A Study on Multi-Class Online Fuzzy Classifiers for Dynamic Environments

Kensuke Ajimoto; Yuma Yamamoto; Yoshifumi Kusunoki; Tomoharu Nakashima

cs.LGFeb 16, 2026

A Study on Multi-Class Online Fuzzy Classifiers for Dynamic Environments

Authors: Kensuke Ajimoto, Yuma Yamamoto, Yoshifumi Kusunoki, Tomoharu Nakashima

Organizations: Graduate School of Informatics, Osaka Metropolitan University, Osaka, Japan

Abstract

This paper proposes a multi-class online fuzzy classifier for dynamic environments. A fuzzy classifier comprises a set of fuzzy if-then rules where human users determine the antecedent fuzzy sets beforehand. In contrast, the consequent real values are determined by learning from training data. In an online framework, not all training dataset patterns are available beforehand. Instead, only a few patterns are available at a time step, and the subsequent patterns become available at the following time steps. The conventional online fuzzy classifier considered only two-class problems. This paper investigates the extension to the conventional fuzzy classifiers for multi-class problems. We evaluate the performance of the multi-class online fuzzy classifiers through numerical experiments on synthetic dynamic data and also several benchmark datasets.

Figures & tables

Fig. 1: Triangular membership functions.

	$x_{1}$	$x_{2}$
Rule 1	small	small
Rule 2	small	large
Rule 3	large	small
Rule 4	large	large

TABLE I: An example of antecedent parts.

Fig. 2: Generated four fuzzy if-then rules in the example.

$j$	$\mu_{j1}(x_{1})$	$\mu_{j2}(x_{2})$	$\mu_{j}(\vec{x})$	$c_{j}$	$\mu_{j}(\vec{x})\cdot c_{j}$
1	0.8	0.6	0.48	0.8	0.384
2	0.8	0.4	0.32	-0.2	-0.064
3	0.2	0.6	0.12	-0.4	-0.048
4	0.2	0.4	0.08	-0.7	-0.056
				Total	0.216

TABLE II: Calculation of membership values.

Fig. 3: Generated boundary by the example fuzzy classifier.

Fig. 4: Don’t Care membership function.

Dataset	Instance	Class	Dim
Speaker accent recognition [ 12 ]	5473	5	10
Crabs [ 13 ]	200	5	4
Glass identification [ 12 ]	214	7	9
Iris [ 12 ]	150	3	4
Page blocks classification [ 12 ]	5473	5	10
Seeds [ 12 ]	210	3	7

TABLE III: Dataset characteristics

dataset	Fuzzy(OvR)	Fuzzy(OvO)	PA(OvR)	PA(OvO)	Delta(OvR)	Delta(OvO)
Speaker accent recognition
accuracy (%)	$\mathbf{52.36}\pm 10.98$	$50.80\pm 13.88$	$41.63\pm 15.96$	$41.63\pm 16.58$	$25.76\pm 7.32$	$45.15\pm 3.33$
time (s)	$314.82\pm 1.68$	$267.50\pm 0.97$	$1.94\pm 0.71$	$\mathbf{1.67}\pm 0.82$	$2.47\pm 0.87$	$2.44\pm 0.92$
Crabs
accuracy (%)	$\mathbf{45.29}\pm 10.09$	$40.66\pm 6.16$	$28.66\pm 4.82$	$32.61\pm 6.56$	$25.00\pm 0.00$	$29.5\pm 4.38$
time (s)	$12.90\pm 1.21$	$9.96\pm 1.24$	$0.76\pm 0.67$	$\mathbf{0.68}\pm 0.67$	$1.04\pm 0.67$	$1.04\pm 0.70$

TABLE IV: Performance comparison of Fuzzy classifier, PA, and Delta learning on benchmark datasets

Fig. 5: Distributions of petal length and petal width of the iris dataset.

Fig. 6: An overview of the area covered by the rules generated from the three-partition membership functions.

Fig. 7: The initial setup of three classes for a dynamic problem.

Fig. 8: Transitions of the important rules in

f_{1}

Fig. 9: Transitions of the important rules in

f_{1,3}

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