A statistic summarizing:

Relative timing in trees

             

Tanja Gernhard

joint work with Erick Matsen, Daniel Ford

Tree statistics

• Investigate the properties of reconstructed trees

• Colless [Colless, 1982] summarizes the information of the shape of the tree (i.e. balanced vs. unbalanced)

• Gamma [Pybus, Harvey 2000] summarizes the time between speciation events (graphical representation in LTT plots)

Relative Timing

Neutrality: Each ranking equally likely

How to summarize rank information?

Each ranking equally likely equivalent to

For each vertex: Each shuffle equally likely

shuffles on a vertex

RRRRLL RRLRRL

Runs summarize shuffle

• Runs: Number of sequences in shuffle with same letter

• Consider (2,4)-shuffle RRLRRL: 4 runs• Run distribuition of (m,n)-shuffle

well-known from statistics:

P 5i=0 xi

PfXm;n = 2k+1g=

¡m¡ 1

k

¢¡n¡ 1k¡ 1

¢+

¡m¡ 1k¡ 1

¢¡n¡ 1

k

¢

¡m+n

m

¢

PfXm;n = 2kg=2¡m¡ 1k¡ 1

¢¡n¡ 1k¡ 1

¢

¡m+n

m

¢

Combining all shuffles in tree

• Runs of a tree, R(T): Sum of runs of all shuffles in tree

• Distribution for runs in tree T:

PfR(T) = kg=kX

i=1

PfXm;n = igk¡ iX

j =0

PfR(L) = jgPfR(R) = k¡ i ¡ jg

Application to Ant lineages

• We investigated the tree of Ants [Moreau et al, 2007]

• LTT plot show substantial increase during Late Cretacceous to Early Eocene

10

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Some comments

• Testing the posterior of Bayesian methods for neutrality (phylogenies or populations)

• Population size might change arbitrary

• All programs available from my homepage in the Python package Casspython Stats.dat YourTrees.tre

Dankeschön