An Improved Similarity Measure for Generalized Trapezoidal Fuzzy Numbers and Its Application in the Classification of EEG Signals

Abstract

The classification of electroencephalogram (EEG) signals plays a key role in detecting brain activities. Fuzzy methods are widely applied in decision-making problems because they are effective tools for handling imprecise and vague data. This paper proposes a modified algorithm to calculate the center of gravity of generalized trapezoidal fuzzy numbers. Accordingly, we introduce a new similarity measure for generalized trapezoidal fuzzy numbers that we use in the classification of EEG signals. This measure combines the height, the center of gravity, the perimeter, the area, and the gyradius of generalized trapezoidal fuzzy numbers to quantify the similarity between generalized trapezoidal fuzzy numbers. We use 16 sets of generalized trapezoidal fuzzy numbers to compare the proposed similarity measure with existing ones. Comparison results indicate that the proposed similarity measure can overcome the drawbacks of existing similarity measures. Finally, an EEG experiment is carried out in laboratory. Experimental results demonstrate that the proposed similarity measure is more effective than other methods in terms of classification of EEG signals.

Appendix

Definition 1

If the membership function of generalized trapezoidal fuzzy number ${\widetilde{A}}=(a_1,a_2,a_3,a_4 ; w)$ is

$$\begin{aligned} \mu _{{\widetilde{A}}}(x) = \left\{ \begin{aligned}&0,\quad { - \infty }< x \le a_1\\&\frac{w\cdot (x - a_1)}{a_2 - a_1},\quad a_1< x< a_2 \\&w,\quad a_2 \le x \le a_3\\&\frac{w\cdot (a_4 - x)}{a_4 -a_3},\quad a_3< x< a_4 \\&0,\quad a_4 \le x < { + \infty } \end{aligned} \right. \end{aligned}$$

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where $a_1,a_2,a_3,a_4$ are real values, $a_1 \le a_2 \le a_3 \le a_4$.$0 \le w \le 1$. Then we call ${\widetilde{A}}$ a generalized trapezoidal fuzzy number.