From 2d7f1d1d5cf24eef2cc182cf8139a3bb82291753 Mon Sep 17 00:00:00 2001 From: mhjensen Date: Mon, 26 Oct 2020 07:48:04 +0100 Subject: [PATCH] update week44 --- doc/pub/week44/html/._week44-bs000.html | 168 +- doc/pub/week44/html/._week44-bs001.html | 208 ++- doc/pub/week44/html/._week44-bs002.html | 173 +-- doc/pub/week44/html/._week44-bs003.html | 210 +-- doc/pub/week44/html/._week44-bs004.html | 181 ++- doc/pub/week44/html/._week44-bs005.html | 266 ++-- doc/pub/week44/html/._week44-bs006.html | 192 ++- doc/pub/week44/html/._week44-bs007.html | 269 ++-- doc/pub/week44/html/._week44-bs008.html | 221 ++- doc/pub/week44/html/._week44-bs009.html | 190 +-- doc/pub/week44/html/._week44-bs010.html | 232 +-- doc/pub/week44/html/._week44-bs011.html | 200 ++- doc/pub/week44/html/._week44-bs012.html | 204 +-- doc/pub/week44/html/._week44-bs013.html | 203 +-- doc/pub/week44/html/._week44-bs014.html | 223 ++- doc/pub/week44/html/._week44-bs015.html | 219 ++- doc/pub/week44/html/._week44-bs016.html 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a/doc/pub/week44/html/._week44-bs000.html b/doc/pub/week44/html/._week44-bs000.html index 73fecd50a..c9eceb6e3 100644 --- a/doc/pub/week44/html/._week44-bs000.html +++ b/doc/pub/week44/html/._week44-bs000.html @@ -41,70 +41,72 @@ Automatically generated HTML file from DocOnce source @@ -142,46 +144,48 @@ MathJax.Hub.Config({
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    • @@ -216,7 +220,7 @@ MathJax.Hub.Config({
    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

    -

    Sep 16, 2020

    +

    Oct 26, 2020


    @@ -240,7 +244,7 @@ MathJax.Hub.Config({

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    Decision trees, overarching aims

    +

    Overview of week 44

    -

    -We start here with the most basic algorithm, the so-called decision -tree. With this basic algorithm we can in turn build more complex -networks, spanning from homogeneous and heterogenous forests (bagging, -random forests and more) to one of the most popular supervised -algorithms nowadays, the extreme gradient boosting, or just -XGBoost. But let us start with the simplest possible ingredient. +

    -

    -Decision trees are supervised learning algorithms used for both, -classification and regression tasks. - -

    -The main idea of decision trees -is to find those descriptive features which contain the most -information regarding the target feature and then split the dataset -along the values of these features such that the target feature values -for the resulting underlying datasets are as pure as possible. - -

    -The descriptive features which reproduce best the target/output features are normally said -to be the most informative ones. The process of finding the most -informative feature is done until we accomplish a stopping criteria -where we then finally end up in so called leaf nodes. - -

    -A decision tree is typically divided into a root node, the interior nodes, -and the final leaf nodes or just leaves. These entities are then connected by so-called branches. - -

    -The leaf nodes -contain the predictions we will make for new query instances presented -to our trained model. This is possible since the model has -learned the underlying structure of the training data and hence can, -given some assumptions, make predictions about the target feature value -(class) of unseen query instances. +Geron's chapter 6 covers decision trees while ensemble models, voting and bagging are discussed in chapter 7. See also lecture from STK-IN4300, lecture 7. Chapter 9.2 of Hastie et al contains also a good discussion.

    @@ -253,7 +227,7 @@ given some assumptions, make predictions about the target feature value

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    A typical Decision Tree with its pertinent Jargon, Classification Problem

    +

    Thursday

    -



    - -

    -This tree was produced using the Wisconsin cancer data (discussed here as well, see code examples below) using Scikit-Learn's decision tree classifier. Here we have used the so-called gini index (see below) to split the various branches. +Overview video, aims and motivations.

    @@ -223,7 +224,7 @@ This tree was produced using the Wisconsin cancer data (discussed here as well,

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    General Features

    +

    Decision trees, overarching aims

    -The overarching approach to decision trees is a top-down approach. +We start here with the most basic algorithm, the so-called decision +tree. With this basic algorithm we can in turn build more complex +networks, spanning from homogeneous and heterogenous forests (bagging, +random forests and more) to one of the most popular supervised +algorithms nowadays, the extreme gradient boosting, or just +XGBoost. But let us start with the simplest possible ingredient. -

    +

    +Decision trees are supervised learning algorithms used for both, +classification and regression tasks. -This process is then repeated for the subtree rooted at the new -node. +

    +The main idea of decision trees +is to find those descriptive features which contain the most +information regarding the target feature and then split the dataset +along the values of these features such that the target feature values +for the resulting underlying datasets are as pure as possible. + +

    +The descriptive features which reproduce best the target/output features are normally said +to be the most informative ones. The process of finding the most +informative feature is done until we accomplish a stopping criteria +where we then finally end up in so called leaf nodes. + +

    +A decision tree is typically divided into a root node, the interior nodes, +and the final leaf nodes or just leaves. These entities are then connected by so-called branches. + +

    +The leaf nodes +contain the predictions we will make for new query instances presented +to our trained model. This is possible since the model has +learned the underlying structure of the training data and hence can, +given some assumptions, make predictions about the target feature value +(class) of unseen query instances.

    @@ -231,7 +259,7 @@ node.

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    How do we set it up?

    +

    A typical Decision Tree with its pertinent Jargon, Classification Problem

    -In simplified terms, the process of training a decision tree and -predicting the target features of query instances is as follows: +



    -
      -
    1. Present a dataset containing of a number of training instances characterized by a number of descriptive features and a target feature
      2. -
    2. 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
      3. -
    3. Grow the tree until we accomplish a stopping criteria create leaf nodes which represent the predictions we want to make for new query instances
      4. -
    4. Show query instances to the tree and run down the tree until we arrive at leaf nodes
      5. -
    - -Then we are essentially done! +

    +This tree was produced using the Wisconsin cancer data (discussed here as well, see code examples below) using Scikit-Learn's decision tree classifier. Here we have used the so-called gini index (see below) to split the various branches.

    @@ -232,7 +229,7 @@ Then we are essentially done!

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    Decision trees and Regression

    +

    General Features

    +

    +The overarching approach to decision trees is a top-down approach. - -

    import numpy as np
-import matplotlib.pyplot as plt
-from sklearn.preprocessing import PolynomialFeatures
-from sklearn.linear_model import LinearRegression
+
      
+
    •  A leaf provides the classification of a given instance.
      • 
+
    •  A node specifies a test of some attribute of the instance.
      • 
+
    •  A branch corresponds to a possible values of 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.
      • 
+
    
 
-steps=250
+This process is then repeated for the subtree rooted at the new
+node.
 
-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()
-

    @@ -311,7 +237,7 @@ plt.show()

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    • @@ -197,30 +201,20 @@ MathJax.Hub.Config({ -

    Building a tree, regression

    +

    How do we set it up?

    -There are mainly two steps +In simplified terms, the process of training a decision tree and +predicting the target features of query instances is as follows:

      -
    1. We split the predictor space (the set of possible values \( x_1,x_2,\dots, x_p \)) into \( J \) distinct and non-non-overlapping regions, \( R_1,R_2,\dots,R_J \).
      2. -
    2. For every observation that falls into the region \( R_j \) , we make the same prediction, which is simply the mean of the response values for the training observations in \( R_j \).
      3. +
    3. Present a dataset containing of a number of training instances characterized by a number of descriptive features and a target feature
      4. +
    4. 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
      5. +
    5. Grow the tree until we accomplish a stopping criteria create leaf nodes which represent the predictions we want to make for new query instances
      6. +
    6. Show query instances to the tree and run down the tree until we arrive at leaf nodes
    -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 - -$$ -\sum_{j=1}^J\sum_{i\in R_j}(y_i-\overline{y}_{R_j})^2, -$$ - -

    -where \( \overline{y}_{R_j} \) is the mean response for the training observations -within box \( j \). +Then we are essentially done!

