567 lines
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567 lines
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<section>
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<!-- ------------------- main content ---------------------- -->
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<center><h1 style="text-align: center;">Data Analysis and Machine Learning: Nearest Neighbors and Decision Trees</h1></center> <!-- document title -->
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<p>
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<!-- author(s): Morten Hjorth-Jensen -->
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<center>
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<b>Morten Hjorth-Jensen</b> [1, 2]
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</center>
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<p> <br>
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<!-- institution(s) -->
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<center>[1] <b>Department of Physics, University of Oslo</b></center>
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<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
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<br>
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<p> <br>
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<center><h4>Nov 1, 2018</h4></center> <!-- date -->
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<br>
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<p>
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<center style="font-size:80%">
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<!-- copyright --> © 1999-2018, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license
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</center>
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</section>
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<section>
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<h2 id="___sec0">Decision trees, overarching aims </h2>
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<div class="alert alert-block alert-block alert-text-normal">
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<b></b>
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<p>
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Decision trees are supervised learning algorithms used for both,
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classification and regression tasks where we will concentrate on
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classification in this first part of our decision tree tutorial.
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Decision trees are assigned to the information based learning
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algorithms which use different measures of information gain for
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learning. We can use decision trees for issues where we have
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continuous but also categorical input and target features.
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</div>
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</section>
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<section>
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<h2 id="___sec1">Nearest Neighbors </h2>
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<p>
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<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
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<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%"><span></span><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">mglearn</span>
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<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
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<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn</span> <span style="color: #8B008B; font-weight: bold">import</span> linear_model
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<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.linear_model</span> <span style="color: #8B008B; font-weight: bold">import</span> LinearRegression
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<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">matplotlib.pyplot</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">plt</span>
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<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.preprocessing</span> <span style="color: #8B008B; font-weight: bold">import</span> PolynomialFeatures
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<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.pipeline</span> <span style="color: #8B008B; font-weight: bold">import</span> Pipeline
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<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.neighbors</span> <span style="color: #8B008B; font-weight: bold">import</span> KNeighborsClassifier
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<span style="color: #228B22"># Generate sample data</span>
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X = np.sort(<span style="color: #B452CD">5</span>*np.random.rand(<span style="color: #B452CD">40</span>,<span style="color: #B452CD">1</span>), axis=<span style="color: #B452CD">0</span>)
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y = X**<span style="color: #B452CD">3</span>
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y=y.ravel()
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<span style="color: #228B22"># Add noise to targets</span>
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X[::<span style="color: #B452CD">4</span>] +=(<span style="color: #B452CD">0.5</span> - np.random.rand(<span style="color: #B452CD">1</span>))
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y[::<span style="color: #B452CD">5</span>] +=(<span style="color: #B452CD">0.5</span> - np.random.rand(<span style="color: #B452CD">8</span>))
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a=np.array(X)
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b=np.array(y)
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X_train=a[:<span style="color: #B452CD">19</span>]
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X_test=a[<span style="color: #B452CD">19</span>:]
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y_train=b[:<span style="color: #B452CD">19</span>]
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y_test=b[<span style="color: #B452CD">19</span>:]
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model=Pipeline([(<span style="color: #CD5555">'poly'</span>, PolynomialFeatures(degree=<span style="color: #B452CD">3</span>)),(<span style="color: #CD5555">'linear'</span>, LinearRegression(fit_intercept=<span style="color: #658b00">False</span>))])
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model=model.fit(X_train, y_train)
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pred=model.predict(X_test)
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poly=PolynomialFeatures(degree=<span style="color: #B452CD">3</span>)
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poly.fit_transform(X_train, y_train)
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plt.scatter(X_test, y_test)
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plt.plot(X_test, pred, color=<span style="color: #CD5555">'green'</span>)
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plt.show()
