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@@ -321,6 +321,43 @@
5.5. Cross-validation
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#exercises-and-projects">
5.6. Exercises and Projects
</a>
<ul class="nav section-nav flex-column">
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#exercise-ordinary-least-square-ols-on-the-franke-function">
5.6.1. Exercise: Ordinary Least Square (OLS) on the Franke function
</a>
</li>
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#exercise-bias-variance-trade-off-and-resampling-techniques">
5.6.2. Exercise: Bias-variance trade-off and resampling techniques
</a>
</li>
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#exercise-cross-validation-as-resampling-techniques-adding-more-complexity">
5.6.3. Exercise: Cross-validation as resampling techniques, adding more complexity
</a>
</li>
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#exercise-ridge-regression-on-the-franke-function-with-resampling">
5.6.4. Exercise: Ridge Regression on the Franke function with resampling
</a>
</li>
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#exercise-lasso-regression-on-the-franke-function-with-resampling">
5.6.5. Exercise: Lasso Regression on the Franke function with resampling
</a>
</li>
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#exercise-analysis-of-real-data">
5.6.6. Exercise: Analysis of real data
</a>
</li>
</ul>
</li>
</ul>
</nav>
@@ -592,10 +629,10 @@ number <span class="math notranslate nohighlight">\(i\)</span> is left out. Usin
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Runtime: 0.1375 sec
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Runtime: 0.137546 sec
Jackknife Statistics :
original bias std. error
100.029 100.019 0.150581
99.9031 99.8931 0.149233
</pre></div>
</div>
</div>
@@ -841,14 +878,14 @@ Error: 0.32149601703519126
Bias^2: 0.3123314713548606
Var: 0.009164545680330616
0.32149601703519126 &gt;= 0.3123314713548606 + 0.009164545680330616 = 0.3214960170351912
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Polynomial degree: 1
Polynomial degree: 1
Error: 0.08426840630693411
Bias^2: 0.07968918676726028
Var: 0.004579219539673833
0.08426840630693411 &gt;= 0.07968918676726028 + 0.004579219539673833 = 0.08426840630693411
Polynomial degree: 2
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Polynomial degree: 2
Error: 0.10398646080125035
Bias^2: 0.10077114273548986
Var: 0.0032153180657605086
@@ -868,21 +905,20 @@ Error: 0.05227921801205707
Bias^2: 0.048187277304303125
Var: 0.004091940707753964
0.05227921801205707 &gt;= 0.048187277304303125 + 0.004091940707753964 = 0.05227921801205709
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Polynomial degree: 6
Polynomial degree: 6
Error: 0.03781367141738898
Bias^2: 0.03365768507152761
Var: 0.004155986345861379
0.03781367141738898 &gt;= 0.03365768507152761 + 0.004155986345861379 = 0.03781367141738899
Polynomial degree: 7
Polynomial degree:
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span> 7
Error: 0.027609773491022498
Bias^2: 0.02299949826036597
Var: 0.004610275230656537
0.027609773491022498 &gt;= 0.02299949826036597 + 0.004610275230656537 = 0.027609773491022505
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Polynomial degree: 8
Polynomial degree: 8
Error: 0.017355848195591973
Bias^2: 0.010331721306655588
Var: 0.007024126888936384
@@ -892,7 +928,10 @@ Error: 0.026605727637189085
Bias^2: 0.010018312644140933
Var: 0.016587414993048166
0.026605727637189085 &gt;= 0.010018312644140933 + 0.016587414993048166 = 0.0266057276371891
Polynomial degree: 10
Polynomial degree:
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span> 10
Error: 0.021592704588043153
Bias^2: 0.010516485576652981
Var: 0.011076219011390184
@@ -902,23 +941,19 @@ Error: 0.07160048164228314
Bias^2: 0.01443680008897583
Var: 0.0571636815533073
0.07160048164228314 &gt;= 0.01443680008897583 + 0.0571636815533073 = 0.07160048164228312
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Polynomial degree: 12
Polynomial degree: 12
Error: 0.1154777721897675
Bias^2: 0.01628578269590588
Var: 0.09919198949386163
0.1154777721897675 &gt;= 0.01628578269590588 + 0.09919198949386163 = 0.11547777218976751
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Polynomial degree: 13
Polynomial degree: 13
Error: 0.22842468702166951
Bias^2: 0.01975416527163567
Var: 0.20867052175003387
0.22842468702166951 &gt;= 0.01975416527163567 + 0.20867052175003387 = 0.22842468702166954
</pre></div>
</div>
<img alt="_images/chapter3_38_6.png" src="_images/chapter3_38_6.png" />
<img alt="_images/chapter3_38_4.png" src="_images/chapter3_38_4.png" />
</div>
</div>
<p>The bias-variance tradeoff summarizes the fundamental tension in
@@ -1139,11 +1174,11 @@ Mean squared error on test data: 123711.53703498
Degree of polynomial: 3
Mean squared error on training data: 9011.85263220
Mean squared error on test data: 10913.84780262
Degree of polynomial: 4
Mean squared error on training data: 303.47610036
