Principal Component Analysis of symmetric fuzzy data

Abstract

Principal Component Analysis (PCA) is a well-known tool often used for the exploratory analysis of a numerical data set. Here an extension of classical PCA is proposed, which deals with fuzzy data (in short PCAF), where the elementary datum cannot be recognized exactly by a specific number but by a center, two spread measures and a membership function. Specifically, two different PCAF methods, associated with different hypotheses of interrelation between parts of the solution, are proposed. In the first method, called Centers-related Spread PCAF (CS-PCAF), the size of the spread measures depends on the size of the centers. In the second method, called Loadings-related Spread PCAF (LS-PCAF), the spreads are not related directly to the sizes of the centers, but indirectly, via the component loadings. To analyze how well PCAF works a simulation study was carried out. On the whole, the PCAF method performed better than or equally well as PCA, except in a few particular conditions. Finally, the application of PCAF to an empirical fuzzy data set is described.

Introduction

There are many practical situations in which numerical observations of I units with respect to J variables cannot be represented by precisely specified values. Examples can be found in economics (daily rate exchanges), in marketing research (assessing public opinions about special events like government elections, launching of new consumer goods), in psychology (the mental state or subjective perceptions), in physics (in a ballistic sphere, the needed velocity of perforation of a plate by projectiles). Another significant situation is related to the attempt to analyze subjective perception or linguistic variables. In each of the above examples an exact numerical coding seems to be very hard to give: thus, the available information can only be represented approximately.

In the above situations the information is vague or fuzzy and can be summarized using interval valued data instead of numerical data. Thus, the score of the generic observation i on the generic variable j is represented by a couple of values: a lower bound and an upper bound which enclose the exact observation. In contrast to the interval valued data framework, the fuzzy data approach offers the possibility to take into account additional information given by the membership function. This function assigns a different “role” to each value of an interval in contrast to the interval valued data approach in which each value has a uniform importance. This distinction will be clearer after defining a fuzzy number, as follows.

A fuzzy number is defined as the triple F=(m,l,r)_LR where m denotes the center and l and r are the left and right spread, respectively, with the following membership function

$$\text{μ}{}_{\mathrm{F}}\text{(α)=}\text{L}\text{m−α}\text{l}\text{,}\text{α⩽m}\text{(l>0),}\text{R}\text{α−m}\text{r}\text{,}\text{α⩾m}\text{(r>0),}$$

where L and R are continuous strictly decreasing functions on [0,1] called shape functions. Moreover these functions must fulfil additional requirements: for example with reference to $\text{L,}\text{L(0)=1}$, L(z)<1 if z>0, L(z)>0 if z<1 and L(1)=0. Further details can be found in Dubois and Prade (1980).

Hereafter, for the sake of simplicity, we consider ‘symmetric’ fuzzy numbers only. For a symmetric fuzzy number l=r and L=R. It follows that a symmetric fuzzy number can be completely identified by the couple F=(m,s)_L=R where s=l=r is the spread. Thus, a fuzzy data set is a set of fuzzy numbers, which represent the scores of I observation units on J symmetric fuzzy variables. By a symmetric fuzzy variable we mean a variable for which each observed datum cannot be quantified exactly but only by using the couple F=(m,s)_L=R.

In this paper we generalize classical Principal Component Analysis (PCA) to deal with fuzzy data sets using a least squares approach. A few proposals for this are available in the literature: Yabuuchi et al. (1997) propose a method to perform PCA on fuzzy data, in which fuzzy eigenvalues and crisp (non-fuzzy) eigenvectors are obtained by solving a linear programming problem. Cazes et al. (1997) and extensions of their approach (see, for example, Palumbo and Lauro, 2002) propose to carry out factorial decompositions of interval valued data as well as a probabilistic generalization thereof. The basic idea consists of performing PCA on the bounds or on the centers of the interval valued data. In their probabilistic generalization, they make use of a coefficient the value of which depends on the probability law at hand that modifies the information taken into account in the factorial decomposition.

The need for generalizing PCA to fuzzy data is based on the assumption that, as well as for single valued data, it can be helpful to synthesize the data without losing relevant information. In the next section, we present a general least squares approach to fuzzy data analysis. In Section 3, we propose two different models that extend PCA to fuzzy data and in Section 4 an alternating least squares algorithm to estimate the solutions will be given. In Section 5, we discuss the representation of each observation unit in the low dimensional subspace obtained by PCAF. In 6 Simulation study, 7 Application to a real data set, the results of a simulation study to compare the PCAF to classical PCA and an application of PCAF to real fuzzy data will be given.