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    A top-down approach, recursive binary splitting

    - +

    Decision trees and Regression

    -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 -

    -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. + +

    import numpy as np
+import matplotlib.pyplot as plt
+from sklearn.preprocessing import PolynomialFeatures
+from sklearn.linear_model import LinearRegression
 
+steps=250
+
+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()
+

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    Making a tree

    +

    Building a tree, regression

    -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 \( R_2 \) -$$ -\left\{X\vert x_j < s\right\}, -$$ +There are mainly two steps -and -$$ -\left\{X\vert x_j \geq s\right\}, -$$ +

      +
    1. We split the predictor space (the set of possible values \( x_1,x_2,\dots, x_p \)) into \( J \) distinct and non-non-overlapping regions, \( R_1,R_2,\dots,R_J \).
      2. +
    2. For every observation that falls into the region \( R_j \) , we make the same prediction, which is simply the mean of the response values for the training observations in \( R_j \).
      3. +
    + +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 -so that we obtain the lowest MSE, that is $$ -\sum_{i:x_i\in R_j}(y_i-\overline{y}_{R_1})^2+\sum_{i:x_i\in R_2}(y_i-\overline{y}_{R_2})^2, +\sum_{j=1}^J\sum_{i\in R_j}(y_i-\overline{y}_{R_j})^2, $$

    -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. - -

    -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. +where \( \overline{y}_{R_j} \) is the mean response for the training observations +within box \( j \).

    @@ -269,7 +250,7 @@ region contains more than five observations.

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    - + -

    Pruning the tree

    +

    A top-down approach, recursive binary splitting

    -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. +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

    -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 \( \alpha \). +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.

    @@ -238,7 +242,7 @@ parameter \( \alpha \).

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    Cost complexity pruning

    -For each value of \( \alpha \) there corresponds a subtree \( T \in T_0 \) such that -$$ -\sum_{m=1}^{\overline{T}}\sum_{i:x_i\in R_m}(y_i-\overline{y}_{R_m})^2+\alpha\overline{T}, -$$ - -is as small as possible. Here \( \overline{T} \) is -the number of terminal nodes of the tree \( T \) , \( R_m \) is the -rectangle (i.e. the subset of predictor space) corresponding to the \( m \)-th terminal node. +

    Making a tree

    -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. +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 +$$ +\sum_{i:x_i\in R_j}(y_i-\overline{y}_{R_1})^2+\sum_{i:x_i\in R_2}(y_i-\overline{y}_{R_2})^2, +$$

    -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 \( \alpha \). +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. + +

    +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.

    @@ -251,7 +275,7 @@ subtree corresponding to \( \alpha \).

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    - + -

    Schematic Regression Procedure

    +

    Pruning the tree

    -

    -
    -

    - -

      -
    1. Use recursive binary splitting to grow a large tree on the training data, stopping only when each terminal node has fewer than some minimum number of observations.
      2. -
    2. Apply cost complexity pruning to the large tree in order to obtain a sequence of best subtrees, as a function of \( \alpha \).
      3. -
    3. Use for example \( K \)-fold cross-validation to choose \( \alpha \). Divide the training observations into \( K \) folds. For each \( k=1,2,\dots,K \) we:
      4. - -
        -
      • repeat steps 1 and 2 on all but the \( k \)-th fold of the training data.
        • -
      • Then we valuate the mean squared prediction error on the data in the left-out \( k \)-th fold, as a function of \( \alpha \).
        • -
      • Finally we average the results for each value of \( \alpha \), and pick \( \alpha \) to minimize the average error.
        • -
      - -
    4. Return the subtree from Step 2 that corresponds to the chosen value of \( \alpha \).
      5. -
    -
    -
    +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. +

    +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 \( \alpha \).

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    A Classification Tree

    +

    Cost complexity pruning

    +For each value of \( \alpha \) there corresponds a subtree \( T \in T_0 \) such that +$$ +\sum_{m=1}^{\overline{T}}\sum_{i:x_i\in R_m}(y_i-\overline{y}_{R_m})^2+\alpha\overline{T}, +$$ + +is as small as possible. Here \( \overline{T} \) is +the number of terminal nodes of the tree \( T \) , \( R_m \) is the +rectangle (i.e. the subset of predictor space) corresponding to the \( m \)-th terminal node.

    -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. +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. + +

    +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 \( \alpha \).

    @@ -239,7 +255,7 @@ fall into that region.

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    Growing a classification tree

    +

    Schematic Regression Procedure

    -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. +

    +
    +

    + +

      +
    1. Use recursive binary splitting to grow a large tree on the training data, stopping only when each terminal node has fewer than some minimum number of observations.
      2. +
    2. Apply cost complexity pruning to the large tree in order to obtain a sequence of best subtrees, as a function of \( \alpha \).
      3. +
    3. Use for example \( K \)-fold cross-validation to choose \( \alpha \). Divide the training observations into \( K \) folds. For each \( k=1,2,\dots,K \) we:
      4. + +
        +
      • repeat steps 1 and 2 on all but the \( k \)-th fold of the training data.
        • +
      • Then we valuate the mean squared prediction error on the data in the left-out \( k \)-th fold, as a function of \( \alpha \).
        • +
      • Finally we average the results for each value of \( \alpha \), and pick \( \alpha \) to minimize the average error.
        • +
      + +
    4. Return the subtree from Step 2 that corresponds to the chosen value of \( \alpha \).
      5. +
    +
    +
    -

    -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.

    @@ -244,7 +251,7 @@ than is the classification error rate.

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    Classification tree, how to split nodes

    +

    A Classification Tree

    -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 - -

    - -$$ -p_{mk} = \frac{1}{N_m}\sum_{x_i\in R_m}I(y_i\ne k) = 1-p_{mk}. -$$ - - - - -$$ -g = \sum_{k=1}^K p_{mk}(1-p_{mk}). -$$ - - - - -$$ -s = -\sum_{k=1}^K p_{mk}\log{p_{mk}}. -$$ +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.

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    Visualizing the Tree, Classification

    +

    Growing a classification tree

    +

    +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. - -

    import os
-from sklearn.datasets import load_breast_cancer
-from sklearn.tree import DecisionTreeClassifier
-from sklearn.model_selection import train_test_split
-from sklearn.metrics import confusion_matrix
-from sklearn.tree import export_graphviz
+
    
+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.
 
-from IPython.display import Image 
-from pydot import graph_from_dot_data
-import pandas as pd
-import numpy as np
-
-
-cancer = load_breast_cancer()
-X = pd.DataFrame(cancer.data, columns=cancer.feature_names)
-print(X)
-y = pd.Categorical.from_codes(cancer.target, cancer.target_names)
-y = pd.get_dummies(y)
-print(y)
-X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=1)
-tree_clf = DecisionTreeClassifier(max_depth=5)
-tree_clf.fit(X_train, y_train)
-
-export_graphviz(
-    tree_clf,
-    out_file="DataFiles/cancer.dot",
-    feature_names=cancer.feature_names,
-    class_names=cancer.target_names,
-    rounded=True,
-    filled=True
-)
-cmd = 'dot -Tpng DataFiles/cancer.dot -o DataFiles/cancer.png'
-os.system(cmd)
-

    @@ -261,7 +248,7 @@ os.system(cmd)

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    • @@ -197,35 +201,53 @@ MathJax.Hub.Config({ -

    Visualizing the Tree, The Moons

    +

    Classification tree, how to split nodes

    +

    +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. - -

    # Common imports
-import numpy as np
-from sklearn.model_selection import  train_test_split 
-from sklearn.tree import DecisionTreeClassifier
-from sklearn.datasets import make_moons
-from sklearn.tree import export_graphviz
-from pydot import graph_from_dot_data
-import pandas as pd
-import os
+
    
+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
 
-np.random.seed(42)
-X, y = make_moons(n_samples=100, noise=0.25, random_state=53)
-X_train, X_test, y_train, y_test = train_test_split(X,y,random_state=0)
-tree_clf = DecisionTreeClassifier(max_depth=5)
-tree_clf.fit(X_train, y_train)
+$$
+p_{mk} = \frac{1}{N_m}\sum_{x_i\in R_m}I(y_i=k).
+$$
+
+
    
+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
+
+
      
+
    •  Misclassification error
      •  
+
    
+
+$$
+p_{mk} = \frac{1}{N_m}\sum_{x_i\in R_m}I(y_i\ne k) = 1-p_{mk}.
+$$
+
+
+
      
+
    •  Gini index \( g \)
      • 
+
    
+
+$$
+g = \sum_{k=1}^K p_{mk}(1-p_{mk}).
+$$
+
+
+
      
+
    •  Information entropy or just entropy \( s \)
      • 
+
    
+
+$$
+s = -\sum_{k=1}^K p_{mk}\log{p_{mk}}.
+$$
 
-export_graphviz(
-    tree_clf,
-    out_file="DataFiles/moons.dot",
-    rounded=True,
-    filled=True
-)
-cmd = 'dot -Tpng DataFiles/moons.dot -o DataFiles/moons.png'
-os.system(cmd)
-