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Mean squared error on test data: 426.30787294
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 4
Mean squared error on training data: 303.47610036
Mean squared error on test data: 426.30787294
Degree of polynomial: 5
Mean squared error on training data: 3.80354994
Mean squared error on test data: 5.98822371
@@ -1175,64 +1210,62 @@ Mean squared error on test data: 0.17446471
Degree of polynomial: 13
Mean squared error on training data: 0.00759119
Mean squared error on test data: 1.08131003
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 14
Degree of polynomial: 14
Mean squared error on training data: 0.00472199
Mean squared error on test data: 0.81333793
Degree of polynomial: 15
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 15
Mean squared error on training data: 0.00410478
Mean squared error on test data: 92.09145189
Degree of polynomial: 16
Mean squared error on training data: 0.00315593
Mean squared error on test data: 234.39716546
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 17
Degree of polynomial: 17
Mean squared error on training data: 0.00242998
Mean squared error on test data: 1271.05295709
Degree of polynomial: 18
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 18
Mean squared error on training data: 0.00228740
Mean squared error on test data: 108.42208194
Degree of polynomial: 19
Mean squared error on training data: 0.00156372
Mean squared error on test data: 1388.41078073
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 20
Degree of polynomial: 20
Mean squared error on training data: 0.00137982
Mean squared error on test data: 1761.43341615
Degree of polynomial: 21
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 21
Mean squared error on training data: 0.00118170
Mean squared error on test data: 15061.31603087
Degree of polynomial: 22
Mean squared error on training data: 0.00092354
Mean squared error on test data: 890.63488525
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 23
Degree of polynomial: 23
Mean squared error on training data: 0.00085887
Mean squared error on test data: 5483.16796929
Degree of polynomial: 24
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 24
Mean squared error on training data: 0.00084589
Mean squared error on test data: 1695.57143061
Degree of polynomial: 25
Mean squared error on training data: 0.00078806
Mean squared error on test data: 131343.30655001
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 26
Degree of polynomial: 26
Mean squared error on training data: 0.00076916
Mean squared error on test data: 17709.14370264
Degree of polynomial: 27
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 27
Mean squared error on training data: 0.00068970
Mean squared error on test data: 2975.38903780
Degree of polynomial: 28
Mean squared error on training data: 0.00062588
Mean squared error on test data: 3848.64522721
</pre></div>
</div>
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Degree of polynomial: 29
Degree of polynomial: 29
Mean squared error on training data: 0.00060728
Mean squared error on test data: 2988.64001211
</pre></div>
@@ -1243,7 +1276,7 @@ Mean squared error on test data: 2988.64001211
plt.plot(polynomial, np.log10(testerror), label=&#39;Test Error&#39;)
</pre></div>
</div>
<img alt="_images/chapter3_41_11.png" src="_images/chapter3_41_11.png" />
<img alt="_images/chapter3_41_10.png" src="_images/chapter3_41_10.png" />
</div>
</div>
</div>
@@ -1525,6 +1558,294 @@ cross-validation (LOOCV).</p>
</div>
</div>
</div>
<div class="section" id="exercises-and-projects">
<h2><span class="section-number">5.6. </span>Exercises and Projects<a class="headerlink" href="#exercises-and-projects" title="Permalink to this headline"></a></h2>
<p>The main aim of this project is to study in more detail various
regression methods, including the Ordinary Least Squares (OLS) method,
The total score is <strong>100</strong> points. Each subtask has its own final score.</p>
<p>We will first study how to fit polynomials to a specific
two-dimensional function called <a class="reference external" href="http://www.dtic.mil/dtic/tr/fulltext/u2/a081688.pdf">Franke’s
function</a>. This
is a function which has been widely used when testing various
interpolation and fitting algorithms. Furthermore, after having
established the model and the method, we will employ resamling
techniques such as cross-validation and/or bootstrap in order to perform a
proper assessment of our models. We will also study in detail the
so-called Bias-Variance trade off.</p>
<p>The Franke function, which is a weighted sum of four exponentials reads as follows</p>
<div class="math notranslate nohighlight">
\[\begin{split}
\begin{align*}
f(x,y) &amp;= \frac{3}{4}\exp{\left(-\frac{(9x-2)^2}{4} - \frac{(9y-2)^2}{4}\right)}+\frac{3}{4}\exp{\left(-\frac{(9x+1)^2}{49}- \frac{(9y+1)}{10}\right)} \\
&amp;+\frac{1}{2}\exp{\left(-\frac{(9x-7)^2}{4} - \frac{(9y-3)^2}{4}\right)} -\frac{1}{5}\exp{\left(-(9x-4)^2 - (9y-7)^2\right) }.
