Task $5$: Regularized Regression

Given the values of an unknown function $f$:$RR$ at some selected points, we try to calculate the

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

$($if ${}_{15}=0)$: a polynomial model function of twelve ${}_i$ parameters:

$(x)={}_0+{}_{1}x+{}_2{x}^2+$cdots$+{}_{12}{x}^{12}.$

$($if ${}_{15}=2)$: a polynomial model function of $\text{'}$oj: parameters: $9$;oj:

$(x)={}_0+{}_{1}x+{}_2{x}^2+$cdots$+{}_{10}{x}^{10}.$

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 $y={}_{16}.$ 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, as these 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.

Assignment $5$

$\backslash$xi $15$ : $2$

$\backslash$xi $16$ : $(12,598584149847\mathrm{.}47),$

$(-3,807319\mathrm{.}82),$

$(19,57488886909204\mathrm{.}66),$

$(10,91197641821\mathrm{.}18),$

$(14,2779602142321\mathrm{.}59),$

$(7,2620242051\mathrm{.}03),$

$(-9,38501926054\mathrm{.}93),$

$(-4,12633971\mathrm{.}08),$

$(11,251877162138\mathrm{.}51),$

$(6,532393003\mathrm{.}02),$

$(-14,3180939109961\mathrm{.}09),$

$(-13,1491073526980\mathrm{.}82),$

$(4,9175113\mathrm{.}77),$

$(9,31076177677\mathrm{.}1),$

$(16,10065022032164\mathrm{.}29),$

$(-8,11548497873\mathrm{.}57),$

$(3,480644\mathrm{.}83),$

$(17,19814624854410\mathrm{.}59),$

$(0,3\mathrm{.}12),$

$(-12,639814153210\mathrm{.}45),$

$(8,9991279634\mathrm{.}81),$

$(-1,27\mathrm{.}17)$