Tropical Balls and Its Applications to K Nearest Neighbor over the Space of Phylogenetic Trees

Ruriko Yoshida

doi:10.3390/math9070779

Abstract

A tropical ball is a ball defined by the tropical metric over the tropical projective torus. In this paper we show several properties of tropical balls over the tropical projective torus and also over the space of phylogenetic trees with a given set of leaf labels. Then we discuss its application to the K nearest neighbors (KNN) algorithm, a supervised learning method used to classify a high-dimensional vector into given categories by looking at a ball centered at the vector, which contains K vectors in the space.

Highlights

A phylogenetic tree with a given set of leaf labels [n] = {1, . . . , n} is a weighted tree whose leaves have labels [n] while their interior nodes do not have labels
1. we show some properties of a tropical ball in the tropical projective torus; 2. we show some properties of a tropical ball in a space of equidistant trees with a given set of leaves [n]; 3. we compare tropical balls with balls defined with L2 norm and L∞ norm; 4. we define a tropical K nearest neighbors (KNN) algorithm; and 5. we applied tropical KNN algorithm to simulated data generated by a multispecies coalescent model
We investigate the tropical ball in terms of the tropical metric in the space of ultrametrics Un

Summary

Introduction

Since we assume that all species in the tree have the same most common ancestor (the root of a phylogenetic tree), a phylogenetic tree of a given set of species has the property that a distance from its root to each leaf is same for all leaves in the tree. We call such a rooted phylogenetic tree an equidistant tree. We often use equidistant trees to analyze genome data since multispecies coalescent processes applied to analyze gene trees and species tree in genome data [1] assume that all gene trees are equidistant trees

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Discussion

Conclusion