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<div class=" highlight hl-ipython3"><pre><span class="n">jupyter</span> <span class="n">nbconvert</span> <span class="o">--</span><span class="n">to</span> <span class="n">slides</span> <span class="o">--</span><span class="n">reveal</span><span class="o">-</span><span class="n">prefix</span> <span class="s">"http://cdn.jsdelivr.net/reveal.js/2.6.2"</span>
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<div class=" highlight hl-ipython3"><pre><span class="kn">import</span> <span class="nn">pods</span>
<span class="kn">from</span> <span class="nn">matplotlib</span> <span class="k">import</span> <span class="n">pyplot</span> <span class="k">as</span> <span class="n">plt</span>
<span class="o">%</span><span class="k">matplotlib</span> inline
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<h1 id="MLAI-Week-3:-Regression">MLAI Week 3: Regression<a class="anchor-link" href="#MLAI-Week-3:-Regression">¶</a></h1><h3 id="Neil-D.-Lawrence">Neil D. Lawrence<a class="anchor-link" href="#Neil-D.-Lawrence">¶</a></h3><h3 id="13th-October-2015">13th October 2015<a class="anchor-link" href="#13th-October-2015">¶</a></h3>
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<h3 id="Review">Review<a class="anchor-link" href="#Review">¶</a></h3><ul>
<li>Last time: Looked at objective functions for movie recommendation.</li>
<li>Minimized sum of squares objective by steepest descent and stochastic gradients.</li>
<li>This time: explore least squares for regression.</li>
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<h3 id="Regression-Examples">Regression Examples<a class="anchor-link" href="#Regression-Examples">¶</a></h3><ul>
<li><p>Predict a real value, $y_i$ given some inputs
$x_i$.</p>
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<li><p>Predict quality of meat given spectral measurements (Tecator data).</p>
</li>
<li><p>Radiocarbon dating, the C14 calibration curve: predict age given
quantity of C14 isotope.</p>
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<li><p>Predict quality of different Go or Backgammon moves given expert
rated training data.</p>
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<h3 id="Olympic-100m-Data">Olympic 100m Data<a class="anchor-link" href="#Olympic-100m-Data">¶</a></h3><ul>
<li>Gold medal times for Olympic 100 m runners since 1896.</li>
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<p><img src="./diagrams/100m_final_start.jpg" alt="image">
Image from Wikimedia Commons <a href="http://bit.ly/191adDC">http://bit.ly/191adDC</a></p>
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<h3 id="Olympic-100m-Data">Olympic 100m Data<a class="anchor-link" href="#Olympic-100m-Data">¶</a></h3>
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<div class=" highlight hl-ipython3"><pre><span class="n">data</span> <span class="o">=</span> <span class="n">pods</span><span class="o">.</span><span class="n">datasets</span><span class="o">.</span><span class="n">olympic_100m_men</span><span class="p">()</span>
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<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">data</span><span class="p">[</span><span class="s">'X'</span><span class="p">],</span> <span class="n">data</span><span class="p">[</span><span class="s">'Y'</span><span class="p">],</span> <span class="s">'rx'</span><span class="p">,</span> <span class="n">markersize</span><span class="o">=</span><span class="mi">10</span><span class="p">)</span>
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<pre>[<matplotlib.lines.Line2D at 0x1092e47f0>]</pre>
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<h3 id="Olympic-Marathon-Data">Olympic Marathon Data<a class="anchor-link" href="#Olympic-Marathon-Data">¶</a></h3><ul>
<li><p>Gold medal times for Olympic Marathon since 1896.</p>
</li>
<li><p>Marathons before 1924 didn’t have a standardised distance.</p>
</li>
<li><p>Present results using pace per km.</p>
</li>
<li><p>In 1904 Marathon was badly organised leading to very slow times.</p>
</li>
</ul>
<p><img src="./diagrams/Stephen_Kiprotich.jpg" alt="image">
Image from Wikimedia Commons <a href="http://bit.ly/16kMKHQ">http://bit.ly/16kMKHQ</a></p>
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<h3 id="Olympic-Marathon-Data">Olympic Marathon Data<a class="anchor-link" href="#Olympic-Marathon-Data">¶</a></h3>
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<div class="prompt input_prompt">In [11]:</div>
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<div class=" highlight hl-ipython3"><pre><span class="n">data</span> <span class="o">=</span> <span class="n">pods</span><span class="o">.</span><span class="n">datasets</span><span class="o">.</span><span class="n">olympic_marathon_men</span><span class="p">()</span>
<span class="n">f</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">7</span><span class="p">,</span><span class="mi">7</span><span class="p">))</span>
<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">data</span><span class="p">[</span><span class="s">'X'</span><span class="p">],</span> <span class="n">data</span><span class="p">[</span><span class="s">'Y'</span><span class="p">],</span> <span class="s">'rx'</span><span class="p">,</span><span class="n">markersize</span><span class="o">=</span><span class="mi">10</span><span class="p">)</span>
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<div class="output_area"><div class="prompt output_prompt">Out[11]:</div>
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<pre>[<matplotlib.lines.Line2D at 0x109612ba8>]</pre>
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<h3 id="What-is-Machine-Learning?">What is Machine Learning?<a class="anchor-link" href="#What-is-Machine-Learning?">¶</a></h3>$$ \text{data} + \text{model} = \text{prediction}$$<ul>
<li><p>$\text{data}$ : observations, could be actively or passively
acquired (meta-data).</p>
</li>
<li><p>$\text{model}$ : assumptions, based on previous experience (other data!