\end{align*}
\end{split}\]</div>
<p>The function will be defined for <span class="math notranslate nohighlight">\(x,y\in [0,1]\)</span>. Our first step will
be to perform an OLS regression analysis of this function, trying out
a polynomial fit with an <span class="math notranslate nohighlight">\(x\)</span> and <span class="math notranslate nohighlight">\(y\)</span> dependence of the form <span class="math notranslate nohighlight">\([x, y,
x^2, y^2, xy, \dots]\)</span>. We will also include bootstrap first as
a resampling technique. After that we will include the cross-validation technique. As in homeworks 1 and 2, we can use a uniform
distribution to set up the arrays of values for <span class="math notranslate nohighlight">\(x\)</span> and <span class="math notranslate nohighlight">\(y\)</span>, or as in
the example below just a set of fixed
values for <span class="math notranslate nohighlight">\(x\)</span> and <span class="math notranslate nohighlight">\(y\)</span> with a given step
size. We will fit a
function (for example a polynomial) of <span class="math notranslate nohighlight">\(x\)</span> and <span class="math notranslate nohighlight">\(y\)</span>. Thereafter we
will repeat much of the same procedure using the Ridge and Lasso
regression methods, introducing thus a dependence on the bias
(penalty) <span class="math notranslate nohighlight">\(\lambda\)</span>.</p>
<p>Finally we are going to use (real) digital terrain data and try to
reproduce these data using the same methods. We will also try to go
beyond the second-order polynomials metioned above and explore
which polynomial fits the data best.</p>
<p>The Python code for the Franke function is included here (it performs also a three-dimensional plot of it)</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">mpl_toolkits.mplot3d</span> <span class="kn">import</span> <span class="n">Axes3D</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">from</span> <span class="nn">matplotlib</span> <span class="kn">import</span> <span class="n">cm</span>
<span class="kn">from</span> <span class="nn">matplotlib.ticker</span> <span class="kn">import</span> <span class="n">LinearLocator</span><span class="p">,</span> <span class="n">FormatStrFormatter</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">from</span> <span class="nn">random</span> <span class="kn">import</span> <span class="n">random</span><span class="p">,</span> <span class="n">seed</span>
<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
<span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">gca</span><span class="p">(</span><span class="n">projection</span><span class="o">=</span><span class="s1">&#39;3d&#39;</span><span class="p">)</span>
<span class="c1"># Make data.</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mf">0.05</span><span class="p">)</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mf">0.05</span><span class="p">)</span>
<span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">meshgrid</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">FrankeFunction</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">):</span>
<span class="n">term1</span> <span class="o">=</span> <span class="mf">0.75</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="p">(</span><span class="mf">0.25</span><span class="o">*</span><span class="p">(</span><span class="mi">9</span><span class="o">*</span><span class="n">x</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> <span class="o">-</span> <span class="mf">0.25</span><span class="o">*</span><span class="p">((</span><span class="mi">9</span><span class="o">*</span><span class="n">y</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
<span class="n">term2</span> <span class="o">=</span> <span class="mf">0.75</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="p">((</span><span class="mi">9</span><span class="o">*</span><span class="n">x</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="mf">49.0</span> <span class="o">-</span> <span class="mf">0.1</span><span class="o">*</span><span class="p">(</span><span class="mi">9</span><span class="o">*</span><span class="n">y</span><span class="o">+</span><span class="mi">1</span><span class="p">))</span>
<span class="n">term3</span> <span class="o">=</span> <span class="mf">0.5</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="p">(</span><span class="mi">9</span><span class="o">*</span><span class="n">x</span><span class="o">-</span><span class="mi">7</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="mf">4.0</span> <span class="o">-</span> <span class="mf">0.25</span><span class="o">*</span><span class="p">((</span><span class="mi">9</span><span class="o">*</span><span class="n">y</span><span class="o">-</span><span class="mi">3</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
