The Rayleigh Quotient and Contrastive Principal Component Analysis II

Area of Science:

• Computational Biology
• Data Science
• Statistical Analysis

Background:

• Contrastive principal component analysis (PCA) is a dimensionality reduction technique that maximizes target dataset variance while minimizing background dataset variance.
• Existing contrastive PCA methods can be framed as generalized eigenvalue problems maximizing specific Rayleigh quotients.

Purpose of the Study:

• To extend contrastive PCA for analyzing spatial and functional data.
• To unify spatial and functional data analysis within a single mathematical framework.

Main Methods:

• Kernel weighting from spatial PCA (k-ρ PCA) to contrast spatial and non-spatial variation.
• Solving the Rayleigh quotient in the space of basis function coefficients (f-ρ PCA) for functional data analysis.

Main Results:

• Demonstrated the utility of k-ρ PCA and f-ρ PCA in genomics.
• Analyzed gene expression data for cancer and immune response to vaccination.

Conclusions:

• The extensions of contrastive PCA broaden its applicability to spatial and functional data.
• These methods offer a unified approach for analyzing diverse datasets in fields like genomics.