Tropical Principal Component Analysis and Its Application to Phylogenetics

Area of Science:

• Tropical geometry
• Computational geometry
• Data analysis

Background:

• Principal component analysis (PCA) is a standard technique for dimensionality reduction in Euclidean spaces.
• High-dimensional data presents challenges for traditional analysis methods.
• Tropical geometry offers a new framework for geometric analysis.

Purpose of the Study:

• To develop and analyze tropical geometry analogues of principal component analysis.
• To explore dimensionality reduction techniques within tropical geometry.
• To apply these novel methods to the field of phylogenetics.

Main Methods:

• Defining and analyzing two tropical analogues of PCA.
• One method involves finding the closest Stiefel tropical linear space.
• The other method involves finding the closest tropical polytope.
• Developing approximative algorithms for both approaches.

Main Results:

• The developed tropical PCA analogues provide new tools for dimensionality reduction.
• Approximative algorithms were successfully implemented.
• The methods were tested on simulated phylogenetic data.
• The methods were applied to an empirical dataset of Apicomplexa genomes.

Conclusions:

• Tropical geometry offers a viable alternative framework for dimensionality reduction.
• The novel tropical PCA methods show potential for applications in phylogenetics.
• Further research can explore the broader applicability of these techniques in data analysis.