Fuzzyfication of principle component analysis for data dimensionalty reduction

Abstract

Principal component analysis (PCA) extracts small uncorrelated data from original high dimensional data space and is widely used for data analysis. The methodology of classical PCA is based on orthogonal projection defined in convex vector space. Thus, a norm between two projected vectors is unavoidably smaller than the norm between any two objects before implementation of PCA. Due to this, in some cases when the PCA cannot capture the data structure, its implementation does not necessarily confirm the real similarity of data in the higher dimensional space, making the results unacceptable. In order to avoid this problem for the purposes of high dimensional data clustering, we propose new Fuzzy PCA (FPCA) algorithm. The novelty is in the extracted similarity structures of objects in high-dimensional space and dissimilarities between objects based on their cluster structure and dimension when the algorithm is implemented in conjunction with known fuzzy clustering algorithms as FCM, GK or GG algorithms. This is done by using the fuzzy membership functions and modification of the classical PCA approach by considering the similarity structures during the construction of projections in smaller dimensional space. The effectiveness of the proposed algorithm is tested on several benchmark data sets. We also evaluate the clustering efficiency by implementing validation measures.