Symbolic Principal Components for Interval-valued Observations

One feature of contemporary datasets is that instead of the single point value in the p-dimensional space < seen in classical data, the data may take interval values thus producing hypercubes in <. This paper extends the methodology of classical principal components to that for interval-valued data. Two methods are proposed, viz., a vertices method which uses all the vertices of the observation’s hypercube, and a centers method which uses the centroid values. Unlike classical data, each symbolic data point has internal variation. For both the vertices and centers methods, we obtain intervalvalued symbolic principal components which recapture the internal variation of the observations, as well as diagnostics such as correlation measures between these principal components and the random variables and/or the observations themselves. We also provide a visualization method that further aids in the interpretation of the methodology. The methods are illustrated in a dataset using measurements of facial characteristics obtained from a study of face recognition patterns for surveillance purposes, and in a dataset of species of bats where the measurements are naturally internal-valued. A comparison with analyses in which classical surrogates replace the intervals, shows how the symbolic analyses give more informative conclusions.