mhjensen
1290 lines
88 KiB
HTML
1290 lines
88 KiB
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<!-- tocinfo
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{'highest level': 2,
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'sections': [('Overview of week 45', 2, None, '___sec0'),
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('Thursday', 2, None, '___sec1'),
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('Why Voting?', 2, None, '___sec2'),
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('Tossing coins', 2, None, '___sec3'),
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('Standard imports first', 2, None, '___sec4'),
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('Simple Voting Example, head or tail', 2, None, '___sec5'),
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('Using the Voting Classifier', 2, None, '___sec6'),
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('Voting and Bagging', 2, None, '___sec7'),
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('Random forests', 2, None, '___sec8'),
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('Random Forest Algorithm', 2, None, '___sec9'),
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('Random Forests Compared with other Methods on the Cancer Data',
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2,
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None,
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('Compare Bagging on Trees with Random Forests',
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("Boosting, a Bird's Eye View", 2, None, '___sec12'),
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('What is boosting? Additive Modelling/Iterative Fitting',
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('Basic Steps of AdaBoost', 2, None, '___sec20'),
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('AdaBoost Examples', 2, None, '___sec21'),
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('The Squared-Error again! Steepest Descent',
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('XGBoost: Extreme Gradient Boosting', 2, None, '___sec28'),
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('Xgboost on the Cancer Data', 2, None, '___sec30')]}
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<!-- ------------------- main content ---------------------- -->
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<center><h1>Week 45: Random Forests and Boosting</h1></center> <!-- document title -->
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<p>
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<!-- author(s): Morten Hjorth-Jensen -->
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<center>
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<b>Morten Hjorth-Jensen</b> [1, 2]
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</center>
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<p>
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<!-- institution(s) -->
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<center>[1] <b>Department of Physics, University of Oslo</b></center>
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<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
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<br>
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<p>
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<center><h4>Nov 12, 2020</h4></center> <!-- date -->
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<br>
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<p>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="___sec0">Overview of week 45 </h2>
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<ul>
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<li> <b>Thursday</b>: Wrapping up from last week. Bagging and Random forests. Boosting methods. <a href="https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h20/forelesningsvideoer/LectureNovember5.mp4?vrtx=view-as-webpage" target="_blank">Link to video of lecture</a></li>
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<li> <b>Friday</b>: Boosting and gradient boosting. <a href="https://www.uio.no/studier/emner/matnat/fys/FYS-STK4155/h20/forelesningsvideoer/LectureNovember6.mp4?vrtx=view-as-webpage" target="_blank">Link to video of lecture</a></li>
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</ul>
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Geron's chapter 7. See also lecture from <a href="https://www.uio.no/studier/emner/matnat/math/STK-IN4300/h20/slides/lecture_9.pdf" target="_blank">STK-IN4300, lecture 9</a>. Chapter 10 (sections 10.1-10.10 are the most relevant ones) of Hastie et al contains also a good discussion.
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<p>
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<a href="https://www.youtube.com/watch?v=wPqtzj5VZus&ab_channel=H2O.ai" target="_blank">Video on boosting methods by Hastie (the textbook author)</a>.
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<p>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="___sec1">Thursday </h2>
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<p>
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Bagging, voting and random forests.
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The material on bagging and voting is a repeat from last week and can be found in the slides from week 44.
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We repeat here the voting approach since this will serve as a motivation for boosting methods later.
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<p>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="___sec2">Why Voting? </h2>
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<p>
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The idea behind boosting, and voting as well can be phrased as follows:
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<b>Can a group of people somehow arrive at highly
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reasoned decisions, despite the weak judgement of the individual
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members?</b>
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<p>
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The aim is to create a good classifier by combining several weak classifiers.
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<b>A weak classifier is a classifier which is able to produce results that are only slightly better than guessing at random.</b>
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<p>
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The basic approach is to apply repeatedly (in boosting this is done in an iterative way) a weak classifier to modifications of the data.
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In voting we simply apply the law of large numbers while in boosting we give more weight to misclassified data in
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each iteration.
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<p>
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Decision trees play an important role as our weak classifier. They serve as the basic method.
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<p>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="___sec3">Tossing coins </h2>
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<p>
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The simplest case is a so-called voting ensemble. To illustrate this,
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think of yourself tossing coins with a biased outcome of 51 per cent
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for heads and 49% for tails. With only few tosses,
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you may not clearly see this distribution for heads and tails. However, after some
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thousands of tosses, there will be a clear majority of heads. With 2000 tosses
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you should see approximately 1020 heads and 980 tails.
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<p>
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We can then state that the outcome is a clear majority of heads. If
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you do this ten thousand times, it is easy to see that there is a 97%
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likelihood of a majority of heads.
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<p>
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Another example would be to collect all polls before an
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election. Different polls may show different likelihoods for a
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candidate winning with say a majority of the popular vote. The majority vote
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would then consist in many polls indicating that this candidate will
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actually win.
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<p>
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The example here shows how we can implement the coin tossing case,
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clealry demostrating that after some tosses we see the <a href="https://en.wikipedia.org/wiki/Law_of_large_numbers" target="_blank">law of large</a>
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numbers kicking in.
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<p>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="___sec4">Standard imports first </h2>
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<p>
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<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #408080; font-style: italic"># Common imports</span>
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">IPython.display</span> <span style="color: #008000; font-weight: bold">import</span> Image
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">pydot</span> <span style="color: #008000; font-weight: bold">import</span> graph_from_dot_data
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">pandas</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">pd</span>
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.tree</span> <span style="color: #008000; font-weight: bold">import</span> DecisionTreeClassifier
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.tree</span> <span style="color: #008000; font-weight: bold">import</span> export_graphviz
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> StandardScaler, OneHotEncoder
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.compose</span> <span style="color: #008000; font-weight: bold">import</span> ColumnTransformer
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">IPython.display</span> <span style="color: #008000; font-weight: bold">import</span> Image
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">pydot</span> <span style="color: #008000; font-weight: bold">import</span> graph_from_dot_data
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">os</span>
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<span style="color: #408080; font-style: italic"># Where to save the figures and data files</span>
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PROJECT_ROOT_DIR <span style="color: #666666">=</span> <span style="color: #BA2121">"Results"</span>
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FIGURE_ID <span style="color: #666666">=</span> <span style="color: #BA2121">"Results/FigureFiles"</span>
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DATA_ID <span style="color: #666666">=</span> <span style="color: #BA2121">"DataFiles/"</span>
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<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(PROJECT_ROOT_DIR):
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os<span style="color: #666666">.</span>mkdir(PROJECT_ROOT_DIR)
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<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(FIGURE_ID):
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os<span style="color: #666666">.</span>makedirs(FIGURE_ID)
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<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(DATA_ID):
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os<span style="color: #666666">.</span>makedirs(DATA_ID)
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<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">image_path</span>(fig_id):
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<span style="color: #008000; font-weight: bold">return</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>join(FIGURE_ID, fig_id)
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<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">data_path</span>(dat_id):
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<span style="color: #008000; font-weight: bold">return</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>join(DATA_ID, dat_id)
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<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">save_fig</span>(fig_id):
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plt<span style="color: #666666">.</span>savefig(image_path(fig_id) <span style="color: #666666">+</span> <span style="color: #BA2121">".png"</span>, <span style="color: #008000">format</span><span style="color: #666666">=</span><span style="color: #BA2121">'png'</span>)
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</pre></div>
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<p>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="___sec5">Simple Voting Example, head or tail </h2>
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<p>
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<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #408080; font-style: italic"># Common imports</span>
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib</span>
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">matplotlib.colors</span> <span style="color: #008000; font-weight: bold">import</span> ListedColormap
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plt<span style="color: #666666">.</span>rcParams[<span style="color: #BA2121">'axes.labelsize'</span>] <span style="color: #666666">=</span> <span style="color: #666666">14</span>
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plt<span style="color: #666666">.</span>rcParams[<span style="color: #BA2121">'xtick.labelsize'</span>] <span style="color: #666666">=</span> <span style="color: #666666">12</span>
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plt<span style="color: #666666">.</span>rcParams[<span style="color: #BA2121">'ytick.labelsize'</span>] <span style="color: #666666">=</span> <span style="color: #666666">12</span>
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heads_proba <span style="color: #666666">=</span> <span style="color: #666666">0.51</span>
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coin_tosses <span style="color: #666666">=</span> (np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>rand(<span style="color: #666666">10000</span>, <span style="color: #666666">10</span>) <span style="color: #666666"><</span> heads_proba)<span style="color: #666666">.</span>astype(np<span style="color: #666666">.</span>int32)
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cumulative_heads_ratio <span style="color: #666666">=</span> np<span style="color: #666666">.</span>cumsum(coin_tosses, axis<span style="color: #666666">=0</span>) <span style="color: #666666">/</span> np<span style="color: #666666">.</span>arange(<span style="color: #666666">1</span>, <span style="color: #666666">10001</span>)<span style="color: #666666">.</span>reshape(<span style="color: #666666">-1</span>, <span style="color: #666666">1</span>)
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plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">8</span>,<span style="color: #666666">3.5</span>))
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plt<span style="color: #666666">.</span>plot(cumulative_heads_ratio)
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plt<span style="color: #666666">.</span>plot([<span style="color: #666666">0</span>, <span style="color: #666666">10000</span>], [<span style="color: #666666">0.51</span>, <span style="color: #666666">0.51</span>], <span style="color: #BA2121">"k--"</span>, linewidth<span style="color: #666666">=2</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">"51%"</span>)
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plt<span style="color: #666666">.</span>plot([<span style="color: #666666">0</span>, <span style="color: #666666">10000</span>], [<span style="color: #666666">0.5</span>, <span style="color: #666666">0.5</span>], <span style="color: #BA2121">"k-"</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">"50%"</span>)
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plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">"Number of coin tosses"</span>)
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plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">"Heads ratio"</span>)
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plt<span style="color: #666666">.</span>legend(loc<span style="color: #666666">=</span><span style="color: #BA2121">"lower right"</span>)
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plt<span style="color: #666666">.</span>axis([<span style="color: #666666">0</span>, <span style="color: #666666">10000</span>, <span style="color: #666666">0.42</span>, <span style="color: #666666">0.58</span>])
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save_fig(<span style="color: #BA2121">"votingsimple"</span>)
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plt<span style="color: #666666">.</span>show()
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</pre></div>
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<p>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="___sec6">Using the Voting Classifier </h2>
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<p>
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We can use the voting classifier on other data sets, here the exciting binary case of two distinct objects using the make moons functionality of <b>Scikit-Learn</b>.
