Homework 2

1. Complete Chapter 3, Problem #1 under “Project: Statistical Analysis in Inverse Problems Using Simulated Data” on pages 58–59 of B&T. Use the same initial conditions as before from Chapter 2.

2. Consider the logistic population growth model

ẋ = ax− bx2, x(0) = x0.

Let K = a b . We will examine the model for the q = (a, b, x0) parameter vectors

(i) q = (0.5, 0.1, 0.1) ⇒ K = 5 (relatively flat curve), (ii) q = (0.7, 0.04, 0.1) ⇒ K = 17.5 (moderately sloped curve),

(iii) q = (0.8, 0.01, 0.1) ⇒ K = 80 (relatively steep curve).

Define the regions R0, R1, and R2 as follows: • R0 is the region where t ∈ [0, 2], • R1 is the region where t ∈ (2, 12], • R2 is the region where t ∈ (12, 16].

For each parameter vector q:

(a) Let n = 15. For i = 0, sample n points from region Ri, distributed uniformly over the interval. Find the qOLS optimized parameters for 3 different initial guesses that are far from the true solution. (You can use the same initial guesses for all regions and all q parameter vectors). Calculate J(qOLS) where J is the cost function of the least squares criterion. Calculate K̂ = â

b̂ . Include all results in a table.

For the optimal qOLS with the lowest cost J(qOLS), plot the solution curve for the true solution and the estimated solution on the same plot with the sampled data points. How do the results compare to the true solution? Determine the standard errors and confidence intervals. Are the true parameters contained within the confidence interval?

Then repeat for i = 1. Then repeat for i = 2.

(b) Repeat problem (a) with n = 50.

(c/d) Repeat problems (a) and (b), but sampling from a uniform distribution over the entire region t ∈ [0, 16] instead of a single Ri region.

(e) When sampling from only region Ri, does increasing the sample size improve the results? How does this vary for i = 0, 1, 2? What if you sample over all three regions?

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