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<span style="color: #8B008B; font-weight: bold">print</span> (model.score(X_test,y_test))
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<span style="color: #8B008B; font-weight: bold">print</span> (<span style="color: #CD5555">"---------K-Nearest Neighbors-------"</span>)
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<span style="color: #CD5555">"""neighbors_settings=range(1,11)</span>
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<span style="color: #CD5555">for n_neighbors in neighbors_settings:</span>
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<span style="color: #CD5555"> clf=KNeighborsClassifier(n_neighbors=n_neighbors)</span>
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<span style="color: #CD5555"> clf.fit(X_train, y_train)</span>
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<span style="color: #CD5555"> training_accuracy.append(clf.score(X_train, y_train))</span>
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<span style="color: #CD5555"> test_accuracy.append(clf.score(X_test, y_test))</span>
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<span style="color: #CD5555">print (mglearn.plots.plot_knn_regression(n_neighbors=3))"""</span>
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<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.neighbors</span> <span style="color: #8B008B; font-weight: bold">import</span> KNeighborsRegressor
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X, y=mglearn.datasets.make_wave(n_samples=<span style="color: #B452CD">40</span>)
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reg = KNeighborsRegressor(n_neighbors=<span style="color: #B452CD">3</span>)
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reg.fit(X_train, y_train)
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</pre></div>
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</section>
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<section>
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<h2 id="___sec2">Decision trees and Regression </h2>
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<p>
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<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
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<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%"><span></span><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
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<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">matplotlib.pyplot</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">plt</span>
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<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.preprocessing</span> <span style="color: #8B008B; font-weight: bold">import</span> PolynomialFeatures
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<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.linear_model</span> <span style="color: #8B008B; font-weight: bold">import</span> LinearRegression
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steps=<span style="color: #B452CD">250</span>
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distance=<span style="color: #B452CD">0</span>
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x=<span style="color: #B452CD">0</span>
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distance_list=[]
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steps_list=[]
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<span style="color: #8B008B; font-weight: bold">while</span> x<steps:
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distance+=np.random.randint(-<span style="color: #B452CD">1</span>,<span style="color: #B452CD">2</span>)
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distance_list.append(distance)
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x+=<span style="color: #B452CD">1</span>
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steps_list.append(x)
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plt.plot(steps_list,distance_list, color=<span style="color: #CD5555">'green'</span>, label=<span style="color: #CD5555">"Random Walk Data"</span>)
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steps_list=np.asarray(steps_list)
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distance_list=np.asarray(distance_list)
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X=steps_list[:,np.newaxis]
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<span style="color: #228B22">#Polynomial fits</span>
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<span style="color: #228B22">#Degree 2</span>
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poly_features=PolynomialFeatures(degree=<span style="color: #B452CD">2</span>, include_bias=<span style="color: #658b00">False</span>)
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X_poly=poly_features.fit_transform(X)
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lin_reg=LinearRegression()
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poly_fit=lin_reg.fit(X_poly,distance_list)
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b=lin_reg.coef_
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c=lin_reg.intercept_
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<span style="color: #8B008B; font-weight: bold">print</span> (<span style="color: #CD5555">"2nd degree coefficients:"</span>)
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<span style="color: #8B008B; font-weight: bold">print</span> (<span style="color: #CD5555">"zero power: "</span>,c)
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<span style="color: #8B008B; font-weight: bold">print</span> (<span style="color: #CD5555">"first power: "</span>, b[<span style="color: #B452CD">0</span>])
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<span style="color: #8B008B; font-weight: bold">print</span> (<span style="color: #CD5555">"second power: "</span>,b[<span style="color: #B452CD">1</span>])
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z = np.arange(<span style="color: #B452CD">0</span>, steps, .<span style="color: #B452CD">01</span>)
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z_mod=b[<span style="color: #B452CD">1</span>]*z**<span style="color: #B452CD">2</span>+b[<span style="color: #B452CD">0</span>]*z+c
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fit_mod=b[<span style="color: #B452CD">1</span>]*X**<span style="color: #B452CD">2</span>+b[<span style="color: #B452CD">0</span>]*X+c
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plt.plot(z, z_mod, color=<span style="color: #CD5555">'r'</span>, label=<span style="color: #CD5555">"2nd Degree Fit"</span>)
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plt.title(<span style="color: #CD5555">"Polynomial Regression"</span>)
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plt.xlabel(<span style="color: #CD5555">"Steps"</span>)
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plt.ylabel(<span style="color: #CD5555">"Distance"</span>)
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<span style="color: #228B22">#Degree 10</span>
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poly_features10=PolynomialFeatures(degree=<span style="color: #B452CD">10</span>, include_bias=<span style="color: #658b00">False</span>)
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X_poly10=poly_features10.fit_transform(X)
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poly_fit10=lin_reg.fit(X_poly10,distance_list)
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y_plot=poly_fit10.predict(X_poly10)
