TY - JOUR

T1 - Principal component histograms from interval-valued observations

AU - Le-Rademacher, J.

AU - Billard, L.

N1 - Funding Information:
The authors wish to thank the Editor, the Associate Editor, and the referees for their thorough review and thoughful comments. Partial support to both authors from NSF grants is gratefully acknowledged.

PY - 2013/10

Y1 - 2013/10

N2 - 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.

AB - 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.

KW - Histogram-valued output data

KW - Interval-valued input data

KW - Linear transformation

KW - Polytopes

KW - Principal component analysis

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U2 - 10.1007/s00180-013-0399-4

DO - 10.1007/s00180-013-0399-4

M3 - Article

AN - SCOPUS:84884702338

SN - 0943-4062

VL - 28

SP - 2117

EP - 2138

JO - Computational Statistics

JF - Computational Statistics

IS - 5

ER -