Computational Statistics

, Volume 28, Issue 5, pp 2117–2138

Principal component histograms from interval-valued observations

Original Paper

DOI: 10.1007/s00180-013-0399-4

Cite this article as:
Le-Rademacher, J. & Billard, L. Comput Stat (2013) 28: 2117. doi:10.1007/s00180-013-0399-4


The focus of this paper is to propose an approach to construct histogram values for the principal components of interval-valued observations. Le-Rademacher and Billard (J Comput Graph Stat 21:413–432, 2012) show that for a principal component analysis on interval-valued observations, the resulting observations in principal component space are polytopes formed by the convex hulls of linearly transformed vertices of the observed hyper-rectangles. In this paper, we propose an algorithm to translate these polytopes into histogram-valued data to provide numerical values for the principal components to be used as input in further analysis. Other existing methods of principal component analysis for interval-valued data construct the principal components, themselves, as intervals which implicitly assume that all values within an observation are uniformly distributed along the principal components axes. However, this assumption is only true in special cases where the variables in the dataset are mutually uncorrelated. Representation of the principal components as histogram values proposed herein more accurately reflects the variation in the internal structure of the observations in a principal component space. As a consequence, subsequent analyses using histogram-valued principal components as input result in improved accuracy.


Interval-valued input data Histogram-valued output data Principal component analysis Linear transformation  Polytopes 

Supplementary material

180_2013_399_MOESM1_ESM.docx (18 kb)
Supplementary material 1 (DOCX 18 KB)
180_2013_399_MOESM1_ESM.txt (17 kb)
Supplementary material 2 (TXT 17 KB)
180_2013_399_MOESM2_ESM.txt (4 kb)
Supplementary material 3 (TXT 4 KB)

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Division of BiostatisticsMedical College of WisconsinMilwaukeeUSA
  2. 2.Department of StatisticsUniversity of GeorgiaAthensUSA