A note on the relaxation time of two Markov chains on rooted phylogenetic tree spaces

2014 ◽  
Vol 84 ◽  
pp. 247-252 ◽  
Author(s):  
David A. Spade ◽  
Radu Herbei ◽  
Laura S. Kubatko
Keyword(s):  
Relaxation Time ◽  
Markov Chains ◽  
Phylogenetic Tree ◽  
Rooted Phylogenetic Tree

scholarly journals On Sackin's original proposal: The variance of the leaves' depths as a phylogenetic balance index

2019 ◽  
Author(s):  
Tomás Martínez Coronado ◽  
Arnau Mir ◽  
Francesc Rossello ◽  
Lucía Rotger
Keyword(s):  
Phylogenetic Tree ◽  
Phylogenetic Trees ◽  
Rooted Tree ◽  
Rooted Trees ◽  
Closed Formula ◽  
Original Proposal ◽  
Yule Model ◽  
Uniform Model ◽  
Rooted Phylogenetic Tree ◽  
Balance Index

Abstract Background: The Sackin index S of a rooted phylogenetic tree, defined as the sum of its leaves' depths, is one of the most popular balance indices in phylogenetics, and Sackin's 1972 paper is usually cited as the source for this index. However, what Sackin actually proposed in his paper as a measure of the imbalance of a rooted tree was not the sum of its leaves' depths, but their "variation". This proposal was later implemented as the variance of the leaves' depths by Kirkpatrick and Slatkin, where moreover they posed the problem of finding a closed formula for its expected value under the Yule model. Nowadays, Sackin's original proposal seems to have passed into oblivion in the phylogenetics literature, replaced by the index bearing his name, which, in fact, was introduced a decade later by Sokal.Results: In this paper we study the properties of the variance of the leaves' depths, V, as a balance index. Firstly, we prove that the rooted trees with n leaves and maximum V value are exactly the combs with n leaves. But although V achieves its minimum value on every space BT_n of bifurcating rooted phylogenetic trees with n< 184 leaves at the so-called "maximally balanced trees" with n leaves, this property fails for almost every n>= 184. We provide then an algorithm that finds in O(n) time the trees in BT_n with minimum V value. Secondly, we obtain closed formulas for the expected V value of a bifurcating rooted tree with any number n of leaves under the Yule and the uniform models and, as a by-product of the computations leading to these formulas, we also obtain closed formulas for the variance of the Sackin index and the total cophenetic indexof a bifurcating rooted tree, as well as of their covariance, under the uniform model, thus filling this gap in the literature.Conclusions: The phylogenetics crowd has been wise in preferring as a balance index the sum S(T) of the leaves’ depths of a phylogenetic tree T over their variance V (T), because the latter does not seem to capture correctly the notion of balance of large bifurcating rooted trees. But for bifurcating trees up to 183 leaves, V is a valid and useful balance index.

scholarly journals On Sackin's original proposal: The variance of the leaves' depths as a phylogenetic balance index

2020 ◽  
Author(s):  
Tomás Martínez Coronado ◽  
Arnau Mir ◽  
Francesc Rossello ◽  
Lucía Rotger
Keyword(s):  
Phylogenetic Tree ◽  
Phylogenetic Trees ◽  
Rooted Tree ◽  
Rooted Trees ◽  
Closed Formula ◽  
Original Proposal ◽  
Yule Model ◽  
Uniform Model ◽  
Rooted Phylogenetic Tree ◽  
Balance Index

Abstract Background. The Sackin index S of a rooted phylogenetic tree, defined as the sum of its leaves' depths, is one of the most popular balance indices in phylogenetics, and Sackin's 1972 paper is usually cited as the source for this index. However, what Sackin actually proposed in his paper as a measure of the imbalance of a rooted tree was not the sum of its leaves' depths, but their ``variation''. This proposal was later implemented as the variance of the leaves' depths by Kirkpatrick and Slatkin in 1993, where they also posed the problem of finding a closed formula for its expected value under the Yule model. Nowadays, Sackin's original proposal seems to have passed into oblivion in the phylogenetics literature, replaced by the index bearing his name, which, in fact, was introduced a decade later by Sokal. Results. In this paper we study the properties of the variance of the leaves' depths, V, as a balance index. Firstly, we prove that the rooted trees with $n$ leaves and maximum V value are exactly the combs with n leaves. But although V achieves its minimum value on every space of bifurcating rooted phylogenetic trees with at most 183 leaves at the so-called ``maximally balanced trees'' with n leaves, this property fails for almost every n larger than 184 We provide then an algorithm that finds the bifurcating rooted trees with n leaves and minimum V value in quasilinear time. Secondly, we obtain closed formulas for the expected V value of a bifurcating rooted tree with any number n of leaves under the Yule and the uniform models and, as a by-product of the computations leading to these formulas, we also obtain closed formulas for the variance under the uniform model of the Sackin index and the total cophenetic index of a bifurcating rooted tree, as well as of their covariance, thus filling this gap in the literature.

