-In simplified terms, the process of training a decision tree and -predicting the target features of query instances is as follows: +Decision trees classify instances by sorting top down. -
@@ -188,7 +192,7 @@ Then we are essentially done!
+In simplified terms, the process of training a decision tree and +predicting the target features of query instances is as follows: - -
import numpy as np
-import matplotlib.pyplot as plt
-from sklearn.preprocessing import PolynomialFeatures
-from sklearn.linear_model import LinearRegression
+
+- Present a dataset containing of a number of training instances characterized by a number of descriptive features and a target feature
+- 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
+- Grow the tree until we accomplish a stopping criteria create leaf nodes which represent the predictions we want to make for new query instances
+- Show query instances to the tree and run down the tree until we arrive at leaf nodes
+
-steps=250
+Then we are essentially done!
-distance=0
-x=0
-distance_list=[]
-steps_list=[]
-while x<steps:
- distance+=np.random.randint(-1,2)
- distance_list.append(distance)
- x+=1
- steps_list.append(x)
-plt.plot(steps_list,distance_list, color='green', label="Random Walk Data")
-
-steps_list=np.asarray(steps_list)
-distance_list=np.asarray(distance_list)
-
-X=steps_list[:,np.newaxis]
-
-#Polynomial fits
-
-#Degree 2
-poly_features=PolynomialFeatures(degree=2, include_bias=False)
-X_poly=poly_features.fit_transform(X)
-
-lin_reg=LinearRegression()
-poly_fit=lin_reg.fit(X_poly,distance_list)
-b=lin_reg.coef_
-c=lin_reg.intercept_
-print ("2nd degree coefficients:")
-print ("zero power: ",c)
-print ("first power: ", b[0])
-print ("second power: ",b[1])
-
-z = np.arange(0, steps, .01)
-z_mod=b[1]*z**2+b[0]*z+c
-
-fit_mod=b[1]*X**2+b[0]*X+c
-plt.plot(z, z_mod, color='r', label="2nd Degree Fit")
-plt.title("Polynomial Regression")
-
-plt.xlabel("Steps")
-plt.ylabel("Distance")
-
-#Degree 10
-poly_features10=PolynomialFeatures(degree=10, include_bias=False)
-X_poly10=poly_features10.fit_transform(X)
-
-poly_fit10=lin_reg.fit(X_poly10,distance_list)
-
-y_plot=poly_fit10.predict(X_poly10)
-plt.plot(X, y_plot, color='black', label="10th Degree Fit")
-
-plt.legend()
-plt.show()
-
-
-#Decision Tree Regression
-from sklearn.tree import DecisionTreeRegressor
-regr_1=DecisionTreeRegressor(max_depth=2)
-regr_2=DecisionTreeRegressor(max_depth=5)
-regr_3=DecisionTreeRegressor(max_depth=7)
-regr_1.fit(X, distance_list)
-regr_2.fit(X, distance_list)
-regr_3.fit(X, distance_list)
-
-X_test = np.arange(0.0, steps, 0.01)[:, np.newaxis]
-y_1 = regr_1.predict(X_test)
-y_2 = regr_2.predict(X_test)
-y_3=regr_3.predict(X_test)
-
-# Plot the results
-plt.figure()
-plt.scatter(X, distance_list, s=2.5, c="black", label="data")
-plt.plot(X_test, y_1, color="red",
- label="max_depth=2", linewidth=2)
-plt.plot(X_test, y_2, color="green", label="max_depth=5", linewidth=2)
-plt.plot(X_test, y_3, color="m", label="max_depth=7", linewidth=2)
-
-plt.xlabel("Data")
-plt.ylabel("Darget")
-plt.title("Decision Tree Regression")
-plt.legend()
-plt.show()
-
@@ -267,7 +193,7 @@ plt.show()
-There are mainly two steps -
import numpy as np
+import matplotlib.pyplot as plt
+from sklearn.preprocessing import PolynomialFeatures
+from sklearn.linear_model import LinearRegression
-How do we construct the regions \( R_1,\dots,R_J \)?
-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 \( R_1,\dots,R_J \)
-that minimize the MSE, given by
-$$
-\sum_{j=1}^J\sum_{i\in R_j}(y_i-\overline{y}_{R_j})^2,
-$$
+steps=250
-where \( \overline{y}_{R_j} \) is the mean response for the training observations
-within box \( j \).
+distance=0
+x=0
+distance_list=[]
+steps_list=[]
+while x<steps:
+ distance+=np.random.randint(-1,2)
+ distance_list.append(distance)
+ x+=1
+ steps_list.append(x)
+plt.plot(steps_list,distance_list, color='green', label="Random Walk Data")
+steps_list=np.asarray(steps_list)
+distance_list=np.asarray(distance_list)
+
+X=steps_list[:,np.newaxis]
+
+#Polynomial fits
+
+#Degree 2
+poly_features=PolynomialFeatures(degree=2, include_bias=False)
+X_poly=poly_features.fit_transform(X)
+
+lin_reg=LinearRegression()
+poly_fit=lin_reg.fit(X_poly,distance_list)
+b=lin_reg.coef_
+c=lin_reg.intercept_
+print ("2nd degree coefficients:")
+print ("zero power: ",c)
+print ("first power: ", b[0])
+print ("second power: ",b[1])
+
+z = np.arange(0, steps, .01)
+z_mod=b[1]*z**2+b[0]*z+c
+
+fit_mod=b[1]*X**2+b[0]*X+c
+plt.plot(z, z_mod, color='r', label="2nd Degree Fit")
+plt.title("Polynomial Regression")
+
+plt.xlabel("Steps")
+plt.ylabel("Distance")
+
+#Degree 10
+poly_features10=PolynomialFeatures(degree=10, include_bias=False)
+X_poly10=poly_features10.fit_transform(X)
+
+poly_fit10=lin_reg.fit(X_poly10,distance_list)
+
+y_plot=poly_fit10.predict(X_poly10)
+plt.plot(X, y_plot, color='black', label="10th Degree Fit")
+
+plt.legend()
+plt.show()
+
+
+#Decision Tree Regression
+from sklearn.tree import DecisionTreeRegressor
+regr_1=DecisionTreeRegressor(max_depth=2)
+regr_2=DecisionTreeRegressor(max_depth=5)
+regr_3=DecisionTreeRegressor(max_depth=7)
+regr_1.fit(X, distance_list)
+regr_2.fit(X, distance_list)
+regr_3.fit(X, distance_list)
+
+X_test = np.arange(0.0, steps, 0.01)[:, np.newaxis]
+y_1 = regr_1.predict(X_test)
+y_2 = regr_2.predict(X_test)
+y_3=regr_3.predict(X_test)
+
+# Plot the results
+plt.figure()
+plt.scatter(X, distance_list, s=2.5, c="black", label="data")
+plt.plot(X_test, y_1, color="red",
+ label="max_depth=2", linewidth=2)
+plt.plot(X_test, y_2, color="green", label="max_depth=5", linewidth=2)
+plt.plot(X_test, y_3, color="m", label="max_depth=7", linewidth=2)
+
+plt.xlabel("Data")
+plt.ylabel("Darget")
+plt.title("Decision Tree Regression")
+plt.legend()
+plt.show()
+
@@ -198,7 +272,7 @@ within box \( j \).
-Unfortunately, it is computationally infeasible to consider every -possible partition of the feature space into \( J \) boxes. -The common strategy is to take a top-down approach +There are mainly two steps -
-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. +
@@ -192,7 +203,7 @@ better tree in some future step.
-In order to implement the recursive binary splitting we start by selecting -the predictor \( x_j \) and a cutpoint \( s \) that splits the predictor space into two regions \( R_1 \) and \( R_2 \) -$$ -\left\{X\vert x_j < s\right\}, -$$ - -and -$$ -\left\{X\vert x_j \geq s\right\}, -$$ - -so that we obtain the lowest MSE, that is -$$ -\sum_{i:x_i\in R_j}(y_i-\overline{y}_{R_1})^2+\sum_{i:x_i\in R_2}(y_i-\overline{y}_{R_2})^2, -$$ - -which we want to minimize by considering all predictors \( x_1,x_2,\dots,x_p \). -We consider also all possible values of \( s \) 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. +Unfortunately, it is computationally infeasible to consider every +possible partition of the feature space into \( J \) boxes. +The common strategy is to take a top-down approach
-For any \( j \) and \( s \), we define the pair of -half-planes where \( \overline{y}_{R_1} \) is the mean response for the training -observations in \( R_1(j,s) \), and \( \overline{y}_{R_2} \) is the mean response for the -training observations in \( R_2(j,s) \). - -
-Finding the values of j and s that -minimize the above equation can be done quite quickly, especially when the number -of features \( p \) is not too large. - -
-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. +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.
@@ -221,7 +197,7 @@ region contains more than five observations.
- + -
-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 \( T_0 \) 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. +In order to implement the recursive binary splitting we start by selecting +the predictor \( x_j \) and a cutpoint \( s \) that splits the predictor space into two regions \( R_1 \) and \( R_2 \) +$$ +\left\{X\vert x_j < s\right\}, +$$ + +and +$$ +\left\{X\vert x_j \geq s\right\}, +$$ + +so that we obtain the lowest MSE, that is +$$ +\sum_{i:x_i\in R_j}(y_i-\overline{y}_{R_1})^2+\sum_{i:x_i\in R_2}(y_i-\overline{y}_{R_2})^2, +$$ + +which we want to minimize by considering all predictors \( x_1,x_2,\dots,x_p \). +We consider also all possible values of \( s \) 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.
-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 \( \alpha \). +For any \( j \) and \( s \), we define the pair of +half-planes where \( \overline{y}_{R_1} \) is the mean response for the training +observations in \( R_1(j,s) \), and \( \overline{y}_{R_2} \) is the mean response for the +training observations in \( R_2(j,s) \). + +
+Finding the values of j and s that +minimize the above equation can be done quite quickly, especially when the number +of features \( p \) is not too large. + +
+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.
