Optimization of Fuzzy C-Means Clustering Algorithm with Combination of Minkowski and Chebyshev Distance Using Principal
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Optimization of Fuzzy C-Means Clustering Algorithm with Combination of Minkowski and Chebyshev Distance Using Principal Component Analysis Sugiyarto Surono1 • Rizki Desia Arindra Putri1
Received: 21 August 2020 / Revised: 1 October 2020 / Accepted: 27 October 2020 Taiwan Fuzzy Systems Association 2020
Abstract Optimization is used to find the maximum or minimum of a function. In this research, optimization is applied to the objective function of the FCM algorithm. FCM is an effective algorithm for grouping data, but it is often trapped in local optimum solutions. Therefore, the similarity measure in the clustering process using FCM is very important. This study uses a new method, which combines the Minkowski distance with the Chebyshev distance which is used as a measure of similarity in the clustering process on FCM. The amount of data that is quite large and complex becomes one of the difficulties in providing analysis of multivariate data. To overcome this, one of the techniques used is dimensional reduction using Principal Component Analysis (PCA). PCA is an algorithm of the dimensional reduction method based on the main components obtained from linear combinations, which can help stabilize cluster analysis measurements. The method used in this research is dimensional reduction using PCA, clustering using FCM with a combination of Minkowski and Chebyshev distances (FCMMC), and clustering evaluation using the Davies Bouldin Index (DBI). The purpose of this research is to minimize the objective function of FCM using new distances, namely, the combination of Minkowski and Chebyshev distances through the assistance of dimensional reduction by PCA. The results showed that the cluster accuracy of the combined application of the PCA and FCMMC algorithms was 1.6468. Besides, the minimum value of the combined objective function of the two methods is also obtained, namely, & Sugiyarto Surono [email protected] 1
0.0373 which is located in the 15th iteration, where this value is the smallest value of the 100 maximum iterations set. Keywords PCA Fuzzy C-Means Minkowski and Chebyshev distance Davies Bouldin Index
1 Introduction Optimization is everywhere, although it can mean different things from another perspective. From basic calculus, optimization can be easily used to find the maximum or minimum of a function [15]. Techniques in optimization with or without using gradient information, depending on the suitability of the problems that arise [15]. Unconstrained optimization is a problem consideration in performing objective functions on real variables without constraints on values. The simple unconstraint optimization problem allows the maxima or minima of a univariate function f ð xÞ to be 1\x\ þ 1 (or the entire real domain R); it can be written [15] as follows: max or min f ð xÞ; x 2 R; for an unconstrained optimization problem, optimally occurs at the critical point given the stationary condition f 0 ð xÞ ¼ 0. However, this stationary state is just a necessary condition, not a sufficient condition.
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