Let's consider data from soft tissue mechanical testing $($Given below$).$ Complete the following curve fitting tasks using PYTHON code.

import numpy as np

$x=np.$array $([1,1\mathrm{.}1,1\mathrm{.}2,1\mathrm{.}3,1\mathrm{.}4,1\mathrm{.}5,1\mathrm{.}6])$ #

$y=$ np$.$array $([20,45,70,150,230,350,500])$ # sigma

Create separate figures for questions $($a$)$ to $($d$).$ Separate the code into sections using $"$ #$\%\%".$ Submit the code and figures as a single pdf file to Blackboard.

$($a$)$ Perform linear regression. Plot data points and the fitted curve. Print coefficient of determination $(R^2)$ value on the plot. The $R^2$ value should be clearly visible on the plot. $(30$ pts $)$

$($b$)$ Perform regression using your personal favorite model equation $($cannot be same as part$($a$)$ or $($c$)).$ However, the model must provide a $R^2$ value better than linear regression. Plot data points and fitted curve. Print coefficient of determination $(R^2),$ value and the model equation with best$-$fit coefficients on the plot. $(30$ pts $)$

$($c$)$ Perform nonlinear regression using the following model. $(40$ pts $)$

$=\frac{k_1}{k_2}$exp$(k_{2}E)-\frac{k_3}{k_4}$exp$(2k_{4}E)$

with upper and lower bounds $($from $0$ to infinity$)$ for all the parameters $k_1,k_2,k_3,$ and $k_4.$ We have shown in class $($Lecture $19)$ that this regression problem could be prone to optimization difficulty known as local optima. To mitigate this issue, we can start with multiple initial guesses.

Use the initial guesses of $k_1=1,5,10,k_2=1,5,10,k_3=1,5,10,$ and $k_4=1,5,10($these lead to $81$ combinations in total$),$ perform regression using all possible combinations of initial parameters, store the regression results $($resulting parameters and optimized error function values, a$.$k$.$a$,$ sum of squares of residuals$)$ of each initial guess. This can be done by using nested for loops.

Then, print the best parameter set with smallest optimized error function value $($consider using numpy.argmin function on the optimized error function values which returns the index of the minimum, the index can be used to retrieve the best parameters$).$ Plot data points with the best fitted curve, and $R^2$ value.