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Bayesian Rugby.slides.html
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<p><h1 class="bigtitle">Rugby Analytics</h1></p>
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<li>Peadar Coyle</li>
<li>Will give this at PyData Berlin 2015</li>
<li>Data Science Meetup Luxembourg May 2015</li>
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<h1 id="Contents:-Probabilistic-Programming-applied-to-Rugby">Contents: Probabilistic Programming applied to Rugby<a class="anchor-link" href="#Contents:-Probabilistic-Programming-applied-to-Rugby">¶</a></h1><ul>
<li>I'll discuss what probabilistic programming is, why should you care and how to use PyMC from Python to implement these methods. </li>
<li>I'll be applying these methods to studying the problem of 'rugby sports analytics' particularly how to model the winning team in the recent Six Nations in Rugby. </li>
<li>I will discuss the framework and how I was able to quickly and easily produce an innovative and powerful model as a non-expert.</li>
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<h2 id="Who-am-I?">Who am I?<a class="anchor-link" href="#Who-am-I?">¶</a></h2><ul>
<li>I'm a Data Analytics Professional based in Luxembourg</li>
<li>I currently work for Vodafone</li>
<li>My intellectual background is in Physics and Mathematics</li>
<li>I've made open source contributions to Pandas and Probabilistic Programming and Bayesian Methods for Hackers. </li>
<li>I've helped companies solve analytics challenges in Supply Chain Management, Air Traffic Analysis and with Customer Analytics</li>
<li>All opinions are my own!</li>
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<h1 id="All-Sports-Commentary!">All Sports Commentary!<a class="anchor-link" href="#All-Sports-Commentary!">¶</a></h1><p><img src="http://imgs.xkcd.com/comics/sports.png" align="middle"></p>
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<li>Attribution: Xkcd</li>
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<h1 id="How-can-statistics-help-with-sports?">How can statistics help with sports?<a class="anchor-link" href="#How-can-statistics-help-with-sports?">¶</a></h1><ul>
<li>Well fundamentally a Rugby game is a predictible event.</li>
<li>How do we generate a model to predict the outcome of a tournament?</li>
<li>How do we quantify our uncertainty in our model?</li>
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<h1 id="What-influenced-me-on-this?">What influenced me on this?<a class="anchor-link" href="#What-influenced-me-on-this?">¶</a></h1><p><img src="http://blog.quantopian.com/wp-content/uploads/2014/12/GraphicalModel.jpg">
Attribution: Quantopian blog</p>
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<h1 id="What's-wrong-with-statistics">What's wrong with statistics<a class="anchor-link" href="#What's-wrong-with-statistics">¶</a></h1><ul>
<li><p>Models should not be built for mathematical convenience (e.g. normality assumption), but to most accurately model the data.</p>
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<li><p>Pre-specified models, like frequentist statistics, make many assumptions that are all to easily violated.</p>
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<h1 id=""The-purpose-of-computation-is-insight,-not-numbers."----Richard-Hamming">"The purpose of computation is insight, not numbers." -- Richard Hamming<a class="anchor-link" href="#"The-purpose-of-computation-is-insight,-not-numbers."----Richard-Hamming">¶</a></h1>
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<h1 id="What-is-Bayesian-Statistics?">What is Bayesian Statistics?<a class="anchor-link" href="#What-is-Bayesian-Statistics?">¶</a></h1><ul>
<li>At the core: formula to update our beliefs after having observed data (Bayes formula)</li>
<li>Implies that we have a prior belief about the world.</li>
<li>Updated beliefs after observing data is called posterior.</li>
<li>Beliefs are represented using random variables.</li>
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<h3 id="So-what-problem-could-I-apply-Bayesian-models-to?">So what problem could I apply Bayesian models to?<a class="anchor-link" href="#So-what-problem-could-I-apply-Bayesian-models-to?">¶</a></h3><ul>
<li>Rugby Analysis!