@@ -194,7 +226,7 @@ parameter \( \alpha \).
- + -
-The tuning parameter \( \alpha \) controls a trade-off between the subtree’s -com- plexity and its fit to the training data. When \( \alpha = 0 \), then the -subtree \( T \) will simply equal \( T_0 \), -because then the above equation just measures the -training error. -However, as \( \alpha \) 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. +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 \( T_0 \) 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.
-It turns out that as we increase \( \alpha \) 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 \( \alpha \) is -easy. We can select a value of \( \alpha \) using a validation set or using -cross-validation. We then return to the full data set and obtain the -subtree corresponding to \( \alpha \). +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 \( \alpha \).
@@ -207,7 +199,7 @@ subtree corresponding to \( \alpha \).
-
- -
+It turns out that as we increase \( \alpha \) 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 \( \alpha \) is +easy. We can select a value of \( \alpha \) using a validation set or using +cross-validation. We then return to the full data set and obtain the +subtree corresponding to \( \alpha \).
@@ -204,7 +212,7 @@ MathJax.Hub.Config({
-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. +
+ +
@@ -197,7 +209,7 @@ fall into that region.
-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 classification -error rate. 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. - -
-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. +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.
@@ -202,7 +201,7 @@ than is the classification error rate.
-If our targets are the outcome of a classification process that takes for example -\( k=1,2,\dots,K \) values, the only thing we need to think of is to set up the splitting criteria for each node. +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 classification +error rate. 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.
-We define a PDF \( p_{mk} \) that represents the number of observations of a class \( k \) in a region \( R_m \) with \( N_m \) observations. We represent this likelihood function in terms of the proportion \( I(y_i=k) \) of observations of this class in the region \( R_m \) as -$$ -p_{mk} = \frac{1}{N_m}\sum_{x_i\in R_m}I(y_i=k). -$$ - -
-We let \( p_{mk} \) represent the majority class of observations in region \( m \). The three most common ways of splitting a node are given by - -
@@ -222,7 +206,7 @@ $$
-More text and code to come here. +If our targets are the outcome of a classification process that takes for example +\( k=1,2,\dots,K \) values, the only thing we need to think of is to set up the splitting criteria for each node. + +
+We define a PDF \( p_{mk} \) that represents the number of observations of a class \( k \) in a region \( R_m \) with \( N_m \) observations. We represent this likelihood function in terms of the proportion \( I(y_i=k) \) of observations of this class in the region \( R_m \) as +$$ +p_{mk} = \frac{1}{N_m}\sum_{x_i\in R_m}I(y_i=k). +$$ + +
+We let \( p_{mk} \) represent the majority class of observations in region \( m \). The three most common ways of splitting a node are given by + +
@@ -186,7 +226,7 @@ More text and code to come here.
+ID3, learns decision trees by constructing +them topdown, beginning with the question which attribute should be tested at the root of the tree? - -
import matplotlib.pyplot as plt
-import numpy as np
-from sklearn.model_selection import train_test_split
-from sklearn.datasets import load_breast_cancer
-from sklearn.svm import SVC
-from sklearn.linear_model import LogisticRegression
-from sklearn.tree import DecisionTreeClassifier
+
+- Each instance attribute is evaluated using a statistical test to determine how well it alone classifies the training examples.
+- The best attribute is selected and used as the test at the root node of the tree.
+- A descendant of the root node is then created for each possible value of this attribute.
+- Training examples are sorted to the appropriate descendant node.
+- 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.
+- This forms a greedy search for an acceptable decision tree, in which the algorithm never backtracks to reconsider earlier choices.
+
-# Load the data
-cancer = load_breast_cancer()
+The ID3 algorithm selects, which attribute to test at each node in the
+tree.
+
+
+We would like to select the attribute that is most useful for classifying
+examples.
+
+
+What is a good quantitative measure of the worth of an attribute?
+
+
+Information gain measures how well a given attribute separates the
+training examples according to their target classification.
+
+
+The ID3 algorithm uses this information gain measure to select among the candidate
+attributes at each step while growing the tree.
-X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
-print(X_train.shape)
-print(X_test.shape)
-# Logistic Regression
-logreg = LogisticRegression(solver='lbfgs')
-logreg.fit(X_train, y_train)
-print("Test set accuracy with Logistic Regression: {:.2f}".format(logreg.score(X_test,y_test)))
-# Support vector machine
-svm = SVC(gamma='auto', C=100)
-svm.fit(X_train, y_train)
-print("Test set accuracy with SVM: {:.2f}".format(svm.score(X_test,y_test)))
-# Decision Trees
-deep_tree_clf = DecisionTreeClassifier(max_depth=None)
-deep_tree_clf.fit(X_train, y_train)
-print("Test set accuracy with Decision Trees: {:.2f}".format(deep_tree_clf.score(X_test,y_test)))
-#now scale the data
-from sklearn.preprocessing import StandardScaler
-scaler = StandardScaler()
-scaler.fit(X_train)
-X_train_scaled = scaler.transform(X_train)
-X_test_scaled = scaler.transform(X_test)
-# Logistic Regression
-logreg.fit(X_train_scaled, y_train)
-print("Test set accuracy Logistic Regression with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
-# Support Vector Machine
-svm.fit(X_train_scaled, y_train)
-print("Test set accuracy SVM with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
-# Decision Trees
-deep_tree_clf.fit(X_train_scaled, y_train)
-print("Test set accuracy with Decision Trees and scaled data: {:.2f}".format(deep_tree_clf.score(X_test_scaled,y_test)))
-
@@ -227,7 +218,7 @@ deep_tree_clf.fit(X_train_scaled, y_train)
+more text to come here, material presented during lecture Friday Oct 25. - -
from __future__ import division, print_function, unicode_literals
-
-# Common imports
-import numpy as np
-import os
-
-# to make this notebook's output stable across runs
-np.random.seed(42)
-
-# To plot pretty figures
-import matplotlib
-import matplotlib.pyplot as plt
-from matplotlib.colors import ListedColormap
-plt.rcParams['axes.labelsize'] = 14
-plt.rcParams['xtick.labelsize'] = 12
-plt.rcParams['ytick.labelsize'] = 12
-
-
-from sklearn.svm import SVC
-from sklearn import datasets
-from sklearn.tree import DecisionTreeClassifier
-from sklearn.datasets import make_moons
-from sklearn.tree import export_graphviz
-
-Xm, ym = make_moons(n_samples=100, noise=0.25, random_state=53)
-
-deep_tree_clf1 = DecisionTreeClassifier(random_state=42)
-deep_tree_clf2 = DecisionTreeClassifier(min_samples_leaf=4, random_state=42)
-deep_tree_clf1.fit(Xm, ym)
-deep_tree_clf2.fit(Xm, ym)
-
-
-def plot_decision_boundary(clf, X, y, axes=[0, 7.5, 0, 3], iris=True, legend=False, plot_training=True):
- x1s = np.linspace(axes[0], axes[1], 100)
- x2s = np.linspace(axes[2], axes[3], 100)
- x1, x2 = np.meshgrid(x1s, x2s)
- X_new = np.c_[x1.ravel(), x2.ravel()]
- y_pred = clf.predict(X_new).reshape(x1.shape)
- custom_cmap = ListedColormap(['#fafab0','#9898ff','#a0faa0'])
- plt.contourf(x1, x2, y_pred, alpha=0.3, cmap=custom_cmap)
- if not iris:
- custom_cmap2 = ListedColormap(['#7d7d58','#4c4c7f','#507d50'])
- plt.contour(x1, x2, y_pred, cmap=custom_cmap2, alpha=0.8)
- if plot_training:
- plt.plot(X[:, 0][y==0], X[:, 1][y==0], "yo", label="Iris-Setosa")
- plt.plot(X[:, 0][y==1], X[:, 1][y==1], "bs", label="Iris-Versicolor")
- plt.plot(X[:, 0][y==2], X[:, 1][y==2], "g^", label="Iris-Virginica")
- plt.axis(axes)
- if iris:
- plt.xlabel("Petal length", fontsize=14)
