FP-Conv-CM: Fuzzy Probabilistic Convolution C-Means

Soft computing models based on fuzzy or probabilistic approaches provide decision system makers with the necessary capabilities to deal with imprecise and incomplete information. Hybrid systems based on different soft computing approaches with complementary qualities and principles have also become popular. On the one hand, fuzzy logic makes its decisions on the basis of the degree of membership but gives no information on the frequency of an event; on the other hand, the probability informs us of the frequency of the event but gives no information on the degree of membership to a set. In this work, we propose a new measure that implements both fuzzy and probabilistic notions (i.e., the degree of membership and the frequency) while exploiting the ability of the convolution operator to combine functions on continuous intervals. This measure evaluates both the degree of membership and the frequency of objects/events in the design of decision support systems. We show, using concrete examples, the drawbacks of fuzzy logic and probability-based approaches taken separately, and we then show how a fuzzy probabilistic convolution measure allows the correction of these drawbacks. Based on this measure, we introduce a new clustering method named Fuzzy-Probabilistic-Convolution-C-Means (FP-Conv-CM). Fuzzy C-Means (FCM), Probabilistic K-Means (PKM), and FP-Conv-CM were tested on multiple datasets and compared on the basis of two performance measures based on the Silhouette metric and the Dunn's Index. FP-Conv-CM was shown to improve on both metrics. In addition, FCM, PKM, and FP-Conv-CM were used for multiple image compression tasks and were compared based on three performance measures: Mean Square Error (MSE), Peak Signal-to-Noise Ratio (PSNR), and Structural SImilarity Index (SSIM). The proposed FP-Conv-CM method shows improvements in all these three measures as well.