<img src="http://the-office-bar.eu/files/2013/06/Rugby-ball.jpg", width="50%" height="50%"/>
Attribution: The-office-bar.eu</li>
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<h1 id="Bayesian-Rugby">Bayesian Rugby<a class="anchor-link" href="#Bayesian-Rugby">¶</a></h1><p>I came across the following blog post on <a href="http://danielweitzenfeld.github.io/passtheroc/blog/2014/10/28/bayes-premier-league/">http://danielweitzenfeld.github.io/passtheroc/blog/2014/10/28/bayes-premier-league/</a></p>
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<li>Based on the work of <a href="http://www.statistica.it/gianluca/Research/BaioBlangiardo.pdf">Baio and Blangiardo</a></li>
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<li>In this talk, I'm going to reproduce the first model described in the paper using pymc.</li>
<li>Since I am a rugby fan I decide to apply the results of the paper Bayesian Football to the Six Nations.</li>
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<p><img src="http://upload.wikimedia.org/wikipedia/de/thumb/0/0b/Zalando_logo.svg/1024px-Zalando_logo.svg.png"></p>
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<h2 id="What-they-did?">What they did?<a class="anchor-link" href="#What-they-did?">¶</a></h2><p>Deriving our new measuring model and verifying that it works took some effort! But it was all worth it, because now we have:</p>
<p>Automatic weight estimations for each Zalando article, which saves workers time
A reliable way to know the accuracy of our estimations
And most importantly: our warehouse workers can now focus on getting your fashion to you as quickly as possible. That’s isn’t just saving money–that’s priceless.</p>
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<h1 id="So-why-Bayesians?">So why Bayesians?<a class="anchor-link" href="#So-why-Bayesians?">¶</a></h1><ul>
<li>Probabilistic Programming is a new paradigm.</li>
<li>Attributions: My friend Thomas Wiecki influenced a lot of my thinking on this.</li>
<li>I'm going to compare Blackbox Machine Learning with scikit-learn</li>
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<p><img src='https://raw.githubusercontent.com/twiecki/pymc3_talk/master/blackbox.png'></p>
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<li>Source: <a href='https://speakerdeck.com/ogrisel/trends-in-machine-learning'>Olivier Grisel's talk on ML</a></li>
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<h1 id="Limitations-of-Machine-learning">Limitations of Machine learning<a class="anchor-link" href="#Limitations-of-Machine-learning">¶</a></h1><ul>
<li>A big limitation of Machine Learning is that most of the models are black boxes.</li>
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<p><img src='https://raw.githubusercontent.com/twiecki/pymc3_talk/master/openbox.png'></p>
<p>Source: <a href='https://speakerdeck.com/ogrisel/trends-in-machine-learning'>Olivier Grisel's talk on ML</a></p>
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<h1 id="Probabilistic-Programming---what's-the-big-deal?">Probabilistic Programming - what's the big deal?<a class="anchor-link" href="#Probabilistic-Programming---what's-the-big-deal?">¶</a></h1><ul>
<li>We are able to use data and our prior beliefs to generate a model. </li>
<li>Generating a model is extremely powerful</li>
<li>We can tell a story, which appeals to our understanding of stories. </li>
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<div class=" highlight hl-ipython3"><pre><span class="kn">import</span> <span class="nn">os</span>
<span class="kn">import</span> <span class="nn">math</span>
<span class="kn">import</span> <span class="nn">warnings</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</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">import</span> <span class="nn">pymc</span> <span class="c"># I know folks are switching to "as pm" but I'm just not there yet</span>
<span class="o">%</span><span class="k">matplotlib</span> inline
<span class="kn">import</span> <span class="nn">seaborn</span> <span class="k">as</span> <span class="nn">sns</span>
<span class="kn">from</span> <span class="nn">IPython.core.pylabtools</span> <span class="k">import</span> <span class="n">figsize</span>
<span class="kn">import</span> <span class="nn">seaborn</span> <span class="k">as</span> <span class="nn">sns</span>
<span class="n">figsize</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">8</span><span class="p">)</span>
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<h1 id="Six-Nations-Rugby">Six Nations Rugby<a class="anchor-link" href="#Six-Nations-Rugby">¶</a></h1><ul>
<li>Rugby is a physical sport popular worldwide.</li>
<li>Six Nations consists of Italy, Ireland, Scotland, England, France and Wales</li>
<li>Game consists of scoring tries (similar to touch downs) or kicking the goal.</li>
<li>Average player is something like 100kg and 1.82m tall.</li>
<li>Paul O'Connell the Irish captain is Height: 6' 6" (1.98 m) Weight: 243 lbs (110 kg)</li>
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<h1 id="They-compete-for-this!">They compete for this!<a class="anchor-link" href="#They-compete-for-this!">¶</a></h1><ul>
<li>Significant this year because the World Cup occurs in 2015.
<img src="http://www.hostelrome.com/wp-content/uploads/2015/01/Rugby_Sei-Nazioni-Uomini.jpg"></li>
<li>Photo: Hostelrome</li>
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<h2 id="Motivation">Motivation<a class="anchor-link" href="#Motivation">¶</a></h2><ul>
<li>Someone in the Sports Analytics community made the point
(I'm paraphrasing here) that your estimate of one team's
strength depends on your estimate of all the others.
