Question $3$: In the following marketing set, we have $9$ years with the sales in $10$ million

euro and the advertising expenditure in million euro.

a$)$ Formulate the response vector $Y,$ which has nine entries.

b$)$ Formulate the data matrix of $x,$ the first column should be all ones corresponding to

the intercept, and the second column should be the predictors. The dimension of $x$ should

be $92.$

c$)$ Write R code to compute $x^{t}x.$

d$)$ Write $R$ code to compute $=(x^{t}x{)}^{-}{x}^{t}Y.$ This is the estimated linear regression

coefficient of the linear model with $Y$ as the response and $x$ as the data matrix.

e$)$Run the linear regression using $Y$ as the response and $x$ as the predictor using $lm$

command in $R$ and compare the output with your own calculation.

f$)$ Now two additional data points arrived. They are Year $10,$ Sales $96,$ and Advertisement

$53$; Year $11,$ Sales $120,$ and Advertisement $63.$ Please use the online algorithm to update

the linear model. Use the two new observations together to perform the sequential learning

and update the model using stochastic gradient descent algorithm using the learning rate

$=0\mathrm{.}0001.$ Note here in the updating scheme hat$({)}^{\text{n}\text{e}\text{w}}=$hat$({)}^{\text{o}\text{l}\text{d}}-grad{E}_n,$ the term $E_n$ will be the

sum of the squared prediction errors over the two new observations:

$E_n=(y_{10}-$hat$(y{)}_{10}{)}^2+(y_{11}-$hat$(y{)}_{11}{)}^2.$

In this example, we implement the online algorithm when the new data come in batches, not

one at a time.