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<p>
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<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.datasets</span> <span style="color: #008000; font-weight: bold">import</span> make_moons
|
|
|
|
X, y <span style="color: #666666">=</span> make_moons(n_samples<span style="color: #666666">=500</span>, noise<span style="color: #666666">=0.30</span>, random_state<span style="color: #666666">=42</span>)
|
|
X_train, X_test, y_train, y_test <span style="color: #666666">=</span> train_test_split(X, y, random_state<span style="color: #666666">=42</span>)
|
|
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.ensemble</span> <span style="color: #008000; font-weight: bold">import</span> RandomForestClassifier
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.ensemble</span> <span style="color: #008000; font-weight: bold">import</span> VotingClassifier
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> LogisticRegression
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.svm</span> <span style="color: #008000; font-weight: bold">import</span> SVC
|
|
|
|
log_clf <span style="color: #666666">=</span> LogisticRegression(solver<span style="color: #666666">=</span><span style="color: #BA2121">"liblinear"</span>, random_state<span style="color: #666666">=42</span>)
|
|
rnd_clf <span style="color: #666666">=</span> RandomForestClassifier(n_estimators<span style="color: #666666">=10</span>, random_state<span style="color: #666666">=42</span>)
|
|
svm_clf <span style="color: #666666">=</span> SVC(gamma<span style="color: #666666">=</span><span style="color: #BA2121">"auto"</span>, random_state<span style="color: #666666">=42</span>)
|
|
|
|
voting_clf <span style="color: #666666">=</span> VotingClassifier(
|
|
estimators<span style="color: #666666">=</span>[(<span style="color: #BA2121">'lr'</span>, log_clf), (<span style="color: #BA2121">'rf'</span>, rnd_clf), (<span style="color: #BA2121">'svc'</span>, svm_clf)],
|
|
voting<span style="color: #666666">=</span><span style="color: #BA2121">'hard'</span>)
|
|
|
|
voting_clf<span style="color: #666666">.</span>fit(X_train, y_train)
|
|
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.metrics</span> <span style="color: #008000; font-weight: bold">import</span> accuracy_score
|
|
|
|
<span style="color: #008000; font-weight: bold">for</span> clf <span style="color: #AA22FF; font-weight: bold">in</span> (log_clf, rnd_clf, svm_clf, voting_clf):
|
|
clf<span style="color: #666666">.</span>fit(X_train, y_train)
|
|
y_pred <span style="color: #666666">=</span> clf<span style="color: #666666">.</span>predict(X_test)
|
|
<span style="color: #008000">print</span>(clf<span style="color: #666666">.</span><span style="color: #19177C">__class__</span><span style="color: #666666">.</span><span style="color: #19177C">__name__</span>, accuracy_score(y_test, y_pred))
|
|
|
|
log_clf <span style="color: #666666">=</span> LogisticRegression(solver<span style="color: #666666">=</span><span style="color: #BA2121">"liblinear"</span>, random_state<span style="color: #666666">=42</span>)
|
|
rnd_clf <span style="color: #666666">=</span> RandomForestClassifier(n_estimators<span style="color: #666666">=10</span>, random_state<span style="color: #666666">=42</span>)
|
|
svm_clf <span style="color: #666666">=</span> SVC(gamma<span style="color: #666666">=</span><span style="color: #BA2121">"auto"</span>, probability<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>, random_state<span style="color: #666666">=42</span>)
|
|
voting_clf <span style="color: #666666">=</span> VotingClassifier(
|
|
estimators<span style="color: #666666">=</span>[(<span style="color: #BA2121">'lr'</span>, log_clf), (<span style="color: #BA2121">'rf'</span>, rnd_clf), (<span style="color: #BA2121">'svc'</span>, svm_clf)],
|
|
voting<span style="color: #666666">=</span><span style="color: #BA2121">'soft'</span>)
|
|
voting_clf<span style="color: #666666">.</span>fit(X_train, y_train)
|
|
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.metrics</span> <span style="color: #008000; font-weight: bold">import</span> accuracy_score
|
|
|
|
<span style="color: #008000; font-weight: bold">for</span> clf <span style="color: #AA22FF; font-weight: bold">in</span> (log_clf, rnd_clf, svm_clf, voting_clf):
|
|
clf<span style="color: #666666">.</span>fit(X_train, y_train)
|
|
y_pred <span style="color: #666666">=</span> clf<span style="color: #666666">.</span>predict(X_test)
|
|
<span style="color: #008000">print</span>(clf<span style="color: #666666">.</span><span style="color: #19177C">__class__</span><span style="color: #666666">.</span><span style="color: #19177C">__name__</span>, accuracy_score(y_test, y_pred))
|
|
</pre></div>
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="___sec7">Voting and Bagging </h2>
|
|
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.datasets</span> <span style="color: #008000; font-weight: bold">import</span> make_moons
|
|
|
|
X, y <span style="color: #666666">=</span> make_moons(n_samples<span style="color: #666666">=500</span>, noise<span style="color: #666666">=0.30</span>, random_state<span style="color: #666666">=42</span>)
|
|
X_train, X_test, y_train, y_test <span style="color: #666666">=</span> train_test_split(X, y, random_state<span style="color: #666666">=42</span>)
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.ensemble</span> <span style="color: #008000; font-weight: bold">import</span> RandomForestClassifier
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.ensemble</span> <span style="color: #008000; font-weight: bold">import</span> VotingClassifier
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> LogisticRegression
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.svm</span> <span style="color: #008000; font-weight: bold">import</span> SVC
|
|
|
|
log_clf <span style="color: #666666">=</span> LogisticRegression(random_state<span style="color: #666666">=42</span>)
|
|
rnd_clf <span style="color: #666666">=</span> RandomForestClassifier(random_state<span style="color: #666666">=42</span>)
|
|
svm_clf <span style="color: #666666">=</span> SVC(random_state<span style="color: #666666">=42</span>)
|
|
|
|
voting_clf <span style="color: #666666">=</span> VotingClassifier(
|
|
estimators<span style="color: #666666">=</span>[(<span style="color: #BA2121">'lr'</span>, log_clf), (<span style="color: #BA2121">'rf'</span>, rnd_clf), (<span style="color: #BA2121">'svc'</span>, svm_clf)],
|
|
voting<span style="color: #666666">=</span><span style="color: #BA2121">'hard'</span>)
|
|
voting_clf<span style="color: #666666">.</span>fit(X_train, y_train)
|
|
</pre></div>
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.metrics</span> <span style="color: #008000; font-weight: bold">import</span> accuracy_score
|
|
|
|
<span style="color: #008000; font-weight: bold">for</span> clf <span style="color: #AA22FF; font-weight: bold">in</span> (log_clf, rnd_clf, svm_clf, voting_clf):
|
|
clf<span style="color: #666666">.</span>fit(X_train, y_train)
|
|
y_pred <span style="color: #666666">=</span> clf<span style="color: #666666">.</span>predict(X_test)
|
|
<span style="color: #008000">print</span>(clf<span style="color: #666666">.</span><span style="color: #19177C">__class__</span><span style="color: #666666">.</span><span style="color: #19177C">__name__</span>, accuracy_score(y_test, y_pred))
|
|
</pre></div>
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>log_clf <span style="color: #666666">=</span> LogisticRegression(random_state<span style="color: #666666">=42</span>)
|
|
rnd_clf <span style="color: #666666">=</span> RandomForestClassifier(random_state<span style="color: #666666">=42</span>)
|
|
svm_clf <span style="color: #666666">=</span> SVC(probability<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>, random_state<span style="color: #666666">=42</span>)
|
|
|
|
voting_clf <span style="color: #666666">=</span> VotingClassifier(
|
|
estimators<span style="color: #666666">=</span>[(<span style="color: #BA2121">'lr'</span>, log_clf), (<span style="color: #BA2121">'rf'</span>, rnd_clf), (<span style="color: #BA2121">'svc'</span>, svm_clf)],
|
|
voting<span style="color: #666666">=</span><span style="color: #BA2121">'soft'</span>)
|
|
voting_clf<span style="color: #666666">.</span>fit(X_train, y_train)
|
|
</pre></div>
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.metrics</span> <span style="color: #008000; font-weight: bold">import</span> accuracy_score
|
|
|
|
<span style="color: #008000; font-weight: bold">for</span> clf <span style="color: #AA22FF; font-weight: bold">in</span> (log_clf, rnd_clf, svm_clf, voting_clf):
|
|
clf<span style="color: #666666">.</span>fit(X_train, y_train)
|
|
y_pred <span style="color: #666666">=</span> clf<span style="color: #666666">.</span>predict(X_test)
|
|
<span style="color: #008000">print</span>(clf<span style="color: #666666">.</span><span style="color: #19177C">__class__</span><span style="color: #666666">.</span><span style="color: #19177C">__name__</span>, accuracy_score(y_test, y_pred))
|
|
</pre></div>
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="___sec8">Random forests </h2>
|
|
|
|
<p>
|
|
Random forests provide an improvement over bagged trees by way of a
|
|
small tweak that decorrelates the trees.
