Morten Hjorth-Jensen
1618 lines
157 KiB
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Applied Data Analysis and Machine Learning, FYS-STK3155/4155 at the University of Oslo, Norway
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Review of Statistics with Resampling Techniques and Linear Algebra
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1. Elements of Probability Theory and Statistical Data Analysis
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2. Linear Algebra, Handling of Arrays and more Python Features
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From Regression to Support Vector Machines
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3. Linear Regression
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4. Ridge and Lasso Regression
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5. Resampling Methods
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6. Logistic Regression
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7. Optimization, the central part of any Machine Learning algortithm
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8. Support Vector Machines, overarching aims
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Decision Trees, Ensemble Methods and Boosting
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9. Decision trees, overarching aims
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10. Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods
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Dimensionality Reduction
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11. Basic ideas of the Principal Component Analysis (PCA)
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Deep Learning Methods
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12. Neural networks
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13. Building a Feed Forward Neural Network
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14. Solving Differential Equations with Deep Learning
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<a class="reference internal nav-link" href="#basics-of-a-tree">
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9.1. Basics of a tree
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<a class="reference internal nav-link" href="#general-features">
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9.2. General Features
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<a class="reference internal nav-link" href="#building-a-tree-regression">
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9.3. Building a tree, regression
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<a class="reference internal nav-link" href="#making-a-tree">
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9.3.1. Making a tree
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9.3.2. Schematic Regression Procedure
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<a class="reference internal nav-link" href="#a-classification-tree">
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9.4. A Classification Tree
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<a class="reference internal nav-link" href="#visualizing-the-tree-classification">
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9.4.1. Visualizing the Tree, Classification
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<li class="toc-h3 nav-item toc-entry">
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<a class="reference internal nav-link" href="#other-ways-of-visualizing-the-trees">
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9.4.2. Other ways of visualizing the trees
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</a>
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</li>
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</ul>
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</li>
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#algorithms-for-setting-up-decision-trees">
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9.5. Algorithms for Setting up Decision Trees
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<li class="toc-h3 nav-item toc-entry">
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<a class="reference internal nav-link" href="#the-cart-algorithm-for-classification">
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9.5.1. The CART algorithm for Classification
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</a>
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</li>
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<li class="toc-h3 nav-item toc-entry">
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<a class="reference internal nav-link" href="#the-cart-algorithm-for-regression">
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9.5.2. The CART algorithm for Regression
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</a>
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<li class="toc-h3 nav-item toc-entry">
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<a class="reference internal nav-link" href="#computing-the-gini-index">
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9.5.3. Computing the Gini index
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<a class="reference internal nav-link" href="#simple-python-code-to-read-in-data-and-perform-classification">
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9.5.4. Simple Python Code to read in Data and perform Classification
|
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</a>
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</li>
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</ul>
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#entropy-and-the-id3-algorithm">
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9.6. Entropy and the ID3 algorithm
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<a class="reference internal nav-link" href="#cancer-data-again-now-with-decision-trees-and-other-methods">
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9.6.1. Cancer Data again now with Decision Trees and other Methods
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9.6.2. Another example, the moons again
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<a class="reference internal nav-link" href="#pros-and-cons-of-trees-pros">
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9.7. Pros and cons of trees, pros
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9.7.1. Disadvantages
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<div class="tex2jax_ignore mathjax_ignore section" id="decision-trees-overarching-aims">
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<h1><span class="section-number">9. </span>Decision trees, overarching aims<a class="headerlink" href="#decision-trees-overarching-aims" title="Permalink to this headline">¶</a></h1>
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<p>We start here with the most basic algorithm, the so-called decision
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tree. With this basic algorithm we can in turn build more complex
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networks, spanning from homogeneous and heterogenous forests (bagging,
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random forests and more) to one of the most popular supervised
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algorithms nowadays, the extreme gradient boosting, or just
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XGBoost. But let us start with the simplest possible ingredient.</p>
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<p>Decision trees are supervised learning algorithms used for both,
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classification and regression tasks.</p>
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<p>The main idea of decision trees
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||
is to find those descriptive features which contain the most
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<strong>information</strong> regarding the target feature and then split the dataset
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along the values of these features such that the target feature values
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for the resulting underlying datasets are as pure as possible.</p>
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||
<p>The descriptive features which reproduce best the target/output features are normally said
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||
to be the most informative ones. The process of finding the <strong>most
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informative</strong> feature is done until we accomplish a stopping criteria
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where we then finally end up in so called <strong>leaf nodes</strong>.</p>
|
||
<div class="section" id="basics-of-a-tree">
|
||
<h2><span class="section-number">9.1. </span>Basics of a tree<a class="headerlink" href="#basics-of-a-tree" title="Permalink to this headline">¶</a></h2>
|
||
<p>A decision tree is typically divided into a <strong>root node</strong>, the <strong>interior nodes</strong>,
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and the final <strong>leaf nodes</strong> or just <strong>leaves</strong>. These entities are then connected by so-called <strong>branches</strong>.</p>
|
||
<p>The leaf nodes
|
||
contain the predictions we will make for new query instances presented
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||
to our trained model. This is possible since the model has
|
||
learned the underlying structure of the training data and hence can,
|
||
given some assumptions, make predictions about the target feature value
|
||
(class) of unseen query instances.</p>
|
||
</div>
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||
<div class="section" id="general-features">
|
||
<h2><span class="section-number">9.2. </span>General Features<a class="headerlink" href="#general-features" title="Permalink to this headline">¶</a></h2>
|
||
<p>The overarching approach to decision trees is a top-down approach.</p>
|
||
<ul class="simple">
|
||
<li><p>A leaf provides the classification of a given instance.</p></li>
|
||
<li><p>A node specifies a test of some attribute of the instance.</p></li>
|
||
<li><p>A branch corresponds to a possible values of an attribute.</p></li>
|
||
<li><p>An instance is classified by starting at the root node of the tree, testing the attribute specified by this node, then moving down the tree branch corresponding to the value of the attribute in the given example.</p></li>
|
||
</ul>
|
||
<p>This process is then repeated for the subtree rooted at the new
|
||
node.</p>
|
||
<p>In simplified terms, the process of training a decision tree and
|
||
predicting the target features of query instances is as follows:</p>
|
||
<ol class="simple">
|
||
<li><p>Present a dataset containing of a number of training instances characterized by a number of descriptive features and a target feature</p></li>
|
||
<li><p>Train the decision tree model by continuously splitting the target feature along the values of the descriptive features using a measure of information gain during the training process</p></li>
|
||
<li><p>Grow the tree until we accomplish a stopping criteria create leaf nodes which represent the <em>predictions</em> we want to make for new query instances</p></li>
|
||
<li><p>Show query instances to the tree and run down the tree until we arrive at leaf nodes</p></li>
|
||
</ol>
|
||
<p>Then we are essentially done!</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="o">%</span><span class="k">matplotlib</span> inline
|
||
|
||
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <span class="n">PolynomialFeatures</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <span class="n">LinearRegression</span>
|
||
|
||
<span class="n">steps</span><span class="o">=</span><span class="mi">250</span>
|
||
|
||
<span class="n">distance</span><span class="o">=</span><span class="mi">0</span>
|
||
<span class="n">x</span><span class="o">=</span><span class="mi">0</span>
|
||
<span class="n">distance_list</span><span class="o">=</span><span class="p">[]</span>
|
||
<span class="n">steps_list</span><span class="o">=</span><span class="p">[]</span>
|
||
<span class="k">while</span> <span class="n">x</span><span class="o"><</span><span class="n">steps</span><span class="p">:</span>
|
||
<span class="n">distance</span><span class="o">+=</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
|
||
<span class="n">distance_list</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">distance</span><span class="p">)</span>
|
||
<span class="n">x</span><span class="o">+=</span><span class="mi">1</span>
|
||
<span class="n">steps_list</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">steps_list</span><span class="p">,</span><span class="n">distance_list</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">'green'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"Random Walk Data"</span><span class="p">)</span>
|
||
|
||
<span class="n">steps_list</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">(</span><span class="n">steps_list</span><span class="p">)</span>
|
||
<span class="n">distance_list</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">(</span><span class="n">distance_list</span><span class="p">)</span>
|
||
|
||
<span class="n">X</span><span class="o">=</span><span class="n">steps_list</span><span class="p">[:,</span><span class="n">np</span><span class="o">.</span><span class="n">newaxis</span><span class="p">]</span>
|
||
|
||
<span class="c1">#Polynomial fits</span>
|
||
|
||
<span class="c1">#Degree 2</span>
|
||
<span class="n">poly_features</span><span class="o">=</span><span class="n">PolynomialFeatures</span><span class="p">(</span><span class="n">degree</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">include_bias</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
|
||
<span class="n">X_poly</span><span class="o">=</span><span class="n">poly_features</span><span class="o">.</span><span class="n">fit_transform</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
|
||
|
||
<span class="n">lin_reg</span><span class="o">=</span><span class="n">LinearRegression</span><span class="p">()</span>
|
||
<span class="n">poly_fit</span><span class="o">=</span><span class="n">lin_reg</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_poly</span><span class="p">,</span><span class="n">distance_list</span><span class="p">)</span>
|
||
<span class="n">b</span><span class="o">=</span><span class="n">lin_reg</span><span class="o">.</span><span class="n">coef_</span>
