On-line elimination of local redundancies in evolving fuzzy systems

Abstract

In this paper, we examine approaches for reducing the complexity of evolving fuzzy systems (EFSs) by eliminating local redundancies during training, evolving the models on on-line data streams. Thus, the complexity reduction steps should support fast incremental single-pass processing steps. In EFSs, such reduction steps are important due to several reasons: (1) originally distinct rules representing distinct local regions in the input/output data space may move together over time and get significantly over-lapping as data samples are filling up the gaps in-between these, (2) two or several fuzzy sets in the fuzzy partitions may become redundant because of projecting high-dimensional clusters onto the single axes, (3) they can be also seen as a first step towards a better readability and interpretability of fuzzy systems, as unnecessary information is discarded and the models are made more transparent. One technique is performing a new rule merging approach directly in the product cluster space using a novel concept for calculating the similarity degree between an updated rule and the remaining ones. Inconsistent rules elicited by comparing the similarity of two redundant rule antecedent parts with the similarity of their consequents are specifically handled in the merging procedure. The second one is operating directly in the fuzzy partition space, where redundant fuzzy sets are merged based on their joint a-cut levels. Redundancy is measured by a novel kernel-based similarity measure. The complexity reduction approaches are evaluated based on high-dimensional noisy real-world measurements and an artificially generated data stream containing 1.2 million samples. Based on this empirical comparison, it will be shown that the novel techniques are (1) fast enough in order to cope with on-line demands and (2) produce fuzzy systems with less structural components while at the same time achieving accuracies similar to EFS not integrating any reduction steps.

Autonomous data stream clustering implementing split-and-merge concepts – Towards a plug-and-play approach

“…This does not only resolve significant overlaps arising over time as clusters may move together [64], but also compensates slight overlaps or touching clusters, which form a reciprocal homogeneity of their shape and direction (see Section 3). Moreover, a splitting strategy resolves the situation that two density areas are crystallized over time within one cluster.…”

SO-ARTIST: Self-Organized ART-2A inspired clustering for online Takagi–Sugeno fuzzy models

“…For this purpose we want to explore more works tackling merging and splitting strategies for fuzzy models. As we have stated in the previous sections, some of the works cited in this paper (Lughofer et al [48], Lughofer [47], Lemos et al [40]) compare the antecedent part for the purpose of merging/splitting more significantly, as we do. Therefore we want to explore the possibilities of hybrid merge-split operations in order to find the most appropriate fuzzy rules to merge, split or discard.…”

“…Lughofer [45] proposed using incremental vector quantization for the organization of rules. This work was later expanded in Lughofer et al [48], where the authors propose new elimination techniques for rule redundancies, including merging of rules. However, they still expect a free parameter (threshold) to be used and adjusted.…”

“…In Lughofer et al [48] the authors also propose a similarity measure in order to determine the merging of rules. However, their strategy is based solely on cluster shapes and positions and thus focuses on the antecedent part.…”

A multi-objective genetic optimization of interpretability-oriented fuzzy rule-based classifiers

“…The modified SPEA2 algorithm was used for optimization. A number of other interesting papers on the discussed issues were published in recent years (e.g., [61], [62], [63], [64], [65], [66]), which proves that the matter is still an open question. The above summary, obviously, does not exhaust the subject.…”