|
|
|
|
<p>
|
|
As in bagging, we build a
|
|
number of decision trees on bootstrapped training samples. But when
|
|
building these decision trees, each time a split in a tree is
|
|
considered, a random sample of \( m \) predictors is chosen as split
|
|
candidates from the full set of \( p \) predictors. The split is allowed to
|
|
use only one of those \( m \) predictors.
|
|
|
|
<p>
|
|
A fresh sample of \( m \) predictors is
|
|
taken at each split, and typically we choose
|
|
|
|
$$
|
|
m\approx \sqrt{p}.
|
|
$$
|
|
|
|
<p>
|
|
In building a random forest, at
|
|
each split in the tree, the algorithm is not even allowed to consider
|
|
a majority of the available predictors.
|
|
|
|
<p>
|
|
The reason for this is rather clever. Suppose that there is one very
|
|
strong predictor in the data set, along with a number of other
|
|
moderately strong predictors. Then in the collection of bagged
|
|
variable importance random forest trees, most or all of the trees will
|
|
use this strong predictor in the top split. Consequently, all of the
|
|
bagged trees will look quite similar to each other. Hence the
|
|
predictions from the bagged trees will be highly correlated.
|
|
Unfortunately, averaging many highly correlated quantities does not
|
|
lead to as large of a reduction in variance as averaging many
|
|
uncorrelated quantities. In particular, this means that bagging will
|
|
not lead to a substantial reduction in variance over a single tree in
|
|
this setting.
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="___sec9">Random Forest Algorithm </h2>
|
|
The algorithm described here can be applied to both classification and regression problems.
|
|
|
|
<p>
|
|
We will grow of forest of say \( B \) trees.
|
|
|
|
<ol>
|
|
<li> For \( b=1:B \)</li>
|
|
|
|
<ul>
|
|
<li> Draw a bootstrap sample from the training data organized in our \( \boldsymbol{X} \) matrix.</li>
|
|
<li> We grow then a random forest tree \( T_b \) based on the bootstrapped data by repeating the steps outlined till we reach the maximum node size is reached</li>
|
|
|
|
<ol>
|
|
<li> we select \( m \le p \) variables at random from the \( p \) predictors/features</li>
|
|
<li> pick the best split point among the \( m \) features using for example the CART algorithm and create a new node</li>
|
|
<li> split the node into daughter nodes</li>
|
|
</ol>
|
|
|
|
</ul>
|
|
|
|
<li> Output then the ensemble of trees \( \{T_b\}_1^{B} \) and make predictions for either a regression type of problem or a classification type of problem.</li>
|
|
</ol>
|
|
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="___sec10">Random Forests Compared with other Methods on the Cancer Data </h2>
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.datasets</span> <span style="color: #008000; font-weight: bold">import</span> load_breast_cancer
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.svm</span> <span style="color: #008000; font-weight: bold">import</span> SVC
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> LogisticRegression
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.tree</span> <span style="color: #008000; font-weight: bold">import</span> DecisionTreeClassifier
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.ensemble</span> <span style="color: #008000; font-weight: bold">import</span> BaggingClassifier
|
|
|
|
<span style="color: #408080; font-style: italic"># Load the data</span>
|
|
cancer <span style="color: #666666">=</span> load_breast_cancer()
|
|
|
|
X_train, X_test, y_train, y_test <span style="color: #666666">=</span> train_test_split(cancer<span style="color: #666666">.</span>data,cancer<span style="color: #666666">.</span>target,random_state<span style="color: #666666">=0</span>)
|
|
<span style="color: #008000">print</span>(X_train<span style="color: #666666">.</span>shape)
|
|
<span style="color: #008000">print</span>(X_test<span style="color: #666666">.</span>shape)
|
|
<span style="color: #408080; font-style: italic"># Logistic Regression</span>
|
|
logreg <span style="color: #666666">=</span> LogisticRegression(solver<span style="color: #666666">=</span><span style="color: #BA2121">'lbfgs'</span>)
|
|
logreg<span style="color: #666666">.</span>fit(X_train, y_train)
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">"Test set accuracy with Logistic Regression: </span><span style="color: #BB6688; font-weight: bold">{:.2f}</span><span style="color: #BA2121">"</span><span style="color: #666666">.</span>format(logreg<span style="color: #666666">.</span>score(X_test,y_test)))
|
|
<span style="color: #408080; font-style: italic"># Support vector machine</span>
|
|
svm <span style="color: #666666">=</span> SVC(gamma<span style="color: #666666">=</span><span style="color: #BA2121">'auto'</span>, C<span style="color: #666666">=100</span>)
|
|
svm<span style="color: #666666">.</span>fit(X_train, y_train)
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">"Test set accuracy with SVM: </span><span style="color: #BB6688; font-weight: bold">{:.2f}</span><span style="color: #BA2121">"</span><span style="color: #666666">.</span>format(svm<span style="color: #666666">.</span>score(X_test,y_test)))
|
|
<span style="color: #408080; font-style: italic"># Decision Trees</span>
|
|
deep_tree_clf <span style="color: #666666">=</span> DecisionTreeClassifier(max_depth<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">None</span>)
|
|
deep_tree_clf<span style="color: #666666">.</span>fit(X_train, y_train)
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">"Test set accuracy with Decision Trees: </span><span style="color: #BB6688; font-weight: bold">{:.2f}</span><span style="color: #BA2121">"</span><span style="color: #666666">.</span>format(deep_tree_clf<span style="color: #666666">.</span>score(X_test,y_test)))
|
|
<span style="color: #408080; font-style: italic">#now scale the data</span>
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> StandardScaler
|
|
scaler <span style="color: #666666">=</span> StandardScaler()
|
|
scaler<span style="color: #666666">.</span>fit(X_train)
|
|
X_train_scaled <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>transform(X_train)
|
|
X_test_scaled <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>transform(X_test)
|
|
<span style="color: #408080; font-style: italic"># Logistic Regression</span>
|
|
logreg<span style="color: #666666">.</span>fit(X_train_scaled, y_train)
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">"Test set accuracy Logistic Regression with scaled data: </span><span style="color: #BB6688; font-weight: bold">{:.2f}</span><span style="color: #BA2121">"</span><span style="color: #666666">.</span>format(logreg<span style="color: #666666">.</span>score(X_test_scaled,y_test)))
|
|
<span style="color: #408080; font-style: italic"># Support Vector Machine</span>
|
|
svm<span style="color: #666666">.</span>fit(X_train_scaled, y_train)
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">"Test set accuracy SVM with scaled data: </span><span style="color: #BB6688; font-weight: bold">{:.2f}</span><span style="color: #BA2121">"</span><span style="color: #666666">.</span>format(logreg<span style="color: #666666">.</span>score(X_test_scaled,y_test)))
|
|
<span style="color: #408080; font-style: italic"># Decision Trees</span>
|
|
deep_tree_clf<span style="color: #666666">.</span>fit(X_train_scaled, y_train)
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">"Test set accuracy with Decision Trees and scaled data: </span><span style="color: #BB6688; font-weight: bold">{:.2f}</span><span style="color: #BA2121">"</span><span style="color: #666666">.</span>format(deep_tree_clf<span style="color: #666666">.</span>score(X_test_scaled,y_test)))
|
|
|
|
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.ensemble</span> <span style="color: #008000; font-weight: bold">import</span> RandomForestClassifier
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> LabelEncoder
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> cross_validate
|
|
<span style="color: #408080; font-style: italic"># Data set not specificied</span>
|
|
<span style="color: #408080; font-style: italic">#Instantiate the model with 500 trees and entropy as splitting criteria</span>
|
|
Random_Forest_model <span style="color: #666666">=</span> RandomForestClassifier(n_estimators<span style="color: #666666">=500</span>,criterion<span style="color: #666666">=</span><span style="color: #BA2121">"entropy"</span>)
|
|
Random_Forest_model<span style="color: #666666">.</span>fit(X_train_scaled, y_train)
|
|
<span style="color: #408080; font-style: italic">#Cross validation</span>
|
|
accuracy <span style="color: #666666">=</span> cross_validate(Random_Forest_model,X_test_scaled,y_test,cv<span style="color: #666666">=10</span>)[<span style="color: #BA2121">'test_score'</span>]
|
|
<span style="color: #008000">print</span>(accuracy)
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">"Test set accuracy with Random Forests and scaled data: </span><span style="color: #BB6688; font-weight: bold">{:.2f}</span><span style="color: #BA2121">"</span><span style="color: #666666">.</span>format(Random_Forest_model<span style="color: #666666">.</span>score(X_test_scaled,y_test)))
|
|
|
|
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">scikitplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">skplt</span>
|
|
y_pred <span style="color: #666666">=</span> Random_Forest_model<span style="color: #666666">.</span>predict(X_test_scaled)
|
|
skplt<span style="color: #666666">.</span>metrics<span style="color: #666666">.</span>plot_confusion_matrix(y_test, y_pred, normalize<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>)
|
|
plt<span style="color: #666666">.</span>show()
|
|
y_probas <span style="color: #666666">=</span> Random_Forest_model<span style="color: #666666">.</span>predict_proba(X_test_scaled)
|
|
skplt<span style="color: #666666">.</span>metrics<span style="color: #666666">.</span>plot_roc(y_test, y_probas)
|
|
plt<span style="color: #666666">.</span>show()
|
|
skplt<span style="color: #666666">.</span>metrics<span style="color: #666666">.</span>plot_cumulative_gain(y_test, y_probas)
|
|
plt<span style="color: #666666">.</span>show()
|
|
</pre></div>
|
|
<p>
|
|
Recall that the cumulative gains curve shows the percentage of the
|
|
overall number of cases in a given category <em>gained</em> by targeting a
|
|
percentage of the total number of cases.