The conclusions you draw from team X beating team Y depends how strong team
Y is, which in turn depends on the conclusions you draw from team Y's other games, which in turn depends on how strong Y's opponents were, etc.</li>
<li>Ireland are a stronger team than Italy for example - but by how much?</li>
<li>Source for Results 2014 are Wikipedia. </li>
<li>I handcrafted these results</li>
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<div class=" highlight hl-ipython3"><pre><span class="n">DATA_DIR</span> <span class="o">=</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">os</span><span class="o">.</span><span class="n">getcwd</span><span class="p">(),</span> <span class="s">'data/'</span><span class="p">)</span>
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<div class=" highlight hl-ipython3"><pre><span class="c">#The results_2014 is a handcrafted results table from Wikipedia</span>
<span class="n">data_file</span> <span class="o">=</span> <span class="n">DATA_DIR</span> <span class="o">+</span> <span class="s">'results_2014.csv'</span>
<span class="n">df</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_csv</span><span class="p">(</span><span class="n">data_file</span><span class="p">,</span> <span class="n">sep</span><span class="o">=</span><span class="s">','</span><span class="p">)</span>
<span class="n">df</span><span class="o">.</span><span class="n">tail</span><span class="p">()</span>
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<th></th>
<th>home_team</th>
<th>away_team</th>
<th>home_score</th>
<th>away_score</th>
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<th>10</th>
<td>Scotland</td>
<td>France</td>
<td>17</td>
<td>19</td>
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<td>England</td>
<td>Wales</td>
<td>29</td>
<td>18</td>
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<th>12</th>
<td>Italy</td>
<td>England</td>
<td>11</td>
<td>52</td>
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<td>Scotland</td>
<td>51</td>
<td>3</td>
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<td>Ireland</td>
<td>20</td>
<td>22</td>
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<div class=" highlight hl-ipython3"><pre><span class="n">teams</span> <span class="o">=</span> <span class="n">df</span><span class="o">.</span><span class="n">home_team</span><span class="o">.</span><span class="n">unique</span><span class="p">()</span>
<span class="n">teams</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">teams</span><span class="p">,</span> <span class="n">columns</span><span class="o">=</span><span class="p">[</span><span class="s">'team'</span><span class="p">])</span>
<span class="n">teams</span><span class="p">[</span><span class="s">'i'</span><span class="p">]</span> <span class="o">=</span> <span class="n">teams</span><span class="o">.</span><span class="n">index</span>
<span class="n">teams</span><span class="o">.</span><span class="n">head</span><span class="p">()</span>
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<table border="1" class="dataframe">
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<th></th>
<th>team</th>
<th>i</th>
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<th>1</th>
<td>France</td>
<td>1</td>
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<th>2</th>
<td>Ireland</td>
<td>2</td>
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<th>3</th>
<td>Scotland</td>
<td>3</td>
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<th>4</th>
<td>Italy</td>
<td>4</td>
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<li>Now we need to do some merging and munging</li>
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<div class=" highlight hl-ipython3"><pre><span class="n">df</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">merge</span><span class="p">(</span><span class="n">df</span><span class="p">,</span> <span class="n">teams</span><span class="p">,</span> <span class="n">left_on</span><span class="o">=</span><span class="s">'home_team'</span><span class="p">,</span> <span class="n">right_on</span><span class="o">=</span><span class="s">'team'</span><span class="p">,</span> <span class="n">how</span><span class="o">=</span><span class="s">'left'</span><span class="p">)</span>
<span class="n">df</span> <span class="o">=</span> <span class="n">df</span><span class="o">.</span><span class="n">rename</span><span class="p">(</span><span class="n">columns</span> <span class="o">=</span> <span class="p">{</span><span class="s">'i'</span><span class="p">:</span> <span class="s">'i_home'</span><span class="p">})</span><span class="o">.</span><span class="n">drop</span><span class="p">(</span><span class="s">'team'</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">df</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">merge</span><span class="p">(</span><span class="n">df</span><span class="p">,</span> <span class="n">teams</span><span class="p">,</span> <span class="n">left_on</span><span class="o">=</span><span class="s">'away_team'</span><span class="p">,</span> <span class="n">right_on</span><span class="o">=</span><span class="s">'team'</span><span class="p">,</span> <span class="n">how</span><span class="o">=</span><span class="s">'left'</span><span class="p">)</span>