|
||
<span class="n">c</span><span class="o">=</span><span class="n">lin_reg</span><span class="o">.</span><span class="n">intercept_</span>
|
||
<span class="nb">print</span> <span class="p">(</span><span class="s2">"2nd degree coefficients:"</span><span class="p">)</span>
|
||
<span class="nb">print</span> <span class="p">(</span><span class="s2">"zero power: "</span><span class="p">,</span><span class="n">c</span><span class="p">)</span>
|
||
<span class="nb">print</span> <span class="p">(</span><span class="s2">"first power: "</span><span class="p">,</span> <span class="n">b</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
|
||
<span class="nb">print</span> <span class="p">(</span><span class="s2">"second power: "</span><span class="p">,</span><span class="n">b</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
|
||
|
||
<span class="n">z</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">steps</span><span class="p">,</span> <span class="mf">.01</span><span class="p">)</span>
|
||
<span class="n">z_mod</span><span class="o">=</span><span class="n">b</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="n">z</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="n">b</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">z</span><span class="o">+</span><span class="n">c</span>
|
||
|
||
<span class="n">fit_mod</span><span class="o">=</span><span class="n">b</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="n">X</span><span class="o">**</span><span class="mi">2</span><span class="o">+</span><span class="n">b</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">X</span><span class="o">+</span><span class="n">c</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">z_mod</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">'r'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"2nd Degree Fit"</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">"Polynomial Regression"</span><span class="p">)</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">"Steps"</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">"Distance"</span><span class="p">)</span>
|
||
|
||
<span class="c1">#Degree 10</span>
|
||
<span class="n">poly_features10</span><span class="o">=</span><span class="n">PolynomialFeatures</span><span class="p">(</span><span class="n">degree</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span> <span class="n">include_bias</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
|
||
<span class="n">X_poly10</span><span class="o">=</span><span class="n">poly_features10</span><span class="o">.</span><span class="n">fit_transform</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
|
||
|
||
<span class="n">poly_fit10</span><span class="o">=</span><span class="n">lin_reg</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_poly10</span><span class="p">,</span><span class="n">distance_list</span><span class="p">)</span>
|
||
|
||
<span class="n">y_plot</span><span class="o">=</span><span class="n">poly_fit10</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_poly10</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y_plot</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">'black'</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"10th Degree Fit"</span><span class="p">)</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
|
||
|
||
<span class="c1">#Decision Tree Regression</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.tree</span> <span class="kn">import</span> <span class="n">DecisionTreeRegressor</span>
|
||
<span class="n">regr_1</span><span class="o">=</span><span class="n">DecisionTreeRegressor</span><span class="p">(</span><span class="n">max_depth</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
|
||
<span class="n">regr_2</span><span class="o">=</span><span class="n">DecisionTreeRegressor</span><span class="p">(</span><span class="n">max_depth</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
|
||
<span class="n">regr_3</span><span class="o">=</span><span class="n">DecisionTreeRegressor</span><span class="p">(</span><span class="n">max_depth</span><span class="o">=</span><span class="mi">7</span><span class="p">)</span>
|
||
<span class="n">regr_1</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">distance_list</span><span class="p">)</span>
|
||
<span class="n">regr_2</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">distance_list</span><span class="p">)</span>
|
||
<span class="n">regr_3</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">distance_list</span><span class="p">)</span>
|
||
|
||
<span class="n">X_test</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mf">0.0</span><span class="p">,</span> <span class="n">steps</span><span class="p">,</span> <span class="mf">0.01</span><span class="p">)[:,</span> <span class="n">np</span><span class="o">.</span><span class="n">newaxis</span><span class="p">]</span>
|
||
<span class="n">y_1</span> <span class="o">=</span> <span class="n">regr_1</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
|
||
<span class="n">y_2</span> <span class="o">=</span> <span class="n">regr_2</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
|
||
<span class="n">y_3</span><span class="o">=</span><span class="n">regr_3</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Plot the results</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">distance_list</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mf">2.5</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s2">"black"</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"data"</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span> <span class="n">y_1</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">"red"</span><span class="p">,</span>
|
||
<span class="n">label</span><span class="o">=</span><span class="s2">"max_depth=2"</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span> <span class="n">y_2</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">"green"</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"max_depth=5"</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span> <span class="n">y_3</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">"m"</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"max_depth=7"</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">"Data"</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">"Darget"</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">"Decision Tree Regression"</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
<div class="cell_output docutils container">
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>2nd degree coefficients:
|
||
zero power: -5.456418966187335
|
||
first power: 0.07099747918547346
|
||
second power: -0.00028369228101198006
|
||
</pre></div>
|
||
</div>
|
||
<img alt="_images/chapter6_1_1.png" src="_images/chapter6_1_1.png" />
|
||
<img alt="_images/chapter6_1_2.png" src="_images/chapter6_1_2.png" />
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="building-a-tree-regression">
|
||
<h2><span class="section-number">9.3. </span>Building a tree, regression<a class="headerlink" href="#building-a-tree-regression" title="Permalink to this headline">¶</a></h2>
|
||
<p>There are mainly two steps</p>
|
||
<ol class="simple">
|
||
<li><p>We split the predictor space (the set of possible values <span class="math notranslate nohighlight">\(x_1,x_2,\dots, x_p\)</span>) into <span class="math notranslate nohighlight">\(J\)</span> distinct and non-non-overlapping regions, <span class="math notranslate nohighlight">\(R_1,R_2,\dots,R_J\)</span>.</p></li>
|
||
<li><p>For every observation that falls into the region <span class="math notranslate nohighlight">\(R_j\)</span> , we make the same prediction, which is simply the mean of the response values for the training observations in <span class="math notranslate nohighlight">\(R_j\)</span>.</p></li>
|
||
</ol>
|
||
<p>How do we construct the regions <span class="math notranslate nohighlight">\(R_1,\dots,R_J\)</span>? In theory, the
|
||
regions could have any shape. However, we choose to divide the
|
||
predictor space into high-dimensional rectangles, or boxes, for
|
||
simplicity and for ease of interpretation of the resulting predictive
|
||
model. The goal is to find boxes <span class="math notranslate nohighlight">\(R_1,\dots,R_J\)</span> that minimize the
|
||
MSE, given by</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\sum_{j=1}^J\sum_{i\in R_j}(y_i-\overline{y}_{R_j})^2,
|
||
\]</div>
|
||
<p>where <span class="math notranslate nohighlight">\(\overline{y}_{R_j}\)</span> is the mean response for the training observations
|
||
within box <span class="math notranslate nohighlight">\(j\)</span>.</p>
|
||
<p>Unfortunately, it is computationally infeasible to consider every
|
||
possible partition of the feature space into <span class="math notranslate nohighlight">\(J\)</span> boxes. The common
|
||
strategy is to take a top-down approach</p>
|
||
<p>The approach is top-down because it begins at the top of the tree (all
|
||
observations belong to a single region) and then successively splits
|
||
the predictor space; each split is indicated via two new branches
|
||
further down on the tree. It is greedy because at each step of the
|
||
tree-building process, the best split is made at that particular step,
|
||
rather than looking ahead and picking a split that will lead to a
|
||
better tree in some future step.</p>
|
||
<div class="section" id="making-a-tree">
|
||
<h3><span class="section-number">9.3.1. </span>Making a tree<a class="headerlink" href="#making-a-tree" title="Permalink to this headline">¶</a></h3>
|
||
<p>In order to implement the recursive binary splitting we start by selecting
|
||
the predictor <span class="math notranslate nohighlight">\(x_j\)</span> and a cutpoint <span class="math notranslate nohighlight">\(s\)</span> that splits the predictor space into two regions <span class="math notranslate nohighlight">\(R_1\)</span> and <span class="math notranslate nohighlight">\(R_2\)</span></p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\left\{X\vert x_j < s\right\},
|
||
\]</div>
|
||
<p>and</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\left\{X\vert x_j \geq s\right\},
|
||
\]</div>
|
||
<p>so that we obtain the lowest MSE, that is</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\sum_{i:x_i\in R_j}(y_i-\overline{y}_{R_1})^2+\sum_{i:x_i\in R_2}(y_i-\overline{y}_{R_2})^2,
|
||
\]</div>
|
||
<p>which we want to minimize by considering all predictors
|
||
<span class="math notranslate nohighlight">\(x_1,x_2,\dots,x_p\)</span>. We consider also all possible values of <span class="math notranslate nohighlight">\(s\)</span> for
|
||
each predictor. These values could be determined by randomly assigned
|
||
numbers or by starting at the midpoint and then proceed till we find
|
||
an optimal value.</p>
|
||
<p>For any <span class="math notranslate nohighlight">\(j\)</span> and <span class="math notranslate nohighlight">\(s\)</span>, we define the pair of half-planes where
|
||
<span class="math notranslate nohighlight">\(\overline{y}_{R_1}\)</span> is the mean response for the training
|
||
observations in <span class="math notranslate nohighlight">\(R_1(j,s)\)</span>, and <span class="math notranslate nohighlight">\(\overline{y}_{R_2}\)</span> is the mean
|
||
response for the training observations in <span class="math notranslate nohighlight">\(R_2(j,s)\)</span>.</p>
|
||
<p>Finding the values of <span class="math notranslate nohighlight">\(j\)</span> and <span class="math notranslate nohighlight">\(s\)</span> that minimize the above equation can be
|
||
done quite quickly, especially when the number of features <span class="math notranslate nohighlight">\(p\)</span> is not
|
||
too large.</p>
|
||
<p>Next, we repeat the process, looking
|
||
for the best predictor and best cutpoint in order to split the data
|
||
further so as to minimize the MSE within each of the resulting
|
||
regions. However, this time, instead of splitting the entire predictor
|
||
space, we split one of the two previously identified regions. We now
|
||
have three regions. Again, we look to split one of these three regions
|
||
further, so as to minimize the MSE. The process continues until a
|
||
stopping criterion is reached; for instance, we may continue until no
|
||
region contains more than five observations.</p>
|
||
<p>The above procedure is rather straightforward, but leads often to
|
||
overfitting and unnecessarily large and complicated trees. The basic
|
||
idea is to grow a large tree <span class="math notranslate nohighlight">\(T_0\)</span> and then prune it back in order to
|
||
obtain a subtree. A smaller tree with fewer splits (fewer regions) can
|
||
lead to smaller variance and better interpretation at the cost of a
|
||
little more bias.</p>
|
||
<p>The so-called Cost complexity pruning algorithm gives us a
|
||
way to do just this. Rather than considering every possible subtree,
|
||
we consider a sequence of trees indexed by a nonnegative tuning
|
||
parameter <span class="math notranslate nohighlight">\(\alpha\)</span>.</p>
|
||
<p>Read more at the following <a class="reference external" href="https://scikit-learn.org/stable/auto_examples/tree/plot_cost_complexity_pruning.html#sphx-glr-auto-examples-tree-plot-cost-complexity-pruning-py">Scikit-Learn link on pruning</a>.</p>
|
||
<p>For each value of <span class="math notranslate nohighlight">\(\alpha\)</span> there corresponds a subtree <span class="math notranslate nohighlight">\(T \in T_0\)</span> such that</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\sum_{m=1}^{\overline{T}}\sum_{i:x_i\in R_m}(y_i-\overline{y}_{R_m})^2+\alpha\overline{T},
|
||
\]</div>
|
||
<p>is as small as possible. Here <span class="math notranslate nohighlight">\(\overline{T}\)</span> is
|
||
the number of terminal nodes of the tree <span class="math notranslate nohighlight">\(T\)</span> , <span class="math notranslate nohighlight">\(R_m\)</span> is the
|
||
rectangle (i.e. the subset of predictor space) corresponding to the <span class="math notranslate nohighlight">\(m\)</span>-th terminal node.</p>
|
||
<p>The tuning parameter <span class="math notranslate nohighlight">\(\alpha\)</span> controls a trade-off between the subtree’s
|
||
complexity and its fit to the training data. When <span class="math notranslate nohighlight">\(\alpha = 0\)</span>, then the
|
||
subtree <span class="math notranslate nohighlight">\(T\)</span> will simply equal <span class="math notranslate nohighlight">\(T_0\)</span>,
|
||
because then the above equation just measures the
|
||
training error.