|
|
|
|
<p>
|
|
Similarly, the receiver operating characteristic curve, or ROC curve,
|
|
displays the diagnostic ability of a binary classifier system as its
|
|
discrimination threshold is varied. It plots the true positive rate against the false positive rate.
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="___sec11">Compare Bagging on Trees with Random Forests </h2>
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>bag_clf <span style="color: #666666">=</span> BaggingClassifier(
|
|
DecisionTreeClassifier(splitter<span style="color: #666666">=</span><span style="color: #BA2121">"random"</span>, max_leaf_nodes<span style="color: #666666">=16</span>, random_state<span style="color: #666666">=42</span>),
|
|
n_estimators<span style="color: #666666">=500</span>, max_samples<span style="color: #666666">=1.0</span>, bootstrap<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>, n_jobs<span style="color: #666666">=-1</span>, random_state<span style="color: #666666">=42</span>)
|
|
</pre></div>
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>bag_clf<span style="color: #666666">.</span>fit(X_train, y_train)
|
|
y_pred <span style="color: #666666">=</span> bag_clf<span style="color: #666666">.</span>predict(X_test)
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.ensemble</span> <span style="color: #008000; font-weight: bold">import</span> RandomForestClassifier
|
|
rnd_clf <span style="color: #666666">=</span> RandomForestClassifier(n_estimators<span style="color: #666666">=500</span>, max_leaf_nodes<span style="color: #666666">=16</span>, n_jobs<span style="color: #666666">=-1</span>, random_state<span style="color: #666666">=42</span>)
|
|
rnd_clf<span style="color: #666666">.</span>fit(X_train, y_train)
|
|
y_pred_rf <span style="color: #666666">=</span> rnd_clf<span style="color: #666666">.</span>predict(X_test)
|
|
np<span style="color: #666666">.</span>sum(y_pred <span style="color: #666666">==</span> y_pred_rf) <span style="color: #666666">/</span> <span style="color: #008000">len</span>(y_pred)
|
|
</pre></div>
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="___sec12">Boosting, a Bird's Eye View </h2>
|
|
|
|
<p>
|
|
The basic idea is to combine weak classifiers in order to create a good
|
|
classifier. With a weak classifier we often intend a classifier which
|
|
produces results which are only slightly better than we would get by
|
|
random guesses.
|
|
|
|
<p>
|
|
This is done by applying in an iterative way a weak (or a standard
|
|
classifier like decision trees) to modify the data. In each iteration
|
|
we emphasize those observations which are misclassified by weighting
|
|
them with a factor.
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="___sec13">What is boosting? Additive Modelling/Iterative Fitting </h2>
|
|
|
|
<p>
|
|
Boosting is a way of fitting an additive expansion in a set of
|
|
elementary basis functions like for example some simple polynomials.
|
|
Assume for example that we have a function
|
|
$$
|
|
f_M(x) = \sum_{i=1}^M \beta_m b(x;\gamma_m),
|
|
$$
|
|
|
|
<p>
|
|
where \( \beta_m \) are the expansion parameters to be determined in a
|
|
minimization process and \( b(x;\gamma_m) \) are some simple functions of
|
|
the multivariable parameter \( x \) which is characterized by the
|
|
parameters \( \gamma_m \).
|
|
|
|
<p>
|
|
As an example, consider the Sigmoid function we used in logistic
|
|
regression. In that case, we can translate the function
|
|
\( b(x;\gamma_m) \) into the Sigmoid function
|
|
|
|
$$
|
|
\sigma(t) = \frac{1}{1+\exp{(-t)}},
|
|
$$
|
|
|
|
<p>
|
|
where \( t=\gamma_0+\gamma_1 x \) and the parameters \( \gamma_0 \) and
|
|
\( \gamma_1 \) were determined by the Logistic Regression fitting
|
|
algorithm.
|
|
|
|
<p>
|
|
As another example, consider the cost function we defined for linear regression
|
|
$$
|
|
C(\boldsymbol{y},\boldsymbol{f}) = \frac{1}{n} \sum_{i=0}^{n-1}(y_i-f(x_i))^2.
|
|
$$
|
|
|
|
<p>
|
|
In this case the function \( f(x) \) was replaced by the design matrix
|
|
\( \boldsymbol{X} \) and the unknown linear regression parameters \( \boldsymbol{\beta} \),
|
|
that is \( \boldsymbol{f}=\boldsymbol{X}\boldsymbol{\beta} \). In linear regression we can
|
|
simply invert a matrix and obtain the parameters \( \beta \) by
|
|
|
|
$$
|
|
\boldsymbol{\beta}=\left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}.
|
|
$$
|
|
|
|
<p>
|
|
In iterative fitting or additive modeling, we minimize the cost function with respect to the parameters \( \beta_m \) and \( \gamma_m \).
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="___sec14">Iterative Fitting, Regression and Squared-error Cost Function </h2>
|
|
|
|
<p>
|
|
The way we proceed is as follows (here we specialize to the squared-error cost function)
|
|
|
|
<ol>
|
|
<li> Establish a cost function, here \( {\cal C}(\boldsymbol{y},\boldsymbol{f}) = \frac{1}{n} \sum_{i=0}^{n-1}(y_i-f_M(x_i))^2 \) with \( f_M(x) = \sum_{i=1}^M \beta_m b(x;\gamma_m) \).</li>
|
|
<li> Initialize with a guess \( f_0(x) \). It could be one or even zero or some random numbers.</li>
|
|
<li> For \( m=1:M \)
|
|
|
|
<ol type="a"></li>
|
|
<li> minimize \( \sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\beta b(x;\gamma))^2 \) wrt \( \gamma \) and \( \beta \)</li>
|
|
<li> This gives the optimal values \( \beta_m \) and \( \gamma_m \)</li>
|
|
<li> Determine then the new values \( f_m(x)=f_{m-1}(x) +\beta_m b(x;\gamma_m) \)</li>
|
|
</ol>
|
|
|
|
</ol>
|
|
|
|
We could use any of the algorithms we have discussed till now. If we
|
|
use trees, \( \gamma \) parameterizes the split variables and split points
|
|
at the internal nodes, and the predictions at the terminal nodes.
