A new approach for tuning interval type2 fuzzy knowledge bases using genetic algorithms
 Praveen Kumar Shukla^{1}Email author and
 Surya Prakash Tripathi^{2}
DOI: 10.1186/2195546824
© Shukla and Tripathi; licensee Springer. 2014
Received: 15 September 2013
Accepted: 13 February 2014
Published: 27 February 2014
Abstract
Fuzzy knowledgebased systems (FKBS) are significantly applicable in the area of control, classification, and modeling, having knowledge in the form of fuzzy ifthen rules. Type2 fuzzy theory is used to make these systems more capable of dealing with inherent uncertainties in realworld 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 type2 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.
Introduction
Fuzzy systems, more specifically fuzzy knowledgebased systems (FKBS) or fuzzy rulebased systems (FRBS), are significantly applicable in areas like control [1], classification [2], and modeling [3]. The essential feature of FKBS is the incorporation of human expert knowledge which is in the form of fuzzy [4] extended ifthen rules. The major components of FKBS are fuzzification interface, inference engine, knowledge base, and defuzzification interface [5]. 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 ifthen 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 [9], bioaerosol detector [10], classification of intrusion attacks from a network traffic data [11], tool wear monitoring [12], smart base isolation system [13], 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’ [14] 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 [15] 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 ‘interpretabilityaccuracy tradeoff’ (IA TradeOff) [16–19]. For the above applications, interpretability as well as accuracy is required to be maintained at the higher level by maintaining a good IA TradeOff.
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 IA TradeOff with higher levels of interpretability as well as accuracy.
The special interest of this paper is the use of interval type2 fuzzy systems (IT2FS) [20]. The membership functions are tuned using GAs, which leads toward a new system, the ‘type2 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 type2 fuzzy systems are discussed in the ‘Type2 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 [15] 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 [21], 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) [24] are two modeling approaches of fuzzy systems. In LFM, fuzzy models are developed by means of linguistic FRBS which are called Mamdanitype FKBS [25] mainly focusing on interpretability. On the other hand, PFM is developed considering the accuracy parameter and called TakagiSugeno FKBS [26]. Accuracy improvement in LFM [15] and interpretability improvement in PFM [14] are carried out to achieve the desired IA TradeOff.
Interpretability in type1 FKBS
Year  Authors  Description of work  Reference 

2000  Y. Jin  Interpretability improvement in highdimensional fuzzy systems  [23] 
2001  S. Guillaume  Automatic rule generation and structure optimization for maintaining interpretability  [27] 
2005  R. Mikut et al.  Maintaining interpretability in databased fuzzy system development along with usercontrollable tradeoff in between interpretability and accuracy  [28] 
2006  R. Alcala et al.  Seven hybrid techniques for developing accurate and interpretable FKBS  [16] 
2008, 2012  J. M. Alonso et al.  Highly interpretable linguistic knowledge (HILK) utilizing the features of LFM and PFM  
2008  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  [31] 
2008  C. Mencar and A. M. Fanelli  Introduction of semantic constraints, distinguishability, coverage, convexity, and normality  [32] 
2009  J. M. Alonso et al.  Conceptual framework for assessing the interpretability based on two issues: ‘description’ and ‘explanation’  [22] 
2011  M. J. Gacto et al.  A proposal of doubleaxis taxonomy: ‘complexity and semantic interpretability’ and ‘rule base and fuzzy partition’  [33] 
2013  M. Fazzolari et al.  IA TradeOff handling with instance selection techniques  [34] 
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 $\mathrm{\text{comp}}=\frac{\mathrm{\text{number}}\phantom{\rule{0.25em}{0ex}}\mathrm{of}\phantom{\rule{0.25em}{0ex}}\mathrm{\text{classes}}}{\mathrm{\text{total}}\phantom{\rule{0.25em}{0ex}}\mathrm{\text{number}}\phantom{\rule{0.25em}{0ex}}\mathrm{of}\phantom{\rule{0.25em}{0ex}}\mathrm{\text{premises}}}$ (it measures the complexity), $\mathrm{\text{part}}=\frac{1}{\mathrm{\text{number}}\phantom{\rule{0.25em}{0ex}}\mathrm{of}\phantom{\rule{0.25em}{0ex}}\mathrm{\text{labels}}1}$ (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 [36]. 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 (nonlinear 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, moreorless, extremely, veryvery, positively, and negatively. Linguistic hedges are playing the role of adjectives and adverbs in the languages responsible for changing the qualitative linguistic statements.
Type2 fuzzy systems
To implement FKBS, type2 fuzzy sets (T2FS) [43, 44] are used having more capacity to deal with inherent uncertainties in the system to be developed. General type2 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, $0\le {\mathit{\mu}}_{{\mathit{A}}^{*}}\left(\mathit{x},\mathit{u}\right)\le 1$; when all ${\mathit{\mu}}_{{\mathit{A}}^{*}}\left(\mathit{x},\mathit{u}\right)=1$, then A* is an interval type2 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 illdefined 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 [52], intervalued fuzzy sets (IVFS) have been used to implement a linguistic fuzzy rulebased classification system based on a new interval fuzzy reasoning method along with a new fuzzy rule learning process, called IVTURSFARC.
In [53], the performance of a fuzzy rulebased classification system is improved using an intervalvalued fuzzy set and a tuning approach using genetic algorithm. The uncertainty is modeled by the function ‘weak ignorance.’
Type2 fuzzy system
Year  Authors  Description  Reference 

