Properties for the Fréchet mean in Billera-Holmes-Vogtmann treespace

Anaya, Maria; Anipchenko-Ulaj, Olga; Ashfaq, Aisha; Chiu, Joyce; Kaiser, Mahedi; Ohsawa, Max Shoji; Owen, Megan; Pavlechko, Ella; John, Katherine St.; Suleria, Shivam; Thompson, Keith; Yap, Corrine

doi:10.1016/j.aam.2020.102072

Cited by 3 publications

(6 citation statements)

References 21 publications

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“…We illustrate the two conditions ( 8) and ( 9) by example from (Anaya et al, 2020, Example 1 and 2). While our condition is stronger, it is still not necessary.…”

Section: Corollary 33 Let T ⊂ T N Be a Finite Collection Of Trees Wit...mentioning

confidence: 99%

“…Example 3.4. (Anaya et al, 2020, Example 1) consider the sample mean µ of four trees T = {T 1 , T 2 , T 3 , T 4 } ⊂ T 4 as shown in Figure 6 where for T 1 the split s 1 has length w > 0. As there is only one tree, namely T 4 ∈ T having splits not in C(s 1 ), ( 9) is equivalent with…”

Section: Corollary 33 Let T ⊂ T N Be a Finite Collection Of Trees Wit...mentioning

confidence: 99%

“…Specific methods developed for statistics in T N include computational methods for Fréchet means in BHV spaces (Anaya et al, 2020), and principal component analysis (Nye, 2011;Feragen et al, 2013;Nye et al, 2017). The Fréchet mean is a generalization of the mean of a probability distribution to metric spaces, defined in the following way.…”

Section: Introductionmentioning

confidence: 99%

“…In practice, a pragmatic approach has been to use an iterative algorithm for computing the Fréchet mean such as Sturm's algorithm, but then to ignore very short edges that come and go as the algorithm proceeds and as the topology repeatedly changes. In a similar way to a criterion due to Anaya et al (2020), in Theorem 3.1 we provide a sufficient condition for the presence of an edge in the Fréchet mean tree, and a related algorithm (Algorithm 2) for finding these edges. Due to decomposition in Theorem 3.7, this criterion involves directional derivatives of the Fréchet function (1) at the star tree.…”

Section: Introductionmentioning

confidence: 99%

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Types of Stickiness in BHV Phylogenetic Tree Spaces and Their Degree

Lars

Van

Nye

et al. 2023

Lecture Notes in Computer Science

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Samples of phylogenetic trees arise in a variety of evolutionary and biomedical applications, and the Fréchet mean in Billera-Holmes-Vogtmann tree space is a summary tree shown to have advantages over other mean or consensus trees. However, use of the Fréchet mean raises computational and statistical issues which we explore in this paper. The Fréchet sample mean is known often to contain fewer internal edges than the trees in the sample, and in this circumstance calculating the mean by iterative schemes can be problematic due to slow convergence. We present new methods for identifying edges which must lie in the Fréchet sample

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“…We illustrate the two conditions ( 8) and ( 9) by example from (Anaya et al, 2020, Example 1 and 2). While our condition is stronger, it is still not necessary.…”

Section: Corollary 33 Let T ⊂ T N Be a Finite Collection Of Trees Wit...mentioning

confidence: 99%

Section: Corollary 33 Let T ⊂ T N Be a Finite Collection Of Trees Wit...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Types of Stickiness in BHV Phylogenetic Tree Spaces and Their Degree

Lars

Van

Nye

et al. 2023

Lecture Notes in Computer Science

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“…e Billera-Holmes-Vogtmann tree space (BHV space), cf. Billera et al (2001), is the rst metric space that serves as a sample space for phylogenetic trees where statistics can be performed, and is under ongoing investigation to this day (Owen & Provan (2011); Nye (2011); Barden et al (2013); Nye (2014); Nye et al (2016); Barden et al (2016); Anaya et al (2020), to name a few).…”

Section: Chapter 1 Introduction and Motivationmentioning

confidence: 99%

Phylogenetic Forest Spaces

Lueg¹

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Tag der mündlichen Prüfung: 04. Oktober 2022 v vi A promising construction that covers the example from above and behaves as biological intuition suggests is our recently introduced Wald Space, cf. Garba et al. (2021a). It is based on the characterization of phylogenetic trees as covariance matrices, which is a byproduct of a generalization of the popular biological substitution models that are used to calculate likelihoods for trees given genetic sequence data, so those substitution models are the backbone of phylogenetic tree estimation. In other words, the Wald Space is a space that is consistent with the tree estimation methods that are currently used, up to the generalizations that have been made. More details can be found in Garba et al. (2021a).In this work, we concentrate on the Wald Space purely from the perspective that it is a mathematical structure and thus we try to enable for a be er understanding of the Wald Space. To this end, we introduce the mathematical structures required: metric spaces, Riemannian manifolds, Riemann strati ed spaces, as well as, for our construction essential, various geometries on the manifold of strictly positive de nite symmetric real matrices. Furthermore, we introduce various possible ways to represent the phylogenetic trees and forests that we consider. Having nished the introduction of the more general and known concepts, we brie y introduce the BHV Space. en we de ne and describe the Wald Space, which is a topological strati ed space, and we investigate its topological features. is part is the core of the thesis. Finally, we equip the Wald Space with a geometry that can be chosen to some degree and nd that these spaces are then Riemann strati ed spaces of type (A). Last but not least, we propose some numerical algorithms to calculate geodesics and distances in the Wald Space equipped with a geometry.ose are not tested in this work, but to some extent in Lueg et al. (2021).

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Poincaré Fréchet mean

Cao

2023

Pattern Recognition

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Properties for the Fréchet mean in Billera-Holmes-Vogtmann treespace

Cited by 3 publications

References 21 publications

Types of Stickiness in BHV Phylogenetic Tree Spaces and Their Degree

Types of Stickiness in BHV Phylogenetic Tree Spaces and Their Degree

Phylogenetic Forest Spaces

Poincaré Fréchet mean

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