New Approaches to Principal Component Analysis for Trees
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Object Oriented Data Analysis is a new area in statistics that studies populations of general data objects. In this article we consider populations of tree-structured objects as our focus of interest. We develop improved analysis tools for data lying in a binary tree space analogous to classical Principal Component Analysis methods in Euclidean space. Our extensions of PCA are analogs of one dimensional subspaces that best fit the data. Previous work was based on the notion of tree-lines.
In this paper, a generalization of the previous tree-line notion is proposed: k-tree-lines. Previously proposed tree-lines are k-tree-lines where k=1. New sub-cases of k-tree-lines studied in this work are the 2-tree-lines and tree-curves, which explain much more variation per principal component than tree-lines. The optimal principal component tree-lines were computable in linear time. Because 2-tree-lines and tree-curves are more complex, they are computationally more expensive, but yield improved data analysis results.
We provide a comparative study of all these methods on a motivating data set consisting of brain vessel structures of 98 subjects.
- New Approaches to Principal Component Analysis for Trees
Statistics in Biosciences
Volume 4, Issue 1 , pp 132-156
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- Print ISSN
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- Binary trees
- Object oriented data analysis
- Principal component analysis
- Vessel structure
- Author Affiliations
- 1. HP Laboratories, 1501 Page Mill Rd MS 1140, Palo Alto, CA, 94304, USA
- 2. UNC at Chapel Hill, 307 Hanes Hall CB 3260, Chapel Hill, NC, 27599, USA
- 3. Colorado State University, 216 Statistics Building, Fort Collins, CO, 80523, USA
- 4. UNC at Chapel Hill, 2160 Bioinformatics #7060, Chapel Hill, NC, 27599, USA
- 5. UNC at Chapel Hill, CB 7062 Department of Neurosurgery UNC-CH, Chapel Hill, NC, 27599, USA
- 6. UNC at Chapel Hill, 352 Hanes Hall CB 3260, Chapel Hill, NC, 27599, USA