    @@ -252,7 +274,7 @@ os.system(cmd)

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    • @@ -197,21 +201,44 @@ MathJax.Hub.Config({ -

    Algorithms for Setting up Decision Trees

    - +

    Visualizing the Tree, Classification

    -Two algorithms stand out in the set up of decision trees: -

      -
    1. The CART (Classification And Regression Tree) algorithm for both classification and regression
      2. -
    2. The ID3 algorithm based on the computation of the information gain for classification
      3. -
    + +
    import os
+from sklearn.datasets import load_breast_cancer
+from sklearn.tree import DecisionTreeClassifier
+from sklearn.model_selection import train_test_split
+from sklearn.metrics import confusion_matrix
+from sklearn.tree import export_graphviz
 
-We discuss both algorithms with applications here. The popular library
-Scikit-Learn uses the CART algorithm. For classification problems
-you can use either the gini index or the entropy to split a tree
-in two branches.
+from IPython.display import Image 
+from pydot import graph_from_dot_data
+import pandas as pd
+import numpy as np
 
+
+cancer = load_breast_cancer()
+X = pd.DataFrame(cancer.data, columns=cancer.feature_names)
+print(X)
+y = pd.Categorical.from_codes(cancer.target, cancer.target_names)
+y = pd.get_dummies(y)
+print(y)
+X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=1)
+tree_clf = DecisionTreeClassifier(max_depth=5)
+tree_clf.fit(X_train, y_train)
+
+export_graphviz(
+    tree_clf,
+    out_file="DataFiles/cancer.dot",
+    feature_names=cancer.feature_names,
+    class_names=cancer.target_names,
+    rounded=True,
+    filled=True
+)
+cmd = 'dot -Tpng DataFiles/cancer.dot -o DataFiles/cancer.png'
+os.system(cmd)
+

    @@ -238,7 +265,7 @@ in two branches.

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    • @@ -197,30 +201,35 @@ MathJax.Hub.Config({ -

    The CART algorithm for Classification

    - +

    Visualizing the Tree, The Moons

    -For classification, the CART algorithm splits the data set in two subsets using a single feature \( k \) and a threshold \( t_k \). -This could be for example a threshold set by a number below a certain circumference of a malign tumor. -

    -How do we find these two quantities? -We search for the pair \( (k,t_k) \) that produces the purest subset using for example the gini factor \( G \). -The cost function it tries to minimize is then -$$ -C(k,t_k) = \frac{m_{\mathrm{left}}}{m}G_{\mathrm{left}}+ \frac{m_{\mathrm{right}}}{m}G_{\mathrm{right}}, -$$ + +

    # Common imports
+import numpy as np
+from sklearn.model_selection import  train_test_split 
+from sklearn.tree import DecisionTreeClassifier
+from sklearn.datasets import make_moons
+from sklearn.tree import export_graphviz
+from pydot import graph_from_dot_data
+import pandas as pd
+import os
 
-where \( G_{\mathrm{left/right}} \) measures the impurity of the left/right subset  and \( m_{\mathrm{left/right}} \)
- is the number of instances in the left/right subset
-
-
    
-Once it has successfully split the training set in two, it splits the subsets using the same logic, then the subsubsets
-and so on, recursively. It stops recursing once it reaches the maximum depth (defined by the
-\( max\_depth \) hyperparameter), or if it cannot find a split that will reduce impurity. A few other
-hyperparameters control additional stopping conditions such as the \( min\_samples\_split \),
-\( min\_samples\_leaf \), \( min\_weight\_fraction\_leaf \), and \( max\_leaf\_nodes \).
+np.random.seed(42)
+X, y = make_moons(n_samples=100, noise=0.25, random_state=53)
+X_train, X_test, y_train, y_test = train_test_split(X,y,random_state=0)
+tree_clf = DecisionTreeClassifier(max_depth=5)
+tree_clf.fit(X_train, y_train)
 
+export_graphviz(
+    tree_clf,
+    out_file="DataFiles/moons.dot",
+    rounded=True,
+    filled=True
+)
+cmd = 'dot -Tpng DataFiles/moons.dot -o DataFiles/moons.png'
+os.system(cmd)
+

    @@ -247,7 +256,7 @@ hyperparameters control additional stopping conditions such as the \( min\_sampl

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    • @@ -197,30 +201,20 @@ MathJax.Hub.Config({ -

    The CART algorithm for Regression

    +

    Algorithms for Setting up Decision Trees

    -The CART algorithm for regression works is similar to the one for classification except that instead of trying to split the -training set in a way that minimizes say the gini or entropy impurity, it now tries to split the training set in a way that minimizes our well-known mean-squared error (MSE). The cost function is now -$$ -C(k,t_k) = \frac{m_{\mathrm{left}}}{m}\mathrm{MSE}_{\mathrm{left}}+ \frac{m_{\mathrm{right}}}{m}\mathrm{MSE}_{\mathrm{right}}. -$$ +Two algorithms stand out in the set up of decision trees: -Here the MSE for a specific node is defined as -$$ -\mathrm{MSE}_{\mathrm{node}}=\frac{1}{m_\mathrm{node}}\sum_{i\in \mathrm{node}}(\overline{y}_{\mathrm{node}}-y_i)^2, -$$ +

      +
    1. The CART (Classification And Regression Tree) algorithm for both classification and regression
      2. +
    2. The ID3 algorithm based on the computation of the information gain for classification
      3. +
    -with -$$ -\overline{y}_{\mathrm{node}}=\frac{1}{m_\mathrm{node}}\sum_{i\in \mathrm{node}}y_i, -$$ - -the mean value of all observations in a specific node. - -

    -Without any regularization, the regression task for decision trees, -just like for classification tasks, is prone to overfitting. +We discuss both algorithms with applications here. The popular library +Scikit-Learn uses the CART algorithm. For classification problems +you can use either the gini index or the entropy to split a tree +in two branches.

    @@ -248,7 +242,7 @@ just like for classification tasks, is prone to overfitting.

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    • @@ -197,48 +201,30 @@ MathJax.Hub.Config({ -

    Computing the Gini index

    +

    The CART algorithm for Classification

    -The example we will look at is a classical one in many Machine -Learning applications. Based on various meteorological features, we -have several so-called attributes which decide whether we at the end -will do some outdoor activity like skiing, going for a bike ride etc -etc. The table here contains the feautures outlook, temperature, -humidity and wind. The target or output is whether we ride -(True=1) or whether we do something else that day (False=0). The -attributes for each feature are then sunny, overcast and rain for the -outlook, hot, cold and mild for temperature, high and normal for -humidity and weak and strong for wind. +For classification, the CART algorithm splits the data set in two subsets using a single feature \( k \) and a threshold \( t_k \). +This could be for example a threshold set by a number below a certain circumference of a malign tumor.

    -The table here summarizes the various attributes and +How do we find these two quantities? +We search for the pair \( (k,t_k) \) that produces the purest subset using for example the gini factor \( G \). +The cost function it tries to minimize is then +$$ +C(k,t_k) = \frac{m_{\mathrm{left}}}{m}G_{\mathrm{left}}+ \frac{m_{\mathrm{right}}}{m}G_{\mathrm{right}}, +$$ + +where \( G_{\mathrm{left/right}} \) measures the impurity of the left/right subset and \( m_{\mathrm{left/right}} \) + is the number of instances in the left/right subset + +

    +Once it has successfully split the training set in two, it splits the subsets using the same logic, then the subsubsets +and so on, recursively. It stops recursing once it reaches the maximum depth (defined by the +\( max\_depth \) hyperparameter), or if it cannot find a split that will reduce impurity. A few other +hyperparameters control additional stopping conditions such as the \( min\_samples\_split \), +\( min\_samples\_leaf \), \( min\_weight\_fraction\_leaf \), and \( max\_leaf\_nodes \). -

    -
    - - - - - - - - - - - - - - - - - - - - -
    Day Outlook Temperature Humidity Wind Ride
    1 Sunny Hot High Weak 0
    2 Sunny Hot High Strong 1
    3 Overcast Hot High Weak 1
    4 Rain Mild High Weak 1
    5 Rain Cool Normal Weak 1
    6 Rain Cool Normal Strong 0
    7 Overcast Cool Normal Strong 1
    8 Sunny Mild High Weak 0
    9 Sunny Cool Normal Weak 1
    10 Rain Mild Normal Weak 1
    11 Sunny Mild Normal Strong 1
    12 Overcast Mild High Strong 1
    13 Overcast Hot Normal Weak 1
    14 Rain Mild High Strong 0
    -
    -

