update book

doc/LectureNotes/_build/html/exercisesweek37.html

@@ -62,7 +62,7 @@
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+    <link rel="next" title="Week 37: Statistical interpretations and Resampling Methods" href="week37.html" />
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@@ -230,6 +230,8 @@
 <li class="toctree-l1"><a class="reference internal" href="exercisesweek36.html">Exercises week 36</a></li>
 <li class="toctree-l1"><a class="reference internal" href="week36.html">Week 36: Linear Regression and Gradient descent</a></li>
 <li class="toctree-l1 current active"><a class="current reference internal" href="#">Exercises week 37</a></li>
+<li class="toctree-l1"><a class="reference internal" href="week37.html">Week 37: Statistical interpretations and Resampling Methods</a></li>
+<li class="toctree-l1"><a class="reference internal" href="exercisesweek38.html">Exercises week 37</a></li>
 </ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Projects</span></p>
 <ul class="nav bd-sidenav">
@@ -470,7 +472,7 @@ y_centered = ?
 </div>
 </div>
 </div>
-<p>Fill in the necessary details.</p>
+<p>Fill in the necessary details. Do we need to center the <span class="math notranslate nohighlight">\(y\)</span>-values?</p>
 <p>After this preprocessing, each column of <span class="math notranslate nohighlight">\(\boldsymbol{X}_{\mathrm{norm}}\)</span> has mean zero and standard deviation <span class="math notranslate nohighlight">\(1\)</span>
 and <span class="math notranslate nohighlight">\(\boldsymbol{y}_{\mathrm{centered}}\)</span> has mean 0. This makes the optimization landscape
 nicer and ensures the regularization penalty <span class="math notranslate nohighlight">\(\lambda \sum_j
@@ -487,7 +489,9 @@ same scale).</p>
 <div class="cell docutils container">
 <div class="cell_input docutils container">
 <div class="highlight-none notranslate"><div class="highlight"><pre><span></span># Set regularization parameter, either a single value or a vector of values
-lambda = ?
+# Note that lambda is a python keyword. The lambda keyword is used to create small, single-expression functions without a formal name. These are often called &quot;anonymous functions&quot; or &quot;lambda functions.&quot;
+lam = ?
+
 
 # Analytical form for OLS and Ridge solution: theta_Ridge = (X^T X + lambda * I)^{-1} X^T y and theta_OLS = (X^T X)^{-1} X^T y
 I = np.eye(n_features)
@@ -533,19 +537,8 @@ num_iters = 1000
 # Initialize weights for gradient descent
 theta = np.zeros(n_features)
 
-# Arrays to store history for plotting
-cost_history = np.zeros(num_iters)
-
 # Gradient descent loop
-m = n_samples  # number of data points
 for t in range(num_iters):
-    # Compute prediction error
-    error = X_norm.dot(theta) - y_centered 
-    # Compute cost for OLS and Ridge (MSE + regularization for Ridge) for monitoring
-    cost_OLS = ?
-    cost_Ridge = ?
-    # You could add a history for both methods (optional)
-    cost_history[t] = ?
     # Compute gradients for OSL and Ridge
     grad_OLS = ?
     grad_Ridge = ?
@@ -564,11 +557,13 @@ print(&quot;Gradient Descent Ridge coefficients:&quot;, theta_gdRidge)
 </div>
 <section id="id2">
 <h3>4a)<a class="headerlink" href="#id2" title="Link to this heading">#</a></h3>
-<p>Discuss the results as function of the learning rate parameters and the number of iterations.</p>
+<p>Write first a gradient descent code for OLS only using the above template.
+Discuss the results as function of the learning rate parameters and the number of iterations</p>
 </section>
 <section id="id3">
 <h3>4b)<a class="headerlink" href="#id3" title="Link to this heading">#</a></h3>
-<p>Try to add a stopping parameter as function of the number iterations and the difference between the new and old <span class="math notranslate nohighlight">\(\theta\)</span> values. How would you define a stopping criterion?</p>
+<p>Write then a similar code for Ridge regression using the above template.
+Try to add a stopping parameter as function of the number iterations and the difference between the new and old <span class="math notranslate nohighlight">\(\theta\)</span> values. How would you define a stopping criterion?</p>
 </section>
 </section>
 <section id="exercise-5-ridge-regression-and-a-new-synthetic-dataset">
@@ -667,11 +662,11 @@ should be in the same ballpark.  Which method (OLS or Ridge) gives the best resu
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