<span class="n">df</span> <span class="o">=</span> <span class="n">df</span><span class="o">.</span><span class="n">rename</span><span class="p">(</span><span class="n">columns</span> <span class="o">=</span> <span class="p">{</span><span class="s">'i'</span><span class="p">:</span> <span class="s">'i_away'</span><span class="p">})</span><span class="o">.</span><span class="n">drop</span><span class="p">(</span><span class="s">'team'</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">df</span><span class="o">.</span><span class="n">head</span><span class="p">()</span>
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<th></th>
<th>home_team</th>
<th>away_team</th>
<th>home_score</th>
<th>away_score</th>
<th>i_home</th>
<th>i_away</th>
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<th>0</th>
<td>Wales</td>
<td>Italy</td>
<td>23</td>
<td>15</td>
<td>0</td>
<td>4</td>
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<th>1</th>
<td>France</td>
<td>England</td>
<td>26</td>
<td>24</td>
<td>1</td>
<td>5</td>
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<th>2</th>
<td>Ireland</td>
<td>Scotland</td>
<td>28</td>
<td>6</td>
<td>2</td>
<td>3</td>
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<th>3</th>
<td>Ireland</td>
<td>Wales</td>
<td>26</td>
<td>3</td>
<td>2</td>
<td>0</td>
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<th>4</th>
<td>Scotland</td>
<td>England</td>
<td>0</td>
<td>20</td>
<td>3</td>
<td>5</td>
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<div class=" highlight hl-ipython3"><pre><span class="n">observed_home_goals</span> <span class="o">=</span> <span class="n">df</span><span class="o">.</span><span class="n">home_score</span><span class="o">.</span><span class="n">values</span>
<span class="n">observed_away_goals</span> <span class="o">=</span> <span class="n">df</span><span class="o">.</span><span class="n">away_score</span><span class="o">.</span><span class="n">values</span>
<span class="n">home_team</span> <span class="o">=</span> <span class="n">df</span><span class="o">.</span><span class="n">i_home</span><span class="o">.</span><span class="n">values</span>
<span class="n">away_team</span> <span class="o">=</span> <span class="n">df</span><span class="o">.</span><span class="n">i_away</span><span class="o">.</span><span class="n">values</span>
<span class="n">num_teams</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">i_home</span><span class="o">.</span><span class="n">drop_duplicates</span><span class="p">())</span>
<span class="n">num_games</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">home_team</span><span class="p">)</span>
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<p>Now we need to prepare the model for PyMC.</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">g</span> <span class="o">=</span> <span class="n">df</span><span class="o">.</span><span class="n">groupby</span><span class="p">(</span><span class="s">'i_away'</span><span class="p">)</span>
<span class="n">att_starting_points</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">g</span><span class="o">.</span><span class="n">away_score</span><span class="o">.</span><span class="n">mean</span><span class="p">())</span>
<span class="n">g</span> <span class="o">=</span> <span class="n">df</span><span class="o">.</span><span class="n">groupby</span><span class="p">(</span><span class="s">'i_home'</span><span class="p">)</span>
<span class="n">def_starting_points</span> <span class="o">=</span> <span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">g</span><span class="o">.</span><span class="n">away_score</span><span class="o">.</span><span class="n">mean</span><span class="p">())</span>
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<h1 id="What-do-we-want-to-infer?">What do we want to infer?<a class="anchor-link" href="#What-do-we-want-to-infer?">¶</a></h1><ul>
<li>We want to infer the latent paremeters (every team's strength) that are generating the data we observe (the scorelines). </li>
<li>Moreover, we know that the scorelines are a noisy measurement of team strength, so ideally, we want a model that makes it easy to quantify our uncertainty about the underlying strengths.</li>
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<h1 id="While-my-MCMC-gently-samples">While my MCMC gently samples<a class="anchor-link" href="#While-my-MCMC-gently-samples">¶</a></h1><ul>
<li>Often we don't know what the Bayesian Model is explicitly, so we have to 'estimate' the Bayesian Model'</li>
<li>If we can't solve something, approximate it.</li>
<li>Markov-Chain Monte Carlo (MCMC) instead draws samples from the posterior.</li>
<li>Fortunately, this algorithm can be applied to almost any model.</li>
<li>Hattip: @twiecki</li>
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<h1 id="What-do-we-want?">What do we want?<a class="anchor-link" href="#What-do-we-want?">¶</a></h1><ul>
<li>We want to quantify our uncertainty</li>
<li>We want to also use this to generate a model</li>
<li>We want the answers as distributions <em>not</em> point estimates</li>
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<h2 id="What-assumptions-do-we-know-for-our-'generative-story'?">What assumptions do we know for our 'generative story'?<a class="anchor-link" href="#What-assumptions-do-we-know-for-our-'generative-story'?">¶</a></h2><ul>
<li>We know that the Six Nations in Rugby only has 6 teams.</li>
<li>We have data from last year!</li>
<li>We also know that in sports scoring is modelled as a Poisson distribution
<img src="http://upload.wikimedia.org/wikipedia/commons/1/16/Poisson_pmf.svg"></li>
<li>Attribution: <em>Wikipedia</em></li>
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