Exercise 15.3

Question 1

exercise-data/data08.csv contains measurements of variables \(x\) and \(y\), and the uncertainty of \(y\), \(\sigma\). The proposed functional relationship between these variables is:

\[ y = a_0 + a_1 e^{-a_2 x} \sin(a_3 x) \]

Part 1

Ignoring the standard deviation for now, find the values of \(a_0\), \(a_1\) and \(a_2\) that minimize the sum of errors squared using the scipy.optimize.least_squares function.

Plot the data points and curve of best fit on the same set of axis to make sure you have indeed found the best fit.

Part 2

Now, taking the uncertainty into account, perform a \(\chi^2\) minimization to find the best fit.

Note that in Part 1 the residual function is:

\[ \epsilon = y - f(a, x) \]

what is the residual when \(\chi^2\) is the objective function (the function to be minimized)?

Part 3

Use scipy.optimize.curve_fit to find the the values of \(a_0\), \(a_1\) and \(a2\) using the data file. Note that the curve_fit function takes an argument for the uncertainty of \(y\) (sigma). Compare your results from this part to Part 2.

Question 2

Repeat Part 2 and Part 3 of the process from Question 1 with exercise-data/data09.tsv (a tab seperated data file), and the proposed functional relation:

\[ y = a_0 \exp\left(-a_1 (x - a_2)^2\right) \sin\left(a_3 (x^2 - a_4)\right) \]