- plt.ylabel("Petal width", fontsize=14)
- else:
- plt.xlabel(r"$x_1$", fontsize=18)
- plt.ylabel(r"$x_2$", fontsize=18, rotation=0)
- if legend:
- plt.legend(loc="lower right", fontsize=14)
-plt.figure(figsize=(11, 4))
-plt.subplot(121)
-plot_decision_boundary(deep_tree_clf1, Xm, ym, axes=[-1.5, 2.5, -1, 1.5], iris=False)
-plt.title("No restrictions", fontsize=16)
-plt.subplot(122)
-plot_decision_boundary(deep_tree_clf2, Xm, ym, axes=[-1.5, 2.5, -1, 1.5], iris=False)
-plt.title("min_samples_leaf = {}".format(deep_tree_clf2.min_samples_leaf), fontsize=14)
-plt.show()
-
@@ -250,7 +190,7 @@ plt.show()
-
np.random.seed(6)
-Xs = np.random.rand(100, 2) - 0.5
-ys = (Xs[:, 0] > 0).astype(np.float32) * 2
+import matplotlib.pyplot as plt
+import numpy as np
+from sklearn.model_selection import train_test_split
+from sklearn.datasets import load_breast_cancer
+from sklearn.svm import SVC
+from sklearn.linear_model import LogisticRegression
+from sklearn.tree import DecisionTreeClassifier
-angle = np.pi / 4
-rotation_matrix = np.array([[np.cos(angle), -np.sin(angle)], [np.sin(angle), np.cos(angle)]])
-Xsr = Xs.dot(rotation_matrix)
+# Load the data
+cancer = load_breast_cancer()
-tree_clf_s = DecisionTreeClassifier(random_state=42)
-tree_clf_s.fit(Xs, ys)
-tree_clf_sr = DecisionTreeClassifier(random_state=42)
-tree_clf_sr.fit(Xsr, ys)
-
-plt.figure(figsize=(11, 4))
-plt.subplot(121)
-plot_decision_boundary(tree_clf_s, Xs, ys, axes=[-0.7, 0.7, -0.7, 0.7], iris=False)
-plt.subplot(122)
-plot_decision_boundary(tree_clf_sr, Xsr, ys, axes=[-0.7, 0.7, -0.7, 0.7], iris=False)
-
-plt.show()
+X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
+print(X_train.shape)
+print(X_test.shape)
+# Logistic Regression
+logreg = LogisticRegression(solver='lbfgs')
+logreg.fit(X_train, y_train)
+print("Test set accuracy with Logistic Regression: {:.2f}".format(logreg.score(X_test,y_test)))
+# Support vector machine
+svm = SVC(gamma='auto', C=100)
+svm.fit(X_train, y_train)
+print("Test set accuracy with SVM: {:.2f}".format(svm.score(X_test,y_test)))
+# Decision Trees
+deep_tree_clf = DecisionTreeClassifier(max_depth=None)
+deep_tree_clf.fit(X_train, y_train)
+print("Test set accuracy with Decision Trees: {:.2f}".format(deep_tree_clf.score(X_test,y_test)))
+#now scale the data
+from sklearn.preprocessing import StandardScaler
+scaler = StandardScaler()
+scaler.fit(X_train)
+X_train_scaled = scaler.transform(X_train)
+X_test_scaled = scaler.transform(X_test)
+# Logistic Regression
+logreg.fit(X_train_scaled, y_train)
+print("Test set accuracy Logistic Regression with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
+# Support Vector Machine
+svm.fit(X_train_scaled, y_train)
+print("Test set accuracy SVM with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
+# Decision Trees
+deep_tree_clf.fit(X_train_scaled, y_train)
+print("Test set accuracy with Decision Trees and scaled data: {:.2f}".format(deep_tree_clf.score(X_test_scaled,y_test)))
@@ -206,7 +231,7 @@ plt.show()
-
# Quadratic training set + noise
+from __future__ import division, print_function, unicode_literals
+
+# Common imports
+import numpy as np
+import os
+
+# to make this notebook's output stable across runs
np.random.seed(42)
-m = 200
-X = np.random.rand(m, 1)
-y = 4 * (X - 0.5) ** 2
-y = y + np.random.randn(m, 1) / 10
-
-
-
-
from sklearn.tree import DecisionTreeRegressor
+# To plot pretty figures
+import matplotlib
+import matplotlib.pyplot as plt
+from matplotlib.colors import ListedColormap
+plt.rcParams['axes.labelsize'] = 14
+plt.rcParams['xtick.labelsize'] = 12
+plt.rcParams['ytick.labelsize'] = 12
-tree_reg = DecisionTreeRegressor(max_depth=2, random_state=42)
-tree_reg.fit(X, y)
+
+from sklearn.svm import SVC
+from sklearn import datasets
+from sklearn.tree import DecisionTreeClassifier
+from sklearn.datasets import make_moons
+from sklearn.tree import export_graphviz
+
+Xm, ym = make_moons(n_samples=100, noise=0.25, random_state=53)
+
+deep_tree_clf1 = DecisionTreeClassifier(random_state=42)
+deep_tree_clf2 = DecisionTreeClassifier(min_samples_leaf=4, random_state=42)
+deep_tree_clf1.fit(Xm, ym)
+deep_tree_clf2.fit(Xm, ym)
+
+
+def plot_decision_boundary(clf, X, y, axes=[0, 7.5, 0, 3], iris=True, legend=False, plot_training=True):
+ x1s = np.linspace(axes[0], axes[1], 100)
+ x2s = np.linspace(axes[2], axes[3], 100)
+ x1, x2 = np.meshgrid(x1s, x2s)
+ X_new = np.c_[x1.ravel(), x2.ravel()]
+ y_pred = clf.predict(X_new).reshape(x1.shape)
+ custom_cmap = ListedColormap(['#fafab0','#9898ff','#a0faa0'])
+ plt.contourf(x1, x2, y_pred, alpha=0.3, cmap=custom_cmap)
+ if not iris:
+ custom_cmap2 = ListedColormap(['#7d7d58','#4c4c7f','#507d50'])
+ plt.contour(x1, x2, y_pred, cmap=custom_cmap2, alpha=0.8)
+ if plot_training:
+ plt.plot(X[:, 0][y==0], X[:, 1][y==0], "yo", label="Iris-Setosa")
+ plt.plot(X[:, 0][y==1], X[:, 1][y==1], "bs", label="Iris-Versicolor")
+ plt.plot(X[:, 0][y==2], X[:, 1][y==2], "g^", label="Iris-Virginica")
+ plt.axis(axes)
+ if iris:
+ plt.xlabel("Petal length", fontsize=14)
+ plt.ylabel("Petal width", fontsize=14)
+ else:
+ plt.xlabel(r"$x_1$", fontsize=18)
+ plt.ylabel(r"$x_2$", fontsize=18, rotation=0)
+ if legend:
+ plt.legend(loc="lower right", fontsize=14)
+plt.figure(figsize=(11, 4))
+plt.subplot(121)
+plot_decision_boundary(deep_tree_clf1, Xm, ym, axes=[-1.5, 2.5, -1, 1.5], iris=False)
+plt.title("No restrictions", fontsize=16)
+plt.subplot(122)
+plot_decision_boundary(deep_tree_clf2, Xm, ym, axes=[-1.5, 2.5, -1, 1.5], iris=False)
+plt.title("min_samples_leaf = {}".format(deep_tree_clf2.min_samples_leaf), fontsize=14)
+plt.show()
@@ -200,7 +254,7 @@ tree_reg.fit(X, y)
-
from sklearn.tree import DecisionTreeRegressor
+np.random.seed(6)
+Xs = np.random.rand(100, 2) - 0.5
+ys = (Xs[:, 0] > 0).astype(np.float32) * 2
-tree_reg1 = DecisionTreeRegressor(random_state=42, max_depth=2)
-tree_reg2 = DecisionTreeRegressor(random_state=42, max_depth=3)
-tree_reg1.fit(X, y)
-tree_reg2.fit(X, y)
+angle = np.pi / 4
+rotation_matrix = np.array([[np.cos(angle), -np.sin(angle)], [np.sin(angle), np.cos(angle)]])
+Xsr = Xs.dot(rotation_matrix)
-def plot_regression_predictions(tree_reg, X, y, axes=[0, 1, -0.2, 1], ylabel="$y$"):
- x1 = np.linspace(axes[0], axes[1], 500).reshape(-1, 1)
- y_pred = tree_reg.predict(x1)
- plt.axis(axes)
- plt.xlabel("$x_1$", fontsize=18)
- if ylabel:
- plt.ylabel(ylabel, fontsize=18, rotation=0)
- plt.plot(X, y, "b.")
- plt.plot(x1, y_pred, "r.-", linewidth=2, label=r"$\hat{y}$")
+tree_clf_s = DecisionTreeClassifier(random_state=42)
+tree_clf_s.fit(Xs, ys)
+tree_clf_sr = DecisionTreeClassifier(random_state=42)
+tree_clf_sr.fit(Xsr, ys)
plt.figure(figsize=(11, 4))
plt.subplot(121)
-plot_regression_predictions(tree_reg1, X, y)
-for split, style in ((0.1973, "k-"), (0.0917, "k--"), (0.7718, "k--")):
- plt.plot([split, split], [-0.2, 1], style, linewidth=2)
-plt.text(0.21, 0.65, "Depth=0", fontsize=15)
-plt.text(0.01, 0.2, "Depth=1", fontsize=13)
-plt.text(0.65, 0.8, "Depth=1", fontsize=13)
-plt.legend(loc="upper center", fontsize=18)
-plt.title("max_depth=2", fontsize=14)
-
+plot_decision_boundary(tree_clf_s, Xs, ys, axes=[-0.7, 0.7, -0.7, 0.7], iris=False)
plt.subplot(122)
-plot_regression_predictions(tree_reg2, X, y, ylabel=None)
-for split, style in ((0.1973, "k-"), (0.0917, "k--"), (0.7718, "k--")):
- plt.plot([split, split], [-0.2, 1], style, linewidth=2)
-for split in (0.0458, 0.1298, 0.2873, 0.9040):
- plt.plot([split, split], [-0.2, 1], "k:", linewidth=1)
-plt.text(0.3, 0.5, "Depth=2", fontsize=13)
-plt.title("max_depth=3", fontsize=14)
-
-plt.show()
-
-
-
-
-
tree_reg1 = DecisionTreeRegressor(random_state=42)
-tree_reg2 = DecisionTreeRegressor(random_state=42, min_samples_leaf=10)
-tree_reg1.fit(X, y)
-tree_reg2.fit(X, y)
-
-x1 = np.linspace(0, 1, 500).reshape(-1, 1)
-y_pred1 = tree_reg1.predict(x1)
-y_pred2 = tree_reg2.predict(x1)
-
-plt.figure(figsize=(11, 4))
-
-plt.subplot(121)
-plt.plot(X, y, "b.")
-plt.plot(x1, y_pred1, "r.-", linewidth=2, label=r"$\hat{y}$")
-plt.axis([0, 1, -0.2, 1.1])
-plt.xlabel("$x_1$", fontsize=18)
-plt.ylabel("$y$", fontsize=18, rotation=0)
-plt.legend(loc="upper center", fontsize=18)
-plt.title("No restrictions", fontsize=14)
-
-plt.subplot(122)
-plt.plot(X, y, "b.")