<span class="n">term4</span> <span class="o">=</span> <span class="o">-</span><span class="mf">0.2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="p">(</span><span class="mi">9</span><span class="o">*</span><span class="n">x</span><span class="o">-</span><span class="mi">4</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="p">(</span><span class="mi">9</span><span class="o">*</span><span class="n">y</span><span class="o">-</span><span class="mi">7</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
<span class="k">return</span> <span class="n">term1</span> <span class="o">+</span> <span class="n">term2</span> <span class="o">+</span> <span class="n">term3</span> <span class="o">+</span> <span class="n">term4</span>
<span class="n">z</span> <span class="o">=</span> <span class="n">FrankeFunction</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
<span class="c1"># Plot the surface.</span>
<span class="n">surf</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">plot_surface</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">cm</span><span class="o">.</span><span class="n">coolwarm</span><span class="p">,</span>
<span class="n">linewidth</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">antialiased</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
<span class="c1"># Customize the z axis.</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_zlim</span><span class="p">(</span><span class="o">-</span><span class="mf">0.10</span><span class="p">,</span> <span class="mf">1.40</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">zaxis</span><span class="o">.</span><span class="n">set_major_locator</span><span class="p">(</span><span class="n">LinearLocator</span><span class="p">(</span><span class="mi">10</span><span class="p">))</span>
<span class="n">ax</span><span class="o">.</span><span class="n">zaxis</span><span class="o">.</span><span class="n">set_major_formatter</span><span class="p">(</span><span class="n">FormatStrFormatter</span><span class="p">(</span><span class="s1">&#39;</span><span class="si">%.02f</span><span class="s1">&#39;</span><span class="p">))</span>
<span class="c1"># Add a color bar which maps values to colors.</span>
<span class="n">fig</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">surf</span><span class="p">,</span> <span class="n">shrink</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">aspect</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
</div>
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<img alt="_images/chapter3_54_0.png" src="_images/chapter3_54_0.png" />
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</div>
<div class="section" id="exercise-ordinary-least-square-ols-on-the-franke-function">
<h3><span class="section-number">5.6.1. </span>Exercise: Ordinary Least Square (OLS) on the Franke function<a class="headerlink" href="#exercise-ordinary-least-square-ols-on-the-franke-function" title="Permalink to this headline"></a></h3>
<p>We will generate our own dataset for a function
<span class="math notranslate nohighlight">\(\mathrm{FrankeFunction}(x,y)\)</span> with <span class="math notranslate nohighlight">\(x,y \in [0,1]\)</span>. The function
<span class="math notranslate nohighlight">\(f(x,y)\)</span> is the Franke function. You should explore also the addition
of an added stochastic noise to this function using the normal
distribution <span class="math notranslate nohighlight">\(N(0,1)\)</span>.</p>
<p><em>Write your own code</em> (using either a matrix inversion or a singular
value decomposition from e.g., <strong>numpy</strong> ) or use your code from
homeworks 1 and 2 and perform a standard least square regression
analysis using polynomials in <span class="math notranslate nohighlight">\(x\)</span> and <span class="math notranslate nohighlight">\(y\)</span> up to fifth order. Find the
<a class="reference external" href="https://en.wikipedia.org/wiki/Confidence_interval">confidence intervals</a> of the parameters (estimators) <span class="math notranslate nohighlight">\(\beta\)</span> by computing their
variances, evaluate the Mean Squared error (MSE)</p>
<div class="math notranslate nohighlight">
\[
MSE(\hat{y},\hat{\tilde{y}}) = \frac{1}{n}
\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2,
\]</div>
<p>and the <span class="math notranslate nohighlight">\(R^2\)</span> score function. If <span class="math notranslate nohighlight">\(\tilde{\hat{y}}_i\)</span> is the predicted
value of the <span class="math notranslate nohighlight">\(i-th\)</span> sample and <span class="math notranslate nohighlight">\(y_i\)</span> is the corresponding true value,
then the score <span class="math notranslate nohighlight">\(R^2\)</span> is defined as</p>
<div class="math notranslate nohighlight">
\[
R^2(\hat{y}, \tilde{\hat{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2},
\]</div>
<p>where we have defined the mean value of <span class="math notranslate nohighlight">\(\hat{y}\)</span> as</p>
<div class="math notranslate nohighlight">
\[
\bar{y} = \frac{1}{n} \sum_{i=0}^{n - 1} y_i.