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="___sec15">Squared-Error Example and Iterative Fitting </h2>
|
|
|
|
<p>
|
|
To better understand what happens, let us develop the steps for the iterative fitting using the above squared error function.
|
|
|
|
<p>
|
|
For simplicity we assume also that our functions \( b(x;\gamma)=1+\gamma x \).
|
|
|
|
<p>
|
|
This means that for every iteration \( m \), we need to optimize
|
|
|
|
$$
|
|
(\beta_m,\gamma_m) = \mathrm{argmin}_{\beta,\lambda}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\beta b(x;\gamma))^2=\sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\beta(1+\gamma x_i))^2.
|
|
$$
|
|
|
|
<p>
|
|
We start our iteration by simply setting \( f_0(x)=0 \).
|
|
Taking the derivatives with respect to \( \beta \) and \( \gamma \) we obtain
|
|
$$
|
|
\frac{\partial {\cal C}}{\partial \beta} = -2\sum_{i}(1+\gamma x_i)(y_i-\beta(1+\gamma x_i))=0,
|
|
$$
|
|
|
|
and
|
|
$$
|
|
\frac{\partial {\cal C}}{\partial \gamma} =-2\sum_{i}\beta x_i(y_i-\beta(1+\gamma x_i))=0.
|
|
$$
|
|
|
|
We can then rewrite these equations as (defining \( \boldsymbol{w}=\boldsymbol{e}+\gamma \boldsymbol{x}) \) with \( \boldsymbol{e} \) being the unit vector)
|
|
$$
|
|
\gamma \boldsymbol{w}^T(\boldsymbol{y}-\beta\gamma \boldsymbol{w})=0,
|
|
$$
|
|
|
|
which gives us \( \beta = \boldsymbol{w}^T\boldsymbol{y}/(\boldsymbol{w}^T\boldsymbol{w}) \). Similarly we have
|
|
$$
|
|
\beta\gamma \boldsymbol{x}^T(\boldsymbol{y}-\beta(1+\gamma \boldsymbol{x}))=0,
|
|
$$
|
|
|
|
<p>
|
|
which leads to \( \gamma =(\boldsymbol{x}^T\boldsymbol{y}-\beta\boldsymbol{x}^T\boldsymbol{e})/(\beta\boldsymbol{x}^T\boldsymbol{x}) \). Inserting
|
|
for \( \beta \) gives us an equation for \( \gamma \). This is a non-linear equation in the unknown \( \gamma \) and has to be solved numerically.
|
|
|
|
<p>
|
|
The solution to these two equations gives us in turn \( \beta_1 \) and \( \gamma_1 \) leading to the new expression for \( f_1(x) \) as
|
|
\( f_1(x) = \beta_1(1+\gamma_1x) \). Doing this \( M \) times results in our final estimate for the function \( f \).
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="___sec16">Iterative Fitting, Classification and AdaBoost </h2>
|
|
|
|
<p>
|
|
Let us consider a binary classification problem with two outcomes \( y_i \in \{-1,1\} \) and \( i=0,1,2,\dots,n-1 \) as our set of
|
|
observations. We define a classification function \( G(x) \) which produces a prediction taking one or the other of the two values
|
|
\( \{-1,1\} \).
|
|
|
|
<p>
|
|
The error rate of the training sample is then
|
|
|
|
$$
|
|
\mathrm{\overline{err}}=\frac{1}{n} \sum_{i=0}^{n-1} I(y_i\ne G(x_i)).
|
|
$$
|
|
|
|
<p>
|
|
The iterative procedure starts with defining a weak classifier whose
|
|
error rate is barely better than random guessing. The iterative
|
|
procedure in boosting is to sequentially apply a weak
|
|
classification algorithm to repeatedly modified versions of the data
|
|
producing a sequence of weak classifiers \( G_m(x) \).
|
|
|
|
<p>
|
|
Here we will express our function \( f(x) \) in terms of \( G(x) \). That is
|
|
$$
|
|
f_M(x) = \sum_{i=1}^M \beta_m b(x;\gamma_m),
|
|
$$
|
|
|
|
will be a function of
|
|
$$
|
|
G_M(x) = \mathrm{sign} \sum_{i=1}^M \alpha_m G_m(x).
|
|
$$
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="___sec17">Adaptive Boosting, AdaBoost </h2>
|
|
|
|
<p>
|
|
In our iterative procedure we define thus
|
|
$$
|
|
f_m(x) = f_{m-1}(x)+\beta_mG_m(x).
|
|
$$
|
|
|
|
<p>
|
|
The simplest possible cost function which leads (also simple from a computational point of view) to the AdaBoost algorithm is the
|
|
exponential cost/loss function defined as
|
|
$$
|
|
C(\boldsymbol{y},\boldsymbol{f}) = \sum_{i=0}^{n-1}\exp{(-y_i(f_{m-1}(x_i)+\beta G(x_i))}.
|
|
$$
|
|
|
|
<p>
|
|
We optimize \( \beta \) and \( G \) for each value of \( m=1:M \) as we did in the regression case.
|
|
This is normally done in two steps. Let us however first rewrite the cost function as
|
|
|
|
$$
|
|
C(\boldsymbol{y},\boldsymbol{f}) = \sum_{i=0}^{n-1}w_i^{m}\exp{(-y_i\beta G(x_i))},
|
|
$$
|
|
|
|
where we have defined \( w_i^m= \exp{(-y_if_{m-1}(x_i))} \).
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="___sec18">Building up AdaBoost </h2>
|
|
|
|
<p>
|
|
First, for any \( \beta > 0 \), we optimize \( G \) by setting
|
|
$$
|
|
G_m(x) = \mathrm{sign} \sum_{i=0}^{n-1} w_i^m I(y_i \ne G_(x_i)),
|
|
$$
|
|
|
|
which is the classifier that minimizes the weighted error rate in predicting \( y \).
|
|
|
|
<p>
|
|
We can do this by rewriting
|
|
$$
|
|
\exp{-(\beta)}\sum_{y_i=G(x_i)}w_i^m+\exp{(\beta)}\sum_{y_i\ne G(x_i)}w_i^m,
|
|
$$
|
|
|
|
which can be rewritten as
|
|
$$
|
|
(\exp{(\beta)}-\exp{-(\beta)})\sum_{i=0}^{n-1}w_i^mI(y_i\ne G(x_i))+\exp{(-\beta)}\sum_{i=0}^{n-1}w_i^m=0,
|
|
$$
|
|
|
|
which leads to
|
|
$$
|
|
\beta_m = \frac{1}{2}\log{\frac{1-\mathrm{\overline{err}}}{\mathrm{\overline{err}}}},
|
|
$$
|
|
|
|
where we have redefined the error as
|
|
$$
|
|
\mathrm{\overline{err}}_m=\frac{1}{n}\frac{\sum_{i=0}^{n-1}w_i^mI(y_i\ne G(x_i)}{\sum_{i=0}^{n-1}w_i^m},
|
|
$$
|
|
|
|
which leads to an update of
|
|
$$
|
|
f_m(x) = f_{m-1}(x) +\beta_m G_m(x).
|
|
$$
|
|
|
|
This leads to the new weights
|
|
$$
|
|
w_i^{m+1} = w_i^m \exp{(-y_i\beta_m G_m(x_i))}
|
|
$$
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="___sec19">Adaptive boosting: AdaBoost, Basic Algorithm </h2>
|
|
|
|
<p>
|
|
The algorithm here is rather straightforward. Assume that our weak
|
|
classifier is a decision tree and we consider a binary set of outputs
|
|
with \( y_i \in \{-1,1\} \) and \( i=0,1,2,\dots,n-1 \) as our set of
|
|
observations. Our design matrix is given in terms of the
|
|
feature/predictor vectors
|
|
\( \boldsymbol{X}=[\boldsymbol{x}_0\boldsymbol{x}_1\dots\boldsymbol{x}_{p-1}] \). Finally, we define also a
|
|
classifier determined by our data via a function \( G(x) \). This function tells us how well we are able to classify our outputs/targets \( \boldsymbol{y} \).
|
|
|
|
<p>
|
|
We have already defined the misclassification error \( \mathrm{err} \) as
|
|
$$
|
|
\mathrm{err}=\frac{1}{n}\sum_{i=0}^{n-1}I(y_i\ne G(x_i)),
|
|
$$
|
|
|
|
where the function \( I() \) is one if we misclassify and zero if we classify correctly.
