Rootlets Hierarchical Principal Component Analysis for Revealing Nested Dependencies in Hierarchical Data
Hierarchical clustering analysis (HCA) is a widely used unsupervised learning method. Limitations of HCA, however, include imposing an artificial hierarchy onto non-hierarchical data and fixed two-way mergers at every level. To address this, the current work describes a novel rootlets hierarchical p...
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2024-12-01
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| author | Korey P. Wylie Jason R. Tregellas |
| author_facet | Korey P. Wylie Jason R. Tregellas |
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| description | Hierarchical clustering analysis (HCA) is a widely used unsupervised learning method. Limitations of HCA, however, include imposing an artificial hierarchy onto non-hierarchical data and fixed two-way mergers at every level. To address this, the current work describes a novel rootlets hierarchical principal component analysis (hPCA). This method extends typical hPCA using multivariate statistics to construct adaptive multiway mergers and Riemannian geometry to visualize nested dependencies. The rootlets hPCA algorithm and its projection onto the Poincaré disk are presented as examples of this extended framework. The algorithm constructs high-dimensional mergers using a single parameter, interpreted as a <i>p</i>-value. It decomposes a similarity matrix from GL(<i>m</i>, ℝ) using a sequence of rotations from SO(<i>k</i>), <i>k</i> << <i>m</i>. Analysis shows that the rootlets algorithm limits the number of distinct eigenvalues for any merger. Nested clusters of arbitrary size but equal correlations are constructed and merged using their leading principal components. The visualization method then maps elements of SO(<i>k</i>) onto a low-dimensional hyperbolic manifold, the Poincaré disk. Rootlets hPCA was validated using simulated datasets with known hierarchical structure, and a neuroimaging dataset with an unknown hierarchy. Experiments demonstrate that rootlets hPCA accurately reconstructs known hierarchies and, unlike HCA, does not impose a hierarchy on data. |
| format | Article |
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| institution | Kabale University |
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| language | English |
| publishDate | 2024-12-01 |
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| spelling | doaj-art-6e6559bbc831491e9e5bfc30c6cf519d2025-01-10T13:18:10ZengMDPI AGMathematics2227-73902024-12-011317210.3390/math13010072Rootlets Hierarchical Principal Component Analysis for Revealing Nested Dependencies in Hierarchical DataKorey P. Wylie0Jason R. Tregellas1Department of Psychiatry, University of Colorado School of Medicine, Anschutz Medical Campus, Anschutz Health Sciences Building, 1890 N Revere Ct, Aurora, CO 80045, USADepartment of Psychiatry, University of Colorado School of Medicine, Anschutz Medical Campus, Anschutz Health Sciences Building, 1890 N Revere Ct, Aurora, CO 80045, USAHierarchical clustering analysis (HCA) is a widely used unsupervised learning method. Limitations of HCA, however, include imposing an artificial hierarchy onto non-hierarchical data and fixed two-way mergers at every level. To address this, the current work describes a novel rootlets hierarchical principal component analysis (hPCA). This method extends typical hPCA using multivariate statistics to construct adaptive multiway mergers and Riemannian geometry to visualize nested dependencies. The rootlets hPCA algorithm and its projection onto the Poincaré disk are presented as examples of this extended framework. The algorithm constructs high-dimensional mergers using a single parameter, interpreted as a <i>p</i>-value. It decomposes a similarity matrix from GL(<i>m</i>, ℝ) using a sequence of rotations from SO(<i>k</i>), <i>k</i> << <i>m</i>. Analysis shows that the rootlets algorithm limits the number of distinct eigenvalues for any merger. Nested clusters of arbitrary size but equal correlations are constructed and merged using their leading principal components. The visualization method then maps elements of SO(<i>k</i>) onto a low-dimensional hyperbolic manifold, the Poincaré disk. Rootlets hPCA was validated using simulated datasets with known hierarchical structure, and a neuroimaging dataset with an unknown hierarchy. Experiments demonstrate that rootlets hPCA accurately reconstructs known hierarchies and, unlike HCA, does not impose a hierarchy on data.https://www.mdpi.com/2227-7390/13/1/72eigendecompositionmultivariate statisticshyperbolic manifoldRiemannian geometrymanifold learning |
| spellingShingle | Korey P. Wylie Jason R. Tregellas Rootlets Hierarchical Principal Component Analysis for Revealing Nested Dependencies in Hierarchical Data Mathematics eigendecomposition multivariate statistics hyperbolic manifold Riemannian geometry manifold learning |
| title | Rootlets Hierarchical Principal Component Analysis for Revealing Nested Dependencies in Hierarchical Data |
| title_full | Rootlets Hierarchical Principal Component Analysis for Revealing Nested Dependencies in Hierarchical Data |
| title_fullStr | Rootlets Hierarchical Principal Component Analysis for Revealing Nested Dependencies in Hierarchical Data |
| title_full_unstemmed | Rootlets Hierarchical Principal Component Analysis for Revealing Nested Dependencies in Hierarchical Data |
| title_short | Rootlets Hierarchical Principal Component Analysis for Revealing Nested Dependencies in Hierarchical Data |
| title_sort | rootlets hierarchical principal component analysis for revealing nested dependencies in hierarchical data |
| topic | eigendecomposition multivariate statistics hyperbolic manifold Riemannian geometry manifold learning |
| url | https://www.mdpi.com/2227-7390/13/1/72 |
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