Typo in ridge matrix in exercise 4 of hw2

doc/Projects/2018/hw2/html/hw2-bs.html

@@ -114,7 +114,7 @@ MathJax.Hub.Config({
 <center><b>Department of Physics, University of Oslo, Norway</b></center>
 <br>
 <p>
-<center><h4>Sep 3, 2018</h4></center> <!-- date -->
+<center><h4>Sep 5, 2018</h4></center> <!-- date -->
 <br>
 <p>
 </div> <!-- end jumbotron -->
@@ -145,7 +145,7 @@ y <span style="color: #666666">=</span> <span style="color: #666666">5*</span>x<
 <ol>
 <li> Write your own code for the Ridge method (see chapter 3.4 of Hastie <em>et al.</em>, equations (3.43) and (3.44)) and compute the parametrization for different values of \( \lambda \). Compare and analyze your results with those from exercise 2. Study the dependence on \( \lambda \) while also varying the strength of the noise in your expression for \( y(x) \).</li> 
 <li> Repeat the above but using the functionality of <b>scikit-learn</b>. Compare your code with the results from <b>scikit-learn</b>. Remember to run with the same random numbers for generating \( x \) and \( y \).</li> 
-<li> Our next step is to study the variance of the parameters \( \beta_1 \) and \( \beta_2 \) (assuming that we are parametrizing our function with a second-order polynomial. We will use standard linear regression and the Ridge regression.  You can now opt for either writing your own function that calculates the variance of these paramaters (recall that this is equal to the diagonal elements of the matrix \( (\hat{X}^T\hat{X})^{-1}+\lambda\hat{I} \)) or use the functionality of <b>scikit-learn</b> and compute their variances. Discuss the results of these variances as functions of \( \lambda \). In particular, try to link your discussion with the discussion in Hastie <em>et al.</em> and their figure 3.11.</li>
+<li> Our next step is to study the variance of the parameters \( \beta_1 \) and \( \beta_2 \) (assuming that we are parametrizing our function with a second-order polynomial. We will use standard linear regression and the Ridge regression.  You can now opt for either writing your own function that calculates the variance of these paramaters (recall that this is equal to the diagonal elements of the matrix \( (\hat{X}^T\hat{X})+\lambda\hat{I})^{-1} \)) or use the functionality of <b>scikit-learn</b> and compute their variances. Discuss the results of these variances as functions of \( \lambda \). In particular, try to link your discussion with the discussion in Hastie <em>et al.</em> and their figure 3.11.</li>
 <li> Repeat the previous step but add now the Lasso method, see equation (3.53) of Hastie <em>et al.</em>. Discuss your results and compare with standard regression and the Ridge regression results. You can write your own code or use the functionality of <b>scikit-learn</b>.</li>  
 <li> Finally, using <b>scikit-learn</b> 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</li>
 </ol>

doc/Projects/2018/hw2/html/hw2.html

@@ -81,7 +81,7 @@ MathJax.Hub.Config({
 <center><b>Department of Physics, University of Oslo, Norway</b></center>
 <br>
 <p>
-<center><h4>Sep 3, 2018</h4></center> <!-- date -->
+<center><h4>Sep 5, 2018</h4></center> <!-- date -->
 <br>
 
 <h2 id="___sec0">Exercise 4  </h2>
@@ -110,7 +110,7 @@ y <span style="color: #666666">=</span> <span style="color: #666666">5*</span>x<
 <ol>
 <li> Write your own code for the Ridge method (see chapter 3.4 of Hastie <em>et al.</em>, equations (3.43) and (3.44)) and compute the parametrization for different values of \( \lambda \). Compare and analyze your results with those from exercise 2. Study the dependence on \( \lambda \) while also varying the strength of the noise in your expression for \( y(x) \).</li> 
 <li> Repeat the above but using the functionality of <b>scikit-learn</b>. Compare your code with the results from <b>scikit-learn</b>. Remember to run with the same random numbers for generating \( x \) and \( y \).</li> 
-<li> Our next step is to study the variance of the parameters \( \beta_1 \) and \( \beta_2 \) (assuming that we are parametrizing our function with a second-order polynomial. We will use standard linear regression and the Ridge regression.  You can now opt for either writing your own function that calculates the variance of these paramaters (recall that this is equal to the diagonal elements of the matrix \( (\hat{X}^T\hat{X})^{-1}+\lambda\hat{I} \)) or use the functionality of <b>scikit-learn</b> and compute their variances. Discuss the results of these variances as functions of \( \lambda \). In particular, try to link your discussion with the discussion in Hastie <em>et al.</em> and their figure 3.11.</li>
+<li> Our next step is to study the variance of the parameters \( \beta_1 \) and \( \beta_2 \) (assuming that we are parametrizing our function with a second-order polynomial. We will use standard linear regression and the Ridge regression.  You can now opt for either writing your own function that calculates the variance of these paramaters (recall that this is equal to the diagonal elements of the matrix \( (\hat{X}^T\hat{X})+\lambda\hat{I})^{-1} \)) or use the functionality of <b>scikit-learn</b> and compute their variances. Discuss the results of these variances as functions of \( \lambda \). In particular, try to link your discussion with the discussion in Hastie <em>et al.</em> and their figure 3.11.</li>
 <li> Repeat the previous step but add now the Lasso method, see equation (3.53) of Hastie <em>et al.</em>. Discuss your results and compare with standard regression and the Ridge regression results. You can write your own code or use the functionality of <b>scikit-learn</b>.</li>  
 <li> Finally, using <b>scikit-learn</b> 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</li>
 </ol>

doc/Projects/2018/hw2/pdf/hw2.p.tex

@@ -155,7 +155,7 @@ Homework 2
 
 % --- begin date ---
 \begin{center}
-Sep 3, 2018
+Sep 5, 2018
 \end{center}
 % --- end date ---
 
@@ -186,7 +186,7 @@ y = 5*x*x+0.1*np.random.randn(100,1)
 
 \item Repeat the above but using the functionality of \textbf{scikit-learn}. Compare your code with the results from \textbf{scikit-learn}. Remember to run with the same random numbers for generating $x$ and $y$. 
 