|
||
However, as <span class="math notranslate nohighlight">\(\alpha\)</span> increases, there is a price to pay for
|
||
having a tree with many terminal nodes. The above equation will
|
||
tend to be minimized for a smaller subtree.</p>
|
||
<p>It turns out that as we increase <span class="math notranslate nohighlight">\(\alpha\)</span> from zero
|
||
branches get pruned from the tree in a nested and predictable fashion,
|
||
so obtaining the whole sequence of subtrees as a function of <span class="math notranslate nohighlight">\(\alpha\)</span> is
|
||
easy. We can select a value of <span class="math notranslate nohighlight">\(\alpha\)</span> using a validation set or using
|
||
cross-validation. We then return to the full data set and obtain the
|
||
subtree corresponding to <span class="math notranslate nohighlight">\(\alpha\)</span>.</p>
|
||
</div>
|
||
<div class="section" id="schematic-regression-procedure">
|
||
<h3><span class="section-number">9.3.2. </span>Schematic Regression Procedure<a class="headerlink" href="#schematic-regression-procedure" title="Permalink to this headline">¶</a></h3>
|
||
<p>Building a Regression Tree</p>
|
||
<ol class="simple">
|
||
<li><p>Use recursive binary splitting to grow a large tree on the training data, stopping only when each terminal node has fewer than some minimum number of observations.</p></li>
|
||
<li><p>Apply cost complexity pruning to the large tree in order to obtain a sequence of best subtrees, as a function of <span class="math notranslate nohighlight">\(\alpha\)</span>.</p></li>
|
||
<li><p>Use for example <span class="math notranslate nohighlight">\(K\)</span>-fold cross-validation to choose <span class="math notranslate nohighlight">\(\alpha\)</span>. Divide the training observations into <span class="math notranslate nohighlight">\(K\)</span> folds. For each <span class="math notranslate nohighlight">\(k=1,2,\dots,K\)</span> we:</p></li>
|
||
</ol>
|
||
<ul class="simple">
|
||
<li><p>repeat steps 1 and 2 on all but the <span class="math notranslate nohighlight">\(k\)</span>-th fold of the training data.</p></li>
|
||
<li><p>Then we valuate the mean squared prediction error on the data in the left-out <span class="math notranslate nohighlight">\(k\)</span>-th fold, as a function of <span class="math notranslate nohighlight">\(\alpha\)</span>.</p></li>
|
||
<li><p>Finally we average the results for each value of <span class="math notranslate nohighlight">\(\alpha\)</span>, and pick <span class="math notranslate nohighlight">\(\alpha\)</span> to minimize the average error.</p></li>
|
||
</ul>
|
||
<ol class="simple">
|
||
<li><p>Return the subtree from Step 2 that corresponds to the chosen value of <span class="math notranslate nohighlight">\(\alpha\)</span>.</p></li>
|
||
</ol>
|
||
<p>!eblock</p>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="a-classification-tree">
|
||
<h2><span class="section-number">9.4. </span>A Classification Tree<a class="headerlink" href="#a-classification-tree" title="Permalink to this headline">¶</a></h2>
|
||
<p>A classification tree is very similar to a regression tree, except
|
||
that it is used to predict a qualitative response rather than a
|
||
quantitative one. Recall that for a regression tree, the predicted
|
||
response for an observation is given by the mean response of the
|
||
training observations that belong to the same terminal node. In
|
||
contrast, for a classification tree, we predict that each observation
|
||
belongs to the most commonly occurring class of training observations
|
||
in the region to which it belongs. In interpreting the results of a
|
||
classification tree, we are often interested not only in the class
|
||
prediction corresponding to a particular terminal node region, but
|
||
also in the class proportions among the training observations that
|
||
fall into that region.</p>
|
||
<p>The task of growing a
|
||
classification tree is quite similar to the task of growing a
|
||
regression tree. Just as in the regression setting, we use recursive
|
||
binary splitting to grow a classification tree. However, in the
|
||
classification setting, the MSE cannot be used as a criterion for making
|
||
the binary splits. A natural alternative to MSE is the <strong>classification
|
||
error rate</strong>. Since we plan to assign an observation in a given region
|
||
to the most commonly occurring error rate class of training
|
||
observations in that region, the classification error rate is simply
|
||
the fraction of the training observations in that region that do not
|
||
belong to the most common class.</p>
|
||
<p>When building a classification tree, either the Gini index or the
|
||
entropy are typically used to evaluate the quality of a particular
|
||
split, since these two approaches are more sensitive to node purity
|
||
than is the classification error rate.</p>
|
||
<p>If our targets are the outcome of a classification process that takes
|
||
for example <span class="math notranslate nohighlight">\(k=1,2,\dots,K\)</span> values, the only thing we need to think of
|
||
is to set up the splitting criteria for each node.</p>
|
||
<p>We define a PDF <span class="math notranslate nohighlight">\(p_{mk}\)</span> that represents the number of observations of
|
||
a class <span class="math notranslate nohighlight">\(k\)</span> in a region <span class="math notranslate nohighlight">\(R_m\)</span> with <span class="math notranslate nohighlight">\(N_m\)</span> observations. We represent
|
||
this likelihood function in terms of the proportion <span class="math notranslate nohighlight">\(I(y_i=k)\)</span> of
|
||
observations of this class in the region <span class="math notranslate nohighlight">\(R_m\)</span> as</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
p_{mk} = \frac{1}{N_m}\sum_{x_i\in R_m}I(y_i=k).
|
||
\]</div>
|
||
<p>We let <span class="math notranslate nohighlight">\(p_{mk}\)</span> represent the majority class of observations in region
|
||
<span class="math notranslate nohighlight">\(m\)</span>. The three most common ways of splitting a node are given by</p>
|
||
<ul class="simple">
|
||
<li><p>Misclassification error</p></li>
|
||
</ul>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
p_{mk} = \frac{1}{N_m}\sum_{x_i\in R_m}I(y_i\ne k) = 1-p_{mk}.
|
||
\]</div>
|
||
<ul class="simple">
|
||
<li><p>Gini index <span class="math notranslate nohighlight">\(g\)</span></p></li>
|
||
</ul>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
g = \sum_{k=1}^K p_{mk}(1-p_{mk}).
|
||
\]</div>
|
||
<ul class="simple">
|
||
<li><p>Information entropy or just entropy <span class="math notranslate nohighlight">\(s\)</span></p></li>
|
||
</ul>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
s = -\sum_{k=1}^K p_{mk}\log{p_{mk}}.
|
||
\]</div>
|
||
<div class="section" id="visualizing-the-tree-classification">
|
||
<h3><span class="section-number">9.4.1. </span>Visualizing the Tree, Classification<a class="headerlink" href="#visualizing-the-tree-classification" title="Permalink to this headline">¶</a></h3>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">os</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">load_breast_cancer</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.tree</span> <span class="kn">import</span> <span class="n">DecisionTreeClassifier</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">confusion_matrix</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.tree</span> <span class="kn">import</span> <span class="n">export_graphviz</span>
|
||
|
||
<span class="kn">from</span> <span class="nn">IPython.display</span> <span class="kn">import</span> <span class="n">Image</span>
|
||
<span class="kn">from</span> <span class="nn">pydot</span> <span class="kn">import</span> <span class="n">graph_from_dot_data</span>
|
||
<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
|
||
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
|
||
|
||
<span class="n">cancer</span> <span class="o">=</span> <span class="n">load_breast_cancer</span><span class="p">()</span>
|
||
<span class="n">X</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">cancer</span><span class="o">.</span><span class="n">data</span><span class="p">,</span> <span class="n">columns</span><span class="o">=</span><span class="n">cancer</span><span class="o">.</span><span class="n">feature_names</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
|
||
<span class="n">y</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">Categorical</span><span class="o">.</span><span class="n">from_codes</span><span class="p">(</span><span class="n">cancer</span><span class="o">.</span><span class="n">target</span><span class="p">,</span> <span class="n">cancer</span><span class="o">.</span><span class="n">target_names</span><span class="p">)</span>
|
||
<span class="n">y</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">get_dummies</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
|
||
<span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
|
||
<span class="n">tree_clf</span> <span class="o">=</span> <span class="n">DecisionTreeClassifier</span><span class="p">(</span><span class="n">max_depth</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
|
||
<span class="n">tree_clf</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
|
||
|
||
<span class="n">export_graphviz</span><span class="p">(</span>
|
||
<span class="n">tree_clf</span><span class="p">,</span>
|
||
<span class="n">out_file</span><span class="o">=</span><span class="s2">"DataFiles/cancer.dot"</span><span class="p">,</span>
|
||
<span class="n">feature_names</span><span class="o">=</span><span class="n">cancer</span><span class="o">.</span><span class="n">feature_names</span><span class="p">,</span>
|
||
<span class="n">class_names</span><span class="o">=</span><span class="n">cancer</span><span class="o">.</span><span class="n">target_names</span><span class="p">,</span>
|
||
<span class="n">rounded</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span>
|
||
<span class="n">filled</span><span class="o">=</span><span class="kc">True</span>
|
||
<span class="p">)</span>
|
||
<span class="n">cmd</span> <span class="o">=</span> <span class="s1">'dot -Tpng DataFiles/cancer.dot -o DataFiles/cancer.png'</span>
|
||
<span class="n">os</span><span class="o">.</span><span class="n">system</span><span class="p">(</span><span class="n">cmd</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
<div class="cell_output docutils container">
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span> mean radius mean texture mean perimeter mean area mean smoothness \
|
||
0 17.99 10.38 122.80 1001.0 0.11840
|
||
1 20.57 17.77 132.90 1326.0 0.08474
|
||
2 19.69 21.25 130.00 1203.0 0.10960
|
||
3 11.42 20.38 77.58 386.1 0.14250
|
||
4 20.29 14.34 135.10 1297.0 0.10030
|
||
.. ... ... ... ... ...
|
||
564 21.56 22.39 142.00 1479.0 0.11100
|
||
565 20.13 28.25 131.20 1261.0 0.09780
|
||
566 16.60 28.08 108.30 858.1 0.08455
|
||
567 20.60 29.33 140.10 1265.0 0.11780
|
||
568 7.76 24.54 47.92 181.0 0.05263
|
||
|
||
mean compactness mean concavity mean concave points mean symmetry \
|
||
0 0.27760 0.30010 0.14710 0.2419
|
||
1 0.07864 0.08690 0.07017 0.1812
|
||
2 0.15990 0.19740 0.12790 0.2069
|
||
3 0.28390 0.24140 0.10520 0.2597
|
||
4 0.13280 0.19800 0.10430 0.1809
|
||
.. ... ... ... ...
|
||
564 0.11590 0.24390 0.13890 0.1726
|
||
565 0.10340 0.14400 0.09791 0.1752
|
||
566 0.10230 0.09251 0.05302 0.1590
|
||
567 0.27700 0.35140 0.15200 0.2397
|
||
568 0.04362 0.00000 0.00000 0.1587
|
||
|
||
mean fractal dimension ... worst radius worst texture \
|
||
0 0.07871 ... 25.380 17.33
|
||
1 0.05667 ... 24.990 23.41
|
||
2 0.05999 ... 23.570 25.53
|
||
3 0.09744 ... 14.910 26.50
|
||
4 0.05883 ... 22.540 16.67
|
||
.. ... ... ... ...
|
||
564 0.05623 ... 25.450 26.40
|
||
565 0.05533 ... 23.690 38.25
|
||
566 0.05648 ... 18.980 34.12
|
||
567 0.07016 ... 25.740 39.42
|
||
568 0.05884 ... 9.456 30.37
|
||
|
||
worst perimeter worst area worst smoothness worst compactness \
|
||
0 184.60 2019.0 0.16220 0.66560
|
||
1 158.80 1956.0 0.12380 0.18660
|
||
2 152.50 1709.0 0.14440 0.42450
|
||
3 98.87 567.7 0.20980 0.86630
|
||
4 152.20 1575.0 0.13740 0.20500
|
||
.. ... ... ... ...
|
||
564 166.10 2027.0 0.14100 0.21130
|
||
565 155.00 1731.0 0.11660 0.19220
|
||
566 126.70 1124.0 0.11390 0.30940
|
||
567 184.60 1821.0 0.16500 0.86810
|
||
568 59.16 268.6 0.08996 0.06444
|
||
|
||
worst concavity worst concave points worst symmetry \
|
||
0 0.7119 0.2654 0.4601
|
||
1 0.2416 0.1860 0.2750
|
||
2 0.4504 0.2430 0.3613
|
||
3 0.6869 0.2575 0.6638
|
||
4 0.4000 0.1625 0.2364
|
||
.. ... ... ...
|
||
564 0.4107 0.2216 0.2060
|
||
565 0.3215 0.1628 0.2572
|
||
566 0.3403 0.1418 0.2218
|
||
567 0.9387 0.2650 0.4087
|
||
568 0.0000 0.0000 0.2871
|
||
|
||
worst fractal dimension
|
||
0 0.11890
|
||
1 0.08902
|
||
2 0.08758
|
||
3 0.17300
|
||
4 0.07678
|
||
.. ...
|
||
564 0.07115
|
||
565 0.06637
|
||
566 0.07820
|
||
567 0.12400
|
||
568 0.07039
|
||
|
||
[569 rows x 30 columns]
|
||
malignant benign
|
||
0 1 0
|
||
1 1 0
|
||
2 1 0
|
||
3 1 0
|
||
4 1 0
|
||
.. ... ...
