Given the values of an unknown function f: R $->$ R at some selected points, we try to calculate the parameters

of a model function using OLS as a distance and a ridge regularization:

a polynomial model function of twelve $\backslash$alpha i parameters:

f$($x$)=\backslash$alpha $0+\backslash$alpha $1$x $+\backslash$alpha $2$x$^2++\backslash$alpha $12$x$^12$

Calculate the OLS estimate, and the OLS ridge$-$regularized estimates for the parameters given the sample points

of the graph of f given that the values are $($x$,$ y$)$ each of the elements of $(-16,-1229002003908364),(8,-402005496217\mathrm{.}89),(13,-127351608763754\mathrm{.}56),(-14,-246331654914533\mathrm{.}9),(-19,-10368384784025250),(18,-6488579997432483),(1,18\mathrm{.}73),(0,4\mathrm{.}95),(-6,-6855593605\mathrm{.}74),(-7,-47193153039\mathrm{.}82),(-2,6538\mathrm{.}38),(-5,-718757302\mathrm{.}93),(19,-12452715998839846),(-9,-1137552710286\mathrm{.}59),(20,-22512170239454896),(-12,-37129942809283\mathrm{.}44),(14,-322158432943680\mathrm{.}9),(2,-29384\mathrm{.}5),(-4,-37242224\mathrm{.}93),(-15,-540051396108246\mathrm{.}06),(10,-5888746171666\mathrm{.}93),(7,-81948575255\mathrm{.}67),(-3,-528889\mathrm{.}82),(12,-52301042281314\mathrm{.}99),(5,-1604903911\mathrm{.}63),(-18,-4917393956387863)$

What weight do you give to the penalties? What are the qualities of each of the solutions?

Remember to include the steps of your computation, which are more important than the actual computations. If you calculate the solution with a program, make sure that you trust and cite the core functions used and that you sketch the mathematical path in a way that is coherent with the program.