$\%$ Load Prof. Billiar's data.

$\%$ Store the loaded data in matrix called C$.$

$\%1$st column of C should be "stretch", expressed as a ratio.

$\%2$nd column of C should be "stress", expressed in kilopascals $($KPa$).$

$\%$ Each row should be an observation

C $=$ load$(\text{'}../$Data$/$billiar$.$csv$\text{'})$;

n $=$ size$($C$,1)$;

p $=$ size$($C$,2)$;

$\%$ As in the in$-$class exercise,

$\%$ divide the data into the independent variable $\text{'}$x$\text{'},$

$\%$ and dependent variable $\text{'}$y$\text{'}.$

$\%$ In this case, the second column of the data

$\%($i$.$e$.,$ stress$)$ will be the dependent variable.

$\%$ We will predict the dependent variable on the basis of the

$\%$ stretch ratio, which is our independent variable.

$\%$ Rename these vectors x and y$,$ accordingly.

y $=\%$

x $=\%$

$\%$ Augment the data, creating a design matrix that will

$\%$ allow us to get the offset $($y$-$intercept$)$ parameter.

$\%$ Call the design matrix $\text{'}$X$\text{'}.$ It should be composed of

$\%$ the values in x $($the first column$),$

$\%$ and a n$-$by$-1$ vector of all ones $($the second column$).$

X $=\%$

$\%$ We're now going to set up to perform a nonlinear regression using many

$\%$ locally$-$linear regression models, as we talked about this week.

$\%$ Like we saw in lecture, the first step in this process is to

$\%$ define a set of reference points in feature space, at which

$\%$ we want to perform a locally$-$linear fit using weighted linear regression.

$\%$ In the present case, the references points are simply values

$\%$ of the independent variable: stretch ratio.

$\%$ Create a variable $\text{'}$m$\text{'},$ describing how many reference points you'd like

$\%$ to consider. Use at least $100$ total locations.

m $=100$;

$\%$ Use Matlab's 'min' and 'max' functions to find the

$\%$ smallest and largest observed value of stretch in the data.

$\%$ Use Matlab's 'linspace' function to create a vector $\text{'}$t$\text{'}$ of

$\%$ evenly$-$spaced reference values.

x$\_$min $=\%$

x$\_$max $=\%$

t $=\%$

$\%$ Augment the data, creating a design matrix called $\text{'}$T$\text{'}.$

$\%$ As with X$,$ above, T should be composed of

$\%$ your reference value vector, just created, and a vector of all ones.

T $=\%$

$\%$ In preparation for the iteration we need to perform

$\%$ in the service of nonlinear regression,

$\%$ create a vector for storing estimated function values

z$\_$hat $=$ zeros$($m$,1)$;

$\%$ In preparation for the iteration we need to perform

$\%$ in the service of nonlinear regression,

$\%$ create a vector for storing regression parameters

b$\_$hat $=$ zeros$($m$,2)$;

$\%$ Iterate over every test point

for itor $=1$:$\%$

$\%$ For each reference point, calculate similarity from current

$\%$ test point to all stretch values, stored in X$.$

$\%$ Store those values on the diagonal of a weight matrix W$.$

$\%$ Use the function 'gaussiankernel', supplied for you,

$\%$ to calculate the similarities.

$\%$ A good value for the kernel parameter value $\text{'}$h$\text{'}$ is $2000.$

W $=$ zeros$($n$,$n$)$;

for jtor $=1$:$\%$

W$($jtor$,$jtor$)=\%$

end

$\%$ Get regression coefficients using your function regress$\_$fit$\_$weighted

b$\_$local $=\%$

$\%$ Store the local coefficient in the vector you created above

$\%$ for that purpose.

b$\_$hat$($itor$,$:$)=\%$

$\%$ Use the regression coefficients just obtained to

$\%$ estimate the stress values at the current point.

$\%$ Store the estimated value in the vector you created above.

z$\_$hat$($itor$)=\%$

end

$\%$ Make a scatter plot of the stretch and stress data

figure

scatter$(\%$

hold on

$\%$ Scatter plot the estimated stress values, obtained above

$\%$

$\%$ Plot a short red line, centered at $1\mathrm{.}6$ stretch ratio.

$\%$ This indicates the locally$-$linear fit at that location.

$\%$ This is tricky, so I'll supply the code for you.

$\%$ Please make an effort to understand what's going on here,

$\%$ because you'll need it in Objective #$4.$

$[$~$,$ ind$]=$ min$(($T$($:$,1)-1\mathrm{.}6).^2)$;

t$\_$min $=$ T$($ind$,1)-0\mathrm{.}1$;

t$\_$max $=$ T$($ind$,1)+0\mathrm{.}1$;

z$\_$hat$\_$min $=[$t$\_$min $1]*$b$\_$hat$($ind$,$:$)\text{'}$;

z$\_$hat$\_$max $=[$t$\_$max $1]*$b$\_$hat$($ind$,$:$)\text{'}$;

plot$([$t$\_$min t$\_$max$],[$z$\_$hat$\_$min z$\_$hat$\_$max$],\text{'}$r$\text{'},$'linewidth',$3)$

$\%$ As a final touch, add a grid, axis labels and title, as before.

$\%$

$\%$

$\%$

$\%$