transfer learning etc), or beliefs about the regularities of
the universe. Inductive bias.</p>
</li>
<li><p>$\text{prediction}$ : an action to be taken or a categorization or a
quality score.</p>
</li>
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<h3 id="Regression:-Linear-Releationship">Regression: Linear Releationship<a class="anchor-link" href="#Regression:-Linear-Releationship">¶</a></h3>$$y_i = m x_i + c$$<ul>
<li><p>$y_i$ : winning time/pace.</p>
</li>
<li><p>$x_i$ : year of Olympics.</p>
</li>
<li><p>$m$ : rate of improvement over time.</p>
</li>
<li><p>$c$ : winning time at year 0.</p>
</li>
</ul>
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<h1 id="Overdetermined-System">Overdetermined System<a class="anchor-link" href="#Overdetermined-System">¶</a></h1><p><img src="diagrams/straightLine7.svg" alt=""></p>
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<h1 id="$y-=-mx-+-c$">$y = mx + c$<a class="anchor-link" href="#$y-=-mx-+-c$">¶</a></h1><p>point 1: $x = 1$, $y=3$ $$3 = m + c$$
point 2: $x = 3$, $y=1$ $$1 = 3m + c$$
point 3: $x = 2$, $y=2.5$ $$2.5 = 2m + c$$</p>
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<p><img src="diagrams/Pierre-Simon_Laplace.png" align=center width=50%></p>
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<h1 id="$y-=-mx-+-c-+-\epsilon$">$y = mx + c + \epsilon$<a class="anchor-link" href="#$y-=-mx-+-c-+-\epsilon$">¶</a></h1><p>point 1: $x = 1$, $y=3$
$$3 = m + c + \epsilon_1$$</p>
<p>point 2: $x = 3$, $y=1$
$$1 = 3m + c + \epsilon_2$$</p>
<p>point 3: $x = 2$, $y=2.5$
$$2.5 = 2m + c + \epsilon_3$$</p>
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<h3 id="The-Gaussian-Density">The Gaussian Density<a class="anchor-link" href="#The-Gaussian-Density">¶</a></h3><ul>
<li>Perhaps the most common probability density.
\begin{align<em>}
p(y| \mu, \sigma^2) & = \frac{1}{\sqrt{2\pi\sigma^2}}\exp\left(-\frac{(y - \mu)^2}{2\sigma^2}\right)\
& \buildrel\triangle\over = \mathcal{N}(y|\mu, \sigma^2)
\end{align</em>}</li>
<li>The Gaussian density.</li>
</ul>
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<div class="prompt input_prompt">In [36]:</div>
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<div class=" highlight hl-ipython3"><pre><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="n">h</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">,</span> <span class="mi">1000</span><span class="p">)</span>
<span class="n">sigma2</span> <span class="o">=</span> <span class="mf">0.0225</span>
<span class="n">mu</span> <span class="o">=</span> <span class="mf">1.7</span>
<span class="n">p</span> <span class="o">=</span> <span class="mf">1.</span><span class="o">/</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="n">sigma2</span><span class="p">)</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="n">h</span><span class="o">-</span><span class="n">mu</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">2</span><span class="o">*</span><span class="n">sigma2</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
<span class="n">f2</span><span class="p">,</span> <span class="n">ax2</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">7</span><span class="p">,</span> <span class="mf">3.5</span><span class="p">))</span>
<span class="n">ax2</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">h</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="s">'b-'</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
<span class="n">ylim</span> <span class="o">=</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
<span class="n">ax2</span><span class="o">.</span><span class="n">vlines</span><span class="p">(</span><span class="n">mu</span><span class="p">,</span> <span class="n">ylim</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">ylim</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">colors</span><span class="o">=</span><span class="s">'r'</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
<span class="n">ax2</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">(</span><span class="n">ylim</span><span class="p">)</span>
<span class="n">ax2</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">(</span><span class="mf">1.4</span><span class="p">,</span> <span class="mf">2.0</span><span class="p">)</span>
<span class="n">ax2</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s">'$h/m$'</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">20</span><span class="p">)</span>
<span class="n">ax2</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s">'$p(h|\mu, \sigma^2)$'</span><span class="p">,</span> <span class="n">fontsize</span> <span class="o">=</span> <span class="mi">20</span><span class="p">)</span>
<span class="n">f2</span><span class="o">.</span><span class="n">savefig</span><span class="p">(</span><span class="s">'./diagrams/gaussianOfHeight.svg'</span><span class="p">)</span>
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