The asymptotic distribution of the $k$-Robinson-Foulds dissimilarity measure on labelled trees
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929651146817536 |
|---|---|
| author | Fuchs, Michael Steel, Mike |
| author_facet | Fuchs, Michael Steel, Mike |
| contents | Motivated by applications in medical bioinformatics, Khayatian et al. (2024) introduced a family of metrics on Cayley trees (the $k$-RF distance, for $k=0, \ldots, n-2$) and explored their distribution on pairs of random Cayley trees via simulations. In this paper, we investigate this distribution mathematically, and derive exact asymptotic descriptions of the distribution of the $k$-RF metric for the extreme values $k=0$ and $k=n-2$, as $n$ becomes large. We show that a linear transform of the $0$-RF metric converges to a Poisson distribution (with mean 2) whereas a similar transform for the $(n-2)$-RF metric leads to a normal distribution (with mean $\sim ne^{-2}$). These results (together with the case $k=1$ which behaves quite differently, and $k=n-3$) shed light on the earlier simulation results, and the predictions made concerning them. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_20012 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The asymptotic distribution of the $k$-Robinson-Foulds dissimilarity measure on labelled trees Fuchs, Michael Steel, Mike Probability Combinatorics Populations and Evolution 05A16 05C05 Motivated by applications in medical bioinformatics, Khayatian et al. (2024) introduced a family of metrics on Cayley trees (the $k$-RF distance, for $k=0, \ldots, n-2$) and explored their distribution on pairs of random Cayley trees via simulations. In this paper, we investigate this distribution mathematically, and derive exact asymptotic descriptions of the distribution of the $k$-RF metric for the extreme values $k=0$ and $k=n-2$, as $n$ becomes large. We show that a linear transform of the $0$-RF metric converges to a Poisson distribution (with mean 2) whereas a similar transform for the $(n-2)$-RF metric leads to a normal distribution (with mean $\sim ne^{-2}$). These results (together with the case $k=1$ which behaves quite differently, and $k=n-3$) shed light on the earlier simulation results, and the predictions made concerning them. |
| title | The asymptotic distribution of the $k$-Robinson-Foulds dissimilarity measure on labelled trees |
| topic | Probability Combinatorics Populations and Evolution 05A16 05C05 |
| url | https://arxiv.org/abs/2412.20012 |