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+TITLE: Exercises week 36
+AUTHOR: September 4-8, 2023 
+DATE: Deadline is Sunday September 10 at midnight
+
+
+=====  Overarching aims of the exercises this week =====
+
+This set of exercises form an important part of the first project. The
+analytical exercises deal with the material covered last week on the
+mathematical interpretations of ordinary least squares and of Ridge
+regression. The numerical exercises can be seen as a continuation of
+exercise 3 from week 35, with the inclusion of Ridge regression. This
+material enters also the discussions of the first project.
+
+===== Exercise: Analytical exercises  =====
+
+The aim here is to derive the expression for the optimal parameters
+using Ridge regression. Furthermore, using the singular value
+decomposition, we will analyze the difference between the ordinary
+least squares approach and Ridge regression.
+
+The expression for the standard Mean Squared Error (MSE) which we used to define our cost function and the equations for the ordinary least squares (OLS) method, that is 
+our optimization problem is
+!bt
+\[
+{\displaystyle \min_{\bm{\beta}\in {\mathbb{R}}^{p}}}\frac{1}{n}\left\{\left(\bm{y}-\bm{X}\bm{\beta}\right)^T\left(\bm{y}-\bm{X}\bm{\beta}\right)\right\}.
+\]
+!et
+or we can state it as
+!bt
+\[
+{\displaystyle \min_{\bm{\beta}\in
+{\mathbb{R}}^{p}}}\frac{1}{n}\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2=\frac{1}{n}\vert\vert \bm{y}-\bm{X}\bm{\beta}\vert\vert_2^2,
+\]
+!et
+where we have used the definition of  a norm-2 vector, that is
+!bt
+\[
+\vert\vert \bm{x}\vert\vert_2 = \sqrt{\sum_i x_i^2}. 
+\]
+!et
+
+By minimizing the above equation with respect to the parameters
+$\bm{\beta}$ we could then obtain an analytical expression for the
+parameters $\bm{\beta}$.
+
+We can add a regularization parameter $\lambda$ by
+defining a new cost function to be optimized, that is
+
+!bt
+\[
+{\displaystyle \min_{\bm{\beta}\in
+{\mathbb{R}}^{p}}}\frac{1}{n}\vert\vert \bm{y}-\bm{X}\bm{\beta}\vert\vert_2^2+\lambda\vert\vert \bm{\beta}\vert\vert_2^2
+\]
+!et
+
+which leads to the Ridge regression minimization problem where we
+require that $\vert\vert \bm{\beta}\vert\vert_2^2\le t$, where $t$ is
+a finite number larger than zero.
+
+=== a) Expression for Ridge regression ===
+Show that the optimal parameters
+!bt
+\[
+\hat{\bm{\beta}}_{\mathrm{Ridge}} = \left(\bm{X}^T\bm{X}+\lambda\bm{I}\right)^{-1}\bm{X}^T\bm{y},
+\]
+!et
+
+with $\bm{I}$ being a $p\times p$ identity matrix with the constraint that
+
+!bt
+\[
+\sum_{i=0}^{p-1} \beta_i^2 \leq t,
+\]
+!et
+
+with $t$ a finite positive number. 
+
+The ordinary least squares result is
+!bt
+\[
+\hat{\bm{\beta}}_{\mathrm{OLS}} = \left(\bm{X}^T\bm{X}\right)^{-1}\bm{X}^T\bm{y},
+\]
+!et
+
+=== b) The singular value decomposition ===
+
+Use the singular value decomposition of an $m\times n$ matrix $\bm{X}$ (our design matrix)
+!bt
+\[
+
+
+discussed in the lecture notes from week 35 to sh
+
+We have already analyzed the OLS solutions in terms of the eigenvectors (the columns) of the right singular value matrix $\bm{U}$ as
+!bt
+\[
+\tilde{\bm{y}}_{\mathrm{OLS}}=\bm{X}\bm{\beta}  =\bm{U}\bm{U}^T\bm{y}.
+\]
+!et
+
+
+For Ridge regression this becomes
+
+!bt
+\[
+\tilde{\bm{y}}_{\mathrm{Ridge}}=\bm{X}\bm{\beta}_{\mathrm{Ridge}} = \bm{U\Sigma V^T}\left(\bm{V}\bm{\Sigma}^2\bm{V}^T+\lambda\bm{I} \right)^{-1}(\bm{U\Sigma V^T})^T\bm{y}=\sum_{j=0}^{p-1}\bm{u}_j\bm{u}_j^T\frac{\sigma_j^2}{\sigma_j^2+\lambda}\bm{y},
+\]
+!et
+
+with the vectors $\bm{u}_j$ being the columns of $\bm{U}$ from the SVD of the matrix $\bm{X}$. 
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+===== Exercise: Adding Ridge Regression  =====
+
+
+This exercise is a continuation of exercise 3 from last week. We will use the same function to
+generate our data set, still staying with a simple function $y(x)$
+which we want to fit using linear regression, but now extending the
+analysis to include the Ridge regression method.
+
+
+
+Write your own code for the Ridge method (see chapter 3.4 of Hastie
+*et al.*, equations (3.43) and (3.44)) and compute the parametrization
+for different values of $\lambda$. Compare and analyze your results
+with those from exercise 3. Study the dependence on $\lambda$ while
+also varying the strength of the noise in your expression for $y(x)$.
+
+
+Repeat the above but using the functionality of
+_Scikit-Learn_. Compare your code with the results from
+_Scikit-Learn_. Remember to run with the same random numbers for
+generating $x$ and $y$.  Observe also that when you compare with _Scikit-Learn_, you need to pay attention to how the intercept is dealt with.
+
+
+
+Finally, using _Scikit-Learn_ or your own code, compute also the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error defined as
+!bt 
+\[ MSE(\hat{y},\hat{\tilde{y}}) = \frac{1}{n}
+\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2, 
+\] 
+!et 
+and the $R^2$ score function.
+If $\tilde{\hat{y}}_i$ is the predicted value of the $i-th$ sample and $y_i$ is the corresponding true value, then the score $R^2$ is defined as
+!bt
+\[
+R^2(\hat{y}, \tilde{\hat{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2},
+\]
+!et
+where we have defined the mean value  of $\hat{y}$ as
+!bt
+\[
+\bar{y} =  \frac{1}{n} \sum_{i=0}^{n - 1} y_i.
+\]
+!et
+Discuss these quantities as functions of the variable $\lambda$ in Ridge regression.
+
+
+
+
+
+
+