\]</div>
<p>Your code has to include a scaling of the data (for example by
subtracting the mean value), and
a split of the data in training and test data. For this exercise you can
either write your own code or use for example the function for
splitting training data provided by the library <strong>Scikit-Learn</strong> (make
sure you have installed it). This function is called
<span class="math notranslate nohighlight">\(train\_test\_split\)</span>. <strong>You should present a critical discussion of why and how you have scaled or not scaled the data</strong>.</p>
<p>It is normal in essentially all Machine Learning studies to split the
data in a training set and a test set (eventually also an additional
validation set). There
is no explicit recipe for how much data should be included as training
data and say test data. An accepted rule of thumb is to use
approximately <span class="math notranslate nohighlight">\(2/3\)</span> to <span class="math notranslate nohighlight">\(4/5\)</span> of the data as training data.</p>
<p>You can easily reuse the solutions to your exercises from week 35 and week 36.</p>
</div>
<div class="section" id="exercise-bias-variance-trade-off-and-resampling-techniques">
<h3><span class="section-number">5.6.2. </span>Exercise: Bias-variance trade-off and resampling techniques<a class="headerlink" href="#exercise-bias-variance-trade-off-and-resampling-techniques" title="Permalink to this headline"></a></h3>
<p>Our aim here is to study the bias-variance trade-off by implementing the <strong>bootstrap</strong> resampling technique.</p>
<p>With a code which does OLS and includes resampling techniques,
we will now discuss the bias-variance trade-off in the context of
continuous predictions such as regression. However, many of the
intuitions and ideas discussed here also carry over to classification
tasks and basically all Machine Learning algorithms.</p>
<p>Before you perform an analysis of the bias-variance trade-off on your test data, make
first a figure similar to Fig. 2.11 of Hastie, Tibshirani, and
Friedman. Figure 2.11 of this reference displays only the test and training MSEs. The test MSE can be used to
indicate possible regions of low/high bias and variance. You will most likely not get an
equally smooth curve!</p>
<p>With this result we move on to the bias-variance trade-off analysis.</p>
<p>Consider a
dataset <span class="math notranslate nohighlight">\(\mathcal{L}\)</span> consisting of the data
<span class="math notranslate nohighlight">\(\mathbf{X}_\mathcal{L}=\{(y_j, \boldsymbol{x}_j), j=0\ldots n-1\}\)</span>.</p>
<p>Let us assume that the true data is generated from a noisy model</p>
<div class="math notranslate nohighlight">
\[
\boldsymbol{y}=f(\boldsymbol{x}) + \boldsymbol{\epsilon}.
\]</div>
<p>Here <span class="math notranslate nohighlight">\(\epsilon\)</span> is normally distributed with mean zero and standard
deviation <span class="math notranslate nohighlight">\(\sigma^2\)</span>.</p>
<p>In our derivation of the ordinary least squares method we defined then
an approximation to the function <span class="math notranslate nohighlight">\(f\)</span> in terms of the parameters
<span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span> and the design matrix <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span> which embody our model,
that is <span class="math notranslate nohighlight">\(\boldsymbol{\tilde{y}}=\boldsymbol{X}\boldsymbol{\beta}\)</span>.</p>
<p>The parameters <span class="math notranslate nohighlight">\(\boldsymbol{\beta}\)</span> are in turn found by optimizing the means
squared error via the so-called cost function</p>
<div class="math notranslate nohighlight">
\[
C(\boldsymbol{X},\boldsymbol{\beta}) =\frac{1}{n}\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2=\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right].