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="___sec20">Basic Steps of AdaBoost </h2>
|
|
|
|
<p>
|
|
With the above definitions we are now ready to set up the algorithm for AdaBoost.
|
|
The basic idea is to set up weights which will be used to scale the correctly classified and the misclassified cases.
|
|
|
|
<ol>
|
|
<li> We start by initializing all weights to \( w_i = 1/n \), with \( i=0,1,2,\dots n-1 \). It is easy to see that we must have \( \sum_{i=0}^{n-1}w_i = 1 \).</li>
|
|
<li> We rewrite the misclassification error as</li>
|
|
</ol>
|
|
|
|
$$
|
|
\mathrm{\overline{err}}_m=\frac{\sum_{i=0}^{n-1}w_i^m I(y_i\ne G(x_i))}{\sum_{i=0}^{n-1}w_i},
|
|
$$
|
|
|
|
|
|
<ol>
|
|
<li> Then we start looping over all attempts at classifying, namely we start an iterative process for \( m=1:M \), where \( M \) is the final number of classifications. Our given classifier could for example be a plain decision tree.
|
|
|
|
<ol type="a"></li>
|
|
<li> Fit then a given classifier to the training set using the weights \( w_i \).</li>
|
|
<li> Compute then \( \mathrm{err} \) and figure out which events are classified properly and which are classified wrongly.</li>
|
|
<li> Define a quantity \( \alpha_{m} = \log{(1-\mathrm{\overline{err}}_m)/\mathrm{\overline{err}}_m} \)</li>
|
|
<li> Set the new weights to \( w_i = w_i\times \exp{(\alpha_m I(y_i\ne G(x_i)} \).</li>
|
|
</ol>
|
|
|
|
<li> Compute the new classifier \( G(x)= \sum_{i=0}^{n-1}\alpha_m I(y_i\ne G(x_i) \).</li>
|
|
</ol>
|
|
|
|
For the iterations with \( m \le 2 \) the weights are modified
|
|
individually at each steps. The observations which were misclassified
|
|
at iteration \( m-1 \) have a weight which is larger than those which were
|
|
classified properly. As this proceeds, the observations which were
|
|
difficult to classifiy correctly are given a larger influence. Each
|
|
new classification step \( m \) is then forced to concentrate on those
|
|
observations that are missed in the previous iterations.
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="___sec21">AdaBoost Examples </h2>
|
|
|
|
<p>
|
|
Using <b>Scikit-Learn</b> it is easy to apply the adaptive boosting algorithm, as done here.
|
|
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.ensemble</span> <span style="color: #008000; font-weight: bold">import</span> AdaBoostClassifier
|
|
|
|
ada_clf <span style="color: #666666">=</span> AdaBoostClassifier(
|
|
DecisionTreeClassifier(max_depth<span style="color: #666666">=1</span>), n_estimators<span style="color: #666666">=200</span>,
|
|
algorithm<span style="color: #666666">=</span><span style="color: #BA2121">"SAMME.R"</span>, learning_rate<span style="color: #666666">=0.5</span>, random_state<span style="color: #666666">=42</span>)
|
|
ada_clf<span style="color: #666666">.</span>fit(X_train, y_train)
|
|
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.ensemble</span> <span style="color: #008000; font-weight: bold">import</span> AdaBoostClassifier
|
|
|
|
ada_clf <span style="color: #666666">=</span> AdaBoostClassifier(
|
|
DecisionTreeClassifier(max_depth<span style="color: #666666">=1</span>), n_estimators<span style="color: #666666">=200</span>,
|
|
algorithm<span style="color: #666666">=</span><span style="color: #BA2121">"SAMME.R"</span>, learning_rate<span style="color: #666666">=0.5</span>, random_state<span style="color: #666666">=42</span>)
|
|
ada_clf<span style="color: #666666">.</span>fit(X_train_scaled, y_train)
|
|
y_pred <span style="color: #666666">=</span> ada_clf<span style="color: #666666">.</span>predict(X_test_scaled)
|
|
skplt<span style="color: #666666">.</span>metrics<span style="color: #666666">.</span>plot_confusion_matrix(y_test, y_pred, normalize<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>)
|
|
plt<span style="color: #666666">.</span>show()
|
|
y_probas <span style="color: #666666">=</span> ada_clf<span style="color: #666666">.</span>predict_proba(X_test_scaled)
|
|
skplt<span style="color: #666666">.</span>metrics<span style="color: #666666">.</span>plot_roc(y_test, y_probas)
|
|
plt<span style="color: #666666">.</span>show()
|
|
skplt<span style="color: #666666">.</span>metrics<span style="color: #666666">.</span>plot_cumulative_gain(y_test, y_probas)
|
|
plt<span style="color: #666666">.</span>show()
|
|
</pre></div>
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="___sec22">Gradient boosting: Basics with Steepest Descent/Functional Gradient Descent </h2>
|
|
|
|
<p>
|
|
Gradient boosting is again a similar technique to Adaptive boosting,
|
|
it combines so-called weak classifiers or regressors into a strong
|
|
method via a series of iterations.
|
|
|
|
<p>
|
|
In order to understand the method, let us illustrate its basics by
|
|
bringing back the essential steps in linear regression, where our cost
|
|
function was the least squares function.
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="___sec23">The Squared-Error again! Steepest Descent </h2>
|
|
|
|
<p>
|
|
We start again with our cost function \( {\cal C}(\boldsymbol{y}m\boldsymbol{f})=\sum_{i=0}^{n-1}{\cal L}(y_i, f(x_i)) \) where we want to minimize
|
|
This means that for every iteration, we need to optimize
|
|
|
|
$$
|
|
(\hat{\boldsymbol{f}}) = \mathrm{argmin}_{\boldsymbol{f}}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i-f(x_i))^2.
|
|
$$
|
|
|
|
<p>
|
|
We define a real function \( h_m(x) \) that defines our final function \( f_M(x) \) as
|
|
$$
|
|
f_M(x) = \sum_{m=0}^M h_m(x).
|
|
$$
|
|
|
|
<p>
|
|
In the steepest decent approach we approximate \( h_m(x) = -\rho_m g_m(x) \), where \( \rho_m \) is a scalar and \( g_m(x) \) the gradient defined as
|
|
$$
|
|
g_m(x_i) = \left[ \frac{\partial {\cal L}(y_i, f(x_i))}{\partial f(x_i)}\right]_{f(x_i)=f_{m-1}(x_i)}.
|
|
$$
|
|
|
|
<p>
|
|
With the new gradient we can update \( f_m(x) = f_{m-1}(x) -\rho_m g_m(x) \). Using the above squared-error function we see that
|
|
the gradient is \( g_m(x_i) = -2(y_i-f(x_i)) \).
|
|
|
|
<p>
|
|
Choosing \( f_0(x)=0 \) we obtain \( g_m(x) = -2y_i \) and inserting this into the minimization problem for the cost function we have
|
|
$$
|
|
(\rho_1) = \mathrm{argmin}_{\rho}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i+2\rho y_i)^2.
|
|
$$
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="___sec24">Steepest Descent Example </h2>
|
|
|
|
<p>
|
|
Optimizing with respect to \( \rho \) we obtain (taking the derivative) that \( \rho_1 = -1/2 \). We have then that
|
|
$$
|
|
f_1(x) = f_{0}(x) -\rho_1 g_1(x)=-y_i.
|
|
$$
|
|
|
|
We can then proceed and compute
|
|
$$
|
|
g_2(x_i) = \left[ \frac{\partial {\cal L}(y_i, f(x_i))}{\partial f(x_i)}\right]_{f(x_i)=f_{1}(x_i)=y_i}=-4y_i,
|
|
$$
|
|
|
|
and find a new value for \( \rho_2=-1/2 \) and continue till we have reached \( m=M \). We can modify the steepest descent method, or steepest boosting, by introducing what is called <b>gradient boosting</b>.
|
|
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="___sec25">Gradient Boosting, algorithm </h2>
|
|
|
|
<p>
|
|
Steepest descent is however not much used, since it only optimizes \( f \) at a fixed set of \( n \) points,
|
|
so we do not learn a function that can generalize. However, we can modify the algorithm by
|
|
fitting a weak learner to approximate the negative gradient signal.
|
|
|
|
<p>
|
|
Suppose we have a cost function \( C(f)=\sum_{i=0}^{n-1}L(y_i, f(x_i)) \) where \( y_i \) is our target and \( f(x_i) \) the function which is meant to model \( y_i \). The above cost function could be our standard squared-error function
|
|
$$
|
|
C(\boldsymbol{y},\boldsymbol{f})=\sum_{i=0}^{n-1}(y_i-f(x_i))^2.