-plt.plot(x1, y_pred2, "r.-", linewidth=2, label=r"$\hat{y}$")
-plt.axis([0, 1, -0.2, 1.1])
-plt.xlabel("$x_1$", fontsize=18)
-plt.title("min_samples_leaf={}".format(tree_reg2.min_samples_leaf), fontsize=14)
+plot_decision_boundary(tree_clf_sr, Xsr, ys, axes=[-0.7, 0.7, -0.7, 0.7], iris=False)
plt.show()
@@ -256,7 +210,7 @@ plt.show()
-
# Quadratic training set + noise
+np.random.seed(42)
+m = 200
+X = np.random.rand(m, 1)
+y = 4 * (X - 0.5) ** 2
+y = y + np.random.randn(m, 1) / 10
++ +
from sklearn.tree import DecisionTreeRegressor
+
+tree_reg = DecisionTreeRegressor(max_depth=2, random_state=42)
+tree_reg.fit(X, y)
+
diff --git a/doc/pub/DecisionTrees/html/._DecisionTrees-bs020.html b/doc/pub/DecisionTrees/html/._DecisionTrees-bs020.html index 9bcb742f8..261df3269 100644 --- a/doc/pub/DecisionTrees/html/._DecisionTrees-bs020.html +++ b/doc/pub/DecisionTrees/html/._DecisionTrees-bs020.html @@ -42,39 +42,41 @@ Automatically generated HTML file from DocOnce source @@ -113,33 +115,35 @@ MathJax.Hub.Config({ Contents @@ -155,20 +159,81 @@ MathJax.Hub.Config({ -
-
from sklearn.tree import DecisionTreeRegressor
-However, by aggregating many decision trees, using methods like bagging, random forests, and boosting, the predictive performance of trees can be substantially improved.
+tree_reg1 = DecisionTreeRegressor(random_state=42, max_depth=2)
+tree_reg2 = DecisionTreeRegressor(random_state=42, max_depth=3)
+tree_reg1.fit(X, y)
+tree_reg2.fit(X, y)
+def plot_regression_predictions(tree_reg, X, y, axes=[0, 1, -0.2, 1], ylabel="$y$"):
+ x1 = np.linspace(axes[0], axes[1], 500).reshape(-1, 1)
+ y_pred = tree_reg.predict(x1)
+ plt.axis(axes)
+ plt.xlabel("$x_1$", fontsize=18)
+ if ylabel:
+ plt.ylabel(ylabel, fontsize=18, rotation=0)
+ plt.plot(X, y, "b.")
+ plt.plot(x1, y_pred, "r.-", linewidth=2, label=r"$\hat{y}$")
+
+plt.figure(figsize=(11, 4))
+plt.subplot(121)
+plot_regression_predictions(tree_reg1, X, y)
+for split, style in ((0.1973, "k-"), (0.0917, "k--"), (0.7718, "k--")):
+ plt.plot([split, split], [-0.2, 1], style, linewidth=2)
+plt.text(0.21, 0.65, "Depth=0", fontsize=15)
+plt.text(0.01, 0.2, "Depth=1", fontsize=13)
+plt.text(0.65, 0.8, "Depth=1", fontsize=13)
+plt.legend(loc="upper center", fontsize=18)
+plt.title("max_depth=2", fontsize=14)
+
+plt.subplot(122)
+plot_regression_predictions(tree_reg2, X, y, ylabel=None)
+for split, style in ((0.1973, "k-"), (0.0917, "k--"), (0.7718, "k--")):
+ plt.plot([split, split], [-0.2, 1], style, linewidth=2)
+for split in (0.0458, 0.1298, 0.2873, 0.9040):
+ plt.plot([split, split], [-0.2, 1], "k:", linewidth=1)
+plt.text(0.3, 0.5, "Depth=2", fontsize=13)
+plt.title("max_depth=3", fontsize=14)
+
+plt.show()
++ + +
tree_reg1 = DecisionTreeRegressor(random_state=42)
+tree_reg2 = DecisionTreeRegressor(random_state=42, min_samples_leaf=10)
+tree_reg1.fit(X, y)
+tree_reg2.fit(X, y)
+
+x1 = np.linspace(0, 1, 500).reshape(-1, 1)
+y_pred1 = tree_reg1.predict(x1)
+y_pred2 = tree_reg2.predict(x1)
+
+plt.figure(figsize=(11, 4))
+
+plt.subplot(121)
+plt.plot(X, y, "b.")
+plt.plot(x1, y_pred1, "r.-", linewidth=2, label=r"$\hat{y}$")
+plt.axis([0, 1, -0.2, 1.1])
+plt.xlabel("$x_1$", fontsize=18)
+plt.ylabel("$y$", fontsize=18, rotation=0)
+plt.legend(loc="upper center", fontsize=18)
+plt.title("No restrictions", fontsize=14)
+
+plt.subplot(122)
+plt.plot(X, y, "b.")
+plt.plot(x1, y_pred2, "r.-", linewidth=2, label=r"$\hat{y}$")
+plt.axis([0, 1, -0.2, 1.1])
+plt.xlabel("$x_1$", fontsize=18)
+plt.title("min_samples_leaf={}".format(tree_reg2.min_samples_leaf), fontsize=14)
+
+plt.show()
+
@@ -193,6 +258,9 @@ However, by aggregating many decision trees, using methods like bagging, random
-The plain decision trees suffer from high -variance. This means that if we split the training data into two parts -at random, and fit a decision tree to both halves, the results that we -get could be quite different. In contrast, a procedure with low -variance will yield similar results if applied repeatedly to distinct -data sets; linear regression tends to have low variance, if the ratio -of \( n \) to \( p \) is moderately large. +
-Bootstrap aggregation, or just bagging, is a -general-purpose procedure for reducing the variance of a statistical -learning method. - -
-Bagging typically results in improved accuracy -over prediction using a single tree. Unfortunately, however, it can be -difficult to interpret the resulting model. Recall that one of the -advantages of decision trees is the attractive and easily interpreted -diagram that results. - -
-However, when we bag a large number of trees, it is no longer -possible to represent the resulting statistical learning procedure -using a single tree, and it is no longer clear which variables are -most important to the procedure. Thus, bagging improves prediction -accuracy at the expense of interpretability. Although the collection -of bagged trees is much more difficult to interpret than a single -tree, one can obtain an overall summary of the importance of each -predictor using the MSE (for bagging regression trees) or the Gini -index (for bagging classification trees). In the case of bagging -regression trees, we can record the total amount that the MSE is -decreased due to splits over a given predictor, averaged over all \( B \) possible -trees. A large value indicates an important predictor. Similarly, in -the context of bagging classification trees, we can add up the total -amount that the Gini index is decreased by splits over a given -predictor, averaged over all \( B \) trees. - -
diff --git a/doc/pub/DecisionTrees/html/._DecisionTrees-bs022.html b/doc/pub/DecisionTrees/html/._DecisionTrees-bs022.html index 2660af6af..c3e65aa61 100644 --- a/doc/pub/DecisionTrees/html/._DecisionTrees-bs022.html +++ b/doc/pub/DecisionTrees/html/._DecisionTrees-bs022.html @@ -42,39 +42,41 @@ Automatically generated HTML file from DocOnce source @@ -113,33 +115,35 @@ MathJax.Hub.Config({ Contents @@ -155,23 +159,20 @@ MathJax.Hub.Config({ -
+
heads_proba = 0.51
-coin_tosses = (np.random.rand(10000, 10) < heads_proba).astype(np.int32)
-cumulative_heads_ratio = np.cumsum(coin_tosses, axis=0) / np.arange(1, 10001).reshape(-1, 1)
-plt.figure(figsize=(8,3.5))
-plt.plot(cumulative_heads_ratio)
-plt.plot([0, 10000], [0.51, 0.51], "k--", linewidth=2, label="51%")
-plt.plot([0, 10000], [0.5, 0.5], "k-", label="50%")
-plt.xlabel("Number of coin tosses")
-plt.ylabel("Heads ratio")
-plt.legend(loc="lower right")
-plt.axis([0, 10000, 0.42, 0.58])
-plt.show()
-
@@ -194,6 +195,8 @@ plt.show()
-Random forests provide an improvement over bagged trees by way of a -small tweak that decorrelates the trees. +The plain decision trees suffer from high +variance. This means that if we split the training data into two parts +at random, and fit a decision tree to both halves, the results that we +get could be quite different. In contrast, a procedure with low +variance will yield similar results if applied repeatedly to distinct +data sets; linear regression tends to have low variance, if the ratio +of \( n \) to \( p \) is moderately large.
-As in bagging, we build a -number of decision trees on bootstrapped training samples. But when -building these decision trees, each time a split in a tree is -considered, a random sample of \( m \) predictors is chosen as split -candidates from the full set of \( p \) predictors. The split is allowed to -use only one of those \( m \) predictors. +Bootstrap aggregation, or just bagging, is a +general-purpose procedure for reducing the variance of a statistical +learning method.
-A fresh sample of \( m \) predictors is -taken at each split, and typically we choose -$$ -m\approx \sqrt{p}. -$$ - -In building a random forest, at -each split in the tree, the algorithm is not even allowed to consider -a majority of the available predictors. +Bagging typically results in improved accuracy +over prediction using a single tree. Unfortunately, however, it can be +difficult to interpret the resulting model. Recall that one of the +advantages of decision trees is the attractive and easily interpreted +diagram that results.
-The reason for this is rather clever. Suppose that there is one very -strong predictor in the data set, along with a number of other -moderately strong predictors. Then in the collection of bagged -variable importance random forest trees, most or all of the trees will -use this strong predictor in the top split. Consequently, all of the -bagged trees will look quite similar to each other. Hence the -predictions from the bagged trees will be highly correlated. -Unfortunately, averaging many highly correlated quantities does not lead -to as large of a reduction in variance as averaging many uncorrelated -quanti- ties. In particular, this means that bagging will not lead to -a substantial reduction in variance over a single tree in this -setting. +However, when we bag a large number of trees, it is no longer +possible to represent the resulting statistical learning procedure +using a single tree, and it is no longer clear which variables are +most important to the procedure. Thus, bagging improves prediction +accuracy at the expense of interpretability. Although the collection +of bagged trees is much more difficult to interpret than a single +tree, one can obtain an overall summary of the importance of each +predictor using the MSE (for bagging regression trees) or the Gini +index (for bagging classification trees). In the case of bagging +regression trees, we can record the total amount that the MSE is +decreased due to splits over a given predictor, averaged over all \( B \) possible +trees. A large value indicates an important predictor. Similarly, in +the context of bagging classification trees, we can add up the total +amount that the Gini index is decreased by splits over a given +predictor, averaged over all \( B \) trees.
