Simple component analysis

Abstract

With a large number of variables measuring different aspects of a same theme, we would like to summarize the information in a limited number of components, i.e. linear combinations of the original variables. Among linear dimension reduction techniques, principal component analysis is optimal in at least two ways: principal components extract the maximum of the variability of the original variables, and they are uncorrelated. Unfortunately, they are often difficult to interpret. Moreover, in most applications, only the first principal component is a 'block component', the remaining components being 'difference components' which are also more difficult to interpret. The goal of simple component analysis is to replace (or to supplement) principal components with suboptimal but better interpretable 'simple components'. We propose a fast algorithm which seeks the optimal system of components under constraints of simplicity. Thus, in contrast with other techniques like 'varimax', this approach always provides a simple solution. The optimal simple system is suboptimal compared with principal components: less variability is extracted and components are correlated. However, if the loss of extracted variability is small, and correlations between components are low, it might be advantageous for practical use. Moreover, our concept of simplicity allows the system to have more than one block component, which also facilitates interpretation. Simplicity is not a guarantee for interpretability. With the help of our algorithm, the user can partly modify an optimal simple system of components to enhance interpretability. In this respect, the ultimate goal of simple component analysis is not to propose a method that leads automatically to a unique solution, but rather to develop tools for assisting the user in his or her choice of an interpretable solution. Finally, we argue that simple components may also make the task of choosing the dimension easier. The methodology is illustrated with a test battery to study the development of neuromotor functions in children and adolescents.