|
||
564 1 0
|
||
565 1 0
|
||
566 1 0
|
||
567 1 0
|
||
568 0 1
|
||
|
||
[569 rows x 2 columns]
|
||
</pre></div>
|
||
</div>
|
||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>sh: dot: command not found
|
||
</pre></div>
|
||
</div>
|
||
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>32512
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Common imports</span>
|
||
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.tree</span> <span class="kn">import</span> <span class="n">DecisionTreeClassifier</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">make_moons</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.tree</span> <span class="kn">import</span> <span class="n">export_graphviz</span>
|
||
<span class="kn">from</span> <span class="nn">pydot</span> <span class="kn">import</span> <span class="n">graph_from_dot_data</span>
|
||
<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
|
||
<span class="kn">import</span> <span class="nn">os</span>
|
||
|
||
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">42</span><span class="p">)</span>
|
||
<span class="n">X</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">make_moons</span><span class="p">(</span><span class="n">n_samples</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">noise</span><span class="o">=</span><span class="mf">0.25</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">53</span><span class="p">)</span>
|
||
<span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">y</span><span class="p">,</span><span class="n">random_state</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
|
||
<span class="n">tree_clf</span> <span class="o">=</span> <span class="n">DecisionTreeClassifier</span><span class="p">(</span><span class="n">max_depth</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
|
||
<span class="n">tree_clf</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
|
||
|
||
<span class="n">export_graphviz</span><span class="p">(</span>
|
||
<span class="n">tree_clf</span><span class="p">,</span>
|
||
<span class="n">out_file</span><span class="o">=</span><span class="s2">"DataFiles/moons.dot"</span><span class="p">,</span>
|
||
<span class="n">rounded</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span>
|
||
<span class="n">filled</span><span class="o">=</span><span class="kc">True</span>
|
||
<span class="p">)</span>
|
||
<span class="n">cmd</span> <span class="o">=</span> <span class="s1">'dot -Tpng DataFiles/moons.dot -o DataFiles/moons.png'</span>
|
||
<span class="n">os</span><span class="o">.</span><span class="n">system</span><span class="p">(</span><span class="n">cmd</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
<div class="cell_output docutils container">
|
||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>sh: dot: command not found
|
||
</pre></div>
|
||
</div>
|
||
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>32512
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="other-ways-of-visualizing-the-trees">
|
||
<h3><span class="section-number">9.4.2. </span>Other ways of visualizing the trees<a class="headerlink" href="#other-ways-of-visualizing-the-trees" title="Permalink to this headline">¶</a></h3>
|
||
<p><strong>Scikit-Learn</strong> has also another way to visualize the trees which is very useful, here with the Iris data.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">load_iris</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn</span> <span class="kn">import</span> <span class="n">tree</span>
|
||
<span class="n">X</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">load_iris</span><span class="p">(</span><span class="n">return_X_y</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
|
||
<span class="n">tree_clf</span> <span class="o">=</span> <span class="n">tree</span><span class="o">.</span><span class="n">DecisionTreeClassifier</span><span class="p">()</span>
|
||
<span class="n">tree_clf</span> <span class="o">=</span> <span class="n">tree_clf</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
|
||
<span class="c1"># and then plot the tree</span>
|
||
<span class="n">tree</span><span class="o">.</span><span class="n">plot_tree</span><span class="p">(</span><span class="n">tree_clf</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
<div class="cell_output docutils container">
|
||
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[Text(167.4, 199.32, 'X[2] <= 2.45\ngini = 0.667\nsamples = 150\nvalue = [50, 50, 50]'),
|
||
Text(141.64615384615385, 163.07999999999998, 'gini = 0.0\nsamples = 50\nvalue = [50, 0, 0]'),
|
||
Text(193.15384615384616, 163.07999999999998, 'X[3] <= 1.75\ngini = 0.5\nsamples = 100\nvalue = [0, 50, 50]'),
|
||
Text(103.01538461538462, 126.83999999999999, 'X[2] <= 4.95\ngini = 0.168\nsamples = 54\nvalue = [0, 49, 5]'),
|
||
Text(51.50769230769231, 90.6, 'X[3] <= 1.65\ngini = 0.041\nsamples = 48\nvalue = [0, 47, 1]'),
|
||
Text(25.753846153846155, 54.359999999999985, 'gini = 0.0\nsamples = 47\nvalue = [0, 47, 0]'),
|
||
Text(77.26153846153846, 54.359999999999985, 'gini = 0.0\nsamples = 1\nvalue = [0, 0, 1]'),
|
||
Text(154.52307692307693, 90.6, 'X[3] <= 1.55\ngini = 0.444\nsamples = 6\nvalue = [0, 2, 4]'),
|
||
Text(128.76923076923077, 54.359999999999985, 'gini = 0.0\nsamples = 3\nvalue = [0, 0, 3]'),
|
||
Text(180.27692307692308, 54.359999999999985, 'X[2] <= 5.45\ngini = 0.444\nsamples = 3\nvalue = [0, 2, 1]'),
|
||
Text(154.52307692307693, 18.119999999999976, 'gini = 0.0\nsamples = 2\nvalue = [0, 2, 0]'),
|
||
Text(206.03076923076924, 18.119999999999976, 'gini = 0.0\nsamples = 1\nvalue = [0, 0, 1]'),
|
||
Text(283.2923076923077, 126.83999999999999, 'X[2] <= 4.85\ngini = 0.043\nsamples = 46\nvalue = [0, 1, 45]'),
|
||
Text(257.53846153846155, 90.6, 'X[1] <= 3.1\ngini = 0.444\nsamples = 3\nvalue = [0, 1, 2]'),
|
||
Text(231.7846153846154, 54.359999999999985, 'gini = 0.0\nsamples = 2\nvalue = [0, 0, 2]'),
|
||
Text(283.2923076923077, 54.359999999999985, 'gini = 0.0\nsamples = 1\nvalue = [0, 1, 0]'),
|
||
Text(309.04615384615386, 90.6, 'gini = 0.0\nsamples = 43\nvalue = [0, 0, 43]')]
|
||
</pre></div>
|
||
</div>
|
||
<img alt="_images/chapter6_24_1.png" src="_images/chapter6_24_1.png" />
|
||
</div>
|
||
</div>
|
||
<p>Alternatively, the tree can also be exported in textual format with the function exporttext.
|
||
This method doesn’t require the installation of external libraries and is more compact:</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">load_iris</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.tree</span> <span class="kn">import</span> <span class="n">DecisionTreeClassifier</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.tree</span> <span class="kn">import</span> <span class="n">export_text</span>
|
||
<span class="n">iris</span> <span class="o">=</span> <span class="n">load_iris</span><span class="p">()</span>
|
||
<span class="n">decision_tree</span> <span class="o">=</span> <span class="n">DecisionTreeClassifier</span><span class="p">(</span><span class="n">random_state</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">max_depth</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
|
||
<span class="n">decision_tree</span> <span class="o">=</span> <span class="n">decision_tree</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">iris</span><span class="o">.</span><span class="n">data</span><span class="p">,</span> <span class="n">iris</span><span class="o">.</span><span class="n">target</span><span class="p">)</span>
|
||
<span class="n">r</span> <span class="o">=</span> <span class="n">export_text</span><span class="p">(</span><span class="n">decision_tree</span><span class="p">,</span> <span class="n">feature_names</span><span class="o">=</span><span class="n">iris</span><span class="p">[</span><span class="s1">'feature_names'</span><span class="p">])</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">r</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
<div class="cell_output docutils container">
|
||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>|--- petal width (cm) <= 0.80
|
||
| |--- class: 0
|
||
|--- petal width (cm) > 0.80
|
||
| |--- petal width (cm) <= 1.75
|
||
| | |--- class: 1
|
||
| |--- petal width (cm) > 1.75
|
||
| | |--- class: 2
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="algorithms-for-setting-up-decision-trees">
|
||
<h2><span class="section-number">9.5. </span>Algorithms for Setting up Decision Trees<a class="headerlink" href="#algorithms-for-setting-up-decision-trees" title="Permalink to this headline">¶</a></h2>
|
||
<p>Two algorithms stand out in the set up of decision trees:</p>
|
||
<ol class="simple">
|
||
<li><p>The CART (Classification And Regression Tree) algorithm for both classification and regression</p></li>
|
||
<li><p>The ID3 algorithm based on the computation of the information gain for classification</p></li>
|
||
</ol>
|
||
<p>We discuss both algorithms with applications here. The popular library
|
||
<strong>Scikit-Learn</strong> uses the CART algorithm. For classification problems
|
||
you can use either the <strong>gini</strong> index or the <strong>entropy</strong> to split a tree
|
||
in two branches.</p>
|
||
<div class="section" id="the-cart-algorithm-for-classification">
|
||
<h3><span class="section-number">9.5.1. </span>The CART algorithm for Classification<a class="headerlink" href="#the-cart-algorithm-for-classification" title="Permalink to this headline">¶</a></h3>
|
||
<p>For classification, the CART algorithm splits the data set in two subsets using a single feature <span class="math notranslate nohighlight">\(k\)</span> and a threshold <span class="math notranslate nohighlight">\(t_k\)</span>.
|
||
This could be for example a threshold set by a number below a certain circumference of a malign tumor.</p>
|
||
<p>How do we find these two quantities?
|
||
We search for the pair <span class="math notranslate nohighlight">\((k,t_k)\)</span> that produces the purest subset using for example the <strong>gini</strong> factor <span class="math notranslate nohighlight">\(G\)</span>.
|
||
The cost function it tries to minimize is then</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
C(k,t_k) = \frac{m_{\mathrm{left}}}{m}G_{\mathrm{left}}+ \frac{m_{\mathrm{right}}}{m}G_{\mathrm{right}},
|
||
\]</div>
|
||
<p>where <span class="math notranslate nohighlight">\(G_{\mathrm{left/right}}\)</span> measures the impurity of the left/right subset and <span class="math notranslate nohighlight">\(m_{\mathrm{left/right}}\)</span>
|
||
is the number of instances in the left/right subset</p>
|
||
<p>Once it has successfully split the training set in two, it splits the subsets using the same logic, then the subsubsets
|
||
and so on, recursively. It stops recursing once it reaches the maximum depth (defined by the
|
||
<span class="math notranslate nohighlight">\(max\_depth\)</span> hyperparameter), or if it cannot find a split that will reduce impurity. A few other
|
||
hyperparameters control additional stopping conditions such as the <span class="math notranslate nohighlight">\(min\_samples\_split\)</span>,
|
||
<span class="math notranslate nohighlight">\(min\_samples\_leaf\)</span>, <span class="math notranslate nohighlight">\(min\_weight\_fraction\_leaf\)</span>, and <span class="math notranslate nohighlight">\(max\_leaf\_nodes\)</span>.</p>
|
||
</div>
|
||
<div class="section" id="the-cart-algorithm-for-regression">
|
||
<h3><span class="section-number">9.5.2. </span>The CART algorithm for Regression<a class="headerlink" href="#the-cart-algorithm-for-regression" title="Permalink to this headline">¶</a></h3>
|
||
<p>The CART algorithm for regression works is similar to the one for classification except that instead of trying to split the
|
||
training set in a way that minimizes say the <strong>gini</strong> or <strong>entropy</strong> impurity, it now tries to split the training set in a way that minimizes our well-known mean-squared error (MSE). The cost function is now</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
C(k,t_k) = \frac{m_{\mathrm{left}}}{m}\mathrm{MSE}_{\mathrm{left}}+ \frac{m_{\mathrm{right}}}{m}\mathrm{MSE}_{\mathrm{right}}.
