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Your challenge for this homework assignment is to simulate data for a single character under a Brownian motion model on the tree shown below.

The model tree

What to do

Assume that the variance per unit time (\(\sigma^2\)) equals 0.05 and that the starting state (\(s_0\)) equals 0.0.

Use Python 3.x to draw the states at nodes above the root from a normal distribution. Here is a template:

from random import seed,normalvariate
from math import sqrt

rate = 0.05 # this is sigma squared in the BM model

s0 = 0.0
s1 = normalvariate(xxx, xxx)
s2 = normalvariate(xxx, xxx)
s3 = normalvariate(xxx, xxx)
s4 = normalvariate(xxx, xxx)
s5 = normalvariate(xxx, xxx)
s6 = normalvariate(xxx, xxx)
s7 = normalvariate(xxx, xxx)
s8 = normalvariate(xxx, xxx)

print("s0 = %12.5f" % s0)
print("s1 = %12.5f" % s1)
print("s2 = %12.5f" % s2)
print("s3 = %12.5f" % s3)
print("s4 = %12.5f" % s4)
print("s5 = %12.5f" % s5)
print("s6 = %12.5f" % s6)
print("s7 = %12.5f" % s7)
print("s8 = %12.5f" % s8)

Replace all the xxx placeholders with either numbers or variable names. The normalvariate function takes two arguments. The first argument is the mean, the second argument is the standard deviation.

You will first need to choose a pseudorandom number seed. This should be a whole number greater than 0. You can use whatever number you like here.

Remember that, in the Brownian motion (BM) model, variance (i.e. uncertainty) accumulates at the rate \(\sigma^2\) per unit time. The trait value at the end of an edge is normally distributed with mean equal to the trait value at the beginning of the edge and variance equal to the length of time represented by the edge multiplied by the rate \(\sigma^2\). Note that I’ve imported the sqrt function so that you can use it to take the square root of the variance (and the square root of the variance is the standard deviation you need to supply to the normalvariate function).

What to turn in

Please send me (via email or slack) your python3 file that you’ve modified from the template above.