Abstract
This paper is motivated by the difference between the classical principal component analysis (PCA) in a Euclidean space and the tropical PCA in a tropical projective torus as follows. In Euclidean space, the projection of the mean point of a given data set on the principle component is the mean point of the projection of the data set. However, in tropical projective torus, it is not guaranteed that the projection of a Fermat-Weber point of a given data set on a tropical polytope is a Fermat-Weber point of the projection of the data set. This is caused by the difference between the Euclidean metric and the tropical metric. In this paper, we focus on the projection on the tropical triangle (the three-point tropical convex hull), and we develop one algorithm and its improved version, such that for a given data set in the tropical projective torus, these algorithms output a tropical triangle, on which the projection of a Fermat-Weber point of the data set is a Fermat-Weber point of the projection of the data set. We implement these algorithms in R language and test how they work with random data sets. We also use R language for numerical computation. The experimental results show that these algorithms are stable and efficient, with a high success rate.
Highlights
Principal component analysis (PCA) is a standard method for dimensionality reduction that analyzes a set of high dimensional data
Zhang proposed the tropical principal component analysis [8], which is of great use in the analysis of phylogenetic trees in Phylogenetics
PCA defined by tropical polytopes, the projection of a tropical mean point of a data set X is not necessarily a Fermat-Weber point of the projection of X
Summary
Principal component analysis (PCA) is a standard method for dimensionality reduction that analyzes a set of high dimensional data. The PCA in the BHV tree space proposed by Nye [16] makes the MSE between a data set X and the projection of X reaches the minimum, and the projected variance reaches the maximum simultaneously. PCA defined by tropical polytopes, the projection of a tropical mean point (in this paper we call it a Fermat-Weber point) of a data set X is not necessarily a Fermat-Weber point of the projection of X (see Example 4). We develop one algorithm (Algorithm 1) and its improved version (Algorithm 2), such that for a given data set X ⊂ Rn/R1, these algorithms output a tropical triangle C , on which the projection of a Fermat-Weber point of X is a Fermat-Weber point of the projection of X. If u(1) , u(2) and u(3) are null return FAIL, otherwise, return u(1) , u(2) , u(3)
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