    @@ -265,7 +251,7 @@ The table here summarizes the various attributes and

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    • @@ -197,79 +201,31 @@ MathJax.Hub.Config({ -

    Simple Python Code to read in Data and perform Classification

    +

    The CART algorithm for Regression

    +The CART algorithm for regression works is similar to the one for classification except that instead of trying to split the +training set in a way that minimizes say the gini or entropy impurity, it now tries to split the training set in a way that minimizes our well-known mean-squared error (MSE). The cost function is now +$$ +C(k,t_k) = \frac{m_{\mathrm{left}}}{m}\mathrm{MSE}_{\mathrm{left}}+ \frac{m_{\mathrm{right}}}{m}\mathrm{MSE}_{\mathrm{right}}. +$$ - -

    # Common imports
-import numpy as np
-import pandas as pd
-import matplotlib.pyplot as plt
-from sklearn.tree import DecisionTreeClassifier
-from sklearn.model_selection import train_test_split
-from sklearn.tree import export_graphviz
-from sklearn.preprocessing import StandardScaler, OneHotEncoder
-from sklearn.compose import ColumnTransformer
-from IPython.display import Image 
-from pydot import graph_from_dot_data
-import os
+Here the MSE for a specific node is defined as
+$$
+\mathrm{MSE}_{\mathrm{node}}=\frac{1}{m_\mathrm{node}}\sum_{i\in \mathrm{node}}(\overline{y}_{\mathrm{node}}-y_i)^2,
+$$
 
-# Where to save the figures and data files
-PROJECT_ROOT_DIR = "Results"
-FIGURE_ID = "Results/FigureFiles"
-DATA_ID = "DataFiles/"
+with
+$$
+\overline{y}_{\mathrm{node}}=\frac{1}{m_\mathrm{node}}\sum_{i\in \mathrm{node}}y_i,
+$$
 
-if not os.path.exists(PROJECT_ROOT_DIR):
-    os.mkdir(PROJECT_ROOT_DIR)
+the mean value of all observations in a specific node.
 
-if not os.path.exists(FIGURE_ID):
-    os.makedirs(FIGURE_ID)
+
    
+Without any regularization, the regression task for decision trees, 
+just like for classification tasks, is  prone to overfitting.
 
-if not os.path.exists(DATA_ID):
-    os.makedirs(DATA_ID)
-
-def image_path(fig_id):
-    return os.path.join(FIGURE_ID, fig_id)
-
-def data_path(dat_id):
-    return os.path.join(DATA_ID, dat_id)
-
-def save_fig(fig_id):
-    plt.savefig(image_path(fig_id) + ".png", format='png')
-
-infile = open(data_path("rideclass.csv"),'r')
-
-# Read the experimental data with Pandas
-from IPython.display import display
-ridedata = pd.read_csv(infile,names = ('Outlook','Temperature','Humidity','Wind','Ride'))
-ridedata = pd.DataFrame(ridedata)
-
-# Features and targets
-X = ridedata.loc[:, ridedata.columns != 'Ride'].values
-y = ridedata.loc[:, ridedata.columns == 'Ride'].values
-
-# Create the encoder.
-encoder = OneHotEncoder(handle_unknown="ignore")
-# Assume for simplicity all features are categorical.
-encoder.fit(X)    
-# Apply the encoder.
-X = encoder.transform(X)
-print(X)
-# Then do a Classification tree
-tree_clf = DecisionTreeClassifier(max_depth=2)
-tree_clf.fit(X, y)
-print("Train set accuracy with Decision Tree: {:.2f}".format(tree_clf.score(X,y)))
-#transfer to a decision tree graph
-export_graphviz(
-    tree_clf,
-    out_file="DataFiles/ride.dot",
-    rounded=True,
-    filled=True
-)
-cmd = 'dot -Tpng DataFiles/cancer.dot -o DataFiles/cancer.png'
-os.system(cmd)
-

    @@ -296,7 +252,7 @@ os.system(cmd)

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    Computing the Gini Factor

    +

    Computing the Gini index

    -The above functions (gini, entropy and misclassification error) are -important components of the so-called CART algorithm. We will discuss -this algorithm below after we have discussed the information gain -algorithm ID3. +The example we will look at is a classical one in many Machine +Learning applications. Based on various meteorological features, we +have several so-called attributes which decide whether we at the end +will do some outdoor activity like skiing, going for a bike ride etc +etc. The table here contains the feautures outlook, temperature, +humidity and wind. The target or output is whether we ride +(True=1) or whether we do something else that day (False=0). The +attributes for each feature are then sunny, overcast and rain for the +outlook, hot, cold and mild for temperature, high and normal for +humidity and weak and strong for wind.

    -In the example here we have converted all our attributes into numerical values \( 0,1,2 \) etc. +The table here summarizes the various attributes and -

    - - -

    # Split a dataset based on an attribute and an attribute value
-def test_split(index, value, dataset):
-	left, right = list(), list()
-	for row in dataset:
-		if row[index] < value:
-			left.append(row)
-		else:
-			right.append(row)
-	return left, right
- 
-# Calculate the Gini index for a split dataset
-def gini_index(groups, classes):
-	# count all samples at split point
-	n_instances = float(sum([len(group) for group in groups]))
-	# sum weighted Gini index for each group
-	gini = 0.0
-	for group in groups:
-		size = float(len(group))
-		# avoid divide by zero
-		if size == 0:
-			continue
-		score = 0.0
-		# score the group based on the score for each class
-		for class_val in classes:
-			p = [row[-1] for row in group].count(class_val) / size
-			score += p * p
-		# weight the group score by its relative size
-		gini += (1.0 - score) * (size / n_instances)
-	return gini
-
-# Select the best split point for a dataset
-def get_split(dataset):
-	class_values = list(set(row[-1] for row in dataset))
-	b_index, b_value, b_score, b_groups = 999, 999, 999, None
-	for index in range(len(dataset[0])-1):
-		for row in dataset:
-			groups = test_split(index, row[index], dataset)
-			gini = gini_index(groups, class_values)
-			print('X%d < %.3f Gini=%.3f' % ((index+1), row[index], gini))
-			if gini < b_score:
-				b_index, b_value, b_score, b_groups = index, row[index], gini, groups
-	return {'index':b_index, 'value':b_value, 'groups':b_groups}
- 
-dataset = [[0,0,0,0,0],
-            [0,0,0,1,1],
-            [1,0,0,0,1],
-            [2,1,0,0,1],
-            [2,2,1,0,1],
-            [2,2,1,1,0],
-            [1,2,1,1,1],
-            [0,1,0,0,0],
-            [0,2,1,0,1],
-            [2,1,1,0,1],
-            [0,1,1,1,1],
-            [1,1,0,1,1],
-            [1,0,1,0,1],
-            [2,1,0,1,0]]
-
-split = get_split(dataset)
-print('Split: [X%d < %.3f]' % ((split['index']+1), split['value']))
-
    +
    +
    + + + + + + + + + + + + + + + + + + + + +
    Day Outlook Temperature Humidity Wind Ride
    1 Sunny Hot High Weak 0
    2 Sunny Hot High Strong 1
    3 Overcast Hot High Weak 1
    4 Rain Mild High Weak 1
    5 Rain Cool Normal Weak 1
    6 Rain Cool Normal Strong 0
    7 Overcast Cool Normal Strong 1
    8 Sunny Mild High Weak 0
    9 Sunny Cool Normal Weak 1
    10 Rain Mild Normal Weak 1
    11 Sunny Mild Normal Strong 1
    12 Overcast Mild High Strong 1
    13 Overcast Hot Normal Weak 1
    14 Rain Mild High Strong 0
    +
    +

    @@ -298,7 +269,7 @@ split = get_split(dataset)

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    Entropy and the ID3 algorithm

    +

    Simple Python Code to read in Data and perform Classification

    -ID3, learns decision trees by constructing -them topdown, beginning with the question which attribute should be tested at the root of the tree? -

      -
    1. Each instance attribute is evaluated using a statistical test to determine how well it alone classifies the training examples.
      2. -
    2. The best attribute is selected and used as the test at the root node of the tree.
      3. -
    3. A descendant of the root node is then created for each possible value of this attribute.
      4. -
    4. Training examples are sorted to the appropriate descendant node.
      5. -
    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.
      6. -
    6. This forms a greedy search for an acceptable decision tree, in which the algorithm never backtracks to reconsider earlier choices.
      7. -
    + +
    # Common imports
+import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+from sklearn.tree import DecisionTreeClassifier
+from sklearn.model_selection import train_test_split
+from sklearn.tree import export_graphviz
+from sklearn.preprocessing import StandardScaler, OneHotEncoder
+from sklearn.compose import ColumnTransformer
+from IPython.display import Image 
+from pydot import graph_from_dot_data
+import os
 