@@ -215,6 +220,8 @@ setting.
-
from sklearn.ensemble import RandomForestClassifier
-from sklearn.preprocessing import LabelEncoder
-from sklearn.model_selection import cross_validate
-# Data set not specificied
-X = dataset.XXX
-Y = dataset.YYY
-#Instantiate the model with 100 trees and entropy as splitting criteria
-Random_Forest_model = RandomForestClassifier(n_estimators=100,criterion="entropy")
-#Cross validation
-accuracy = cross_validate(Random_Forest_model,X,Y,cv=10)['test_score']
+heads_proba = 0.51
+coin_tosses = (np.random.rand(10000, 10) < heads_proba).astype(np.int32)
+cumulative_heads_ratio = np.cumsum(coin_tosses, axis=0) / np.arange(1, 10001).reshape(-1, 1)
+plt.figure(figsize=(8,3.5))
+plt.plot(cumulative_heads_ratio)
+plt.plot([0, 10000], [0.51, 0.51], "k--", linewidth=2, label="51%")
+plt.plot([0, 10000], [0.5, 0.5], "k-", label="50%")
+plt.xlabel("Number of coin tosses")
+plt.ylabel("Heads ratio")
+plt.legend(loc="lower right")
+plt.axis([0, 10000, 0.42, 0.58])
+plt.show()
@@ -190,6 +196,8 @@ accuracy = cross_validate(Random_Forest_mode
+Random forests provide an improvement over bagged trees by way of a +small tweak that decorrelates the trees. - -
from sklearn.model_selection import train_test_split
-from sklearn.datasets import make_moons
-
-X, y = make_moons(n_samples=500, noise=0.30, random_state=42)
-X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=42)
-from sklearn.ensemble import RandomForestClassifier
-from sklearn.ensemble import VotingClassifier
-from sklearn.linear_model import LogisticRegression
-from sklearn.svm import SVC
-
-log_clf = LogisticRegression(random_state=42)
-rnd_clf = RandomForestClassifier(random_state=42)
-svm_clf = SVC(random_state=42)
-
-voting_clf = VotingClassifier(
- estimators=[('lr', log_clf), ('rf', rnd_clf), ('svc', svm_clf)],
- voting='hard')
-voting_clf.fit(X_train, y_train)
-+As in bagging, we build a +number of decision trees on bootstrapped training samples. But when +building these decision trees, each time a split in a tree is +considered, a random sample of \( m \) predictors is chosen as split +candidates from the full set of \( p \) predictors. The split is allowed to +use only one of those \( m \) predictors. - -
from sklearn.metrics import accuracy_score
-
-for clf in (log_clf, rnd_clf, svm_clf, voting_clf):
- clf.fit(X_train, y_train)
- y_pred = clf.predict(X_test)
- print(clf.__class__.__name__, accuracy_score(y_test, y_pred))
-+A fresh sample of \( m \) predictors is +taken at each split, and typically we choose +$$ +m\approx \sqrt{p}. +$$ - -
log_clf = LogisticRegression(random_state=42)
-rnd_clf = RandomForestClassifier(random_state=42)
-svm_clf = SVC(probability=True, random_state=42)
+In building a random forest, at
+each split in the tree, the algorithm is not even allowed to consider
+a majority of the available predictors.
-voting_clf = VotingClassifier(
- estimators=[('lr', log_clf), ('rf', rnd_clf), ('svc', svm_clf)],
- voting='soft')
-voting_clf.fit(X_train, y_train)
-+The reason for this is rather clever. Suppose that there is one very +strong predictor in the data set, along with a number of other +moderately strong predictors. Then in the collection of bagged +variable importance random forest trees, most or all of the trees will +use this strong predictor in the top split. Consequently, all of the +bagged trees will look quite similar to each other. Hence the +predictions from the bagged trees will be highly correlated. +Unfortunately, averaging many highly correlated quantities does not lead +to as large of a reduction in variance as averaging many uncorrelated +quanti- ties. In particular, this means that bagging will not lead to +a substantial reduction in variance over a single tree in this +setting. - -
from sklearn.metrics import accuracy_score
-
-for clf in (log_clf, rnd_clf, svm_clf, voting_clf):
- clf.fit(X_train, y_train)
- y_pred = clf.predict(X_test)
- print(clf.__class__.__name__, accuracy_score(y_test, y_pred))
-
@@ -229,6 +217,8 @@ voting_clf.fit(X_train, y_train)
-
from sklearn.ensemble import BaggingClassifier
-from sklearn.tree import DecisionTreeClassifier
-
-bag_clf = BaggingClassifier(
- DecisionTreeClassifier(random_state=42), n_estimators=500,
- max_samples=100, bootstrap=True, n_jobs=-1, random_state=42)
-bag_clf.fit(X_train, y_train)
-y_pred = bag_clf.predict(X_test)
-- - -
from sklearn.metrics import accuracy_score
-print(accuracy_score(y_test, y_pred))
-- - -
tree_clf = DecisionTreeClassifier(random_state=42)
-tree_clf.fit(X_train, y_train)
-y_pred_tree = tree_clf.predict(X_test)
-print(accuracy_score(y_test, y_pred_tree))
-- - -
from matplotlib.colors import ListedColormap
-
-def plot_decision_boundary(clf, X, y, axes=[-1.5, 2.5, -1, 1.5], alpha=0.5, contour=True):
- x1s = np.linspace(axes[0], axes[1], 100)
- x2s = np.linspace(axes[2], axes[3], 100)
- x1, x2 = np.meshgrid(x1s, x2s)
- X_new = np.c_[x1.ravel(), x2.ravel()]
- y_pred = clf.predict(X_new).reshape(x1.shape)
- custom_cmap = ListedColormap(['#fafab0','#9898ff','#a0faa0'])
- plt.contourf(x1, x2, y_pred, alpha=0.3, cmap=custom_cmap)
- if contour:
- custom_cmap2 = ListedColormap(['#7d7d58','#4c4c7f','#507d50'])
- plt.contour(x1, x2, y_pred, cmap=custom_cmap2, alpha=0.8)
- plt.plot(X[:, 0][y==0], X[:, 1][y==0], "yo", alpha=alpha)
- plt.plot(X[:, 0][y==1], X[:, 1][y==1], "bs", alpha=alpha)
- plt.axis(axes)
- plt.xlabel(r"$x_1$", fontsize=18)
- plt.ylabel(r"$x_2$", fontsize=18, rotation=0)
-plt.figure(figsize=(11,4))
-plt.subplot(121)
-plot_decision_boundary(tree_clf, X, y)
-plt.title("Decision Tree", fontsize=14)
-plt.subplot(122)
-plot_decision_boundary(bag_clf, X, y)
-plt.title("Decision Trees with Bagging", fontsize=14)
-plt.show()
+from sklearn.ensemble import RandomForestClassifier
+from sklearn.preprocessing import LabelEncoder
+from sklearn.model_selection import cross_validate
+# Data set not specificied
+X = dataset.XXX
+Y = dataset.YYY
+#Instantiate the model with 100 trees and entropy as splitting criteria
+Random_Forest_model = RandomForestClassifier(n_estimators=100,criterion="entropy")
+#Cross validation
+accuracy = cross_validate(Random_Forest_model,X,Y,cv=10)['test_score']
@@ -231,6 +192,8 @@ plt.show()
-
bag_clf = BaggingClassifier(
- DecisionTreeClassifier(splitter="random", max_leaf_nodes=16, random_state=42),
- n_estimators=500, max_samples=1.0, bootstrap=True, n_jobs=-1, random_state=42)
+from sklearn.model_selection import train_test_split
+from sklearn.datasets import make_moons
+
+X, y = make_moons(n_samples=500, noise=0.30, random_state=42)
+X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=42)
+from sklearn.ensemble import RandomForestClassifier
+from sklearn.ensemble import VotingClassifier
+from sklearn.linear_model import LogisticRegression
+from sklearn.svm import SVC
+
+log_clf = LogisticRegression(random_state=42)
+rnd_clf = RandomForestClassifier(random_state=42)
+svm_clf = SVC(random_state=42)
+
+voting_clf = VotingClassifier(
+ estimators=[('lr', log_clf), ('rf', rnd_clf), ('svc', svm_clf)],
+ voting='hard')
+voting_clf.fit(X_train, y_train)
-
bag_clf.fit(X_train, y_train)
-y_pred = bag_clf.predict(X_test)
-from sklearn.ensemble import RandomForestClassifier
-rnd_clf = RandomForestClassifier(n_estimators=500, max_leaf_nodes=16, n_jobs=-1, random_state=42)
-rnd_clf.fit(X_train, y_train)
-y_pred_rf = rnd_clf.predict(X_test)
-np.sum(y_pred == y_pred_rf) / len(y_pred)
+from sklearn.metrics import accuracy_score
+
+for clf in (log_clf, rnd_clf, svm_clf, voting_clf):
+ clf.fit(X_train, y_train)
+ y_pred = clf.predict(X_test)
+ print(clf.__class__.__name__, accuracy_score(y_test, y_pred))
+
+
+
+
+
log_clf = LogisticRegression(random_state=42)
+rnd_clf = RandomForestClassifier(random_state=42)
+svm_clf = SVC(probability=True, random_state=42)
+
+voting_clf = VotingClassifier(
+ estimators=[('lr', log_clf), ('rf', rnd_clf), ('svc', svm_clf)],
+ voting='soft')
+voting_clf.fit(X_train, y_train)
+
+
+
+
+
from sklearn.metrics import accuracy_score
+
+for clf in (log_clf, rnd_clf, svm_clf, voting_clf):
+ clf.fit(X_train, y_train)
+ y_pred = clf.predict(X_test)
+ print(clf.__class__.__name__, accuracy_score(y_test, y_pred))
@@ -191,6 +231,8 @@ np.sum(y_pred =
27
28
29
+ 30
+ 31
»
diff --git a/doc/pub/DecisionTrees/html/._DecisionTrees-bs028.html b/doc/pub/DecisionTrees/html/._DecisionTrees-bs028.html
index 0988bec6c..935838dd3 100644
--- a/doc/pub/DecisionTrees/html/._DecisionTrees-bs028.html
+++ b/doc/pub/DecisionTrees/html/._DecisionTrees-bs028.html
@@ -42,39 +42,41 @@ Automatically generated HTML file from DocOnce source
@@ -113,33 +115,35 @@ MathJax.Hub.Config({
Contents
@@ -153,13 +157,67 @@ MathJax.Hub.Config({
-
+
-Boosting and more
-More material to come here.
