Symbolic Covariance Principal Component Analysis and Visualization for Interval-Valued Data

Abstract

This article proposes a new approach to principal component analysis (PCA) for interval-valued data. Unlike classical observations, which are represented by single points in p-dimensional space ℜ^p, interval-valued observations are represented by hyper-rectangles in ℜ^p, and as such, have an internal structure that does not exist in classical observations. As a consequence, statistical methods for classical data must be modified to account for the structure of the hyper-rectangles before they can be applied to interval-valued data. This article extends the classical PCA method to interval-valued data by using the so-called symbolic covariance to determine the principal component (PC) space to reflect the total variation of interval-valued data. The article also provides a new approach to constructing the observations in a PC space for better visualization. This new representation of the observations reflects their true structure in the PC space. Supplementary materials for this article are available online.

Key Words:

• Convex hull
• Linear transformation
• Polytopes
• Symbolic data analysis