-The ID3 algorithm selects, which attribute to test at each node in the
-tree.
+# Where to save the figures and data files
+PROJECT_ROOT_DIR = "Results"
+FIGURE_ID = "Results/FigureFiles"
+DATA_ID = "DataFiles/"
 
-
    
-We would like to select the attribute that is most useful for classifying
-examples.
+if not os.path.exists(PROJECT_ROOT_DIR):
+    os.mkdir(PROJECT_ROOT_DIR)
 
-
    
-What is a good quantitative measure of the worth of an attribute?
+if not os.path.exists(FIGURE_ID):
+    os.makedirs(FIGURE_ID)
 
-
    
-Information gain measures how well a given attribute separates the
-training examples according to their target classification.
+if not os.path.exists(DATA_ID):
+    os.makedirs(DATA_ID)
 
-
    
-The ID3 algorithm uses this information gain measure to select among the candidate
-attributes at each step while growing the tree.
+def image_path(fig_id):
+    return os.path.join(FIGURE_ID, fig_id)
 
+def data_path(dat_id):
+    return os.path.join(DATA_ID, dat_id)
+
+def save_fig(fig_id):
+    plt.savefig(image_path(fig_id) + ".png", format='png')
+
+infile = open(data_path("rideclass.csv"),'r')
+
+# Read the experimental data with Pandas
+from IPython.display import display
+ridedata = pd.read_csv(infile,names = ('Outlook','Temperature','Humidity','Wind','Ride'))
+ridedata = pd.DataFrame(ridedata)
+
+# Features and targets
+X = ridedata.loc[:, ridedata.columns != 'Ride'].values
+y = ridedata.loc[:, ridedata.columns == 'Ride'].values
+
+# Create the encoder.
+encoder = OneHotEncoder(handle_unknown="ignore")
+# Assume for simplicity all features are categorical.
+encoder.fit(X)    
+# Apply the encoder.
+X = encoder.transform(X)
+print(X)
+# Then do a Classification tree
+tree_clf = DecisionTreeClassifier(max_depth=2)
+tree_clf.fit(X, y)
+print("Train set accuracy with Decision Tree: {:.2f}".format(tree_clf.score(X,y)))
+#transfer to a decision tree graph
+export_graphviz(
+    tree_clf,
+    out_file="DataFiles/ride.dot",
+    rounded=True,
+    filled=True
+)
+cmd = 'dot -Tpng DataFiles/cancer.dot -o DataFiles/cancer.png'
+os.system(cmd)
+

    @@ -256,7 +300,7 @@ attributes at each step while growing the tree.

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    Implementing the ID3 Algorithm

    +

    Computing the Gini Factor

    + +

    +The above functions (gini, entropy and misclassification error) are +important components of the so-called CART algorithm. We will discuss +this algorithm below after we have discussed the information gain +algorithm ID3. + +

    +In the example here we have converted all our attributes into numerical values \( 0,1,2 \) etc.

    -

    import re
-import math
-from collections import deque
+
    # Split a dataset based on an attribute and an attribute value
+def test_split(index, value, dataset):
+	left, right = list(), list()
+	for row in dataset:
+		if row[index] < value:
+			left.append(row)
+		else:
+			right.append(row)
+	return left, right
+ 
+# Calculate the Gini index for a split dataset
+def gini_index(groups, classes):
+	# count all samples at split point
+	n_instances = float(sum([len(group) for group in groups]))
+	# sum weighted Gini index for each group
+	gini = 0.0
+	for group in groups:
+		size = float(len(group))
+		# avoid divide by zero
+		if size == 0:
+			continue
+		score = 0.0
+		# score the group based on the score for each class
+		for class_val in classes:
+			p = [row[-1] for row in group].count(class_val) / size
+			score += p * p
+		# weight the group score by its relative size
+		gini += (1.0 - score) * (size / n_instances)
+	return gini
 
-# x is examples in training set
-# y is set of targets
-# label is target attributes
-# Node is a class which has properties values, childs, and next
-# root is top node in the decision tree
+# Select the best split point for a dataset
+def get_split(dataset):
+	class_values = list(set(row[-1] for row in dataset))
+	b_index, b_value, b_score, b_groups = 999, 999, 999, None
+	for index in range(len(dataset[0])-1):
+		for row in dataset:
+			groups = test_split(index, row[index], dataset)
+			gini = gini_index(groups, class_values)
+			print('X%d < %.3f Gini=%.3f' % ((index+1), row[index], gini))
+			if gini < b_score:
+				b_index, b_value, b_score, b_groups = index, row[index], gini, groups
+	return {'index':b_index, 'value':b_value, 'groups':b_groups}
+ 
+dataset = [[0,0,0,0,0],
+            [0,0,0,1,1],
+            [1,0,0,0,1],
+            [2,1,0,0,1],
+            [2,2,1,0,1],
+            [2,2,1,1,0],
+            [1,2,1,1,1],
+            [0,1,0,0,0],
+            [0,2,1,0,1],
+            [2,1,1,0,1],
+            [0,1,1,1,1],
+            [1,1,0,1,1],
+            [1,0,1,0,1],
+            [2,1,0,1,0]]
 
