1. … a tricky task !? Tanja Gernhard, Klaas Hartmann
Simulating trees…
… a tricky task !?
Tanja Gernhard, Klaas Hartmann
             

2. Even simple models hard to analyze analytically!
Why simulations?
How do typical trees look like under a specific model of speciation?
expected Colless value?
expected gamma value, LTT plot?
p-values for statistical testing?
Even simple models hard to analyze analytically!

3. Aspects of simulations
Tree distribution ill-defined (expected age of the tree is infinite)
Condition on age of tree (simulations straightforward)
Condition on number of species in the tree (assuming a uniform prior for the age of the tree)

4. Standard method Simulate until n species are obtained.
Stop at following speciation or extinction event.
Example for n=5 species:
Later periods with n species disregarded!
Pendant edges are too long!
Each simulation makes same contribution!

5. General evolutionary model
Simulate a tree, t, until n* species or extinction is reached
Find the expected number of trees to sample from t:
For each sample required:
Randomly choose an interval, i, according to the weights
Choose the pendant edge length uniformly at random from (0, )
Repeat from step 1 until the required number of samples has been obtained = P k i 1 Á Á i Á i

6. Special models
Yule model: Pure birth process. Individuals evolve independently. Speciation occurs at a constant rate.
Constant rate birth-death model: Add constant death rate to the Yule model.
Coalescent: Time between successive coalescent events is exponentially distributed

7. Memoryless pure birth models
Time between sn and sn+1 does not depend on the current tree (e.g. Yule model, Coalescent).
Correct simple approach:
Simulate until
Determine s n : :

8. Determining . h ( ) / Z 1 j n g d = Z 1 g ( n ) d

9. Yule model / Coalescenth ( ) / Z 1 g n d = f e
Simulating until the (n+1)-th speciation event yields the desired results!

10. Birth-death model f ( s j t = ; n ) ¸ ¡ ¹ e 1 q ( t j n ) = ¸ ¡ ¹ 1 e
reconstructed tree
tor
point process s3 s1 s2 s4 1 2 3 4 5 f ( s i j t o r = ; n ) 2 e 1 q o r ( t j n ) = 2 1 e +

11. Available Program
Tool TreeSample implemented in Pearl by Klaas Hartmann. Available as a stand-alone application or as a Pearl script.
Demonstration:

12. Dankeschön
Dennis Wong
Mike Steel
Erick Matsen
Klaas Hartmann