Principal component analysis for interval data

Principal component analysis for classical data is a method used frequently to reduce the effective dimension underlying a data set from p random variables to s << p linear functions of those p random variables and their observed values. With contemporary large data sets, it is often the case that the data are aggregated in some meaningful scientific way such that the resulting data are symbolic data (such as lists, intervals, histograms, and the like); though symbolic data can and do occur naturally and in smaller data sets. Since symbolic data have internal variations along with the familiar (between observations) variation of classical data, direct application of classical methods to symbolic data will ignore much of the information contained in the data. Our focus is to describe and illustrate principal component methodology for interval data. The significance of symbolic data in general and of this article in particular is illustrated by its applicability for our analysis of three key 21st century challengers: networks, security data, and translational medicine. It is relatively easy to visualize the applicability to security data and translational medicine, though less easy to visualize its applicability to networks. Since an interval is typically denoted by (a, b), in a network interval, we let a be a pair of nodes and b be their edge with characteristics c and d, respectively. If this representation of a network interval is valid, then we can more easily visualize its applicability to networks also. (C) 2012 Wiley Periodicals, Inc.