-class Node(object):
-	def __init__(self):
-		self.value = None
-		self.next = None
-		self.childs = None
-
-# Simple class of Decision Tree
-# Aimed for who want to learn Decision Tree, so it is not optimized
-class DecisionTree(object):
-	def __init__(self, sample, attributes, labels):
-		self.sample = sample
-		self.attributes = attributes
-		self.labels = labels
-		self.labelCodes = None
-		self.labelCodesCount = None
-		self.initLabelCodes()
-		# print(self.labelCodes)
-		self.root = None
-		self.entropy = self.getEntropy([x for x in range(len(self.labels))])
-
-	def initLabelCodes(self):
-		self.labelCodes = []
-		self.labelCodesCount = []
-		for l in self.labels:
-			if l not in self.labelCodes:
-				self.labelCodes.append(l)
-				self.labelCodesCount.append(0)
-			self.labelCodesCount[self.labelCodes.index(l)] += 1
-
-	def getLabelCodeId(self, sampleId):
-		return self.labelCodes.index(self.labels[sampleId])
-
-	def getAttributeValues(self, sampleIds, attributeId):
-		vals = []
-		for sid in sampleIds:
-			val = self.sample[sid][attributeId]
-			if val not in vals:
-				vals.append(val)
-		# print(vals)
-		return vals
-
-	def getEntropy(self, sampleIds):
-		entropy = 0
-		labelCount = [0] * len(self.labelCodes)
-		for sid in sampleIds:
-			labelCount[self.getLabelCodeId(sid)] += 1
-		# print("-ge", labelCount)
-		for lv in labelCount:
-			# print(lv)
-			if lv != 0:
-				entropy += -lv/len(sampleIds) * math.log(lv/len(sampleIds), 2)
-			else:
-				entropy += 0
-		return entropy
-
-	def getDominantLabel(self, sampleIds):
-		labelCodesCount = [0] * len(self.labelCodes)
-		for sid in sampleIds:
-			labelCodesCount[self.labelCodes.index(self.labels[sid])] += 1
-		return self.labelCodes[labelCodesCount.index(max(labelCodesCount))]
-
-	def getInformationGain(self, sampleIds, attributeId):
-		gain = self.getEntropy(sampleIds)
-		attributeVals = []
-		attributeValsCount = []
-		attributeValsIds = []
-		for sid in sampleIds:
-			val = self.sample[sid][attributeId]
-			if val not in attributeVals:
-				attributeVals.append(val)
-				attributeValsCount.append(0)
-				attributeValsIds.append([])
-			vid = attributeVals.index(val)
-			attributeValsCount[vid] += 1
-			attributeValsIds[vid].append(sid)
-		# print("-gig", self.attributes[attributeId])
-		for vc, vids in zip(attributeValsCount, attributeValsIds):
-			# print("-gig", vids)
-			gain -= vc/len(sampleIds) * self.getEntropy(vids)
-		return gain
-
-	def getAttributeMaxInformationGain(self, sampleIds, attributeIds):
-		attributesEntropy = [0] * len(attributeIds)
-		for i, attId in zip(range(len(attributeIds)), attributeIds):
-			attributesEntropy[i] = self.getInformationGain(sampleIds, attId)
-		maxId = attributeIds[attributesEntropy.index(max(attributesEntropy))]
-		return self.attributes[maxId], maxId
-
-	def isSingleLabeled(self, sampleIds):
-		label = self.labels[sampleIds[0]]
-		for sid in sampleIds:
-			if self.labels[sid] != label:
-				return False
-		return True
-
-	def getLabel(self, sampleId):
-		return self.labels[sampleId]
-
-	def id3(self):
-		sampleIds = [x for x in range(len(self.sample))]
-		attributeIds = [x for x in range(len(self.attributes))]
-		self.root = self.id3Recv(sampleIds, attributeIds, self.root)
-
-	def id3Recv(self, sampleIds, attributeIds, root):
-		root = Node() # Initialize current root
-		if self.isSingleLabeled(sampleIds):
-			root.value = self.labels[sampleIds[0]]
-			return root
-		# print(attributeIds)
-		if len(attributeIds) == 0:
-			root.value = self.getDominantLabel(sampleIds)
-			return root
-		bestAttrName, bestAttrId = self.getAttributeMaxInformationGain(
-			sampleIds, attributeIds)
-		# print(bestAttrName)
-		root.value = bestAttrName
-		root.childs = []  # Create list of children
-		for value in self.getAttributeValues(sampleIds, bestAttrId):
-			# print(value)
-			child = Node()
-			child.value = value
-			root.childs.append(child)  # Append new child node to current
-									   # root
-			childSampleIds = []
-			for sid in sampleIds:
-				if self.sample[sid][bestAttrId] == value:
-					childSampleIds.append(sid)
-			if len(childSampleIds) == 0:
-				child.next = self.getDominantLabel(sampleIds)
-			else:
-				# print(bestAttrName, bestAttrId)
-				# print(attributeIds)
-				if len(attributeIds) > 0 and bestAttrId in attributeIds:
-					toRemove = attributeIds.index(bestAttrId)
-					attributeIds.pop(toRemove)
-				child.next = self.id3Recv(
-					childSampleIds, attributeIds, child.next)
-		return root
-
-	def printTree(self):
-		if self.root:
-			roots = deque()
-			roots.append(self.root)
-			while len(roots) > 0:
-				root = roots.popleft()
-				print(root.value)
-				if root.childs:
-					for child in root.childs:
-						print('({})'.format(child.value))
-						roots.append(child.next)
-				elif root.next:
-					print(root.next)
-
-
-def test():
-	f = open('DataFiles/rideclass.csv')
-	attributes = f.readline().split(',')
-	attributes = attributes[1:len(attributes)-1]
-	print(attributes)
-	sample = f.readlines()
-	f.close()
-	for i in range(len(sample)):
-		sample[i] = re.sub('\d+,', '', sample[i])
-		sample[i] = sample[i].strip().split(',')
-	labels = []
-	for s in sample:
-		labels.append(s.pop())
-	# print(sample)
-	# print(labels)
-	decisionTree = DecisionTree(sample, attributes, labels)
-	print("System entropy {}".format(decisionTree.entropy))
-	decisionTree.id3()
-	decisionTree.printTree()
-
-
-if __name__ == '__main__':
-	test()
+split = get_split(dataset)
+print('Split: [X%d < %.3f]' % ((split['index']+1), split['value']))
 
    
 
    
 
    
@@ -416,7 +302,7 @@ MathJax.Hub.Config({
   
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diff --git a/doc/pub/week44/html/._week44-bs025.html b/doc/pub/week44/html/._week44-bs025.html
index 4c30c4e9d..ddc31cb93 100644
--- a/doc/pub/week44/html/._week44-bs025.html
+++ b/doc/pub/week44/html/._week44-bs025.html
@@ -41,70 +41,72 @@ Automatically generated HTML file from DocOnce source
 
 
 
 
@@ -142,46 +144,48 @@ MathJax.Hub.Config({
       
  • 
         Contents 
         
       
    • 
@@ -197,52 +201,39 @@ MathJax.Hub.Config({
 
 
 
-
    Cancer Data again now with Decision Trees and other Methods 
    
+
    Entropy and the ID3 algorithm 
    
+
 
    
+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
+
      
+
    1.  Each instance attribute is evaluated using a statistical test to determine how well it alone classifies the training examples.
      2. 
+
    2.  The best attribute is selected and used as the test at the root node of the tree.
      3. 
+
    3.  A descendant of the root node is then created for each possible value of this attribute.
      4. 
+
    4.  Training examples are sorted to the appropriate descendant node.
      5. 
+
    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.
      6. 
+
    6.  This forms a greedy search for an acceptable decision tree, in which the algorithm never backtracks to reconsider earlier choices.
      7.  
+
    
 
-# 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)))
-
    
 
    
 
    
 
@@ -269,7 +260,7 @@ deep_tree_clf.fit(X_train_scaled, y_train)
   
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diff --git a/doc/pub/week44/html/._week44-bs026.html b/doc/pub/week44/html/._week44-bs026.html
index e50d7c888..9c2f27e83 100644
--- a/doc/pub/week44/html/._week44-bs026.html
+++ b/doc/pub/week44/html/._week44-bs026.html
@@ -41,70 +41,72 @@ Automatically generated HTML file from DocOnce source
 
 
 
 
@@ -142,46 +144,48 @@ MathJax.Hub.Config({
       
  • 
         Contents 
         
       
    • 
@@ -197,74 +201,198 @@ MathJax.Hub.Config({
 
 
 
-
    Another example, the moons again 
    
+
    Implementing the ID3 Algorithm 
    
+
 
    
 
 
-
    from __future__ import division, print_function, unicode_literals
+
    import re
+import math
+from collections import deque
 
-# Common imports
-import numpy as np
-import os
+# x is examples in training set
+# y is set of targets
+# label is target attributes
+# Node is a class which has properties values, childs, and next
+# root is top node in the decision tree
 
-# to make this notebook's output stable across runs
-np.random.seed(42)
+class Node(object):
+	def __init__(self):
+		self.value = None
+		self.next = None
+		self.childs = None
 