+Bagging examples
+
+
from sklearn.ensemble import BaggingClassifier
+from sklearn.tree import DecisionTreeClassifier
+
+bag_clf = BaggingClassifier(
+ DecisionTreeClassifier(random_state=42), n_estimators=500,
+ max_samples=100, bootstrap=True, n_jobs=-1, random_state=42)
+bag_clf.fit(X_train, y_train)
+y_pred = bag_clf.predict(X_test)
+
+
+
+
+
from sklearn.metrics import accuracy_score
+print(accuracy_score(y_test, y_pred))
+
+
+
+
+
tree_clf = DecisionTreeClassifier(random_state=42)
+tree_clf.fit(X_train, y_train)
+y_pred_tree = tree_clf.predict(X_test)
+print(accuracy_score(y_test, y_pred_tree))
+
+
+
+
+
from matplotlib.colors import ListedColormap
+
+def plot_decision_boundary(clf, X, y, axes=[-1.5, 2.5, -1, 1.5], alpha=0.5, contour=True):
+ x1s = np.linspace(axes[0], axes[1], 100)
+ x2s = np.linspace(axes[2], axes[3], 100)
+ x1, x2 = np.meshgrid(x1s, x2s)
+ X_new = np.c_[x1.ravel(), x2.ravel()]
+ y_pred = clf.predict(X_new).reshape(x1.shape)
+ custom_cmap = ListedColormap(['#fafab0','#9898ff','#a0faa0'])
+ plt.contourf(x1, x2, y_pred, alpha=0.3, cmap=custom_cmap)
+ if contour:
+ custom_cmap2 = ListedColormap(['#7d7d58','#4c4c7f','#507d50'])
+ plt.contour(x1, x2, y_pred, cmap=custom_cmap2, alpha=0.8)
+ plt.plot(X[:, 0][y==0], X[:, 1][y==0], "yo", alpha=alpha)
+ plt.plot(X[:, 0][y==1], X[:, 1][y==1], "bs", alpha=alpha)
+ plt.axis(axes)
+ plt.xlabel(r"$x_1$", fontsize=18)
+ plt.ylabel(r"$x_2$", fontsize=18, rotation=0)
+plt.figure(figsize=(11,4))
+plt.subplot(121)
+plot_decision_boundary(tree_clf, X, y)
+plt.title("Decision Tree", fontsize=14)
+plt.subplot(122)
+plot_decision_boundary(bag_clf, X, y)
+plt.title("Decision Trees with Bagging", fontsize=14)
+plt.show()
+
+
diff --git a/doc/pub/DecisionTrees/html/._DecisionTrees-bs029.html b/doc/pub/DecisionTrees/html/._DecisionTrees-bs029.html
index b2684c0b1..861de541c 100644
--- a/doc/pub/DecisionTrees/html/._DecisionTrees-bs029.html
+++ b/doc/pub/DecisionTrees/html/._DecisionTrees-bs029.html
@@ -42,40 +42,41 @@ Automatically generated HTML file from DocOnce source
@@ -114,34 +115,35 @@ MathJax.Hub.Config({
Contents
@@ -155,13 +157,28 @@ MathJax.Hub.Config({
-
-
-Boosting and more
-More material to come here.
+
+Then random forests
+
+
bag_clf = BaggingClassifier(
+ DecisionTreeClassifier(splitter="random", max_leaf_nodes=16, random_state=42),
+ n_estimators=500, max_samples=1.0, bootstrap=True, n_jobs=-1, random_state=42)
+
+
+
+
+
bag_clf.fit(X_train, y_train)
+y_pred = bag_clf.predict(X_test)
+from sklearn.ensemble import RandomForestClassifier
+rnd_clf = RandomForestClassifier(n_estimators=500, max_leaf_nodes=16, n_jobs=-1, random_state=42)
+rnd_clf.fit(X_train, y_train)
+y_pred_rf = rnd_clf.predict(X_test)
+np.sum(y_pred == y_pred_rf) / len(y_pred)
+
+
diff --git a/doc/pub/DecisionTrees/html/._DecisionTrees-bs030.html b/doc/pub/DecisionTrees/html/._DecisionTrees-bs030.html
index c4795b4f2..99821f748 100644
--- a/doc/pub/DecisionTrees/html/._DecisionTrees-bs030.html
+++ b/doc/pub/DecisionTrees/html/._DecisionTrees-bs030.html
@@ -42,9 +42,9 @@ Automatically generated HTML file from DocOnce source
Decision trees, overarching aims
- How do we set it up?
- Decision trees and Regression
- Maxwell-Boltzmann velocity distribution
+ General Features
+ How do we set it up?
+ Decision trees and Regression
Building a tree, regression
A top-down approach, recursive binary splitting
Making a tree
Pruning the tree
Cost complexity pruning
- A schematic procedure
- A classification tree
+ Schematic Regression Procedure
+ A Classification Tree
Growing a classification tree
Classification tree, how to split nodes
Entropy and the ID3 algorithm
- Writing your own code for a classification tree
- Back to moons again
- Playing around with regions
- Regression trees
- Final regressor code
- Classification again: The zoo data
+ Implementing the ID3 Algorithm
+ Cancer Data again now with Decision Trees
+ Another example, the moons again
+ Playing around with regions
+ Regression trees
+ Final regressor code
Pros and cons of trees, pros
Disadvantages
Bagging
diff --git a/doc/pub/DecisionTrees/html/DecisionTrees-bs.html b/doc/pub/DecisionTrees/html/DecisionTrees-bs.html
index a1f2bc91d..60b829003 100644
--- a/doc/pub/DecisionTrees/html/DecisionTrees-bs.html
+++ b/doc/pub/DecisionTrees/html/DecisionTrees-bs.html
@@ -42,39 +42,41 @@ Automatically generated HTML file from DocOnce source
@@ -113,33 +115,35 @@ MathJax.Hub.Config({
Contents
@@ -198,7 +202,7 @@ MathJax.Hub.Config({
9
10
...
- 29
+ 31
»
diff --git a/doc/pub/DecisionTrees/html/DecisionTrees-reveal.html b/doc/pub/DecisionTrees/html/DecisionTrees-reveal.html
index 791a5279f..7cf28ce07 100644
--- a/doc/pub/DecisionTrees/html/DecisionTrees-reveal.html
+++ b/doc/pub/DecisionTrees/html/DecisionTrees-reveal.html
@@ -193,7 +193,26 @@ and leaf nodes which are then connected by branches.
-How do we set it up?
+General Features
+
+
+Decision trees classify instances by sorting top down.
+
+
+- A leaf provides the classification of the instance.
+- A node specifies a test of some attribute of the instance.
+- A branch corresponds to a possible values an attribute.
+- 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.
+
+
+
+This processis then repeated for the subtree rooted at the new
+node.
+
+
+
+
+How do we set it up?
In simplified terms, the process of training a decision tree and
@@ -212,7 +231,7 @@ Then we are essentially done!
-Decision trees and Regression
+Decision trees and Regression
@@ -309,7 +328,7 @@ plt.show()
-Building a tree, regression
+Building a tree, regression
There are mainly two steps
@@ -339,7 +358,7 @@ within box \( j \).
-A top-down approach, recursive binary splitting
+A top-down approach, recursive binary splitting
Unfortunately, it is computationally infeasible to consider every
@@ -358,7 +377,7 @@ better tree in some future step.
-Making a tree
+Making a tree
In order to implement the recursive binary splitting we start by selecting
@@ -411,7 +430,7 @@ region contains more than five observations.
-Pruning the tree
+Pruning the tree
The above procedure is rather straightforward, but leads often to
@@ -430,7 +449,7 @@ parameter \( \alpha \).
-Cost complexity pruning
+Cost complexity pruning
For each value of \( \alpha \) there corresponds a subtree \( T \in T_0 \) such that
$$
@@ -463,7 +482,7 @@ subtree corresponding to \( \alpha \).
-Schematic Regression Procedure
+Schematic Regression Procedure
@@ -488,7 +507,7 @@ subtree corresponding to \( \alpha \).
-A Classification Tree
+A Classification Tree
A classification tree is very similar to a regression tree, except
@@ -507,7 +526,7 @@ fall into that region.
-Growing a classification tree
+Growing a classification tree
The task of growing a
@@ -531,7 +550,7 @@ than is the classification error rate.
-Classification tree, how to split nodes
+Classification tree, how to split nodes
If our targets are the outcome of a classification process that takes for example
@@ -580,15 +599,52 @@ $$
-Entropy and the ID3 algorithm
+Entropy and the ID3 algorithm
-More text and code to come here.
+ID3, learns decision trees by constructing
+them topdown, beginning with the question which attribute should be tested at the root of the tree?
+
+
+- Each instance attribute is evaluated using a statistical test to determine how well it alone classifies the training examples.
+- The best attribute is selected and used as the test at the root node of the tree.
+- A descendant of the root node is then created for each possible value of this attribute.
+- Training examples are sorted to the appropriate descendant node.
+- 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.
+- This forms a greedy search for an acceptable decision tree, in which the algorithm never backtracks to reconsider earlier choices.
+
+
+
+The ID3 algorithm selects, which attribute to test at each node in the
+tree.
+
+
+We would like to select the attribute that is most useful for classifying
+examples.