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plt.plot(X, y_plot, color=<span style="color: #CD5555">'black'</span>, label=<span style="color: #CD5555">"10th Degree Fit"</span>)
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plt.legend()
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plt.show()
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<span style="color: #228B22">#Decision Tree Regression</span>
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<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.tree</span> <span style="color: #8B008B; font-weight: bold">import</span> DecisionTreeRegressor
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regr_1=DecisionTreeRegressor(max_depth=<span style="color: #B452CD">2</span>)
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regr_2=DecisionTreeRegressor(max_depth=<span style="color: #B452CD">5</span>)
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regr_3=DecisionTreeRegressor(max_depth=<span style="color: #B452CD">7</span>)
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regr_1.fit(X, distance_list)
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regr_2.fit(X, distance_list)
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regr_3.fit(X, distance_list)
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X_test = np.arange(<span style="color: #B452CD">0.0</span>, steps, <span style="color: #B452CD">0.01</span>)[:, np.newaxis]
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y_1 = regr_1.predict(X_test)
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y_2 = regr_2.predict(X_test)
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y_3=regr_3.predict(X_test)
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<span style="color: #228B22"># Plot the results</span>
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plt.figure()
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plt.scatter(X, distance_list, s=<span style="color: #B452CD">2.5</span>, c=<span style="color: #CD5555">"black"</span>, label=<span style="color: #CD5555">"data"</span>)
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plt.plot(X_test, y_1, color=<span style="color: #CD5555">"red"</span>,
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label=<span style="color: #CD5555">"max_depth=2"</span>, linewidth=<span style="color: #B452CD">2</span>)
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plt.plot(X_test, y_2, color=<span style="color: #CD5555">"green"</span>, label=<span style="color: #CD5555">"max_depth=5"</span>, linewidth=<span style="color: #B452CD">2</span>)
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plt.plot(X_test, y_3, color=<span style="color: #CD5555">"m"</span>, label=<span style="color: #CD5555">"max_depth=7"</span>, linewidth=<span style="color: #B452CD">2</span>)
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plt.xlabel(<span style="color: #CD5555">"Data"</span>)
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plt.ylabel(<span style="color: #CD5555">"Darget"</span>)
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plt.title(<span style="color: #CD5555">"Decision Tree Regression"</span>)
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plt.legend()
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plt.show()
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</pre></div>
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<p>
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<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
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<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%"><span></span><span style="color: #228B22"># Program to test the Metropolis algorithm with one particle at given temp in</span>
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<span style="color: #228B22"># one dimension</span>
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<span style="color: #228B22">#!/usr/bin/env python</span>
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<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
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<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">matplotlib.mlab</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">mlab</span>
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<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">matplotlib.pyplot</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">plt</span>
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<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">random</span>
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<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">math</span> <span style="color: #8B008B; font-weight: bold">import</span> sqrt, exp, log
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<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.preprocessing</span> <span style="color: #8B008B; font-weight: bold">import</span> PolynomialFeatures
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<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.linear_model</span> <span style="color: #8B008B; font-weight: bold">import</span> LinearRegression
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<span style="color: #228B22"># initialize the rng with a seed</span>
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random.seed()
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<span style="color: #228B22"># Hard coding of input parameters</span>
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MCcycles = <span style="color: #B452CD">100000</span>
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Temperature = <span style="color: #B452CD">2.0</span>
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beta = <span style="color: #B452CD">1.</span>/Temperature
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InitialVelocity = -<span style="color: #B452CD">2.0</span>
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CurrentVelocity = InitialVelocity
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Energy = <span style="color: #B452CD">0.5</span>*InitialVelocity*InitialVelocity
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VelocityRange = <span style="color: #B452CD">10</span>*sqrt(Temperature)
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VelocityStep = <span style="color: #B452CD">2</span>*VelocityRange/<span style="color: #B452CD">10.</span>
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AverageEnergy = Energy
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AverageEnergy2 = Energy*Energy
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VelocityValues = np.zeros(MCcycles)
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<span style="color: #228B22"># The Monte Carlo sampling with Metropolis starts here</span>
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<span style="color: #8B008B; font-weight: bold">for</span> i <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span> (<span style="color: #B452CD">1</span>, MCcycles, <span style="color: #B452CD">1</span>):
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TrialVelocity = CurrentVelocity + (<span style="color: #B452CD">2.0</span>*random.random() - <span style="color: #B452CD">1.0</span>)*VelocityStep
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EnergyChange = <span style="color: #B452CD">0.5</span>*(TrialVelocity*TrialVelocity -CurrentVelocity*CurrentVelocity);
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<span style="color: #8B008B; font-weight: bold">if</span> random.random() <= exp(-beta*EnergyChange):