-# 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
+# Simple class of Decision Tree
+# Aimed for who want to learn Decision Tree, so it is not optimized
+class DecisionTree(object):
+	def __init__(self, sample, attributes, labels):
+		self.sample = sample
+		self.attributes = attributes
+		self.labels = labels
+		self.labelCodes = None
+		self.labelCodesCount = None
+		self.initLabelCodes()
+		# print(self.labelCodes)
+		self.root = None
+		self.entropy = self.getEntropy([x for x in range(len(self.labels))])
+
+	def initLabelCodes(self):
+		self.labelCodes = []
+		self.labelCodesCount = []
+		for l in self.labels:
+			if l not in self.labelCodes:
+				self.labelCodes.append(l)
+				self.labelCodesCount.append(0)
+			self.labelCodesCount[self.labelCodes.index(l)] += 1
+
+	def getLabelCodeId(self, sampleId):
+		return self.labelCodes.index(self.labels[sampleId])
+
+	def getAttributeValues(self, sampleIds, attributeId):
+		vals = []
+		for sid in sampleIds:
+			val = self.sample[sid][attributeId]
+			if val not in vals:
+				vals.append(val)
+		# print(vals)
+		return vals
+
+	def getEntropy(self, sampleIds):
+		entropy = 0
+		labelCount = [0] * len(self.labelCodes)
+		for sid in sampleIds:
+			labelCount[self.getLabelCodeId(sid)] += 1
+		# print("-ge", labelCount)
+		for lv in labelCount:
+			# print(lv)
+			if lv != 0:
+				entropy += -lv/len(sampleIds) * math.log(lv/len(sampleIds), 2)
+			else:
+				entropy += 0
+		return entropy
+
+	def getDominantLabel(self, sampleIds):
+		labelCodesCount = [0] * len(self.labelCodes)
+		for sid in sampleIds:
+			labelCodesCount[self.labelCodes.index(self.labels[sid])] += 1
+		return self.labelCodes[labelCodesCount.index(max(labelCodesCount))]
+
+	def getInformationGain(self, sampleIds, attributeId):
+		gain = self.getEntropy(sampleIds)
+		attributeVals = []
+		attributeValsCount = []
+		attributeValsIds = []
+		for sid in sampleIds:
+			val = self.sample[sid][attributeId]
+			if val not in attributeVals:
+				attributeVals.append(val)
+				attributeValsCount.append(0)
+				attributeValsIds.append([])
+			vid = attributeVals.index(val)
+			attributeValsCount[vid] += 1
+			attributeValsIds[vid].append(sid)
+		# print("-gig", self.attributes[attributeId])
+		for vc, vids in zip(attributeValsCount, attributeValsIds):
+			# print("-gig", vids)
+			gain -= vc/len(sampleIds) * self.getEntropy(vids)
+		return gain
+
+	def getAttributeMaxInformationGain(self, sampleIds, attributeIds):
+		attributesEntropy = [0] * len(attributeIds)
+		for i, attId in zip(range(len(attributeIds)), attributeIds):
+			attributesEntropy[i] = self.getInformationGain(sampleIds, attId)
+		maxId = attributeIds[attributesEntropy.index(max(attributesEntropy))]
+		return self.attributes[maxId], maxId
+
+	def isSingleLabeled(self, sampleIds):
+		label = self.labels[sampleIds[0]]
+		for sid in sampleIds:
+			if self.labels[sid] != label:
+				return False
+		return True
+
+	def getLabel(self, sampleId):
+		return self.labels[sampleId]
+
+	def id3(self):
+		sampleIds = [x for x in range(len(self.sample))]
+		attributeIds = [x for x in range(len(self.attributes))]
+		self.root = self.id3Recv(sampleIds, attributeIds, self.root)
+
+	def id3Recv(self, sampleIds, attributeIds, root):
+		root = Node() # Initialize current root
+		if self.isSingleLabeled(sampleIds):
+			root.value = self.labels[sampleIds[0]]
+			return root
+		# print(attributeIds)
+		if len(attributeIds) == 0:
+			root.value = self.getDominantLabel(sampleIds)
+			return root
+		bestAttrName, bestAttrId = self.getAttributeMaxInformationGain(
+			sampleIds, attributeIds)
+		# print(bestAttrName)
+		root.value = bestAttrName
+		root.childs = []  # Create list of children
+		for value in self.getAttributeValues(sampleIds, bestAttrId):
+			# print(value)
+			child = Node()
+			child.value = value
+			root.childs.append(child)  # Append new child node to current
+									   # root
+			childSampleIds = []
+			for sid in sampleIds:
+				if self.sample[sid][bestAttrId] == value:
+					childSampleIds.append(sid)
+			if len(childSampleIds) == 0:
+				child.next = self.getDominantLabel(sampleIds)
+			else:
+				# print(bestAttrName, bestAttrId)
+				# print(attributeIds)
+				if len(attributeIds) > 0 and bestAttrId in attributeIds:
+					toRemove = attributeIds.index(bestAttrId)
+					attributeIds.pop(toRemove)
+				child.next = self.id3Recv(
+					childSampleIds, attributeIds, child.next)
+		return root
+
+	def printTree(self):
+		if self.root:
+			roots = deque()
+			roots.append(self.root)
+			while len(roots) > 0:
+				root = roots.popleft()
+				print(root.value)
+				if root.childs:
+					for child in root.childs:
+						print('({})'.format(child.value))
+						roots.append(child.next)
+				elif root.next:
+					print(root.next)
 
 
-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 test():
+	f = open('DataFiles/rideclass.csv')
+	attributes = f.readline().split(',')
+	attributes = attributes[1:len(attributes)-1]
+	print(attributes)
+	sample = f.readlines()
+	f.close()
+	for i in range(len(sample)):
+		sample[i] = re.sub('\d+,', '', sample[i])
+		sample[i] = sample[i].strip().split(',')
+	labels = []
+	for s in sample:
+		labels.append(s.pop())
+	# print(sample)
+	# print(labels)
+	decisionTree = DecisionTree(sample, attributes, labels)
+	print("System entropy {}".format(decisionTree.entropy))
+	decisionTree.id3()
+	decisionTree.printTree()
 
 
-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()
+if __name__ == '__main__':
+	test()
 
    
 
    
 
    
@@ -292,7 +420,7 @@ plt.show()
   
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diff --git a/doc/pub/week44/html/._week44-bs027.html b/doc/pub/week44/html/._week44-bs027.html
index 22e1b312d..391e5ce6a 100644
--- a/doc/pub/week44/html/._week44-bs027.html
+++ b/doc/pub/week44/html/._week44-bs027.html
@@ -41,70 +41,72 @@ Automatically generated HTML file from DocOnce source
 
 
 
 
@@ -142,46 +144,48 @@ MathJax.Hub.Config({
       
  • 
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    • 
@@ -197,30 +201,51 @@ MathJax.Hub.Config({
 
 
 
-
    Playing around with regions 
    
+
    Cancer Data again now with Decision Trees and other Methods 
    
 
    
 
 
-
    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)))
 
    
 
    
 
    
@@ -248,7 +273,7 @@ plt.show()
   
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diff --git a/doc/pub/week44/html/._week44-bs028.html b/doc/pub/week44/html/._week44-bs028.html
index 030073d03..fb1a7203d 100644
--- a/doc/pub/week44/html/._week44-bs028.html
+++ b/doc/pub/week44/html/._week44-bs028.html
@@ -41,70 +41,72 @@ Automatically generated HTML file from DocOnce source
 
 
 
 
@@ -142,46 +144,48 @@ MathJax.Hub.Config({
       
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    • 
@@ -197,24 +201,74 @@ MathJax.Hub.Config({
 
 
 
-
    Regression trees 
    
+
    Another example, the moons again 
    
 
    
 
 
-
    # 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()
 
    
 
    
 
    
@@ -242,7 +296,7 @@ tree_reg.fit(X, y)
   
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diff --git a/doc/pub/week44/html/._week44-bs029.html b/doc/pub/week44/html/._week44-bs029.html
index 804c6cd0a..792509cbd 100644
--- a/doc/pub/week44/html/._week44-bs029.html
+++ b/doc/pub/week44/html/._week44-bs029.html
@@ -41,70 +41,72 @@ Automatically generated HTML file from DocOnce source
 
 
 
 
@@ -142,46 +144,48 @@ MathJax.Hub.Config({
       
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    • 
@@ -197,78 +201,28 @@ MathJax.Hub.Config({
 
 
 
-
    Final regressor code 
    
+
    Playing around with regions 
    
 
    
 
 
-
    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()
 
    
@@ -298,7 +252,7 @@ plt.show()
   
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diff --git a/doc/pub/week44/html/._week44-bs030.html b/doc/pub/week44/html/._week44-bs030.html
index 25c77f8e5..d845d8a66 100644
--- a/doc/pub/week44/html/._week44-bs030.html
+++ b/doc/pub/week44/html/._week44-bs030.html
@@ -41,70 +41,72 @@ Automatically generated HTML file from DocOnce source
 
 
 
 
@@ -142,46 +144,48 @@ MathJax.Hub.Config({
       
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    • 
@@ -197,18 +201,26 @@ MathJax.Hub.Config({
 
 
 
-
    Pros and cons of trees, pros 
    
+
    Regression trees 
    
+
    
 
-
      
-
    •  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)
      • 
-
    •  Trees are very easy to explain to people. In fact, they are even easier to explain than linear regression!
      • 
-
    •  No feature normalization needed
      • 
-
    •  Tree models can handle both continuous and categorical data (Classification and Regression Trees)
      • 
-
    •  Can model nonlinear relationships
      • 
-
    •  Can model interactions between the different descriptive features
      • 
-
    •  Trees can be displayed graphically, and are easily interpreted even by a non-expert (especially if they are small)
      • 
-
    
+
+
    # 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)
+
    
+
    
 
    
 
 
      
@@ -234,7 +246,7 @@ MathJax.Hub.Config({
   
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diff --git a/doc/pub/week44/html/._week44-bs031.html b/doc/pub/week44/html/._week44-bs031.html
index d9eb5b881..644e96a16 100644
--- a/doc/pub/week44/html/._week44-bs031.html
+++ b/doc/pub/week44/html/._week44-bs031.html
@@ -41,70 +41,72 @@ Automatically generated HTML file from DocOnce source
 
 
 
 
@@ -142,46 +144,48 @@ MathJax.Hub.Config({
       
  • 
         Contents 
         
       
    • 
@@ -197,22 +201,81 @@ MathJax.Hub.Config({
 
 
 
-
    Disadvantages 
    
+
    Final regressor code 
    
+
    
 
-
      
-
    •  Unfortunately, trees generally do not have the same level of predictive accuracy as some of the other regression and classification approaches
      • 
-
    •  If continuous features are used the tree may become quite large and hence less interpretable
      • 
-
    •  Decision trees are prone to overfit the training data and hence do not well generalize the data if no stopping criteria or improvements like pruning, boosting or bagging are implemented
      • 
-
    •  Small changes in the data may lead to a completely different tree. This issue can be addressed by using ensemble methods like bagging, boosting or random forests
      • 
-
    •  Unbalanced datasets where some target feature values occur much more frequently than others may lead to biased trees since the frequently occurring feature values are preferred over the less frequently occurring ones.
      •  
-
    •  If the number of features is relatively large (high dimensional) and the number of instances is relatively low, the tree might overfit the data
      • 
-
    •  Features with many levels may be preferred over features with less levels since for them it is more easy to split the dataset such that the sub datasets only contain pure target feature values. This issue can be addressed by preferring for instance the information gain ratio as splitting criteria over information gain
      • 
-
    
+
+
    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()
+
    
 
    
 
    
 
@@ -238,6 +301,8 @@ trees can be substantially improved.
   
