# Stochastic Safety Radius on UPGMA
## 1 Introduction  
Phylogenetics is one of the oldest fields in biology to study the evolutionary history of organisms using a phylogenetic tree, which is a tree representation of evolutionary history among species (or taxa). In order to reconstruct a phylogenetic tree from genetic data, researchers develop many statistical methods including maximum likelihood estimators, Bayesian inference, and distance-based methods [1]. A distance-based method is one of the most popular methods to reconstruct a phylogenetic tree for its computational speed and relatively simple two-step procedure: (1) computing all pairwise distances between all possible pair of sequences from the input alignment; and (2) reconstructing a phylogenetic tree from all pairwise distances of sequences computed in Step (1) using combinatorics. Maximum likelihood estimators under evolutionary models produce pairwise distances between all possible pairs of sequences and they form as a _distance matrix_. A distance matrix is an input for a distance-based method to reconstruct a phylogenetic tree and we can consider them as a multivariate random variable and these distance-based and probabilistic methods do not always return the true phylogenetic tree topology. Therefore, we have to measure the robustness of a method and one metric of the robustness of a distance-based method for phylogenetic tree reconstruction is called the _safety radius_ of the method. A safety radius is a radius of all distance matrices such that a given distance-based method returns the \"true tree topology\" of a phylogenetic tree. This means that all distance matrices within the safety radius satisfy a precise combinatorial condition so that the distance-based method is guaranteed to return the true tree topology [2]. In 2015, [PERSON] introduced a notion of _Stochastic safety radius_ in [2] for analyzing the probability for a distance-based method to return the true tree topology from a given distance matrix. In 2017, [PERSON] et al. worked on developing a stochastic safety radius using the neighbor-joining (NJ) method and balance minimal evolution method for trees with number of leaves equal to 4 or 5 [3]. Phylogenomics is a new field, which applies tools from phylogenetics to genome data. In phylogenomics, we often conduct the species tree and gene trees analyses using themulti-species coalescent model [4]. Under the multi-species coalescent model, we assume that all gene trees, phylogenetic trees reconstructed from genes, are _equidistant trees_[4]. An equidistant tree is a rooted tree whose total branch length, from its root to each leaf, is the same for all leaves (an example of an equidistant tree with three leaves is shown in Figure 1). In the past five years, there has been much work on the _space of all equidistant trees_ for phylogenomics [5; 6; 7; 8; 9; 10; 11]. However, these recent studies assume that all given equidistant trees are true trees or close to true trees, which is often not true. [PERSON] and Sullivant worked on variability of a distance-based method to reconstruct an equidistant tree from all pairwise distances, called the _Unweighted Pair Group Method with Arithmetic Mean (UPGMA)_ using polyhedral geometry [12]. They study how UPGMA project a given distance matrix to the space of all equidistant trees so that their result is from the view of polyhedral geometry and deterministic. In this paper, therefore, we focus on the stochastic safety radius of the Unweighted Pair Group Method with Arithmetic Mean (UPGMA), one of the most popular distance-based methods and a lower bound of its probability that the UPGMA method returns the true tree topology from a random input distance matrix with noise distributed from the Gaussian distribution. Note that UPGMA is a hierarchical clustering method that builds a dendrogram from a distance matrix which records pairwise \"distances\" defined by a user input [PERSON] between all pairs of observations in a higher [PERSON] vector space [13].