This study is concerned with the augmentation of rule-based models or fuzzy rule-based models by associating the numeric results produced by them with granular characterization in the form of prediction intervals.
The role of prediction intervals encountered in regression analysis is well emphasized in the literature, especially for “monolithic” models such as linear regression or neural networks. However, there have not been comprehensive and algorithmically complete studies devoted to the conceptualization and determination of prediction intervals for rule-based models, Boolean (two-valued) and fuzzy ones. While the results generated by rule-based models are formed by aggregating partial outcomes resulting from the individual rules, the prediction intervals adhere to the same way of aggregation. In this sense, one can regard the rule-based model augmented by the associated prediction intervals as a granular rule-based model. The construction of rule-based models is, in a nutshell, the optimization process being commonly guided by the sum of squared errors (say, RMSE or alike), and the quality of the granular counterpart of the rule-based models is assessed by looking at the quality of prediction intervals and introducing the two pertinent quality measures focused on the granular nature of the obtained results, namely coverage and specificity along with their combined index. In this study, we analyze an impact of the rules and the parameters of fuzzy clustering on the quality of the numeric and granular performance of the models. A series of experimental results is presented to help quantify the performance of granular outputs (prediction intervals) constructed for rule-based models.
Rule-based model Clustering Information granules Prediction interval Coverage Specificity Optimization
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Support from the Natural Sciences and Engineering Research Council (NSERC) and Canada Research Chair (CRC) Program is gratefully acknowledged.
This study was funded by NSERC and CRC.
Compliance with ethical standards
Conflict of interest
All the authors have no conflict of interest.
This article does not contain any studies with human participants or animals performed by any of the authors.
Ak R, Li YF, Vitelli V, Zio E (2018) Adequacy assessment of a wind-integrated system using neural network-based interval predictions of wind power generation and load. Electr Power Energy Syst 95:213–226CrossRefGoogle Scholar
Beran RJ (1992) Introduction to Efron (1979) bootstrap methods: another look at the Jackknife. Springer, New YorkGoogle Scholar
Box GEP, Jenkins GM, Reinsel GC (1994) Time series analysis: forecasting and control. Englewood Cliffs, Prentice-HallzbMATHGoogle Scholar
Kerr-Wilson J, Pedrycz W (2017) Some new qualitative insights into quality of fuzzy rule-based models. Fuzzy Sets Syst 307:29–49MathSciNetCrossRefGoogle Scholar
Khosravi A, Nahavandi S, Creighton D (2010) A prediction interval-based approach to determine optimal structures of neural network metamodels. Expert Syst Appl 37:2377–2387CrossRefGoogle Scholar
Khosravi A, Nahavandi S, Creighton D, Atiya AF (2011a) Comprehensive review of neural network-based prediction intervals and new advances. IEEE Trans Neural Netw 22:1341–1356CrossRefGoogle Scholar
Khosravi A, Nahavandi S, Creighton D, Atiya AF (2011b) Lower upper bound estimation method for construction of neural network-based prediction intervals. IEEE Trans Neural Netw 22:337–346CrossRefGoogle Scholar
Lee YS, Scholtes S (2014) Empirical prediction intervals revisited. Int J Forecast 30:217–234CrossRefGoogle Scholar
Li K, Wang R, Lei H, Zhang T, Liu Y, Zheng X (2018a) Interval prediction of solar power using an improved bootstrap method. Sol Energy 159:97–112CrossRefGoogle Scholar
Li J, Yang L, Qu Y et al (2018b) An extended Takagi–Sugeno–Kang inference system (TSK+) with fuzzy interpolation and its rule base generation. Soft Comput 22(10):3155–3170CrossRefGoogle Scholar
Moore RE (1966) Interval analysis. Prentice-Hall, Englewood CliffsGoogle Scholar
Pedrycz W (2013) Granular computing: analysis and design of intelligent systems. CRC Press/Francis Taylor, Boca RatonCrossRefGoogle Scholar
Pedrycz W, Lu W, Liu X et al (2014) Human-centric analysis and interpretation of time series: a perspective of granular computing. Soft Comput 18(12):2397–2411CrossRefGoogle Scholar