scholarly journals On Sackin's original proposal: The variance of the leaves' depths as a phylogenetic balance index

2020 ◽  
Author(s):  
Tomás Martínez Coronado ◽  
Arnau Mir ◽  
Francesc Rossello ◽  
Lucía Rotger
Keyword(s):  
Phylogenetic Tree ◽  
Phylogenetic Trees ◽  
Rooted Tree ◽  
Rooted Trees ◽  
Closed Formula ◽  
Original Proposal ◽  
Yule Model ◽  
Uniform Model ◽  
Rooted Phylogenetic Tree ◽  
Balance Index

Abstract Background. The Sackin index S of a rooted phylogenetic tree, defined as the sum of its leaves' depths, is one of the most popular balance indices in phylogenetics, and Sackin's 1972 paper is usually cited as the source for this index. However, what Sackin actually proposed in his paper as a measure of the imbalance of a rooted tree was not the sum of its leaves' depths, but their ``variation''. This proposal was later implemented as the variance of the leaves' depths by Kirkpatrick and Slatkin in 1993, where they also posed the problem of finding a closed formula for its expected value under the Yule model. Nowadays, Sackin's original proposal seems to have passed into oblivion in the phylogenetics literature, replaced by the index bearing his name, which, in fact, was introduced a decade later by Sokal. Results. In this paper we study the properties of the variance of the leaves' depths, V, as a balance index. Firstly, we prove that the rooted trees with $n$ leaves and maximum V value are exactly the combs with n leaves. But although V achieves its minimum value on every space of bifurcating rooted phylogenetic trees with at most 183 leaves at the so-called ``maximally balanced trees'' with n leaves, this property fails for almost every n larger than 184 We provide then an algorithm that finds the bifurcating rooted trees with n leaves and minimum V value in quasilinear time. Secondly, we obtain closed formulas for the expected V value of a bifurcating rooted tree with any number n of leaves under the Yule and the uniform models and, as a by-product of the computations leading to these formulas, we also obtain closed formulas for the variance under the uniform model of the Sackin index and the total cophenetic index of a bifurcating rooted tree, as well as of their covariance, thus filling this gap in the literature.

scholarly journals Monotone measures of ergodicity for Markov chains

1998 ◽  
Vol 11 (3) ◽  
pp. 283-288
Author(s):  
J. Keilson ◽  
O. A. Vasicek
Keyword(s):  
Markov Chain ◽  
Relaxation Time ◽  
Markov Chains ◽  
Small Subset ◽  
Finite Markov Chain ◽  
Time Reversibility

The following paper, first written in 1974, was never published other than as part of an internal research series. Its lack of publication is unrelated to the merits of the paper and the paper is of current importance by virtue of its relation to the relaxation time. A systematic discussion is provided of the approach of a finite Markov chain to ergodicity by proving the monotonicity of an important set of norms, each measures of egodicity, whether or not time reversibility is present. The paper is of particular interest because the discussion of the relaxation time of a finite Markov chain [2] has only been clean for time reversible chains, a small subset of the chains of interest. This restriction is not present here. Indeed, a new relaxation time quoted quantifies the relaxation time for all finite ergodic chains (cf. the discussion of Q1(t) below Equation (1.7)]. This relaxation time was developed by Keilson with A. Roy in his thesis [6], yet to be published.

Markov models—Markov chains

2019 ◽  
Vol 16 (8) ◽  
pp. 663-664 ◽  
Author(s):  
Jasleen K. Grewal ◽  
Martin Krzywinski ◽  
Naomi Altman
Keyword(s):  
Markov Chains ◽  
Markov Models

NUCLEAR SPIN-LATTICE RELAXATION TIME IN TIN

1978 ◽  
Vol 39 (C6) ◽  
pp. C6-1215-C6-1216
Author(s):  
H. Ahola ◽  
G.J. Ehnholm ◽  
S.T. Islander ◽  
B. Rantala
Keyword(s):  
Relaxation Time ◽  
Nuclear Spin ◽  
Lattice Relaxation ◽  
Lattice Relaxation Time ◽  
Spin Lattice Relaxation ◽  
Spin Lattice Relaxation Time ◽  
Spin Lattice ◽  
Nuclear Spin Lattice Relaxation
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