TY - JOUR

T1 - Empirical Fuzzy Sets

AU - Angelov, Plamen Parvanov

AU - Gu, Xiaowei

N1 - This is the peer reviewed version of the following article: Angelov, P. P. and Gu, X. (2018), Empirical Fuzzy Sets. Int. J. Intell. Syst., 33: 362–395. doi:10.1002/int.21935 which has been published in final form at http://onlinelibrary.wiley.com/doi/10.1002/int.21935/abstract This article may be used for non-commercial purposes in accordance With Wiley Terms and Conditions for self-archiving.

PY - 2018/2

Y1 - 2018/2

N2 - In this paper, we introduce a new form of describing fuzzy sets (FSs) and a new form of fuzzy rule-based (FRB) systems, namely, empirical fuzzy sets (εFSs) and empirical fuzzy rule-based (εFRB) systems. Traditionally, the membership functions (MFs), which are the key mathematical representation of FSs, are designed subjectively or extracted from the data by clustering projections or defined subjectively. εFSs, on the contrary, are described by the empirically derived membership functions (εMFs). The new proposal made in this paper is based on the recently introduced Empirical Data Analytics (EDA) computational framework and is closely linked with the density of the data. This allows to keep and improve the link between the objective data and the subjective labels, linguistic terms and classes definition. Furthermore, εFSs can deal with heterogeneous data combining categorical with continuous and/or discrete data in a natural way. εFRB systems can be extracted from data including data streams and can have dynamically evolving structure. However, they can also be used as a tool to represent expert knowledge. The main difference with the traditional FSs and FRB systems is that the expert does not need to define the MF per variable; instead, possibly multimodal, densities will be extracted automatically from the data and used as εMFs in a vector form for all numerical variables. This is done in a seamless way whereby the human involvement is only required to label the classes and linguistic terms. Moreover, even this intervention is optional. Thus, the proposed new approach to define and design the FSs and FRB systems puts the human “in the driving seat”. Instead of asking experts to define features and MFs correspondingly, to parameterize them, to define algorithm parameters, to choose types of MFs or to label each individual item, it only requires (optionally) to select prototypes from data and (again, optionally) to label them. Numerical examples as well as a naïve empirical fuzzy (εF) classifier are presented with an illustrative purpose. Due to the very fundamental nature of the proposal it can have a very wide area of applications resulting in a series of new algorithms such as εF classifiers, εF predictors, εF controllers, etc. This is left for the future research.

AB - In this paper, we introduce a new form of describing fuzzy sets (FSs) and a new form of fuzzy rule-based (FRB) systems, namely, empirical fuzzy sets (εFSs) and empirical fuzzy rule-based (εFRB) systems. Traditionally, the membership functions (MFs), which are the key mathematical representation of FSs, are designed subjectively or extracted from the data by clustering projections or defined subjectively. εFSs, on the contrary, are described by the empirically derived membership functions (εMFs). The new proposal made in this paper is based on the recently introduced Empirical Data Analytics (EDA) computational framework and is closely linked with the density of the data. This allows to keep and improve the link between the objective data and the subjective labels, linguistic terms and classes definition. Furthermore, εFSs can deal with heterogeneous data combining categorical with continuous and/or discrete data in a natural way. εFRB systems can be extracted from data including data streams and can have dynamically evolving structure. However, they can also be used as a tool to represent expert knowledge. The main difference with the traditional FSs and FRB systems is that the expert does not need to define the MF per variable; instead, possibly multimodal, densities will be extracted automatically from the data and used as εMFs in a vector form for all numerical variables. This is done in a seamless way whereby the human involvement is only required to label the classes and linguistic terms. Moreover, even this intervention is optional. Thus, the proposed new approach to define and design the FSs and FRB systems puts the human “in the driving seat”. Instead of asking experts to define features and MFs correspondingly, to parameterize them, to define algorithm parameters, to choose types of MFs or to label each individual item, it only requires (optionally) to select prototypes from data and (again, optionally) to label them. Numerical examples as well as a naïve empirical fuzzy (εF) classifier are presented with an illustrative purpose. Due to the very fundamental nature of the proposal it can have a very wide area of applications resulting in a series of new algorithms such as εF classifiers, εF predictors, εF controllers, etc. This is left for the future research.

KW - membership functions

KW - AnYa type fuzzy rule-based systems

KW - empirical data analytics

KW - naïve empirical fuzzy rule-based classifier

KW - non-parametric

U2 - 10.1002/int.21935

DO - 10.1002/int.21935

M3 - Journal article

VL - 33

SP - 362

EP - 395

JO - International Journal of Intelligent Systems

JF - International Journal of Intelligent Systems

SN - 0884-8173

IS - 2

ER -