Q1. Consider a fish population that eats kelp. It grows according to the model
Rt+1 = rRte1−Rt/k, (1)
where Rt is the biomass at time t, for t ∈ {0,1,…,n}, with carrying capacity k
and a parameter related to the growth of the population, r. Assume that k has
been estimated independently by biologists, who know exactly how much kelp is
in the ocean. So you can consider k as fixed.
(a) What is proliferation in this model? In other words, compute F′(0).
(b) Given data, {R0,R1,…,Rn}, for population biomass in years t = 0,1,…n.
Derive the least squares estimate of r. Call it ˆrLS. Be sure to verify it is a
least squares estimate.
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Q2. Consider the model from Q1 but slightly modified,
Rt+1 = Rteρ−Rt/k. (2)
(a) Write down an expression for ρ in terms of r, such that the model from Q1
is mathematically equivalent to the model above.
(b) Now we introduce noise, in the following way. Consider
Rt+1 = ztRteρ−Rt/k, (3)
where zt is a random variable such that log (zt) ∼ N(0,σ2). Assume that k
is fixed. Find the maximum likelihood estimate of ρ, in terms of k and the
given time series data, {R0,R1,…,Rn}.
(c) From parts (a) and (b), write down an estimate for r. Call it ˆrMLE.
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Q3. In this question you will explore whether ˆrLS and ˆrMLE are biased.
(a) In Ror a programming language of your choice, simulate time series data,
{R0,R1,…,Rn}, using the model
Rt+1 = ztrRte1−Rt/k, (4)
with r = 1.3, k = 100, σ = 0.1, starting at R0 = 5 and n = 50, where zt is
described as in Q2. Plot your time series data with t on the x-axis and Rt
on the y-axis
(b) Given your time series data, compute the values of ˆrLS, and ˆrMLE.
(c) Plot your data with Rt on the x-axis and Rt+1 on the y-axis. On top of this
data plot two curves, the Ricker function, rRte1−Rt/k, with r = ˆrMLE and
r = ˆrLS. Which curve looks like it fits the data better?
(d) Determine which of your estimators r = ˆrMLE and r = ˆrLS has greater bias?
To do this you need to simulate 10,000 sets of time series (e.g. repeat part(a)
10,000 times with new values of zt each time. You can store the time series
in a 10,000 by n matrix. You may want to consult the week 5 practical.
(e) Which estimator is more biased? Can you guess in words why one is more
biased than the other? Note you can take a guess at this question even if
you were unable to do the programming for parts (a-d).
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Q4. Consider a species where individuals survive each year with probability θ. Say
you monitored a sample of n1 individuals, meaning you were able to observe all n1
individuals you were monitoring exactly. You discovered that n2 total individuals,
out of the n1 you monitored, were alive at the end of the year.
(a) Compute the posterior distribution of θ given your data n1, n2. Assume a
Beta distributed prior for θ, with parameters α and β. That is assume the
prior,
p(θ) = θα−1(1 −θ)β−1
B(α,β) , (5)
where,
B(α,β) =
∫1
0
tα−1(1 −t)β−1 dt.
(b) Let α = 1, and β = 1. Plot the pdf of the prior. In words, what are we
assuming about θ in the absence of data? Plot the pdf of the posterior if n2 =
8 and n1 = 10. How did observing 8 surviving individuals out of 10 change
our belief in the survival probability? You can use Rto do the plots (it is
good practice with R) but if you are in a rush you can also screenshot this app
for the plots. http://eurekastatistics.com/beta-distribution-pdf-grapher/
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