Least squares fitting using Python

Week $3-$ Assignment

Begin the day by going through the $h5p$ workbook on loop. Today you will learn how to use python to perform linear regression, and least square fitting, and as python's fitting functions report uncertainties of all fitted parameters you will also learn how to visualise the uncertainty intervals of fitting parameters, and how this links to the concept of error progression.

This week you will not be working on new problems but re$-$analyse sets from Week $1$ and Week $2.$

Task $1$: Repeat the fitting of the position tracking for the runners from Week $1$ using python. Use error weighted fitting and provide a plot extrapolating the fit to $t=0$ including shaded confidence intervals for the reported $1$ confidence as well as the $95\%$ interval. Comment how this plot makes it easier to see if a runner was cheating.

Task $2$: Repeat the fitting of the exponential decay problem from Week $2.$ Report on the uncertainty of the fitted parameter and likewise provide a plot extrapolating the fit to a three week including shaded confidence intervals for the reported $1$ confidence as well as the $95\%$ interval. Comment if and how this analysis makes it safer to test if a minimum residual activity of $200B\frac{q}{m^2}$

Task $3$: Do a full least square fit on the damped oscillator problem $($PID controller$)$ from Week $2.$ Report on the uncertainty of the fitted parameter and likewise provide a plot including the $1$ confidence as well as the $95\%$ interval.

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