|
||
\]</div>
|
||
<p>Here the MSE for a specific node is defined as</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\mathrm{MSE}_{\mathrm{node}}=\frac{1}{m_\mathrm{node}}\sum_{i\in \mathrm{node}}(\overline{y}_{\mathrm{node}}-y_i)^2,
|
||
\]</div>
|
||
<p>with</p>
|
||
<div class="math notranslate nohighlight">
|
||
\[
|
||
\overline{y}_{\mathrm{node}}=\frac{1}{m_\mathrm{node}}\sum_{i\in \mathrm{node}}y_i,
|
||
\]</div>
|
||
<p>the mean value of all observations in a specific node.</p>
|
||
<p>Without any regularization, the regression task for decision trees,
|
||
just like for classification tasks, is prone to overfitting.</p>
|
||
</div>
|
||
<div class="section" id="computing-the-gini-index">
|
||
<h3><span class="section-number">9.5.3. </span>Computing the Gini index<a class="headerlink" href="#computing-the-gini-index" title="Permalink to this headline">¶</a></h3>
|
||
<p>The example we will look at is a classical one in many Machine
|
||
Learning applications. Based on various meteorological features, we
|
||
have several so-called attributes which decide whether we at the end
|
||
will do some outdoor activity like skiing, going for a bike ride etc
|
||
etc. The table here contains the feautures <strong>outlook</strong>, <strong>temperature</strong>,
|
||
<strong>humidity</strong> and <strong>wind</strong>. The target or output is whether we ride
|
||
(True=1) or whether we do something else that day (False=0). The
|
||
attributes for each feature are then sunny, overcast and rain for the
|
||
outlook, hot, cold and mild for temperature, high and normal for
|
||
humidity and weak and strong for wind.</p>
|
||
<p>The table here summarizes the various attributes and</p>
|
||
<table border="1">
|
||
<thead>
|
||
<tr><th align="center">Day</th> <th align="center">Outlook </th> <th align="center">Temperature</th> <th align="center">Humidity</th> <th align="center"> Wind </th> <th align="center">Ride</th> </tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr><td align="center"> 1 </td> <td align="center"> Sunny </td> <td align="center"> Hot </td> <td align="center"> High </td> <td align="center"> Weak </td> <td align="center"> 0 </td> </tr>
|
||
<tr><td align="center"> 2 </td> <td align="center"> Sunny </td> <td align="center"> Hot </td> <td align="center"> High </td> <td align="center"> Strong </td> <td align="center"> 1 </td> </tr>
|
||
<tr><td align="center"> 3 </td> <td align="center"> Overcast </td> <td align="center"> Hot </td> <td align="center"> High </td> <td align="center"> Weak </td> <td align="center"> 1 </td> </tr>
|
||
<tr><td align="center"> 4 </td> <td align="center"> Rain </td> <td align="center"> Mild </td> <td align="center"> High </td> <td align="center"> Weak </td> <td align="center"> 1 </td> </tr>
|
||
<tr><td align="center"> 5 </td> <td align="center"> Rain </td> <td align="center"> Cool </td> <td align="center"> Normal </td> <td align="center"> Weak </td> <td align="center"> 1 </td> </tr>
|
||
<tr><td align="center"> 6 </td> <td align="center"> Rain </td> <td align="center"> Cool </td> <td align="center"> Normal </td> <td align="center"> Strong </td> <td align="center"> 0 </td> </tr>
|
||
<tr><td align="center"> 7 </td> <td align="center"> Overcast </td> <td align="center"> Cool </td> <td align="center"> Normal </td> <td align="center"> Strong </td> <td align="center"> 1 </td> </tr>
|
||
<tr><td align="center"> 8 </td> <td align="center"> Sunny </td> <td align="center"> Mild </td> <td align="center"> High </td> <td align="center"> Weak </td> <td align="center"> 0 </td> </tr>
|
||
<tr><td align="center"> 9 </td> <td align="center"> Sunny </td> <td align="center"> Cool </td> <td align="center"> Normal </td> <td align="center"> Weak </td> <td align="center"> 1 </td> </tr>
|
||
<tr><td align="center"> 10 </td> <td align="center"> Rain </td> <td align="center"> Mild </td> <td align="center"> Normal </td> <td align="center"> Weak </td> <td align="center"> 1 </td> </tr>
|
||
<tr><td align="center"> 11 </td> <td align="center"> Sunny </td> <td align="center"> Mild </td> <td align="center"> Normal </td> <td align="center"> Strong </td> <td align="center"> 1 </td> </tr>
|
||
<tr><td align="center"> 12 </td> <td align="center"> Overcast </td> <td align="center"> Mild </td> <td align="center"> High </td> <td align="center"> Strong </td> <td align="center"> 1 </td> </tr>
|
||
<tr><td align="center"> 13 </td> <td align="center"> Overcast </td> <td align="center"> Hot </td> <td align="center"> Normal </td> <td align="center"> Weak </td> <td align="center"> 1 </td> </tr>
|
||
<tr><td align="center"> 14 </td> <td align="center"> Rain </td> <td align="center"> Mild </td> <td align="center"> High </td> <td align="center"> Strong </td> <td align="center"> 0 </td> </tr>
|
||
</tbody>
|
||
</table>
|
||
</div>
|
||
<div class="section" id="simple-python-code-to-read-in-data-and-perform-classification">
|
||
<h3><span class="section-number">9.5.4. </span>Simple Python Code to read in Data and perform Classification<a class="headerlink" href="#simple-python-code-to-read-in-data-and-perform-classification" title="Permalink to this headline">¶</a></h3>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Common imports</span>
|
||
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
|
||
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.tree</span> <span class="kn">import</span> <span class="n">DecisionTreeClassifier</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.tree</span> <span class="kn">import</span> <span class="n">export_graphviz</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <span class="n">StandardScaler</span><span class="p">,</span> <span class="n">OneHotEncoder</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.compose</span> <span class="kn">import</span> <span class="n">ColumnTransformer</span>
|
||
<span class="kn">from</span> <span class="nn">IPython.display</span> <span class="kn">import</span> <span class="n">Image</span>
|
||
<span class="kn">from</span> <span class="nn">pydot</span> <span class="kn">import</span> <span class="n">graph_from_dot_data</span>
|
||
<span class="kn">import</span> <span class="nn">os</span>
|
||
|
||
<span class="c1"># Where to save the figures and data files</span>
|
||
<span class="n">PROJECT_ROOT_DIR</span> <span class="o">=</span> <span class="s2">"Results"</span>
|
||
<span class="n">FIGURE_ID</span> <span class="o">=</span> <span class="s2">"Results/FigureFiles"</span>
|
||
<span class="n">DATA_ID</span> <span class="o">=</span> <span class="s2">"DataFiles/"</span>
|
||
|
||
<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="n">PROJECT_ROOT_DIR</span><span class="p">):</span>
|
||
<span class="n">os</span><span class="o">.</span><span class="n">mkdir</span><span class="p">(</span><span class="n">PROJECT_ROOT_DIR</span><span class="p">)</span>
|
||
|
||
<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="n">FIGURE_ID</span><span class="p">):</span>
|
||
<span class="n">os</span><span class="o">.</span><span class="n">makedirs</span><span class="p">(</span><span class="n">FIGURE_ID</span><span class="p">)</span>
|
||
|
||
<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="n">DATA_ID</span><span class="p">):</span>
|
||
<span class="n">os</span><span class="o">.</span><span class="n">makedirs</span><span class="p">(</span><span class="n">DATA_ID</span><span class="p">)</span>
|
||
|
||
<span class="k">def</span> <span class="nf">image_path</span><span class="p">(</span><span class="n">fig_id</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">FIGURE_ID</span><span class="p">,</span> <span class="n">fig_id</span><span class="p">)</span>
|
||
|
||
<span class="k">def</span> <span class="nf">data_path</span><span class="p">(</span><span class="n">dat_id</span><span class="p">):</span>
|
||
<span class="k">return</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">DATA_ID</span><span class="p">,</span> <span class="n">dat_id</span><span class="p">)</span>
|
||
|
||
<span class="k">def</span> <span class="nf">save_fig</span><span class="p">(</span><span class="n">fig_id</span><span class="p">):</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">savefig</span><span class="p">(</span><span class="n">image_path</span><span class="p">(</span><span class="n">fig_id</span><span class="p">)</span> <span class="o">+</span> <span class="s2">".png"</span><span class="p">,</span> <span class="nb">format</span><span class="o">=</span><span class="s1">'png'</span><span class="p">)</span>
|
||
|
||
<span class="n">infile</span> <span class="o">=</span> <span class="nb">open</span><span class="p">(</span><span class="n">data_path</span><span class="p">(</span><span class="s2">"rideclass.csv"</span><span class="p">),</span><span class="s1">'r'</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Read the experimental data with Pandas</span>
|
||
<span class="kn">from</span> <span class="nn">IPython.display</span> <span class="kn">import</span> <span class="n">display</span>
|
||
<span class="n">ridedata</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_csv</span><span class="p">(</span><span class="n">infile</span><span class="p">,</span><span class="n">names</span> <span class="o">=</span> <span class="p">(</span><span class="s1">'Outlook'</span><span class="p">,</span><span class="s1">'Temperature'</span><span class="p">,</span><span class="s1">'Humidity'</span><span class="p">,</span><span class="s1">'Wind'</span><span class="p">,</span><span class="s1">'Ride'</span><span class="p">))</span>
|
||
<span class="n">ridedata</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">ridedata</span><span class="p">)</span>
|
||
|
||
<span class="c1"># Features and targets</span>
|
||
<span class="n">X</span> <span class="o">=</span> <span class="n">ridedata</span><span class="o">.</span><span class="n">loc</span><span class="p">[:,</span> <span class="n">ridedata</span><span class="o">.</span><span class="n">columns</span> <span class="o">!=</span> <span class="s1">'Ride'</span><span class="p">]</span><span class="o">.</span><span class="n">values</span>
|
||
<span class="n">y</span> <span class="o">=</span> <span class="n">ridedata</span><span class="o">.</span><span class="n">loc</span><span class="p">[:,</span> <span class="n">ridedata</span><span class="o">.</span><span class="n">columns</span> <span class="o">==</span> <span class="s1">'Ride'</span><span class="p">]</span><span class="o">.</span><span class="n">values</span>
|
||
|
||
<span class="c1"># Create the encoder.</span>
|
||
<span class="n">encoder</span> <span class="o">=</span> <span class="n">OneHotEncoder</span><span class="p">(</span><span class="n">handle_unknown</span><span class="o">=</span><span class="s2">"ignore"</span><span class="p">)</span>
|
||
<span class="c1"># Assume for simplicity all features are categorical.</span>
|
||
<span class="n">encoder</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
|
||
<span class="c1"># Apply the encoder.</span>
|
||
<span class="n">X</span> <span class="o">=</span> <span class="n">encoder</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
|
||
<span class="c1"># Then do a Classification tree</span>
|
||
<span class="n">tree_clf</span> <span class="o">=</span> <span class="n">DecisionTreeClassifier</span><span class="p">(</span><span class="n">max_depth</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
|
||
<span class="n">tree_clf</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Train set accuracy with Decision Tree: </span><span class="si">{:.2f}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">tree_clf</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X</span><span class="p">,</span><span class="n">y</span><span class="p">)))</span>
|
||
<span class="c1">#transfer to a decision tree graph</span>
|
||
<span class="n">export_graphviz</span><span class="p">(</span>
|
||
<span class="n">tree_clf</span><span class="p">,</span>
|
||
<span class="n">out_file</span><span class="o">=</span><span class="s2">"DataFiles/ride.dot"</span><span class="p">,</span>
|
||
<span class="n">rounded</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span>
|
||
<span class="n">filled</span><span class="o">=</span><span class="kc">True</span>
|
||
<span class="p">)</span>
|
||
<span class="n">cmd</span> <span class="o">=</span> <span class="s1">'dot -Tpng DataFiles/cancer.dot -o DataFiles/cancer.png'</span>
|
||
<span class="n">os</span><span class="o">.</span><span class="n">system</span><span class="p">(</span><span class="n">cmd</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
<div class="cell_output docutils container">
|
||
<div class="output traceback highlight-ipythontb notranslate"><div class="highlight"><pre><span></span><span class="gt">---------------------------------------------------------------------------</span>
|
||
<span class="ne">FileNotFoundError</span><span class="g g-Whitespace"> </span>Traceback (most recent call last)
|
||
<span class="o">/</span><span class="n">var</span><span class="o">/</span><span class="n">folders</span><span class="o">/</span><span class="n">jy</span><span class="o">/</span><span class="n">g42mrgv128v34gnnhxwk9nrc0000gp</span><span class="o">/</span><span class="n">T</span><span class="o">/</span><span class="n">ipykernel_42541</span><span class="o">/</span><span class="mf">4192423635.</span><span class="n">py</span> <span class="ow">in</span> <span class="o"><</span><span class="n">module</span><span class="o">></span>
|
||
<span class="g g-Whitespace"> </span><span class="mi">35</span> <span class="n">plt</span><span class="o">.</span><span class="n">savefig</span><span class="p">(</span><span class="n">image_path</span><span class="p">(</span><span class="n">fig_id</span><span class="p">)</span> <span class="o">+</span> <span class="s2">".png"</span><span class="p">,</span> <span class="nb">format</span><span class="o">=</span><span class="s1">'png'</span><span class="p">)</span>
|
||
<span class="g g-Whitespace"> </span><span class="mi">36</span>
|
||
<span class="ne">---> </span><span class="mi">37</span> <span class="n">infile</span> <span class="o">=</span> <span class="nb">open</span><span class="p">(</span><span class="n">data_path</span><span class="p">(</span><span class="s2">"rideclass.csv"</span><span class="p">),</span><span class="s1">'r'</span><span class="p">)</span>
|
||
<span class="g g-Whitespace"> </span><span class="mi">38</span>
|
||
<span class="g g-Whitespace"> </span><span class="mi">39</span> <span class="c1"># Read the experimental data with Pandas</span>
|
||
|
||
<span class="ne">FileNotFoundError</span>: [Errno 2] No such file or directory: 'DataFiles/rideclass.csv'
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<p>The above functions (gini, entropy and misclassification error) are
|
||
important components of the so-called CART algorithm. We will discuss
|
||
this algorithm below after we have discussed the information gain
|
||
algorithm ID3.</p>
|
||
<p>In the example here we have converted all our attributes into numerical values <span class="math notranslate nohighlight">\(0,1,2\)</span> etc.</p>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Split a dataset based on an attribute and an attribute value</span>
|
||
