A statistic summarizing: Relative timing in trees

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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)

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### 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

RRRRLL

RRLRRL

shuffles on a vertex

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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:

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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:
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