|
|
$$
|
|
|
|
<p>
|
|
The way we proceed in an iterative fashion is to
|
|
|
|
<ol>
|
|
<li> Initialize our estimate \( f_0(x) \).</li>
|
|
<li> For \( m=1:M \), we
|
|
|
|
<ol type="a"></li>
|
|
<li> compute the negative gradient vector \( \boldsymbol{u}_m = -\partial C(\boldsymbol{y},\boldsymbol{f})/\partial \boldsymbol{f}(x) \) at \( f(x) = f_{m-1}(x) \);</li>
|
|
<li> fit the so-called base-learner to the negative gradient \( h_m(u_m,x) \);</li>
|
|
<li> update the estimate \( f_m(x) = f_{m-1}(x)+h_m(u_m,x) \);</li>
|
|
</ol>
|
|
|
|
<li> The final estimate is then \( f_M(x) = \sum_{m=1}^M h_m(u_m,x) \).</li>
|
|
</ol>
|
|
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="___sec26">Gradient Boosting, Examples of Regression </h2>
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.ensemble</span> <span style="color: #008000; font-weight: bold">import</span> GradientBoostingRegressor
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> StandardScaler
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">scikitplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">skplt</span>
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.metrics</span> <span style="color: #008000; font-weight: bold">import</span> mean_squared_error
|
|
|
|
n <span style="color: #666666">=</span> <span style="color: #666666">100</span>
|
|
maxdegree <span style="color: #666666">=</span> <span style="color: #666666">6</span>
|
|
|
|
<span style="color: #408080; font-style: italic"># Make data set.</span>
|
|
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(<span style="color: #666666">-3</span>, <span style="color: #666666">3</span>, n)<span style="color: #666666">.</span>reshape(<span style="color: #666666">-1</span>, <span style="color: #666666">1</span>)
|
|
y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>x<span style="color: #666666">**2</span>) <span style="color: #666666">+</span> <span style="color: #666666">1.5</span> <span style="color: #666666">*</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>(x<span style="color: #666666">-2</span>)<span style="color: #666666">**2</span>)<span style="color: #666666">+</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>normal(<span style="color: #666666">0</span>, <span style="color: #666666">0.1</span>, x<span style="color: #666666">.</span>shape)
|
|
|
|
error <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(maxdegree)
|
|
bias <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(maxdegree)
|
|
variance <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(maxdegree)
|
|
polydegree <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(maxdegree)
|
|
X_train, X_test, y_train, y_test <span style="color: #666666">=</span> train_test_split(x, y, test_size<span style="color: #666666">=0.2</span>)
|
|
scaler <span style="color: #666666">=</span> StandardScaler()
|
|
scaler<span style="color: #666666">.</span>fit(X_train)
|
|
X_train_scaled <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>transform(X_train)
|
|
X_test_scaled <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>transform(X_test)
|
|
|
|
<span style="color: #008000; font-weight: bold">for</span> degree <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #666666">1</span>,maxdegree):
|
|
model <span style="color: #666666">=</span> GradientBoostingRegressor(max_depth<span style="color: #666666">=</span>degree, n_estimators<span style="color: #666666">=100</span>, learning_rate<span style="color: #666666">=1.0</span>)
|
|
model<span style="color: #666666">.</span>fit(X_train_scaled,y_train)
|
|
y_pred <span style="color: #666666">=</span> model<span style="color: #666666">.</span>predict(X_test_scaled)
|
|
polydegree[degree] <span style="color: #666666">=</span> degree
|
|
error[degree] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean( np<span style="color: #666666">.</span>mean((y_test <span style="color: #666666">-</span> y_pred)<span style="color: #666666">**2</span>) )
|
|
bias[degree] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean( (y_test <span style="color: #666666">-</span> np<span style="color: #666666">.</span>mean(y_pred))<span style="color: #666666">**2</span> )
|
|
variance[degree] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean( np<span style="color: #666666">.</span>var(y_pred) )
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">'Max depth:'</span>, degree)
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">'Error:'</span>, error[degree])
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">'Bias^2:'</span>, bias[degree])
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">'Var:'</span>, variance[degree])
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">'</span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121"> >= </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121"> + </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121"> = </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121">'</span><span style="color: #666666">.</span>format(error[degree], bias[degree], variance[degree], bias[degree]<span style="color: #666666">+</span>variance[degree]))
|
|
|
|
plt<span style="color: #666666">.</span>xlim(<span style="color: #666666">1</span>,maxdegree<span style="color: #666666">-1</span>)
|
|
plt<span style="color: #666666">.</span>plot(polydegree, error, label<span style="color: #666666">=</span><span style="color: #BA2121">'Error'</span>)
|
|
plt<span style="color: #666666">.</span>plot(polydegree, bias, label<span style="color: #666666">=</span><span style="color: #BA2121">'bias'</span>)
|
|
plt<span style="color: #666666">.</span>plot(polydegree, variance, label<span style="color: #666666">=</span><span style="color: #BA2121">'Variance'</span>)
|
|
plt<span style="color: #666666">.</span>legend()
|
|
save_fig(<span style="color: #BA2121">"gdregression"</span>)
|
|
plt<span style="color: #666666">.</span>show()
|
|
</pre></div>
|
|
<p>
|
|
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
|
|
|
<h2 id="___sec27">Gradient Boosting, Classification Example </h2>
|
|
<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
|
|
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.datasets</span> <span style="color: #008000; font-weight: bold">import</span> load_breast_cancer
|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">scikitplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">skplt</span>
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.ensemble</span> <span style="color: #008000; font-weight: bold">import</span> GradientBoostingClassifier
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> cross_validate
|
|
|
|
<span style="color: #408080; font-style: italic"># Load the data</span>
|
|
cancer <span style="color: #666666">=</span> load_breast_cancer()
|
|
|
|
X_train, X_test, y_train, y_test <span style="color: #666666">=</span> train_test_split(cancer<span style="color: #666666">.</span>data,cancer<span style="color: #666666">.</span>target,random_state<span style="color: #666666">=0</span>)
|
|
<span style="color: #008000">print</span>(X_train<span style="color: #666666">.</span>shape)
|
|
<span style="color: #008000">print</span>(X_test<span style="color: #666666">.</span>shape)
|
|
<span style="color: #408080; font-style: italic">#now scale the data</span>
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> StandardScaler
|
|
scaler <span style="color: #666666">=</span> StandardScaler()
|
|
scaler<span style="color: #666666">.</span>fit(X_train)
|
|
X_train_scaled <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>transform(X_train)
|
|
X_test_scaled <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>transform(X_test)
|
|
|
|
gd_clf <span style="color: #666666">=</span> GradientBoostingClassifier(max_depth<span style="color: #666666">=3</span>, n_estimators<span style="color: #666666">=100</span>, learning_rate<span style="color: #666666">=1.0</span>)
|
|
gd_clf<span style="color: #666666">.</span>fit(X_train_scaled, y_train)
|
|
<span style="color: #408080; font-style: italic">#Cross validation</span>
|
|
accuracy <span style="color: #666666">=</span> cross_validate(gd_clf,X_test_scaled,y_test,cv<span style="color: #666666">=10</span>)[<span style="color: #BA2121">'test_score'</span>]
|
|
<span style="color: #008000">print</span>(accuracy)
|
|
<span style="color: #008000">print</span>(<span style="color: #BA2121">"Test set accuracy with Random Forests and scaled data: </span><span style="color: #BB6688; font-weight: bold">{:.2f}</span><span style="color: #BA2121">"</span><span style="color: #666666">.</span>format(gd_clf<span style="color: #666666">.</span>score(X_test_scaled,y_test)))
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">scikitplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">skplt</span>
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y_pred <span style="color: #666666">=</span> gd_clf<span style="color: #666666">.</span>predict(X_test_scaled)
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skplt<span style="color: #666666">.</span>metrics<span style="color: #666666">.</span>plot_confusion_matrix(y_test, y_pred, normalize<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>)
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save_fig(<span style="color: #BA2121">"gdclassiffierconfusion"</span>)
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plt<span style="color: #666666">.</span>show()
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y_probas <span style="color: #666666">=</span> gd_clf<span style="color: #666666">.</span>predict_proba(X_test_scaled)
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skplt<span style="color: #666666">.</span>metrics<span style="color: #666666">.</span>plot_roc(y_test, y_probas)
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save_fig(<span style="color: #BA2121">"gdclassiffierroc"</span>)
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plt<span style="color: #666666">.</span>show()
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skplt<span style="color: #666666">.</span>metrics<span style="color: #666666">.</span>plot_cumulative_gain(y_test, y_probas)
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save_fig(<span style="color: #BA2121">"gdclassiffiercgain"</span>)
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plt<span style="color: #666666">.</span>show()
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</pre></div>
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<p>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="___sec28">XGBoost: Extreme Gradient Boosting </h2>
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<p>
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<a href="https://github.com/dmlc/xgboost" target="_blank">XGBoost</a> or Extreme Gradient
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Boosting, is an optimized distributed gradient boosting library
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designed to be highly efficient, flexible and portable. It implements
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machine learning algorithms under the Gradient Boosting
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framework. XGBoost provides a parallel tree boosting that solve many
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data science problems in a fast and accurate way. See the <a href="https://arxiv.org/abs/1603.02754" target="_blank">article by Chen and Guestrin</a>.
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<p>
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The authors design and build a highly scalable end-to-end tree
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boosting system. It has a theoretically justified weighted quantile
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sketch for efficient proposal calculation. It introduces a novel sparsity-aware algorithm for parallel tree learning and an effective cache-aware block structure for out-of-core tree learning.
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<p>
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It is now the algorithm which wins essentially all ML competitions!!!