2006  D. Wu and W. W. Tan  Less computational expensive type2 FLC is developed for realtime applications  [54] 
2006  D. Wu and W. W. Tan  GAs are used to evolve type2 FLC  [55] 
2007  R. Sepulveda et al.  Feedback control systems for a nonlinear plant using type1 and type2 fuzzy logic controllers  [56] 
2009  R. Martinez et al.  Type2 fuzzy systems and GAs are used to implement track controller for unicycle mobile robot  [57] 
2009  M. H. F. Zarandi et al.  An interval type2 fuzzy system has been developed for stock price analysis  [58] 
2011  O. Castillo et al.  An interval type2 fuzzy logic controller has been developed using evolutionary algorithms  [59] 
2012  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  [60] 
2012  D. Hidalgo et al.  A footprint of uncertainty (FoU)based type2 fuzzy system optimization has been developed  [61] 
2012  O. Castillo and P. Mellin  A review on the optimization methods of type2 fuzzy systems using bioinspired computing  [62] 
2012  R. Hosseini et al.  Automatic tuning and learning approach for type2 fuzzy systems has been proposed applied to lung CAD classification system  [63] 
New proposed KB representation using GA
Encoding scheme
where CR_{M} encodes the membership function and CR_{T} encodes the tuning information for the membership function.
Fitness 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 }.
GA operators
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 twopoint 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 userspecified ranges.
Experiments and results
The RB generation methods used in the experiments are the decision tree (DT) method, WangMendel method [64], and fast prototyping algorithms. The experiments are supported by the openaccess free software tool ‘Guaje’ [29, 65] for type1 fuzzy system implementation.
Description of data set
Serial number  Characteristics  Value 

1  Type  Classification 
2  Number of attributes  3 
3  Number of instances  306 
4  Attribute characteristics  Integer 
Type1 fuzzy system implementation
Accuracy and interpretability measures
Parameter  E1  E2  E3  E4 

Accuracy  
MSE  0.121  0.112  0.117  0.092 
RMSE  0.491  0.474  0.483  0.428 
Interpretability  
NI  0.016  0.016  0.009  0.016 
ARL  2.839  2.773  2.69  2.652 
NOR  31  22  42  23 
TRL  88  61  113  61 
AIFR  3.902  4.964  6.199  4.075 

Experiment 1 (E1)

Fuzzy partition method: hierarchical fuzzy partition (HFP) and rule generation method: WangMendel method

Experiment 2 (E2)

Fuzzy partition method: strong fuzzy partition (SFP) and rule generation method: WangMendel 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: WangMendel 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 WangMendel 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
Serial number  Variable name  α value  β value 