\]</div>
<p>Here the expected value <span class="math notranslate nohighlight">\(\mathbb{E}\)</span> is the sample value.</p>
<p>Show that you can rewrite this as</p>
<div class="math notranslate nohighlight">
\[
\mathbb{E}\left[(\boldsymbol{y}-\boldsymbol{\tilde{y}})^2\right]=\frac{1}{n}\sum_i(f_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2+\frac{1}{n}\sum_i(\tilde{y}_i-\mathbb{E}\left[\boldsymbol{\tilde{y}}\right])^2+\sigma^2.
\]</div>
<p>Explain what the terms mean, which one is the bias and which one is
the variance and discuss their interpretations.</p>
<p>Perform then a bias-variance analysis of the Franke function by
studying the MSE value as function of the complexity of your model.</p>
<p>Discuss the bias and variance trade-off as function
of your model complexity (the degree of the polynomial) and the number
of data points, and possibly also your training and test data using the <strong>bootstrap</strong> resampling method.</p>
<p>Note also that when you calculate the bias, in all applications you don’t know the function values <span class="math notranslate nohighlight">\(f_i\)</span>. You would hence replace them with the actual data points <span class="math notranslate nohighlight">\(y_i\)</span>.</p>
</div>
<div class="section" id="exercise-cross-validation-as-resampling-techniques-adding-more-complexity">
<h3><span class="section-number">5.6.3. </span>Exercise: Cross-validation as resampling techniques, adding more complexity<a class="headerlink" href="#exercise-cross-validation-as-resampling-techniques-adding-more-complexity" title="Permalink to this headline"></a></h3>
<p>The aim here is to write your own code for another widely popular
resampling technique, the so-called cross-validation method. Again,
before you start with cross-validation approach, you should scale your
data.</p>
<p>Implement the <span class="math notranslate nohighlight">\(k\)</span>-fold cross-validation algorithm (write your own
code) and evaluate again the MSE function resulting
from the test folds. You can compare your own code with that from
<strong>Scikit-Learn</strong> if needed.</p>
<p>Compare the MSE you get from your cross-validation code with the one
you got from your <strong>bootstrap</strong> code. Comment your results. Try <span class="math notranslate nohighlight">\(5-10\)</span>
folds. You can also compare your own cross-validation code with the
one provided by <strong>Scikit-Learn</strong>.</p>
</div>
<div class="section" id="exercise-ridge-regression-on-the-franke-function-with-resampling">
<h3><span class="section-number">5.6.4. </span>Exercise: Ridge Regression on the Franke function with resampling<a class="headerlink" href="#exercise-ridge-regression-on-the-franke-function-with-resampling" title="Permalink to this headline"></a></h3>
<p>Write your own code for the Ridge method, either using matrix
inversion or the singular value decomposition as done in the previous
exercise. Perform the same bootstrap analysis as in the
Exercise 2 (for the same polynomials) and the cross-validation in exercise 3 but now for different values of <span class="math notranslate nohighlight">\(\lambda\)</span>. Compare and
analyze your results with those obtained in exercises 1-3. Study the
dependence on <span class="math notranslate nohighlight">\(\lambda\)</span>.</p>
<p>Study also the bias-variance trade-off as function of various values of
the parameter <span class="math notranslate nohighlight">\(\lambda\)</span>. For the bias-variance trade-off, use the <strong>bootstrap</strong> resampling method. Comment your results.</p>
</div>
<div class="section" id="exercise-lasso-regression-on-the-franke-function-with-resampling">
<h3><span class="section-number">5.6.5. </span>Exercise: Lasso Regression on the Franke function with resampling<a class="headerlink" href="#exercise-lasso-regression-on-the-franke-function-with-resampling" title="Permalink to this headline"></a></h3>
<p>This exercise is essentially a repeat of the previous two ones, but now
with Lasso regression. Write either your own code (difficult and optional) or, in this case,
you can also use the functionalities of <strong>Scikit-Learn</strong> (recommended).
