Kernel: SageMath 10.4

C11 Parameter Estimation

Atlantic Cod Harvesting Effort

This is the Atlantic cod harvesting effort data from Table 1.2 in the text.

tdata = list(range(1978,2008)) udata = [0.18847, 0.14974, 0.21921, 0.17678, 0.28203, 0.34528, 0.20655, 0.33819, 0.14724, 0.19757, 0.23154, 0.2086, 0.33565, 0.29534, 0.33185, 0.35039, 0.2827, 0.19928, 0.18781, 0.19357, 0.18953, 0.17011, 0.1566, 0.28179, 0.25287, 0.25542, 0.08103, 0.0874, 0.08195, 0.10518] data = list(zip(tdata, udata)) dplt = scatter_plot(data, axes = False, frame = True, axes_labels = ["Year", "Harvesting Effort (per year)"], axes_labels_size = 1) dplt.show(figsize = 4)

Exercise 1. Find best fit constant, linear, and quadratic models for this data. By "best fit" we mean that parameters should be chosen to minimize the sum of the squared residuals. Compare the three models in terms of the sum of squared residuals and the number of parameters.

Usain Bolt's Olympic Victory

Table 1.1 reports race splits (seconds) every 10 meters for Usain Bolt's 2008 Olympic gold medal final 100 meter race. The following code draws a scatter plot of this data.

tdata = [0.165, 1.85, 2.87, 3.78, 4.65, 5.50, 6.32, 7.14, 7.96, 8.79, 9.68] xdata = range(0, 110, 10) data = list(zip(tdata, xdata)) dplot = scatter_plot(data, axes = False, frame = True, axes_labels_size = 1, axes_labels = ["Time (seconds)", "Position (meters)"]) dplot.show(figsize = 4)

Exercise 2. Solve the Hill-Keller differential equation v(t)=Pkv(t),x(t)=v(t)v'(t) = P - kv(t), x'(t) = v(t) with race initial condition v(t0)=0,x(t0)=0v(t_0) = 0, x(t_0) = 0.

Exercise 3. Assume t0=0.165t_0 = 0.165 and P=11P = 11. Find the best fit value of kk to the Usain Bolt data.

Exercise 4. Assume t0=0.165t_0 = 0.165. Find the best fit values of kk and PP to the Usain Bolt data. Compare with the results of the previous exercise