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<p>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="___sec29">Regression Case </h2>
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<p>
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<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">xgboost</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">xgb</span>
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> StandardScaler
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">scikitplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">skplt</span>
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.metrics</span> <span style="color: #008000; font-weight: bold">import</span> mean_squared_error
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n <span style="color: #666666">=</span> <span style="color: #666666">100</span>
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maxdegree <span style="color: #666666">=</span> <span style="color: #666666">6</span>
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<span style="color: #408080; font-style: italic"># Make data set.</span>
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x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(<span style="color: #666666">-3</span>, <span style="color: #666666">3</span>, n)<span style="color: #666666">.</span>reshape(<span style="color: #666666">-1</span>, <span style="color: #666666">1</span>)
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y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>x<span style="color: #666666">**2</span>) <span style="color: #666666">+</span> <span style="color: #666666">1.5</span> <span style="color: #666666">*</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>(x<span style="color: #666666">-2</span>)<span style="color: #666666">**2</span>)<span style="color: #666666">+</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>normal(<span style="color: #666666">0</span>, <span style="color: #666666">0.1</span>, x<span style="color: #666666">.</span>shape)
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error <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(maxdegree)
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bias <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(maxdegree)
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variance <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(maxdegree)
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polydegree <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros(maxdegree)
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X_train, X_test, y_train, y_test <span style="color: #666666">=</span> train_test_split(x, y, test_size<span style="color: #666666">=0.2</span>)
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scaler <span style="color: #666666">=</span> StandardScaler()
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scaler<span style="color: #666666">.</span>fit(X_train)
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X_train_scaled <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>transform(X_train)
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X_test_scaled <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>transform(X_test)
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<span style="color: #008000; font-weight: bold">for</span> degree <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(maxdegree):
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model <span style="color: #666666">=</span> xgb<span style="color: #666666">.</span>XGBRegressor(objective <span style="color: #666666">=</span><span style="color: #BA2121">'reg:squarederror'</span>, colsaobjective <span style="color: #666666">=</span><span style="color: #BA2121">'reg:squarederror'</span>, colsample_bytree <span style="color: #666666">=</span> <span style="color: #666666">0.3</span>, learning_rate <span style="color: #666666">=</span> <span style="color: #666666">0.1</span>,max_depth <span style="color: #666666">=</span> degree, alpha <span style="color: #666666">=</span> <span style="color: #666666">10</span>, n_estimators <span style="color: #666666">=</span> <span style="color: #666666">200</span>)
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model<span style="color: #666666">.</span>fit(X_train_scaled,y_train)
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y_pred <span style="color: #666666">=</span> model<span style="color: #666666">.</span>predict(X_test_scaled)
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polydegree[degree] <span style="color: #666666">=</span> degree
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error[degree] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean( np<span style="color: #666666">.</span>mean((y_test <span style="color: #666666">-</span> y_pred)<span style="color: #666666">**2</span>) )
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bias[degree] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean( (y_test <span style="color: #666666">-</span> np<span style="color: #666666">.</span>mean(y_pred))<span style="color: #666666">**2</span> )
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variance[degree] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>mean( np<span style="color: #666666">.</span>var(y_pred) )
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<span style="color: #008000">print</span>(<span style="color: #BA2121">'Max depth:'</span>, degree)
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<span style="color: #008000">print</span>(<span style="color: #BA2121">'Error:'</span>, error[degree])
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<span style="color: #008000">print</span>(<span style="color: #BA2121">'Bias^2:'</span>, bias[degree])
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<span style="color: #008000">print</span>(<span style="color: #BA2121">'Var:'</span>, variance[degree])
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<span style="color: #008000">print</span>(<span style="color: #BA2121">'</span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121"> >= </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121"> + </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121"> = </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121">'</span><span style="color: #666666">.</span>format(error[degree], bias[degree], variance[degree], bias[degree]<span style="color: #666666">+</span>variance[degree]))
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|
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plt<span style="color: #666666">.</span>xlim(<span style="color: #666666">1</span>,maxdegree<span style="color: #666666">-1</span>)
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plt<span style="color: #666666">.</span>plot(polydegree, error, label<span style="color: #666666">=</span><span style="color: #BA2121">'Error'</span>)
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plt<span style="color: #666666">.</span>plot(polydegree, bias, label<span style="color: #666666">=</span><span style="color: #BA2121">'bias'</span>)
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|
plt<span style="color: #666666">.</span>plot(polydegree, variance, label<span style="color: #666666">=</span><span style="color: #BA2121">'Variance'</span>)
|
|
plt<span style="color: #666666">.</span>legend()
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|
plt<span style="color: #666666">.</span>show()
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</pre></div>
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<p>
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<!-- !split --><br><br><br><br><br><br><br><br><br><br>
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<h2 id="___sec30">Xgboost on the Cancer Data </h2>
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|
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<p>
|
|
As you will see from the confusion matrix below, XGBoots does an excellent job on the Wisconsin cancer data and outperforms essentially all agorithms we have discussed till now.
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<p>
|
|
|
|
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
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|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
|
|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
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|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.datasets</span> <span style="color: #008000; font-weight: bold">import</span> load_breast_cancer
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> LabelEncoder
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> cross_validate
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">scikitplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">skplt</span>
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|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">xgboost</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">xgb</span>
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|
<span style="color: #408080; font-style: italic"># Load the data</span>
|
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cancer <span style="color: #666666">=</span> load_breast_cancer()
|
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|
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X_train, X_test, y_train, y_test <span style="color: #666666">=</span> train_test_split(cancer<span style="color: #666666">.</span>data,cancer<span style="color: #666666">.</span>target,random_state<span style="color: #666666">=0</span>)
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<span style="color: #008000">print</span>(X_train<span style="color: #666666">.</span>shape)
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<span style="color: #008000">print</span>(X_test<span style="color: #666666">.</span>shape)
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<span style="color: #408080; font-style: italic">#now scale the data</span>
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|
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> StandardScaler
|
|
scaler <span style="color: #666666">=</span> StandardScaler()
|
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scaler<span style="color: #666666">.</span>fit(X_train)
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X_train_scaled <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>transform(X_train)
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X_test_scaled <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>transform(X_test)
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xg_clf <span style="color: #666666">=</span> xgb<span style="color: #666666">.</span>XGBClassifier()
|
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xg_clf<span style="color: #666666">.</span>fit(X_train_scaled,y_train)
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y_test <span style="color: #666666">=</span> xg_clf<span style="color: #666666">.</span>predict(X_test_scaled)
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|
<span style="color: #008000">print</span>(<span style="color: #BA2121">"Test set accuracy with Random Forests and scaled data: </span><span style="color: #BB6688; font-weight: bold">{:.2f}</span><span style="color: #BA2121">"</span><span style="color: #666666">.</span>format(xg_clf<span style="color: #666666">.</span>score(X_test_scaled,y_test)))
|
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|
|
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">scikitplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">skplt</span>
|
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y_pred <span style="color: #666666">=</span> xg_clf<span style="color: #666666">.</span>predict(X_test_scaled)
|
|
skplt<span style="color: #666666">.</span>metrics<span style="color: #666666">.</span>plot_confusion_matrix(y_test, y_pred, normalize<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">True</span>)
|
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save_fig(<span style="color: #BA2121">"xdclassiffierconfusion"</span>)
|
|
plt<span style="color: #666666">.</span>show()
|
|
y_probas <span style="color: #666666">=</span> xg_clf<span style="color: #666666">.</span>predict_proba(X_test_scaled)
|
|
skplt<span style="color: #666666">.</span>metrics<span style="color: #666666">.</span>plot_roc(y_test, y_probas)
|
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save_fig(<span style="color: #BA2121">"xdclassiffierroc"</span>)
|
|
plt<span style="color: #666666">.</span>show()
|
|
skplt<span style="color: #666666">.</span>metrics<span style="color: #666666">.</span>plot_cumulative_gain(y_test, y_probas)
|
|
save_fig(<span style="color: #BA2121">"gdclassiffiercgain"</span>)
|
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plt<span style="color: #666666">.</span>show()
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|
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xgb<span style="color: #666666">.</span>plot_tree(xg_clf,num_trees<span style="color: #666666">=0</span>)
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plt<span style="color: #666666">.</span>rcParams[<span style="color: #BA2121">'figure.figsize'</span>] <span style="color: #666666">=</span> [<span style="color: #666666">50</span>, <span style="color: #666666">10</span>]
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save_fig(<span style="color: #BA2121">"xgtree"</span>)
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plt<span style="color: #666666">.</span>show()
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xgb<span style="color: #666666">.</span>plot_importance(xg_clf)
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plt<span style="color: #666666">.</span>rcParams[<span style="color: #BA2121">'figure.figsize'</span>] <span style="color: #666666">=</span> [<span style="color: #666666">5</span>, <span style="color: #666666">5</span>]
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save_fig(<span style="color: #BA2121">"xgparams"</span>)
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plt<span style="color: #666666">.</span>show()
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</pre></div>
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<p>
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<!-- ------------------- end of main content --------------- -->
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<center style="font-size:80%">
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<!-- copyright --> © 1999-2020, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license
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</center>
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</body>
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</html>
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