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CurrentVelocity = TrialVelocity
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Energy += EnergyChange
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VelocityValues[i] = CurrentVelocity
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AverageEnergy += Energy
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AverageEnergy2 += Energy*Energy
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<span style="color: #228B22">#Final averages</span>
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AverageEnergy = AverageEnergy/MCcycles
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AverageEnergy2 = AverageEnergy2/MCcycles
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Variance = AverageEnergy2 - AverageEnergy*AverageEnergy
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<span style="color: #8B008B; font-weight: bold">print</span>(AverageEnergy, Variance)
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n, bins, patches = plt.hist(VelocityValues, <span style="color: #B452CD">400</span>, facecolor=<span style="color: #CD5555">'green'</span>)
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plt.xlabel(<span style="color: #CD5555">'$v$'</span>)
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plt.ylabel(<span style="color: #CD5555">'Velocity distribution P(v)'</span>)
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plt.title(<span style="color: #CD5555">r'Velocity histogram at $k_BT=2$'</span>)
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plt.axis([-<span style="color: #B452CD">5</span>, <span style="color: #B452CD">5</span>, <span style="color: #B452CD">0</span>, <span style="color: #B452CD">600</span>])
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plt.grid(<span style="color: #658b00">True</span>)
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<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">collections</span> <span style="color: #8B008B; font-weight: bold">import</span> Counter
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<span style="color: #228B22">#print (Counter(VelocityValues))</span>
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<span style="color: #8B008B; font-weight: bold">print</span> (VelocityValues[:<span style="color: #B452CD">20</span>])
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VelocityValues=<span style="color: #658b00">list</span>(Counter(VelocityValues).keys())
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d=<span style="color: #658b00">list</span>(Counter(VelocityValues).values())
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VelocityValues=np.asarray(VelocityValues)[:, np.newaxis]
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d=np.asarray(d)
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<span style="color: #8B008B; font-weight: bold">print</span> (VelocityValues.shape, d.shape)
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plt.scatter(VelocityValues, d)
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plt.show()
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<span style="color: #228B22">#2nd Degree Polynomial</span>
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poly_feat=PolynomialFeatures(degree=<span style="color: #B452CD">20</span>, include_bias=<span style="color: #658b00">False</span>)
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X_poly=poly_feat.fit_transform(VelocityValues)
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lin_reg=LinearRegression()
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poly_fit=lin_reg.fit(X_poly,d)
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y_plot=poly_fit.predict(X_poly)
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plt.title(<span style="color: #CD5555">"Polynomial Fit"</span>)
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plt.plot(VelocityValues, y_plot, color=<span style="color: #CD5555">'black'</span>, label=<span style="color: #CD5555">"Fit"</span>)
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plt.show()
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<span style="color: #228B22">#Decision Trees</span>
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<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.tree</span> <span style="color: #8B008B; font-weight: bold">import</span> DecisionTreeRegressor
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regr_1=DecisionTreeRegressor(max_depth=<span style="color: #B452CD">2</span>)
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regr_2=DecisionTreeRegressor(max_depth=<span style="color: #B452CD">5</span>)
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regr_3=DecisionTreeRegressor(max_depth=<span style="color: #B452CD">7</span>)
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regr_1.fit(VelocityValues, d)
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regr_2.fit(VelocityValues, d)
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regr_3.fit(VelocityValues, d)
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X_test = np.arange(<span style="color: #B452CD">0.0</span>, MCcycles, <span style="color: #B452CD">0.01</span>)[:, np.newaxis]
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y_1=regr_1.predict(X_test)
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y_2=regr_2.predict(X_test)
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y_3=regr_3.predict(X_test)
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plt.title(<span style="color: #CD5555">"Decision Tree"</span>)
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plt.plot(X_test, y_1, color=<span style="color: #CD5555">"red"</span>, label=<span style="color: #CD5555">"max_depth=2"</span>, linewidth=<span style="color: #B452CD">2</span>)
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plt.plot(X_test, y_2, color=<span style="color: #CD5555">"green"</span>, label=<span style="color: #CD5555">"max_depth=5"</span>, linewidth=<span style="color: #B452CD">2</span>)
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plt.plot(X_test, y_3, color=<span style="color: #CD5555">"m"</span>, label=<span style="color: #CD5555">"max_depth=7"</span>, linewidth=<span style="color: #B452CD">2</span>)
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plt.show()
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<span style="color: #228B22">#Separate each frequency not in one specific velocity, but in a range of values,</span>
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<span style="color: #228B22">#i.e. frequency of all velocities in range -5 to -4.9, -4.9 to -4.8, etc...</span>
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</pre></div>
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</section>
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transition: 'default', // default/cube/page/concave/zoom/linear/fade/none
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//parallaxBackgroundImage: '', // e.g. "'https://s3.amazonaws.com/hakim-static/reveal-js/reveal-parallax-1.jpg'"
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Reveal.initialize({
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dependencies: [
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{ src: 'reveal.js/lib/js/classList.js', condition: function() { return !document.body.classList; } },
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//{ src: 'reveal.js/plugin/math/math.js', async: true }
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margin: 0.1,
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