Evolving fuzzy systems from data streams in real-time

An approach to real-time generation of fuzzy rule-base systems of eXtended Takagi-Sugeno (xTS) type from data streams is proposed in the paper: The xTS fuzzy system combines both zero and first order Takagi-Sugeno (TS) type systems. The fuzzy rule-base (system structure) evolves starting 'from scratch' based on the data distribution in the joint input/output data space. An incremental clustering procedure that takes into account the non-stationary nature of the data pattern and generates clusters that are used to form fuzzy rule based systems antecedent part in on-line mode is used as a first stage of the non-iterative learning process. This structure proved to be computationally efficient and powerful to represent in a transparent way complex non-linear relationships. The decoupling of the learning task into a non-iterative, recursive (thus computationally very efficient and applicable in real-time) clustering with a modified version of the well known recursive parameter estimation technique leads to a very powerful construct - evolving xTS (exTS). It is transparent and linguistically interpretable. The contributions of this paper are: i) introduction of an adaptive recursively updated radius of the clusters (zone of influence of the fuzzy rules) that learns the data distribution/variance/scatter in each cluster; ii) a new condition to replace clusters that excludes contradictory rules; iii) an extended formulation that includes both zero order TS and simplified Mamdani multi-input-multi-output (MIMO) systems; iv) new improved formulation of the membership functions, which closer resembles the normal Gaussian distribution; v) introduction of measures of clusters quality that are used to form the antecedent parts of respective fuzzy rules, namely their age and support; vi) experimental results with a well known benchmark problem as well as with real experimental data of concentration of exhaust gases (NOx) in on-line modeling of car engine test rigs.