-\item Our next step is to study the variance of the parameters $\beta_1$ and $\beta_2$ (assuming that we are parametrizing our function with a second-order polynomial. We will use standard linear regression and the Ridge regression.  You can now opt for either writing your own function that calculates the variance of these paramaters (recall that this is equal to the diagonal elements of the matrix $(\hat{X}^T\hat{X})^{-1}+\lambda\hat{I}$) or use the functionality of \textbf{scikit-learn} and compute their variances. Discuss the results of these variances as functions of $\lambda$. In particular, try to link your discussion with the discussion in Hastie \emph{et al.} and their figure 3.11.
+\item Our next step is to study the variance of the parameters $\beta_1$ and $\beta_2$ (assuming that we are parametrizing our function with a second-order polynomial. We will use standard linear regression and the Ridge regression.  You can now opt for either writing your own function that calculates the variance of these paramaters (recall that this is equal to the diagonal elements of the matrix $(\hat{X}^T\hat{X})+\lambda\hat{I})^{-1}$) or use the functionality of \textbf{scikit-learn} and compute their variances. Discuss the results of these variances as functions of $\lambda$. In particular, try to link your discussion with the discussion in Hastie \emph{et al.} and their figure 3.11.
 
 \item Repeat the previous step but add now the Lasso method, see equation (3.53) of Hastie \emph{et al.}. Discuss your results and compare with standard regression and the Ridge regression results. You can write your own code or use the functionality of \textbf{scikit-learn}.  
 

doc/Projects/2018/hw2/pdf/hw2.tex

@@ -125,7 +125,7 @@ Homework 2
 
 % --- begin date ---
 \begin{center}
-Sep 3, 2018
+Sep 5, 2018
 \end{center}
 % --- end date ---
 
@@ -156,7 +156,7 @@ y = 5*x*x+0.1*np.random.randn(100,1)
 
 \item Repeat the above but using the functionality of \textbf{scikit-learn}. Compare your code with the results from \textbf{scikit-learn}. Remember to run with the same random numbers for generating $x$ and $y$. 
 
-\item Our next step is to study the variance of the parameters $\beta_1$ and $\beta_2$ (assuming that we are parametrizing our function with a second-order polynomial. We will use standard linear regression and the Ridge regression.  You can now opt for either writing your own function that calculates the variance of these paramaters (recall that this is equal to the diagonal elements of the matrix $(\hat{X}^T\hat{X})^{-1}+\lambda\hat{I}$) or use the functionality of \textbf{scikit-learn} and compute their variances. Discuss the results of these variances as functions of $\lambda$. In particular, try to link your discussion with the discussion in Hastie \emph{et al.} and their figure 3.11.
+\item Our next step is to study the variance of the parameters $\beta_1$ and $\beta_2$ (assuming that we are parametrizing our function with a second-order polynomial. We will use standard linear regression and the Ridge regression.  You can now opt for either writing your own function that calculates the variance of these paramaters (recall that this is equal to the diagonal elements of the matrix $(\hat{X}^T\hat{X})+\lambda\hat{I})^{-1}$) or use the functionality of \textbf{scikit-learn} and compute their variances. Discuss the results of these variances as functions of $\lambda$. In particular, try to link your discussion with the discussion in Hastie \emph{et al.} and their figure 3.11.
 
 \item Repeat the previous step but add now the Lasso method, see equation (3.53) of Hastie \emph{et al.}. Discuss your results and compare with standard regression and the Ridge regression results. You can write your own code or use the functionality of \textbf{scikit-learn}.  
 

doc/src/Projects/2018/Exercises/hw2.do.txt

@@ -26,7 +26,7 @@ o Write your own code for the Ridge method (see chapter 3.4 of Hastie *et al.*,
 
 o 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 $y$. 
 
-o Our next step is to study the variance of the parameters $\beta_1$ and $\beta_2$ (assuming that we are parametrizing our function with a second-order polynomial. We will use standard linear regression and the Ridge regression.  You can now opt for either writing your own function that calculates the variance of these paramaters (recall that this is equal to the diagonal elements of the matrix $(\hat{X}^T\hat{X})^{-1}+\lambda\hat{I}$) or use the functionality of _scikit-learn_ and compute their variances. Discuss the results of these variances as functions of $\lambda$. In particular, try to link your discussion with the discussion in Hastie *et al.* and their figure 3.11.
+o Our next step is to study the variance of the parameters $\beta_1$ and $\beta_2$ (assuming that we are parametrizing our function with a second-order polynomial. We will use standard linear regression and the Ridge regression.  You can now opt for either writing your own function that calculates the variance of these paramaters (recall that this is equal to the diagonal elements of the matrix $(\hat{X}^T\hat{X})+\lambda\hat{I})^{-1}$) or use the functionality of _scikit-learn_ and compute their variances. Discuss the results of these variances as functions of $\lambda$. In particular, try to link your discussion with the discussion in Hastie *et al.* and their figure 3.11.
 
 o Repeat the previous step but add now the Lasso method, see equation (3.53) of Hastie *et al.*. Discuss your results and compare with standard regression and the Ridge regression results. You can write your own code or use the functionality of _scikit-learn_.