+
+
+What is a good quantitative measure of the worth of an attribute?
+
+
+Information gain measures how well a given attribute separates the
+training examples according to their target classification.
+
+
+The ID3 algorithm uses this information gain measure to select among the candidate
+attributes at each step while growing the tree.
-Cancer Data again now with Decision Trees
+Implementing the ID3 Algorithm
+
+
+more text to come here, material presented during lecture Friday Oct 25.
+
+
+
+
+Cancer Data again now with Decision Trees
@@ -638,7 +694,7 @@ deep_tree_clf.fit(X_train_scaled, y_train)
-Another example, the moons again
+Another example, the moons again
@@ -711,7 +767,7 @@ plt.show()
-Playing around with regions
+Playing around with regions
@@ -740,7 +796,7 @@ plt.show()
-Regression trees
+Regression trees
@@ -763,7 +819,7 @@ tree_reg.fit(X, y)
-Final regressor code
+Final regressor code
@@ -842,7 +898,7 @@ plt.show()
-Pros and cons of trees, pros
+Pros and cons of trees, pros
- 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)
@@ -857,7 +913,7 @@ plt.show()
-Disadvantages
+Disadvantages
- Unfortunately, trees generally do not have the same level of predictive accuracy as some of the other regression and classification approaches
@@ -875,7 +931,7 @@ However, by aggregating many decision trees, using methods like bagging, random
-Bagging
+Bagging
The plain decision trees suffer from high
@@ -918,7 +974,7 @@ predictor, averaged over all \( B \) trees.
-Simple example, head or tail
+Simple example, head or tail
@@ -939,7 +995,7 @@ plt.show()
-Random forests
+Random forests
Random forests provide an improvement over bagged trees by way of a
@@ -983,7 +1039,7 @@ setting.
-A simple scikit-learn example
+A simple scikit-learn example
@@ -1002,7 +1058,7 @@ accuracy = cross_validate(Random_Forest_model,X,Y,cv=Please, not the moons again!
+Please, not the moons again!
@@ -1061,7 +1117,7 @@ voting_clf.fit(X_train, y_train)
-Bagging examples
+Bagging examples
@@ -1123,7 +1179,7 @@ plt.show()
-Then random forests
+Then random forests
@@ -1146,7 +1202,7 @@ np.sum(y_pred == y_pred_rf) / len(y_pred)
-Boosting and more
+Boosting and more
More material to come here.
diff --git a/doc/pub/DecisionTrees/html/DecisionTrees-solarized.html b/doc/pub/DecisionTrees/html/DecisionTrees-solarized.html
index 73ecfd695..1d7217057 100644
--- a/doc/pub/DecisionTrees/html/DecisionTrees-solarized.html
+++ b/doc/pub/DecisionTrees/html/DecisionTrees-solarized.html
@@ -62,39 +62,41 @@ div { text-align: justify; text-justify: inter-word; }
@@ -175,7 +177,25 @@ and leaf nodes which are then connected by branches.
-
How do we set it up?
+General Features
+
+
+Decision trees classify instances by sorting top down.
+
+
+- A leaf provides the classification of the instance.
+- A node specifies a test of some attribute of the instance.
+- A branch corresponds to a possible values an attribute.
+- 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.
+
+
+This processis then repeated for the subtree rooted at the new
+node.
+
+
+
+
+
How do we set it up?
In simplified terms, the process of training a decision tree and
@@ -193,7 +213,7 @@ Then we are essentially done!
-
Decision trees and Regression
+Decision trees and Regression
@@ -289,7 +309,7 @@ plt.show()
-
Building a tree, regression
+Building a tree, regression
There are mainly two steps
@@ -315,7 +335,7 @@ within box \( j \).
-
A top-down approach, recursive binary splitting
+A top-down approach, recursive binary splitting
Unfortunately, it is computationally infeasible to consider every
@@ -334,7 +354,7 @@ better tree in some future step.
-
Making a tree
+Making a tree
In order to implement the recursive binary splitting we start by selecting
@@ -381,7 +401,7 @@ region contains more than five observations.
-
Pruning the tree
+Pruning the tree
The above procedure is rather straightforward, but leads often to
@@ -400,7 +420,7 @@ parameter \( \alpha \).
-
Cost complexity pruning
+Cost complexity pruning
For each value of \( \alpha \) there corresponds a subtree \( T \in T_0 \) such that
$$
\sum_{m=1}^{\overline{T}}\sum_{i:x_i\in R_m}(y_i-\overline{y}_{R_m})^2+\alpha\overline{T},
@@ -431,7 +451,7 @@ subtree corresponding to \( \alpha \).
-
Schematic Regression Procedure
+Schematic Regression Procedure
@@ -457,7 +477,7 @@ subtree corresponding to \( \alpha \).
-
A Classification Tree
+A Classification Tree
A classification tree is very similar to a regression tree, except
@@ -476,7 +496,7 @@ fall into that region.
-
Growing a classification tree
+Growing a classification tree
The task of growing a
@@ -500,7 +520,7 @@ than is the classification error rate.
-
Classification tree, how to split nodes
+Classification tree, how to split nodes
If our targets are the outcome of a classification process that takes for example
@@ -544,15 +564,51 @@ $$
-
Entropy and the ID3 algorithm
+Entropy and the ID3 algorithm
-More text and code to come here.
+ID3, learns decision trees by constructing
+them topdown, beginning with the question which attribute should be tested at the root of the tree?
+
+
+- Each instance attribute is evaluated using a statistical test to determine how well it alone classifies the training examples.
+- The best attribute is selected and used as the test at the root node of the tree.
+- A descendant of the root node is then created for each possible value of this attribute.
+- Training examples are sorted to the appropriate descendant node.
+- 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.
+- This forms a greedy search for an acceptable decision tree, in which the algorithm never backtracks to reconsider earlier choices.
+
+
+The ID3 algorithm selects, which attribute to test at each node in the
+tree.
+
+
+We would like to select the attribute that is most useful for classifying
+examples.
+
+
+What is a good quantitative measure of the worth of an attribute?
+
+
+Information gain measures how well a given attribute separates the
+training examples according to their target classification.
+
+
+The ID3 algorithm uses this information gain measure to select among the candidate
+attributes at each step while growing the tree.
-
Cancer Data again now with Decision Trees
+Implementing the ID3 Algorithm
+
+
+more text to come here, material presented during lecture Friday Oct 25.
+
+
+
+
+
Cancer Data again now with Decision Trees
@@ -601,7 +657,7 @@ deep_tree_clf.fit(X_train_scaled, y_train)
-
Another example, the moons again
+Another example, the moons again
@@ -673,7 +729,7 @@ plt.show()
-
Playing around with regions
+Playing around with regions
@@ -701,7 +757,7 @@ plt.show()
-
Regression trees
+Regression trees
@@ -723,7 +779,7 @@ tree_reg.fit(X, y)
-
Final regressor code
+Final regressor code
@@ -801,7 +857,7 @@ plt.show()
-
Pros and cons of trees, pros
+Pros and cons of trees, pros
- 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)
@@ -815,7 +871,7 @@ plt.show()
-Disadvantages
+Disadvantages
- Unfortunately, trees generally do not have the same level of predictive accuracy as some of the other regression and classification approaches
@@ -832,7 +888,7 @@ However, by aggregating many decision trees, using methods like bagging, random
-
Bagging
+Bagging
The plain decision trees suffer from high
@@ -875,7 +931,7 @@ predictor, averaged over all \( B \) trees.
-
Simple example, head or tail
+Simple example, head or tail
@@ -895,7 +951,7 @@ plt.show()
-
Random forests
+Random forests
Random forests provide an improvement over bagged trees by way of a
@@ -937,7 +993,7 @@ setting.
-
A simple scikit-learn example
+A simple scikit-learn example
@@ -955,7 +1011,7 @@ accuracy = cross_validate(Random_Forest_model,X,Y,cv=Please, not the moons again!
+Please, not the moons again!
@@ -1013,7 +1069,7 @@ voting_clf.fit(X_train, y_train)
-
Bagging examples
+Bagging examples
@@ -1074,7 +1130,7 @@ plt.show()
-
Then random forests
+Then random forests
@@ -1096,7 +1152,7 @@ np.sum(y_pred == y_pred_rf) / len(y_pred)
-
Boosting and more
+Boosting and more
More material to come here.
diff --git a/doc/pub/DecisionTrees/html/DecisionTrees.html b/doc/pub/DecisionTrees/html/DecisionTrees.html
index f0657dd24..99778a107 100644
--- a/doc/pub/DecisionTrees/html/DecisionTrees.html
+++ b/doc/pub/DecisionTrees/html/DecisionTrees.html
@@ -67,39 +67,41 @@ div { text-align: justify; text-justify: inter-word; }
@@ -180,7 +182,25 @@ and leaf nodes which are then connected by branches.
-
How do we set it up?
+General Features
+
+
+Decision trees classify instances by sorting top down.
+
+
+- A leaf provides the classification of the instance.
+- A node specifies a test of some attribute of the instance.
+- A branch corresponds to a possible values an attribute.
+- 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.
+
+
+This processis then repeated for the subtree rooted at the new
+node.
+
+
+
+
+
How do we set it up?
In simplified terms, the process of training a decision tree and
@@ -198,7 +218,7 @@ Then we are essentially done!
-
Decision trees and Regression
+Decision trees and Regression
@@ -294,7 +314,7 @@ plt.show()
-
Building a tree, regression
+Building a tree, regression
There are mainly two steps
@@ -320,7 +340,7 @@ within box \( j \).
-
A top-down approach, recursive binary splitting
+A top-down approach, recursive binary splitting
Unfortunately, it is computationally infeasible to consider every
@@ -339,7 +359,7 @@ better tree in some future step.
-
Making a tree
+Making a tree
In order to implement the recursive binary splitting we start by selecting
@@ -386,7 +406,7 @@ region contains more than five observations.
-
Pruning the tree
+Pruning the tree
The above procedure is rather straightforward, but leads often to
@@ -405,7 +425,7 @@ parameter \( \alpha \).