<span class="k">def</span> <span class="nf">test_split</span><span class="p">(</span><span class="n">index</span><span class="p">,</span> <span class="n">value</span><span class="p">,</span> <span class="n">dataset</span><span class="p">):</span>
|
||
<span class="n">left</span><span class="p">,</span> <span class="n">right</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(),</span> <span class="nb">list</span><span class="p">()</span>
|
||
<span class="k">for</span> <span class="n">row</span> <span class="ow">in</span> <span class="n">dataset</span><span class="p">:</span>
|
||
<span class="k">if</span> <span class="n">row</span><span class="p">[</span><span class="n">index</span><span class="p">]</span> <span class="o"><</span> <span class="n">value</span><span class="p">:</span>
|
||
<span class="n">left</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">row</span><span class="p">)</span>
|
||
<span class="k">else</span><span class="p">:</span>
|
||
<span class="n">right</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">row</span><span class="p">)</span>
|
||
<span class="k">return</span> <span class="n">left</span><span class="p">,</span> <span class="n">right</span>
|
||
|
||
<span class="c1"># Calculate the Gini index for a split dataset</span>
|
||
<span class="k">def</span> <span class="nf">gini_index</span><span class="p">(</span><span class="n">groups</span><span class="p">,</span> <span class="n">classes</span><span class="p">):</span>
|
||
<span class="c1"># count all samples at split point</span>
|
||
<span class="n">n_instances</span> <span class="o">=</span> <span class="nb">float</span><span class="p">(</span><span class="nb">sum</span><span class="p">([</span><span class="nb">len</span><span class="p">(</span><span class="n">group</span><span class="p">)</span> <span class="k">for</span> <span class="n">group</span> <span class="ow">in</span> <span class="n">groups</span><span class="p">]))</span>
|
||
<span class="c1"># sum weighted Gini index for each group</span>
|
||
<span class="n">gini</span> <span class="o">=</span> <span class="mf">0.0</span>
|
||
<span class="k">for</span> <span class="n">group</span> <span class="ow">in</span> <span class="n">groups</span><span class="p">:</span>
|
||
<span class="n">size</span> <span class="o">=</span> <span class="nb">float</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">group</span><span class="p">))</span>
|
||
<span class="c1"># avoid divide by zero</span>
|
||
<span class="k">if</span> <span class="n">size</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
|
||
<span class="k">continue</span>
|
||
<span class="n">score</span> <span class="o">=</span> <span class="mf">0.0</span>
|
||
<span class="c1"># score the group based on the score for each class</span>
|
||
<span class="k">for</span> <span class="n">class_val</span> <span class="ow">in</span> <span class="n">classes</span><span class="p">:</span>
|
||
<span class="n">p</span> <span class="o">=</span> <span class="p">[</span><span class="n">row</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">row</span> <span class="ow">in</span> <span class="n">group</span><span class="p">]</span><span class="o">.</span><span class="n">count</span><span class="p">(</span><span class="n">class_val</span><span class="p">)</span> <span class="o">/</span> <span class="n">size</span>
|
||
<span class="n">score</span> <span class="o">+=</span> <span class="n">p</span> <span class="o">*</span> <span class="n">p</span>
|
||
<span class="c1"># weight the group score by its relative size</span>
|
||
<span class="n">gini</span> <span class="o">+=</span> <span class="p">(</span><span class="mf">1.0</span> <span class="o">-</span> <span class="n">score</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">size</span> <span class="o">/</span> <span class="n">n_instances</span><span class="p">)</span>
|
||
<span class="k">return</span> <span class="n">gini</span>
|
||
|
||
<span class="c1"># Select the best split point for a dataset</span>
|
||
<span class="k">def</span> <span class="nf">get_split</span><span class="p">(</span><span class="n">dataset</span><span class="p">):</span>
|
||
<span class="n">class_values</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">set</span><span class="p">(</span><span class="n">row</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">row</span> <span class="ow">in</span> <span class="n">dataset</span><span class="p">))</span>
|
||
<span class="n">b_index</span><span class="p">,</span> <span class="n">b_value</span><span class="p">,</span> <span class="n">b_score</span><span class="p">,</span> <span class="n">b_groups</span> <span class="o">=</span> <span class="mi">999</span><span class="p">,</span> <span class="mi">999</span><span class="p">,</span> <span class="mi">999</span><span class="p">,</span> <span class="kc">None</span>
|
||
<span class="k">for</span> <span class="n">index</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">dataset</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span><span class="o">-</span><span class="mi">1</span><span class="p">):</span>
|
||
<span class="k">for</span> <span class="n">row</span> <span class="ow">in</span> <span class="n">dataset</span><span class="p">:</span>
|
||
<span class="n">groups</span> <span class="o">=</span> <span class="n">test_split</span><span class="p">(</span><span class="n">index</span><span class="p">,</span> <span class="n">row</span><span class="p">[</span><span class="n">index</span><span class="p">],</span> <span class="n">dataset</span><span class="p">)</span>
|
||
<span class="n">gini</span> <span class="o">=</span> <span class="n">gini_index</span><span class="p">(</span><span class="n">groups</span><span class="p">,</span> <span class="n">class_values</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s1">'X</span><span class="si">%d</span><span class="s1"> < </span><span class="si">%.3f</span><span class="s1"> Gini=</span><span class="si">%.3f</span><span class="s1">'</span> <span class="o">%</span> <span class="p">((</span><span class="n">index</span><span class="o">+</span><span class="mi">1</span><span class="p">),</span> <span class="n">row</span><span class="p">[</span><span class="n">index</span><span class="p">],</span> <span class="n">gini</span><span class="p">))</span>
|
||
<span class="k">if</span> <span class="n">gini</span> <span class="o"><</span> <span class="n">b_score</span><span class="p">:</span>
|
||
<span class="n">b_index</span><span class="p">,</span> <span class="n">b_value</span><span class="p">,</span> <span class="n">b_score</span><span class="p">,</span> <span class="n">b_groups</span> <span class="o">=</span> <span class="n">index</span><span class="p">,</span> <span class="n">row</span><span class="p">[</span><span class="n">index</span><span class="p">],</span> <span class="n">gini</span><span class="p">,</span> <span class="n">groups</span>
|
||
<span class="k">return</span> <span class="p">{</span><span class="s1">'index'</span><span class="p">:</span><span class="n">b_index</span><span class="p">,</span> <span class="s1">'value'</span><span class="p">:</span><span class="n">b_value</span><span class="p">,</span> <span class="s1">'groups'</span><span class="p">:</span><span class="n">b_groups</span><span class="p">}</span>
|
||
|
||
<span class="n">dataset</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">],</span>
|
||
<span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span>
|
||
<span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span>
|
||
<span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span>
|
||
<span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span>
|
||
<span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">],</span>
|
||
<span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span>
|
||
<span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">],</span>
|
||
<span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span>
|
||
<span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span>
|
||
<span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span>
|
||
<span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span>
|
||
<span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span>
|
||
<span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">]]</span>
|
||
|
||
<span class="n">split</span> <span class="o">=</span> <span class="n">get_split</span><span class="p">(</span><span class="n">dataset</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s1">'Split: [X</span><span class="si">%d</span><span class="s1"> < </span><span class="si">%.3f</span><span class="s1">]'</span> <span class="o">%</span> <span class="p">((</span><span class="n">split</span><span class="p">[</span><span class="s1">'index'</span><span class="p">]</span><span class="o">+</span><span class="mi">1</span><span class="p">),</span> <span class="n">split</span><span class="p">[</span><span class="s1">'value'</span><span class="p">]))</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="entropy-and-the-id3-algorithm">
|
||
<h2><span class="section-number">9.6. </span>Entropy and the ID3 algorithm<a class="headerlink" href="#entropy-and-the-id3-algorithm" title="Permalink to this headline">¶</a></h2>
|
||
<p>The ID3 algorithm learns decision trees by constructing
|
||
them in a top down way, beginning with the question <strong>which attribute should be tested at the root of the tree</strong>?</p>
|
||
<ol class="simple">
|
||
<li><p>Each instance attribute is evaluated using a statistical test to determine how well it alone classifies the training examples.</p></li>
|
||
<li><p>The best attribute is selected and used as the test at the root node of the tree.</p></li>
|
||
<li><p>A descendant of the root node is then created for each possible value of this attribute.</p></li>
|
||
<li><p>Training examples are sorted to the appropriate descendant node.</p></li>
|
||
<li><p>The entire process is then repeated using the training examples associated with each descendant node to select the best attribute to test at that point in the tree.</p></li>
|
||
<li><p>This forms a greedy search for an acceptable decision tree, in which the algorithm never backtracks to reconsider earlier choices.</p></li>
|
||
</ol>
|
||
<p>The ID3 algorithm selects which attribute to test at each node in the
|
||
tree.</p>
|
||
<p>We would like to select the attribute that is most useful for classifying
|
||
examples.</p>
|
||
<p>What is a good quantitative measure of the worth of an attribute?</p>
|
||
<p>Information gain measures how well a given attribute separates the
|
||
training examples according to their target classification.</p>
|
||
<p>The ID3 algorithm uses this information gain measure to select among the candidate
|
||
attributes at each step while growing the tree.</p>
|
||
<div class="section" id="cancer-data-again-now-with-decision-trees-and-other-methods">
|
||
<h3><span class="section-number">9.6.1. </span>Cancer Data again now with Decision Trees and other Methods<a class="headerlink" href="#cancer-data-again-now-with-decision-trees-and-other-methods" title="Permalink to this headline">¶</a></h3>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
|
||
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">load_breast_cancer</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.svm</span> <span class="kn">import</span> <span class="n">SVC</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <span class="n">LogisticRegression</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.tree</span> <span class="kn">import</span> <span class="n">DecisionTreeClassifier</span>
|
||
|
||
<span class="c1"># Load the data</span>
|
||
<span class="n">cancer</span> <span class="o">=</span> <span class="n">load_breast_cancer</span><span class="p">()</span>
|
||
|
||
<span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span><span class="n">cancer</span><span class="o">.</span><span class="n">data</span><span class="p">,</span><span class="n">cancer</span><span class="o">.</span><span class="n">target</span><span class="p">,</span><span class="n">random_state</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">X_train</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="n">X_test</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
|
||
<span class="c1"># Logistic Regression</span>
|
||
<span class="n">logreg</span> <span class="o">=</span> <span class="n">LogisticRegression</span><span class="p">(</span><span class="n">solver</span><span class="o">=</span><span class="s1">'lbfgs'</span><span class="p">)</span>
|
||
<span class="n">logreg</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Test set accuracy with Logistic Regression: </span><span class="si">{:.2f}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">logreg</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span><span class="n">y_test</span><span class="p">)))</span>
|
||
<span class="c1"># Support vector machine</span>
|
||
<span class="n">svm</span> <span class="o">=</span> <span class="n">SVC</span><span class="p">(</span><span class="n">gamma</span><span class="o">=</span><span class="s1">'auto'</span><span class="p">,</span> <span class="n">C</span><span class="o">=</span><span class="mi">100</span><span class="p">)</span>
|
||
<span class="n">svm</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Test set accuracy with SVM: </span><span class="si">{:.2f}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">svm</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span><span class="n">y_test</span><span class="p">)))</span>
|
||
<span class="c1"># Decision Trees</span>
|
||
<span class="n">deep_tree_clf</span> <span class="o">=</span> <span class="n">DecisionTreeClassifier</span><span class="p">(</span><span class="n">max_depth</span><span class="o">=</span><span class="kc">None</span><span class="p">)</span>
|
||
<span class="n">deep_tree_clf</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Test set accuracy with Decision Trees: </span><span class="si">{:.2f}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">deep_tree_clf</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span><span class="n">y_test</span><span class="p">)))</span>
|
||
<span class="c1">#now scale the data</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <span class="n">StandardScaler</span>
|
||
<span class="n">scaler</span> <span class="o">=</span> <span class="n">StandardScaler</span><span class="p">()</span>
|
||
<span class="n">scaler</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
|
||
<span class="n">X_train_scaled</span> <span class="o">=</span> <span class="n">scaler</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
|
||
<span class="n">X_test_scaled</span> <span class="o">=</span> <span class="n">scaler</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
|
||
<span class="c1"># Logistic Regression</span>
|
||
<span class="n">logreg</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train_scaled</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Test set accuracy Logistic Regression with scaled data: </span><span class="si">{:.2f}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">logreg</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_test_scaled</span><span class="p">,</span><span class="n">y_test</span><span class="p">)))</span>