1  Age  2.785  1.12 
2  Year of operation  0.184  0.918 
3  Number of auxiliary nodes  13.75  3.42 
Results of experiment 5
Parameter  Value 

Accuracy  
MSE  0.094 
RMSE  0.432 
Interpretability  
NI  0.019 
ARL  18 
NOR  54 
TRL  2.98 
AIFR  3.876 
Results of experiment 6
Parameter  Value 

Accuracy  
MSE  0.081 
RMSE  0.398 
Interpretability  
NI  0.029 
ARL  2.615 
NOR  13 
TRL  34 
AIFR  2.02 
The genetic algorithm parameters are the same as those in experiment 5.
Comparative results
Method  Number of rules  Total number of instances  Number of wrong classification  MSE 

WM  18  100  22  0.11 
FDT  14  100  20  0.10 
WM + LDEC  24  100  16  0.08 
FDT + LDEC  22  100  14  0.07 
Conclusions
Type2 fuzzy systems are strongly capable of modeling uncertainties in FKBS than type1 fuzzy systems using threedimensional membership function representation. General type2 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 type2 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.
Abbreviations
 DB:

data base
 FKBS:

fuzzy knowledgebased system
 GAs:

genetic algorithms
 GFS:

genetic fuzzy systems
 KB:

knowledge base
 MFs:

membership functions
 RB:

rule base
 SFs:

scaling functions.
Declarations
Authors’ Affiliations
References
 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/s1206500700015View 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 ultraprecision 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 geneticallygenerated fuzzy knowledge bases. Eng. Appl. Artif. Intel. 2002, 15: 303–314. 10.1016/S09521976(02)000714View 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, AFdez J, Cassilas J, Cordon O, Herrera F: Hybrid learning models to get the interpretabilityaccuracy tradeoff in fuzzy modeling. Soft. Comput. 2006, 10: 717–734. 10.1007/s0050000500021View ArticleGoogle Scholar
 Shukla PK, Tripathi SP: A survey on interpretabilityaccuracy (IA) tradeoff 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 interpretabilityaccuracy tradeoff in evolutionary multiobjective fuzzy systems (EMOFS). Information 2012, 3(3):256–277.View ArticleGoogle Scholar
 Shukla PK, Tripathi SP: Interpretability issues in evolutionary multiobjective fuzzy knowledge base systems. In Proceedings of 7th International Conference on Bioinspired Computing: Theories and Applications (BICTA 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 type2 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, GonzalezRodriguez 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 interpretabilityaccuracy 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/s0050001006285View 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 datadriven 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: accuracycomplexity tradeoff. Knowledge Based Syst 2013, 54: 32–41.View ArticleGoogle Scholar
 Nauck DD: Measuring interpretability in rule based classification systems. In Proceedings of FUZZIEEE. 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/S00199958(76)800113MathSciNetView ArticleMATHGoogle Scholar
 Mendel JM, John RIB: Type2 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 type2 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 type2 and type1 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 type2 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 type2 Gaussian fuzzy sets and genetic algorithms. IEEE Trans. Fuzzy Syst. 2013, 21(3):412–425.View ArticleGoogle Scholar
 Karnik NN, Mendel JM: Type2 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. AddisonWesley, 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 type2 fuzzy logic controller for real time control. ISA Trans. 2006, 45(4):503–516. 10.1016/S00190578(07)602286View ArticleGoogle Scholar
 Wu D, Tan WW: Genetic learning and performance evaluation of interval type2 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 type1 and type2 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 type2 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 type2 fuzzy rulebased 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 type2 fuzzy logic controllers using evolutionary algorithms. Soft. Comput. 2011, 15(6):1145–1160. 10.1007/s0050001005889View ArticleGoogle Scholar
 Castillo O, MMarroquin R, Melin P, Valdez F, Soria J: Comparative study of bioinspired algorithms applied to optimization of type1 and type2 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 type2 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 type2 fuzzy systems based on bioinspired 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 type2 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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