Give a
critical discussion of the three methods and a judgement of which
model fits the data best. Perform here as well an analysis of the bias-variance trade-off using the <strong>bootstrap</strong> resampling technique and an analysis of the mean squared error using cross-validation.</p>
</div>
<div class="section" id="exercise-analysis-of-real-data">
<h3><span class="section-number">5.6.6. </span>Exercise: Analysis of real data<a class="headerlink" href="#exercise-analysis-of-real-data" title="Permalink to this headline"></a></h3>
<p>With our codes functioning and having been tested properly on a
simpler function we are now ready to look at real data. We will
essentially repeat in this exercise what was done in exercises 1-5. However, we
need first to download the data and prepare properly the inputs to our
codes. We are going to download digital terrain data from the website
<a class="reference external" href="https://earthexplorer.usgs.gov/">https://earthexplorer.usgs.gov/</a>,</p>
<p>Or, if you prefer, we have placed selected datafiles at <a class="reference external" href="https://github.com/CompPhysics/MachineLearning/tree/master/doc/Projects/2021/Project1/DataFiles">https://github.com/CompPhysics/MachineLearning/tree/master/doc/Projects/2021/Project1/DataFiles</a></p>
<p>In order to obtain data for a specific region, you need to register as
a user (free) at this website and then decide upon which area you want
to fetch the digital terrain data from. In order to be able to read
the data properly, you need to specify that the format should be <strong>SRTM
Arc-Second Global</strong> and download the data as a <strong>GeoTIF</strong> file. The
files are then stored in <em>tif</em> format which can be imported into a
Python program using</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">scipy</span><span class="o">.</span><span class="n">misc</span><span class="o">.</span><span class="n">imread</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output traceback highlight-ipythontb notranslate"><div class="highlight"><pre><span></span><span class="gt">---------------------------------------------------------------------------</span>
<span class="ne">NameError</span><span class="g g-Whitespace"> </span>Traceback (most recent call last)
<span class="o">&lt;</span><span class="n">ipython</span><span class="o">-</span><span class="nb">input</span><span class="o">-</span><span class="mi">10</span><span class="o">-</span><span class="n">d985fb40c43d</span><span class="o">&gt;</span> <span class="ow">in</span> <span class="o">&lt;</span><span class="n">module</span><span class="o">&gt;</span>
<span class="ne">----&gt; </span><span class="mi">1</span> <span class="n">scipy</span><span class="o">.</span><span class="n">misc</span><span class="o">.</span><span class="n">imread</span>
<span class="ne">NameError</span>: name &#39;scipy&#39; is not defined
</pre></div>
</div>
</div>
</div>
<p>Here is a simple part of a Python code which reads and plots the data
from such files</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">from</span> <span class="nn">imageio</span> <span class="kn">import</span> <span class="n">imread</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">from</span> <span class="nn">mpl_toolkits.mplot3d</span> <span class="kn">import</span> <span class="n">Axes3D</span>
<span class="kn">from</span> <span class="nn">matplotlib</span> <span class="kn">import</span> <span class="n">cm</span>
<span class="c1"># Load the terrain</span>
<span class="n">terrain1</span> <span class="o">=</span> <span class="n">imread</span><span class="p">(</span><span class="s1">&#39;SRTM_data_Norway_1.tif&#39;</span><span class="p">)</span>
<span class="c1"># Show the terrain</span>
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">&#39;Terrain over Norway 1&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">terrain1</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="s1">&#39;gray&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;X&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;Y&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
</div>
</div>
<p>If you should have problems in downloading the digital terrain data,
we provide two examples under the data folder of project 1. One is
from a region close to Stavanger in Norway and the other Møsvatn
Austfjell, again in Norway.
Feel free to produce your own terrain data.</p>
<p>Alternatively, if you would like to use another data set, feel free to do so. This could be data close to your reseach area or simply a data set you found interesting. See for example <a class="reference external" href="https://www.kaggle.com/datasets">kaggle.com</a> for examples.</p>
<p>Our final part deals with the parameterization of your digital terrain
data (or your own data). We will apply all three methods for linear regression, the same type (or higher order) of polynomial
approximation and cross-validation as resampling technique to evaluate which
model fits the data best.</p>
<p>At the end, you should present a critical evaluation of your results
and discuss the applicability of these regression methods to the type
of data presented here (either the terrain data we propose or other data sets).</p>
</div>
</div>
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