|
||
<span class="c1"># Support Vector Machine</span>
|
||
<span class="n">svm</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train_scaled</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Test set accuracy SVM with scaled data: </span><span class="si">{:.2f}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">logreg</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_test_scaled</span><span class="p">,</span><span class="n">y_test</span><span class="p">)))</span>
|
||
<span class="c1"># Decision Trees</span>
|
||
<span class="n">deep_tree_clf</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train_scaled</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
|
||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Test set accuracy with Decision Trees and scaled data: </span><span class="si">{:.2f}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">deep_tree_clf</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_test_scaled</span><span class="p">,</span><span class="n">y_test</span><span class="p">)))</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="another-example-the-moons-again">
|
||
<h3><span class="section-number">9.6.2. </span>Another example, the moons again<a class="headerlink" href="#another-example-the-moons-again" title="Permalink to this headline">¶</a></h3>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">__future__</span> <span class="kn">import</span> <span class="n">division</span><span class="p">,</span> <span class="n">print_function</span><span class="p">,</span> <span class="n">unicode_literals</span>
|
||
|
||
<span class="c1"># Common imports</span>
|
||
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||
<span class="kn">import</span> <span class="nn">os</span>
|
||
|
||
<span class="c1"># to make this notebook's output stable across runs</span>
|
||
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">42</span><span class="p">)</span>
|
||
|
||
<span class="c1"># To plot pretty figures</span>
|
||
<span class="kn">import</span> <span class="nn">matplotlib</span>
|
||
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
|
||
<span class="kn">from</span> <span class="nn">matplotlib.colors</span> <span class="kn">import</span> <span class="n">ListedColormap</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">rcParams</span><span class="p">[</span><span class="s1">'axes.labelsize'</span><span class="p">]</span> <span class="o">=</span> <span class="mi">14</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">rcParams</span><span class="p">[</span><span class="s1">'xtick.labelsize'</span><span class="p">]</span> <span class="o">=</span> <span class="mi">12</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">rcParams</span><span class="p">[</span><span class="s1">'ytick.labelsize'</span><span class="p">]</span> <span class="o">=</span> <span class="mi">12</span>
|
||
|
||
|
||
<span class="kn">from</span> <span class="nn">sklearn.svm</span> <span class="kn">import</span> <span class="n">SVC</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn</span> <span class="kn">import</span> <span class="n">datasets</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.tree</span> <span class="kn">import</span> <span class="n">DecisionTreeClassifier</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">make_moons</span>
|
||
<span class="kn">from</span> <span class="nn">sklearn.tree</span> <span class="kn">import</span> <span class="n">export_graphviz</span>
|
||
|
||
<span class="n">Xm</span><span class="p">,</span> <span class="n">ym</span> <span class="o">=</span> <span class="n">make_moons</span><span class="p">(</span><span class="n">n_samples</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">noise</span><span class="o">=</span><span class="mf">0.25</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">53</span><span class="p">)</span>
|
||
|
||
<span class="n">deep_tree_clf1</span> <span class="o">=</span> <span class="n">DecisionTreeClassifier</span><span class="p">(</span><span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">)</span>
|
||
<span class="n">deep_tree_clf2</span> <span class="o">=</span> <span class="n">DecisionTreeClassifier</span><span class="p">(</span><span class="n">min_samples_leaf</span><span class="o">=</span><span class="mi">4</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">)</span>
|
||
<span class="n">deep_tree_clf1</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">Xm</span><span class="p">,</span> <span class="n">ym</span><span class="p">)</span>
|
||
<span class="n">deep_tree_clf2</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">Xm</span><span class="p">,</span> <span class="n">ym</span><span class="p">)</span>
|
||
|
||
|
||
<span class="k">def</span> <span class="nf">plot_decision_boundary</span><span class="p">(</span><span class="n">clf</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">axes</span><span class="o">=</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mf">7.5</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="n">iris</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">legend</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">plot_training</span><span class="o">=</span><span class="kc">True</span><span class="p">):</span>
|
||
<span class="n">x1s</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="mi">100</span><span class="p">)</span>
|
||
<span class="n">x2s</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">],</span> <span class="n">axes</span><span class="p">[</span><span class="mi">3</span><span class="p">],</span> <span class="mi">100</span><span class="p">)</span>
|
||
<span class="n">x1</span><span class="p">,</span> <span class="n">x2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">meshgrid</span><span class="p">(</span><span class="n">x1s</span><span class="p">,</span> <span class="n">x2s</span><span class="p">)</span>
|
||
<span class="n">X_new</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">c_</span><span class="p">[</span><span class="n">x1</span><span class="o">.</span><span class="n">ravel</span><span class="p">(),</span> <span class="n">x2</span><span class="o">.</span><span class="n">ravel</span><span class="p">()]</span>
|
||
<span class="n">y_pred</span> <span class="o">=</span> <span class="n">clf</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_new</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">x1</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
|
||
<span class="n">custom_cmap</span> <span class="o">=</span> <span class="n">ListedColormap</span><span class="p">([</span><span class="s1">'#fafab0'</span><span class="p">,</span><span class="s1">'#9898ff'</span><span class="p">,</span><span class="s1">'#a0faa0'</span><span class="p">])</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">contourf</span><span class="p">(</span><span class="n">x1</span><span class="p">,</span> <span class="n">x2</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.3</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">custom_cmap</span><span class="p">)</span>
|
||
<span class="k">if</span> <span class="ow">not</span> <span class="n">iris</span><span class="p">:</span>
|
||
<span class="n">custom_cmap2</span> <span class="o">=</span> <span class="n">ListedColormap</span><span class="p">([</span><span class="s1">'#7d7d58'</span><span class="p">,</span><span class="s1">'#4c4c7f'</span><span class="p">,</span><span class="s1">'#507d50'</span><span class="p">])</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">contour</span><span class="p">(</span><span class="n">x1</span><span class="p">,</span> <span class="n">x2</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">,</span> <span class="n">cmap</span><span class="o">=</span><span class="n">custom_cmap2</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.8</span><span class="p">)</span>
|
||
<span class="k">if</span> <span class="n">plot_training</span><span class="p">:</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">][</span><span class="n">y</span><span class="o">==</span><span class="mi">0</span><span class="p">],</span> <span class="n">X</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">][</span><span class="n">y</span><span class="o">==</span><span class="mi">0</span><span class="p">],</span> <span class="s2">"yo"</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"Iris-Setosa"</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">][</span><span class="n">y</span><span class="o">==</span><span class="mi">1</span><span class="p">],</span> <span class="n">X</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">][</span><span class="n">y</span><span class="o">==</span><span class="mi">1</span><span class="p">],</span> <span class="s2">"bs"</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"Iris-Versicolor"</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">][</span><span class="n">y</span><span class="o">==</span><span class="mi">2</span><span class="p">],</span> <span class="n">X</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">][</span><span class="n">y</span><span class="o">==</span><span class="mi">2</span><span class="p">],</span> <span class="s2">"g^"</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"Iris-Virginica"</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="n">axes</span><span class="p">)</span>
|
||
<span class="k">if</span> <span class="n">iris</span><span class="p">:</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">"Petal length"</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">"Petal width"</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>
|
||
<span class="k">else</span><span class="p">:</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="sa">r</span><span class="s2">"$x_1$"</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">18</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="sa">r</span><span class="s2">"$x_2$"</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">18</span><span class="p">,</span> <span class="n">rotation</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
|
||
<span class="k">if</span> <span class="n">legend</span><span class="p">:</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s2">"lower right"</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">11</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">121</span><span class="p">)</span>
|
||
<span class="n">plot_decision_boundary</span><span class="p">(</span><span class="n">deep_tree_clf1</span><span class="p">,</span> <span class="n">Xm</span><span class="p">,</span> <span class="n">ym</span><span class="p">,</span> <span class="n">axes</span><span class="o">=</span><span class="p">[</span><span class="o">-</span><span class="mf">1.5</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">],</span> <span class="n">iris</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">"No restrictions"</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">122</span><span class="p">)</span>
|
||
<span class="n">plot_decision_boundary</span><span class="p">(</span><span class="n">deep_tree_clf2</span><span class="p">,</span> <span class="n">Xm</span><span class="p">,</span> <span class="n">ym</span><span class="p">,</span> <span class="n">axes</span><span class="o">=</span><span class="p">[</span><span class="o">-</span><span class="mf">1.5</span><span class="p">,</span> <span class="mf">2.5</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">],</span> <span class="n">iris</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">"min_samples_leaf = </span><span class="si">{}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">deep_tree_clf2</span><span class="o">.</span><span class="n">min_samples_leaf</span><span class="p">),</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">6</span><span class="p">)</span>
|
||
<span class="n">Xs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">100</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span> <span class="o">-</span> <span class="mf">0.5</span>
|
||
<span class="n">ys</span> <span class="o">=</span> <span class="p">(</span><span class="n">Xs</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">></span> <span class="mi">0</span><span class="p">)</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">float32</span><span class="p">)</span> <span class="o">*</span> <span class="mi">2</span>
|
||
|
||
<span class="n">angle</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">/</span><span class="mi">4</span>
|
||
<span class="n">rotation_matrix</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">angle</span><span class="p">),</span> <span class="o">-</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">angle</span><span class="p">)],</span> <span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">angle</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">angle</span><span class="p">)]])</span>
|
||
<span class="n">Xsr</span> <span class="o">=</span> <span class="n">Xs</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">rotation_matrix</span><span class="p">)</span>
|
||
|
||
<span class="n">tree_clf_s</span> <span class="o">=</span> <span class="n">DecisionTreeClassifier</span><span class="p">(</span><span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">)</span>
|
||
<span class="n">tree_clf_s</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">Xs</span><span class="p">,</span> <span class="n">ys</span><span class="p">)</span>
|
||
<span class="n">tree_clf_sr</span> <span class="o">=</span> <span class="n">DecisionTreeClassifier</span><span class="p">(</span><span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">)</span>
|
||
<span class="n">tree_clf_sr</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">Xsr</span><span class="p">,</span> <span class="n">ys</span><span class="p">)</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">11</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">121</span><span class="p">)</span>
|
||
<span class="n">plot_decision_boundary</span><span class="p">(</span><span class="n">tree_clf_s</span><span class="p">,</span> <span class="n">Xs</span><span class="p">,</span> <span class="n">ys</span><span class="p">,</span> <span class="n">axes</span><span class="o">=</span><span class="p">[</span><span class="o">-</span><span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">],</span> <span class="n">iris</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">122</span><span class="p">)</span>
|
||
<span class="n">plot_decision_boundary</span><span class="p">(</span><span class="n">tree_clf_sr</span><span class="p">,</span> <span class="n">Xsr</span><span class="p">,</span> <span class="n">ys</span><span class="p">,</span> <span class="n">axes</span><span class="o">=</span><span class="p">[</span><span class="o">-</span><span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">],</span> <span class="n">iris</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Quadratic training set + noise</span>
|
||
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">42</span><span class="p">)</span>
|
||
<span class="n">m</span> <span class="o">=</span> <span class="mi">200</span>