-
Cost complexity pruning
+Cost complexity pruning
For each value of \( \alpha \) there corresponds a subtree \( T \in T_0 \) such that
$$
\sum_{m=1}^{\overline{T}}\sum_{i:x_i\in R_m}(y_i-\overline{y}_{R_m})^2+\alpha\overline{T},
@@ -436,7 +456,7 @@ subtree corresponding to \( \alpha \).
-
Schematic Regression Procedure
+Schematic Regression Procedure
@@ -462,7 +482,7 @@ subtree corresponding to \( \alpha \).
-
A Classification Tree
+A Classification Tree
A classification tree is very similar to a regression tree, except
@@ -481,7 +501,7 @@ fall into that region.
-
Growing a classification tree
+Growing a classification tree
The task of growing a
@@ -505,7 +525,7 @@ than is the classification error rate.
-
Classification tree, how to split nodes
+Classification tree, how to split nodes
If our targets are the outcome of a classification process that takes for example
@@ -549,15 +569,51 @@ $$
-
Entropy and the ID3 algorithm
+Entropy and the ID3 algorithm
-More text and code to come here.
+ID3, learns decision trees by constructing
+them topdown, beginning with the question which attribute should be tested at the root of the tree?
+
+
+- Each instance attribute is evaluated using a statistical test to determine how well it alone classifies the training examples.
+- The best attribute is selected and used as the test at the root node of the tree.
+- A descendant of the root node is then created for each possible value of this attribute.
+- Training examples are sorted to the appropriate descendant node.
+- 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.
+- This forms a greedy search for an acceptable decision tree, in which the algorithm never backtracks to reconsider earlier choices.
+
+
+The ID3 algorithm selects, which attribute to test at each node in the
+tree.
+
+
+We would like to select the attribute that is most useful for classifying
+examples.
+
+
+What is a good quantitative measure of the worth of an attribute?
+
+
+Information gain measures how well a given attribute separates the
+training examples according to their target classification.
+
+
+The ID3 algorithm uses this information gain measure to select among the candidate
+attributes at each step while growing the tree.
-
Cancer Data again now with Decision Trees
+Implementing the ID3 Algorithm
+
+
+more text to come here, material presented during lecture Friday Oct 25.
+
+
+
+
+
Cancer Data again now with Decision Trees
@@ -606,7 +662,7 @@ deep_tree_clf.fit(X_train_scaled, y_train)
-
Another example, the moons again
+Another example, the moons again
@@ -678,7 +734,7 @@ plt.show()
-
Playing around with regions
+Playing around with regions
@@ -706,7 +762,7 @@ plt.show()
-
Regression trees
+Regression trees
@@ -728,7 +784,7 @@ tree_reg.fit(X, y)
-
Final regressor code
+Final regressor code
@@ -806,7 +862,7 @@ plt.show()
-
Pros and cons of trees, pros
+Pros and cons of trees, pros
- 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)
@@ -820,7 +876,7 @@ plt.show()
-Disadvantages
+Disadvantages
- Unfortunately, trees generally do not have the same level of predictive accuracy as some of the other regression and classification approaches
@@ -837,7 +893,7 @@ However, by aggregating many decision trees, using methods like bagging, random
-
Bagging
+Bagging
The plain decision trees suffer from high
@@ -880,7 +936,7 @@ predictor, averaged over all \( B \) trees.
-
Simple example, head or tail
+Simple example, head or tail
@@ -900,7 +956,7 @@ plt.show()
-
Random forests
+Random forests
Random forests provide an improvement over bagged trees by way of a
@@ -942,7 +998,7 @@ setting.
-
A simple scikit-learn example
+A simple scikit-learn example
@@ -960,7 +1016,7 @@ accuracy = cross_validate(Random_Forest_mode
-
Please, not the moons again!
+Please, not the moons again!
@@ -1018,7 +1074,7 @@ voting_clf.fit(X_train, y_train)
-
Bagging examples
+Bagging examples
@@ -1079,7 +1135,7 @@ plt.show()
-
Then random forests
+Then random forests
@@ -1101,7 +1157,7 @@ np.sum(y_pred =
-
Boosting and more
+Boosting and more
More material to come here.
diff --git a/doc/pub/DecisionTrees/ipynb/DecisionTrees.ipynb b/doc/pub/DecisionTrees/ipynb/DecisionTrees.ipynb
index 2f31deb81..a7101164c 100644
--- a/doc/pub/DecisionTrees/ipynb/DecisionTrees.ipynb
+++ b/doc/pub/DecisionTrees/ipynb/DecisionTrees.ipynb
@@ -47,6 +47,22 @@
"and **leaf nodes** which are then connected by **branches**.\n",
"\n",
"\n",
+ "## General Features\n",
+ "\n",
+ "Decision trees classify instances by sorting top down.\n",
+ "\n",
+ "* A leaf provides the classification of the instance.\n",
+ "\n",
+ "* A node specifies a test of some attribute of the instance.\n",
+ "\n",
+ "* A branch corresponds to a possible values an attribute.\n",
+ "\n",
+ "* 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.\n",
+ "\n",
+ "This processis then repeated for the subtree rooted at the new\n",
+ "node.\n",
+ "\n",
+ "\n",
"## How do we set it up?\n",
"\n",
"\n",
@@ -475,7 +491,38 @@
"source": [
"## Entropy and the ID3 algorithm\n",
"\n",
- "More text and code to come here.\n",
+ "ID3, learns decision trees by constructing\n",
+ "them topdown, beginning with the question **which attribute should be tested at the root of the tree**?\n",
+ "\n",
+ "1. Each instance attribute is evaluated using a statistical test to determine how well it alone classifies the training examples.\n",
+ "\n",
+ "2. The best attribute is selected and used as the test at the root node of the tree.\n",
+ "\n",
+ "3. A descendant of the root node is then created for each possible value of this attribute.\n",
+ "\n",
+ "4. Training examples are sorted to the appropriate descendant node.\n",
+ "\n",
+ "5. 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.\n",
+ "\n",
+ "6. This forms a greedy search for an acceptable decision tree, in which the algorithm never backtracks to reconsider earlier choices. \n",
+ "\n",
+ "The ID3 algorithm selects, which attribute to test at each node in the\n",
+ "tree.\n",
+ "\n",
+ "We would like to select the attribute that is most useful for classifying\n",
+ "examples.\n",
+ "\n",
+ "What is a good quantitative measure of the worth of an attribute?\n",
+ "\n",
+ "Information gain measures how well a given attribute separates the\n",
+ "training examples according to their target classification.\n",
+ "\n",
+ "The ID3 algorithm uses this information gain measure to select among the candidate\n",
+ "attributes at each step while growing the tree.\n",
+ "\n",
+ "## Implementing the ID3 Algorithm\n",
+ "\n",
+ "**more text to come here**, material presented during lecture Friday Oct 25.\n",
"\n",
"## Cancer Data again now with Decision Trees"
]
diff --git a/doc/pub/DecisionTrees/ipynb/ipynb-DecisionTrees-src.tar.gz b/doc/pub/DecisionTrees/ipynb/ipynb-DecisionTrees-src.tar.gz
index a59095945..d737091ae 100644
Binary files a/doc/pub/DecisionTrees/ipynb/ipynb-DecisionTrees-src.tar.gz and b/doc/pub/DecisionTrees/ipynb/ipynb-DecisionTrees-src.tar.gz differ
diff --git a/doc/pub/DecisionTrees/pdf/DecisionTrees-minted.pdf b/doc/pub/DecisionTrees/pdf/DecisionTrees-minted.pdf
index 7c7c78e72..489ec1ea1 100644
Binary files a/doc/pub/DecisionTrees/pdf/DecisionTrees-minted.pdf and b/doc/pub/DecisionTrees/pdf/DecisionTrees-minted.pdf differ
diff --git a/doc/src/DecisionTrees/DecisionTrees.do.txt b/doc/src/DecisionTrees/DecisionTrees.do.txt
index 44c4fda5e..a5dc32152 100644
--- a/doc/src/DecisionTrees/DecisionTrees.do.txt
+++ b/doc/src/DecisionTrees/DecisionTrees.do.txt
@@ -34,6 +34,21 @@ A decision tree mainly contains of a _root node_, _interior nodes_,
and _leaf nodes_ which are then connected by _branches_.
+!split
+===== General Features =====
+
+Decision trees classify instances by sorting top down.
+
+* A leaf provides the classification of the instance.
+* A node specifies a test of some attribute of the instance.
+* A branch corresponds to a possible values an attribute.
+* 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.
+
+
+This processis then repeated for the subtree rooted at the new
+node.
+
+
!split
===== How do we set it up? =====
@@ -364,7 +379,34 @@ s = -\sum_{k=1}^K p_{mk}\log{p_{mk}}.
!split
===== Entropy and the ID3 algorithm =====
-More text and code to come here.
+ID3, learns decision trees by constructing
+them topdown, beginning with the question _which attribute should be tested at the root of the tree_?
+
+o Each instance attribute is evaluated using a statistical test to determine how well it alone classifies the training examples.
+o The best attribute is selected and used as the test at the root node of the tree.
+o A descendant of the root node is then created for each possible value of this attribute.
+o Training examples are sorted to the appropriate descendant node.
+o 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.
+o This forms a greedy search for an acceptable decision tree, in which the algorithm never backtracks to reconsider earlier choices.
+
+The ID3 algorithm selects, which attribute to test at each node in the
+tree.
+
+We would like to select the attribute that is most useful for classifying
+examples.
+
+What is a good quantitative measure of the worth of an attribute?
+
+Information gain measures how well a given attribute separates the
+training examples according to their target classification.
+
+The ID3 algorithm uses this information gain measure to select among the candidate
+attributes at each step while growing the tree.
+
+!split
+===== Implementing the ID3 Algorithm =====
+
+_more text to come here_, material presented during lecture Friday Oct 25.
!split
===== Cancer Data again now with Decision Trees =====