“Spaghetti” PCA analysis: An extension of principal components analysis to time dependent interval data

Abstract

Interval data are generally defined by the upper and the lower value assumed by a unit for a continuous variable. In our approach we introduce a special type of interval description depending on time. Each observation is characterized by an oriented interval of values with a starting and an ending value for each period of observation: for example, the opening and the closing price of a stock in a financial market in a day or a week, the initial and the final expression of a gene at the beginning and at the ending of an experiment. Several factorial techniques have been developed in order to treat interval data, but not yet for oriented intervals. In this paper we present an extension of principal component analysis to time dependent interval data, or, in general, to oriented intervals. From a geometrical point of view, the proposed approach can be considered as an analysis of oriented segments (nicely called “ spaghetti”) defined in a multidimensional space identified by periods. We introduce the formulas for the standardization of data, the calculation of matrices and the interpretation of the results.