|
||
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
|
||
<span class="n">y</span> <span class="o">=</span> <span class="mi">4</span> <span class="o">*</span> <span class="p">(</span><span class="n">X</span> <span class="o">-</span> <span class="mf">0.5</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span>
|
||
<span class="n">y</span> <span class="o">=</span> <span class="n">y</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span> <span class="o">/</span> <span class="mi">10</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.tree</span> <span class="kn">import</span> <span class="n">DecisionTreeRegressor</span>
|
||
|
||
<span class="n">tree_reg</span> <span class="o">=</span> <span class="n">DecisionTreeRegressor</span><span class="p">(</span><span class="n">max_depth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">)</span>
|
||
<span class="n">tree_reg</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.tree</span> <span class="kn">import</span> <span class="n">DecisionTreeRegressor</span>
|
||
|
||
<span class="n">tree_reg1</span> <span class="o">=</span> <span class="n">DecisionTreeRegressor</span><span class="p">(</span><span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">,</span> <span class="n">max_depth</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
|
||
<span class="n">tree_reg2</span> <span class="o">=</span> <span class="n">DecisionTreeRegressor</span><span class="p">(</span><span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">,</span> <span class="n">max_depth</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
|
||
<span class="n">tree_reg1</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
|
||
<span class="n">tree_reg2</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
|
||
|
||
<span class="k">def</span> <span class="nf">plot_regression_predictions</span><span class="p">(</span><span class="n">tree_reg</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">axes</span><span class="o">=</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.2</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">ylabel</span><span class="o">=</span><span class="s2">"$y$"</span><span class="p">):</span>
|
||
<span class="n">x1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="mi">500</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
|
||
<span class="n">y_pred</span> <span class="o">=</span> <span class="n">tree_reg</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">x1</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="n">axes</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">"$x_1$"</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">18</span><span class="p">)</span>
|
||
<span class="k">if</span> <span class="n">ylabel</span><span class="p">:</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="n">ylabel</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">18</span><span class="p">,</span> <span class="n">rotation</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="s2">"b."</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x1</span><span class="p">,</span> <span class="n">y_pred</span><span class="p">,</span> <span class="s2">"r.-"</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="sa">r</span><span class="s2">"$\hat</span><span class="si">{y}</span><span class="s2">$"</span><span class="p">)</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">11</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">121</span><span class="p">)</span>
|
||
<span class="n">plot_regression_predictions</span><span class="p">(</span><span class="n">tree_reg1</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
|
||
<span class="k">for</span> <span class="n">split</span><span class="p">,</span> <span class="n">style</span> <span class="ow">in</span> <span class="p">((</span><span class="mf">0.1973</span><span class="p">,</span> <span class="s2">"k-"</span><span class="p">),</span> <span class="p">(</span><span class="mf">0.0917</span><span class="p">,</span> <span class="s2">"k--"</span><span class="p">),</span> <span class="p">(</span><span class="mf">0.7718</span><span class="p">,</span> <span class="s2">"k--"</span><span class="p">)):</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">([</span><span class="n">split</span><span class="p">,</span> <span class="n">split</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mf">0.2</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">style</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="mf">0.21</span><span class="p">,</span> <span class="mf">0.65</span><span class="p">,</span> <span class="s2">"Depth=0"</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="mf">0.01</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">,</span> <span class="s2">"Depth=1"</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">13</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="mf">0.65</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="s2">"Depth=1"</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">13</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s2">"upper center"</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">18</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">"max_depth=2"</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">122</span><span class="p">)</span>
|
||
<span class="n">plot_regression_predictions</span><span class="p">(</span><span class="n">tree_reg2</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">ylabel</span><span class="o">=</span><span class="kc">None</span><span class="p">)</span>
|
||
<span class="k">for</span> <span class="n">split</span><span class="p">,</span> <span class="n">style</span> <span class="ow">in</span> <span class="p">((</span><span class="mf">0.1973</span><span class="p">,</span> <span class="s2">"k-"</span><span class="p">),</span> <span class="p">(</span><span class="mf">0.0917</span><span class="p">,</span> <span class="s2">"k--"</span><span class="p">),</span> <span class="p">(</span><span class="mf">0.7718</span><span class="p">,</span> <span class="s2">"k--"</span><span class="p">)):</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">([</span><span class="n">split</span><span class="p">,</span> <span class="n">split</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mf">0.2</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">style</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
|
||
<span class="k">for</span> <span class="n">split</span> <span class="ow">in</span> <span class="p">(</span><span class="mf">0.0458</span><span class="p">,</span> <span class="mf">0.1298</span><span class="p">,</span> <span class="mf">0.2873</span><span class="p">,</span> <span class="mf">0.9040</span><span class="p">):</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">([</span><span class="n">split</span><span class="p">,</span> <span class="n">split</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mf">0.2</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="s2">"k:"</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">text</span><span class="p">(</span><span class="mf">0.3</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="s2">"Depth=2"</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">13</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">"max_depth=3"</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="cell docutils container">
|
||
<div class="cell_input docutils container">
|
||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">tree_reg1</span> <span class="o">=</span> <span class="n">DecisionTreeRegressor</span><span class="p">(</span><span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">)</span>
|
||
<span class="n">tree_reg2</span> <span class="o">=</span> <span class="n">DecisionTreeRegressor</span><span class="p">(</span><span class="n">random_state</span><span class="o">=</span><span class="mi">42</span><span class="p">,</span> <span class="n">min_samples_leaf</span><span class="o">=</span><span class="mi">10</span><span class="p">)</span>
|
||
<span class="n">tree_reg1</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
|
||
<span class="n">tree_reg2</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
|
||
|
||
<span class="n">x1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">500</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
|
||
<span class="n">y_pred1</span> <span class="o">=</span> <span class="n">tree_reg1</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">x1</span><span class="p">)</span>
|
||
<span class="n">y_pred2</span> <span class="o">=</span> <span class="n">tree_reg2</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">x1</span><span class="p">)</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">11</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">121</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="s2">"b."</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x1</span><span class="p">,</span> <span class="n">y_pred1</span><span class="p">,</span> <span class="s2">"r.-"</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="sa">r</span><span class="s2">"$\hat</span><span class="si">{y}</span><span class="s2">$"</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.2</span><span class="p">,</span> <span class="mf">1.1</span><span class="p">])</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">"$x_1$"</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">18</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">"$y$"</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">18</span><span class="p">,</span> <span class="n">rotation</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s2">"upper center"</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">18</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">"No restrictions"</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">122</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="s2">"b."</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x1</span><span class="p">,</span> <span class="n">y_pred2</span><span class="p">,</span> <span class="s2">"r.-"</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="sa">r</span><span class="s2">"$\hat</span><span class="si">{y}</span><span class="s2">$"</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.2</span><span class="p">,</span> <span class="mf">1.1</span><span class="p">])</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">"$x_1$"</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">18</span><span class="p">)</span>
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">"min_samples_leaf=</span><span class="si">{}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">tree_reg2</span><span class="o">.</span><span class="n">min_samples_leaf</span><span class="p">),</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">14</span><span class="p">)</span>
|
||
|
||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||
</pre></div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<div class="section" id="pros-and-cons-of-trees-pros">
|
||
<h2><span class="section-number">9.7. </span>Pros and cons of trees, pros<a class="headerlink" href="#pros-and-cons-of-trees-pros" title="Permalink to this headline">¶</a></h2>
|
||
<ul class="simple">
|
||
<li><p>White box, easy to interpret model. Some people believe that decision trees more closely mirror human decision-making than do the regression and classification approaches discussed earlier (think of support vector machines)</p></li>
|
||
<li><p>Trees are very easy to explain to people. In fact, they are even easier to explain than linear regression!</p></li>
|
||
<li><p>No feature normalization needed</p></li>
|
||
<li><p>Tree models can handle both continuous and categorical data (Classification and Regression Trees)</p></li>
|
||
<li><p>Can model nonlinear relationships</p></li>
|
||
<li><p>Can model interactions between the different descriptive features</p></li>
|
||
<li><p>Trees can be displayed graphically, and are easily interpreted even by a non-expert (especially if they are small)</p></li>
|
||
</ul>
|
||
<div class="section" id="disadvantages">
|
||
<h3><span class="section-number">9.7.1. </span>Disadvantages<a class="headerlink" href="#disadvantages" title="Permalink to this headline">¶</a></h3>
|
||
<ul class="simple">
|
||
<li><p>Unfortunately, trees generally do not have the same level of predictive accuracy as some of the other regression and classification approaches</p></li>
|
||
<li><p>If continuous features are used the tree may become quite large and hence less interpretable</p></li>
|
||
<li><p>Decision trees are prone to overfit the training data and hence do not well generalize the data if no stopping criteria or improvements like pruning, boosting or bagging are implemented</p></li>
|
||
<li><p>Small changes in the data may lead to a completely different tree. This issue can be addressed by using ensemble methods like bagging, boosting or random forests</p></li>
|
||
<li><p>Unbalanced datasets where some target feature values occur much more frequently than others may lead to biased trees since the frequently occurring feature values are preferred over the less frequently occurring ones.</p></li>
|
||
<li><p>If the number of features is relatively large (high dimensional) and the number of instances is relatively low, the tree might overfit the data</p></li>
|
||
<li><p>Features with many levels may be preferred over features with less levels since for them it is <em>more easy</em> to split the dataset such that the sub datasets only contain pure target feature values. This issue can be addressed by preferring for instance the information gain ratio as splitting criteria over information gain</p></li>
|
||
</ul>
|
||
<p>However, by aggregating many decision trees, using methods like
|
||
bagging, random forests, and boosting, the predictive performance of
|
||
trees can be substantially improved.</p>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
|
||
<script type="text/x-thebe-config">
|
||
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|
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|
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|
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|
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|
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|
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|
||
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||
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|
||
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|
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|
||
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|
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|
||
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|
||
</script>
|
||
<script>kernelName = 'python3'</script>
|
||
|
||
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|
||
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|
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<p class="prevnext-title"><span class="section-number">10. </span>Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods</p>
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