions of linguistic hedges. Proceedings of Joint 9th IFSA World Congress & 20th NAFIPS, Vancouver, 25–28 July 2001
- Zadeh LA: A fuzzy set theoretic interpretation of linguistic hedges. J Cybernetics 1972, 2(3):4–34. 10.1080/01969727208542910MathSciNetView ArticleGoogle Scholar
- Cox E: The Fuzzy Systems Handbook. AP Professional, New York; 1998.Google Scholar
- Mizumoto M, Tanaka K: Some properties of fuzzy sets of type 2. Inf. Control. 1976, 31: 312–340. 10.1016/S0019-9958(76)80011-3MathSciNetView ArticleMATHGoogle Scholar
- Mendel JM, John RIB: Type-2 fuzzy sets made simple. IEEE Trans. Fuzzy Syst. 2002, 10(2):117–127. 10.1109/91.995115View ArticleGoogle Scholar
- Wu H, Mendel JM: Uncertainty bounds and their use in the design of interval type-2 fuzzy logic systems. IEEE Trans. Fuzzy Syst. 2002, 10(5):622–639. 10.1109/TFUZZ.2002.803496View ArticleGoogle Scholar
- Wu D: On the fundamental differences between interval type-2 and type-1 fuzzy logic controllers. IEEE Trans. Fuzzy Syst. 2012, 20(5):832–848.View ArticleGoogle Scholar
- Wu D: Approaches for reducing the computational cost of interval type-2 fuzzy logic systems: overview and comparison. IEEE Trans. Fuzzy Syst. 2013, 21(1):80–99.View ArticleGoogle Scholar
- Chen SM, Chang YC, Pan JS: Fuzzy rules interpolation for sparse fuzzy rule based systems based on interval type-2 Gaussian fuzzy sets and genetic algorithms. IEEE Trans. Fuzzy Syst. 2013, 21(3):412–425.View ArticleGoogle Scholar
- Karnik NN, Mendel JM: Type-2 fuzzy logic systems. IEEE Trans. Fuzzy Syst. 1999, 7(6):643–658. 10.1109/91.811231View ArticleMATHGoogle Scholar
- Goldberg DE: Genetic Algorithms in Search, Optimization and Machine Learning. Addison-Wesley, Reading; 1989.MATHGoogle Scholar
- Michalewicz Z: Genetic Algorithms + Data Structures = Evolution Programs. Springer Verlag, Berlin; 1996.View ArticleMATHGoogle Scholar
- Sanz JA, Fernandez A, Bustince H: IVTURS: A linguistic fuzzy rule based classification system based on a new interval valued fuzzy reasoning method with tuning and rule selection. IEEE Trans. Fuzzy Syst. 2013, 21(3):399–411.View ArticleGoogle Scholar
- Sanz J, Fernandez A, Bustince H, Herrera F: A genetic tuning to improve the performance of fuzzy rule based classification systems with interval valued fuzzy sets: degree of ignorance and lateral position. Int. J. Approx. Reason. 2011, 52(6):751–766. 10.1016/j.ijar.2011.01.011View ArticleGoogle Scholar
- Wu D, Tan WW: A simplified type-2 fuzzy logic controller for real time control. ISA Trans. 2006, 45(4):503–516. 10.1016/S0019-0578(07)60228-6View ArticleGoogle Scholar
- Wu D, Tan WW: Genetic learning and performance evaluation of interval type-2 fuzzy logic controllers. Eng. Appl. Artif. Intel. 2006, 19(8):829–841. 10.1016/j.engappai.2005.12.011View ArticleGoogle Scholar
- Sepulveda R, Castillo O, Melin P, R–Diaz A, Montiel O: Exponential study of intelligent controllers under uncertainty using type-1 and type-2 fuzzy logic. Inform. Sci. 2007, 177(10):2023–2048. 10.1016/j.ins.2006.10.004View ArticleGoogle Scholar
- Martinez R, Castillo O, Aguilar LT: Optimization of interval type-2 fuzzy logic controllers for a perturbed autonomous wheeled mobile robot using genetic algorithms. Inform. Sci. 2009, 179(13):2158–2174. 10.1016/j.ins.2008.12.028View ArticleMATHGoogle Scholar
- Zarandi MHF, Rezaee B, Turksen IB, Neshat E: A type-2 fuzzy rule-based expert system model for stock price analysis. Expert Syst. Appl. 2009, 36(1):139–154. 10.1016/j.eswa.2007.09.034View ArticleGoogle Scholar
- Castillo O, Melin P, Alanis A, Montiel O, Sepulveda R: Optimization of interval type-2 fuzzy logic controllers using evolutionary algorithms. Soft. Comput. 2011, 15(6):1145–1160. 10.1007/s00500-010-0588-9View ArticleGoogle Scholar
- Castillo O, M-Marroquin R, Melin P, Valdez F, Soria J: Comparative study of bio-inspired algorithms applied to optimization of type-1 and type-2 fuzzy controllers for an autonomous mobile robot. Inform. Sci. 2012, 192(1):19–38.View ArticleGoogle Scholar
- Hidalgo D, Melin P, Castillo O: An optimization method for designing type-2 fuzzy inference systems based on the footprint of uncertainty using genetic algorithms. Expert Syst. Appl. 2012, 39(4):4590–4598. 10.1016/j.eswa.2011.10.003View ArticleGoogle Scholar
- Castillo O, Melin P: Optimization of type-2 fuzzy systems based on bio-inspired methods: a concise review. Inform. Sci. 2012, 205(1):1–19.View ArticleGoogle Scholar
- Hosseini R, Qanadli SD, Barman S, Mazinani M, Ellis T, Dehmeshki J: An automatic approach for learning and tuning Gaussian interval type-2 fuzzy membership functions applied to lung CAD classification system. IEEE Trans. Fuzzy Syst. 2012, 20(2):224–234.View ArticleGoogle Scholar
- Wang LX, Mendel JM: Generating fuzzy rule by learning from examples. IEEE Trans. Syst. Man Cybern. 1992, 22(6):1414–1427. 10.1109/21.199466MathSciNetView ArticleGoogle Scholar
- Alonso JM, Magdalena L: Generating understandable and accurate fuzzy rule based systems in a Java environment. In Fuzzy Logic and Applications, 9th International Workshop, WILF 2011, Trani, Italy, August 29–31, 2011. Lecture Notes in Artificial Intelligence, vol. 6857. Edited by: Fanelli AM, Pedrycz W, Petrosino A. Springer, Berlin; 2011:212–219.Google Scholar
- Bache K, Lichman M: UCI Machine Learning Repository. School of Information and Computer Science, University of California, Irvine, CA; (2013). Accessed 15 June 2013 http://archive.ics.uci.edu/ml
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