Only smooth rule bases can generate monotone MamdaniAssilian models under centerofgravity defuzzification
 Author
 Ester Van Broekhoven and Bernard De Baets (UGent)
 Organization
 Abstract
 At first sight, it seems that ordered linguistic values for all input variables and the output variable and a set of rules describing a monotone system are all that is needed for a monotone model. However, this is not the case. In this study, we show that the choice of the mathematical operators used when calculating the model output and the properties of the membership functions in the output domain are also of crucial importance to obtain a monotone inputoutput behavior. In the MamdaniAssilian models considered in this study, the linguistic values of the input variables, as well as the output variable, are described by trapezoidal membership functions that form a fuzzy partition, the rule base is monotone, and the crisp output is obtained by the centerofgravity (COG) defuzzification method. It is verified that for each of the three basic tnorms, i.e., the minimum TM, the product TP, and the Lukasiewicz tnorm TL, a monotone inputoutput behavior is obtained for any monotone rule base, or at least for any monotone smooth rule base. The outcome of this study is a guideline for designers of monotone linguistic fuzzy models. For the tnorms TM and TL, models with a single input variable show a monotone inputoutput behavior for any monotone rule base when the linguistic output values in the consequents of the rules are defined by trapezoidal or triangular membership functions with intervals of changing membership degrees of equal length. The latter restriction can easily be bypassed by an auxiliary interpolation procedure. For the tnorm TP, models with a single input variable show a monotone inputoutput behavior for any monotone rule base and any fuzzy output partition. When designing a monotone model with more than one input variable, one should opt for the tnorm TP and use a monotone smooth rule base. It is shown that monotonicity of models with two input variables that apply TP is guaranteed for any monotone smooth rule base and any fuzzy output partition. Finally, it is proved that a monotone inputoutput behavior is always obtained for models with three input variables that apply TP and a monotone smooth rule base when the linguistic output values in the consequents of the rules are defined by trapezoidal or triangular membership functions of identical shape.
 Keywords
 linguistic fuzzy models, monotonicity, Centerofgravity (COG) defuzzification method, trapezoidal membership functions, FUZZY SYSTEM, ALGORITHMS, PARAMETER, LOGIC
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Please use this url to cite or link to this publication: http://hdl.handle.net/1854/LU780611
 MLA
 Van Broekhoven, Ester, and Bernard De Baets. “Only Smooth Rule Bases Can Generate Monotone MamdaniAssilian Models Under Centerofgravity Defuzzification.” IEEE TRANSACTIONS ON FUZZY SYSTEMS 17.5 (2009): 1157–1174. Print.
 APA
 Van Broekhoven, E., & De Baets, B. (2009). Only smooth rule bases can generate monotone MamdaniAssilian models under centerofgravity defuzzification. IEEE TRANSACTIONS ON FUZZY SYSTEMS, 17(5), 1157–1174.
 Chicago authordate
 Van Broekhoven, Ester, and Bernard De Baets. 2009. “Only Smooth Rule Bases Can Generate Monotone MamdaniAssilian Models Under Centerofgravity Defuzzification.” Ieee Transactions on Fuzzy Systems 17 (5): 1157–1174.
 Chicago authordate (all authors)
 Van Broekhoven, Ester, and Bernard De Baets. 2009. “Only Smooth Rule Bases Can Generate Monotone MamdaniAssilian Models Under Centerofgravity Defuzzification.” Ieee Transactions on Fuzzy Systems 17 (5): 1157–1174.
 Vancouver
 1.Van Broekhoven E, De Baets B. Only smooth rule bases can generate monotone MamdaniAssilian models under centerofgravity defuzzification. IEEE TRANSACTIONS ON FUZZY SYSTEMS. 2009;17(5):1157–74.
 IEEE
 [1]E. Van Broekhoven and B. De Baets, “Only smooth rule bases can generate monotone MamdaniAssilian models under centerofgravity defuzzification,” IEEE TRANSACTIONS ON FUZZY SYSTEMS, vol. 17, no. 5, pp. 1157–1174, 2009.
@article{780611, abstract = {{At first sight, it seems that ordered linguistic values for all input variables and the output variable and a set of rules describing a monotone system are all that is needed for a monotone model. However, this is not the case. In this study, we show that the choice of the mathematical operators used when calculating the model output and the properties of the membership functions in the output domain are also of crucial importance to obtain a monotone inputoutput behavior. In the MamdaniAssilian models considered in this study, the linguistic values of the input variables, as well as the output variable, are described by trapezoidal membership functions that form a fuzzy partition, the rule base is monotone, and the crisp output is obtained by the centerofgravity (COG) defuzzification method. It is verified that for each of the three basic tnorms, i.e., the minimum TM, the product TP, and the Lukasiewicz tnorm TL, a monotone inputoutput behavior is obtained for any monotone rule base, or at least for any monotone smooth rule base. The outcome of this study is a guideline for designers of monotone linguistic fuzzy models. For the tnorms TM and TL, models with a single input variable show a monotone inputoutput behavior for any monotone rule base when the linguistic output values in the consequents of the rules are defined by trapezoidal or triangular membership functions with intervals of changing membership degrees of equal length. The latter restriction can easily be bypassed by an auxiliary interpolation procedure. For the tnorm TP, models with a single input variable show a monotone inputoutput behavior for any monotone rule base and any fuzzy output partition. When designing a monotone model with more than one input variable, one should opt for the tnorm TP and use a monotone smooth rule base. It is shown that monotonicity of models with two input variables that apply TP is guaranteed for any monotone smooth rule base and any fuzzy output partition. Finally, it is proved that a monotone inputoutput behavior is always obtained for models with three input variables that apply TP and a monotone smooth rule base when the linguistic output values in the consequents of the rules are defined by trapezoidal or triangular membership functions of identical shape.}}, author = {{Van Broekhoven, Ester and De Baets, Bernard}}, issn = {{10636706}}, journal = {{IEEE TRANSACTIONS ON FUZZY SYSTEMS}}, keywords = {{linguistic fuzzy models,monotonicity,Centerofgravity (COG) defuzzification method,trapezoidal membership functions,FUZZY SYSTEM,ALGORITHMS,PARAMETER,LOGIC}}, language = {{eng}}, number = {{5}}, pages = {{11571174}}, title = {{Only smooth rule bases can generate monotone MamdaniAssilian models under centerofgravity defuzzification}}, url = {{http://dx.doi.org/10.1109/TFUZZ.2009.2023328}}, volume = {{17}}, year = {{2009}}, }
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