Principal Components Analysis for Trapezoidal Fuzzy Numbers

Abstract

Scientists in many disciplines face the problem of interpretation of complex structures such as the symbolic data extracted from databases with a significant amount of records; or containing fuzzy numbers based on expert knowledge or partial knowledge originating from incomplete records. Principal Components Analysis (PCA) is most often used to interpret complex patterns as it allows reducing dimensionality and extracting the main characteristics of the data sample, as well as visualization in a two-dimensional plane and in a correlation circle. There is a need to extend this widely used method to the above mentioned data types. A new method called PCA-TF is proposed that allows performing PCA on data sets of trapezoidal (or triangular) fuzzy numbers, that may contain also real numbers and intervals. The approach is an extension to fuzzy numbers of the algorithm by Rodriguez (2000). A group of orthogonal axes is found that permits the projection of the maximum variance of a real numbers’ matrix, where each number represents a trapezoidal fuzzy number. The initial matrix of fuzzy numbers is projected to these axes by means of fuzzy number arithmetic, which yields Principal Components and they are also fuzzy numbers. Based on these components it is possible to produce graphs of the individuals in the two-dimensional plane. It is also possible to evaluate the shape of the ordered pairs of fuzzy numbers and visualize the membership function for each point on the z axis over the two-dimensional xy plane. The application is demonstrated on a data sample of students’ grades in Denœux and Masson (2004) and is compared to the results of the principal component analysis of fuzzy data using associative neural networks proposed by Denœux & Masson (D&M-PCA). Also, an important relation between the arithmetics of the intervals and projection formulas for the interval data type is demonstrated.