Suppose you are interested in predicting a one-dimensional quantitative random variable $Y$ (outcome) in terms of a two-dimensional quantitative random variable $X$ (covariates). You have a learning set $\left(x_1, y_1\right), \ldots,\left(x_3, y_3\right)$ of three observations, given in the table.
| $i$ | $x_i$ | $y_i$ |
| 1 | $\left[\begin{array}{l}
0 \\
3
\end{array}\right]$ | -3 |
| 2 | $\left[\begin{array}{l}
0 \\
3
\end{array}\right]$ | 0 |
| 3 | $\left[\begin{array}{l}
6 \\
0
\end{array}\right]$ | 3 |
Select the one that expresses the regularized empirical risk (mean squared error) for ridge regression for this dataset in terms of $\beta_1$ and $\beta_2$ when $\lambda=1$.
- $16 \beta_1^2+12 \beta_1+4 \beta_2^2-6 \beta_2+6$
- $16 \beta_1^2-12 \beta_1+4 \beta_2^2+6 \beta_2+6$
- $16 \beta_1^2+12 \beta_1-4 \beta_2^2-6 \beta_2+6$
- $16 \beta_1^2-12 \beta_1-4 \beta_2^2-6 \beta_2+6$