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diff --git a/doc/pub/week44/html/._week44-bs032.html b/doc/pub/week44/html/._week44-bs032.html
index ca05b421f..ceb4b8fe8 100644
--- a/doc/pub/week44/html/._week44-bs032.html
+++ b/doc/pub/week44/html/._week44-bs032.html
@@ -41,70 +41,72 @@ Automatically generated HTML file from DocOnce source
 
 
 
 
@@ -142,46 +144,48 @@ MathJax.Hub.Config({
       
  • 
         Contents 
         
       
    • 
@@ -197,31 +201,18 @@ MathJax.Hub.Config({
 
 
 
-
    Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods 
    
+
    Pros and cons of trees, pros 
    
 
-
    
-As stated above and seen in many of the examples discussed here about
-a single decision tree, we often end up overfitting our training
-data. This normally means that we have a high variance. Can we reduce
-the variance of a statistical learning method?
+
      
+
    •  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)
      • 
+
    •  Trees are very easy to explain to people. In fact, they are even easier to explain than linear regression!
      • 
+
    •  No feature normalization needed
      • 
+
    •  Tree models can handle both continuous and categorical data (Classification and Regression Trees)
      • 
+
    •  Can model nonlinear relationships
      • 
+
    •  Can model interactions between the different descriptive features
      • 
+
    •  Trees can be displayed graphically, and are easily interpreted even by a non-expert (especially if they are small)
      • 
+
    
 
-
    
-This leads us to a set of different methods that can combine different
-machine learning algorithms or just use one of them to construct
-forests and jungles of trees, homogeneous ones or heterogenous
-ones. These methods are recognized by different names which we will
-try to explain here. These are
-
-
      
-
    1.  Voting classifiers
      2. 
-
    2.  Bagging and Pasting
      3. 
-
    3.  Random forests
      4. 
-
    4.  Boosting methods, from adaptive to Extreme Gradient Boosting (XGBoost)
      5. 
-
    
-
-We discuss these methods here.
-
-
    
 
    
 
 
      
@@ -245,6 +236,9 @@ We discuss these methods here.
   
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diff --git a/doc/pub/week44/html/._week44-bs033.html b/doc/pub/week44/html/._week44-bs033.html
index aff8b6070..835e0560d 100644
--- a/doc/pub/week44/html/._week44-bs033.html
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@@ -41,70 +41,72 @@ Automatically generated HTML file from DocOnce source
 
 
 
 
@@ -142,46 +144,48 @@ MathJax.Hub.Config({
       
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-
    An Overview of Ensemble Methods  
    
+
    Disadvantages 
    
 
-
    
-



    
+
      
+
    •  Unfortunately, trees generally do not have the same level of predictive accuracy as some of the other regression and classification approaches
      • 
+
    •  If continuous features are used the tree may become quite large and hence less interpretable
      • 
+
    •  Decision trees are prone to overfit the training data and hence do not well generalize the data if no stopping criteria or improvements like pruning, boosting or bagging are implemented
      • 
+
    •  Small changes in the data may lead to a completely different tree. This issue can be addressed by using ensemble methods like bagging, boosting or random forests
      • 
+
    •  Unbalanced datasets where some target feature values occur much more frequently than others may lead to biased trees since the frequently occurring feature values are preferred over the less frequently occurring ones.
      •  
+
    •  If the number of features is relatively large (high dimensional) and the number of instances is relatively low, the tree might overfit the data
      • 
+
    •  Features with many levels may be preferred over features with less levels since for them it is more easy to split the dataset such that the sub datasets only contain pure target feature values. This issue can be addressed by preferring for instance the information gain ratio as splitting criteria over information gain
      • 
+
    
+
+However, by aggregating many decision trees, using methods like
+bagging, random forests, and boosting, the predictive performance of
+trees can be substantially improved.
 
 
    
 
    
@@ -225,6 +240,8 @@ MathJax.Hub.Config({
   
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diff --git a/doc/pub/week44/html/._week44-bs034.html b/doc/pub/week44/html/._week44-bs034.html
index 094616fd3..d70346235 100644
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@@ -142,46 +144,48 @@ MathJax.Hub.Config({
       
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    • 
@@ -197,21 +201,29 @@ MathJax.Hub.Config({
 
 
 
-
    Bagging 
    
+
    Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods 
    
 
 
    
-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 stated above and seen in many of the examples discussed here about
+a single decision tree, we often end up overfitting our training
+data. This normally means that we have a high variance. Can we reduce
+the variance of a statistical learning method?
 
 
    
-Bootstrap aggregation, or just bagging, is a
-general-purpose procedure for reducing the variance of a statistical
-learning method.
+This leads us to a set of different methods that can combine different
+machine learning algorithms or just use one of them to construct
+forests and jungles of trees, homogeneous ones or heterogenous
+ones. These methods are recognized by different names which we will
+try to explain here. These are
+
+
      
+
    1.  Voting classifiers
      2. 
+
    2.  Bagging and Pasting
      3. 
+
    3.  Random forests
      4. 
+
    4.  Boosting methods, from adaptive to Extreme Gradient Boosting (XGBoost)
      5. 
+
    
+
+We discuss these methods here.
 
 
    
 
    
@@ -235,6 +247,8 @@ learning method.
   
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diff --git a/doc/pub/week44/html/._week44-bs035.html b/doc/pub/week44/html/._week44-bs035.html
index 5b19ed079..bc0bb7b51 100644
--- a/doc/pub/week44/html/._week44-bs035.html
+++ b/doc/pub/week44/html/._week44-bs035.html
@@ -41,70 +41,72 @@ Automatically generated HTML file from DocOnce source
 
 
 
 
@@ -142,46 +144,48 @@ MathJax.Hub.Config({
       
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@@ -197,31 +201,10 @@ MathJax.Hub.Config({
 
 
 
-
    More bagging 
    
+
    An Overview of Ensemble Methods  
    
 
 
    
-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.
+



    
 
 
    
 
    
@@ -244,6 +227,8 @@ predictor, averaged over all \( B \) trees.
   
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diff --git a/doc/pub/week44/html/._week44-bs036.html b/doc/pub/week44/html/._week44-bs036.html
index 951944fda..9f955c8b4 100644
--- a/doc/pub/week44/html/._week44-bs036.html
+++ b/doc/pub/week44/html/._week44-bs036.html
@@ -41,70 +41,72 @@ Automatically generated HTML file from DocOnce source
 
 
 
 
@@ -142,46 +144,48 @@ MathJax.Hub.Config({
       
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         Contents 
         
       
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@@ -197,24 +201,22 @@ MathJax.Hub.Config({
 
 
 
-
    Simple Voting Example, head or tail 
    
-
    
+
    Bagging 
    
+
+
    
+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.
 
-
-
    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])
-save_fig("votingsimple")
-plt.show()
-
    
 
    
 
    
 
@@ -235,6 +237,8 @@ plt.show()
   
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diff --git a/doc/pub/week44/html/._week44-bs037.html b/doc/pub/week44/html/._week44-bs037.html
index f489f7235..ae76eae96 100644
--- a/doc/pub/week44/html/._week44-bs037.html
+++ b/doc/pub/week44/html/._week44-bs037.html
@@ -41,70 +41,72 @@ Automatically generated HTML file from DocOnce source
 
 
 
 
@@ -142,46 +144,48 @@ MathJax.Hub.Config({
       
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@@ -197,54 +201,32 @@ MathJax.Hub.Config({
 
 
 
-
    Using the Voting Classifier 
    
+
    More bagging 
    
+
 
    
+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.
 
-
-
    from sklearn.model_selection import train_test_split
-from sklearn.datasets import make_moons
+
    
+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.
 
-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(solver="liblinear", random_state=42)
-rnd_clf = RandomForestClassifier(n_estimators=10, random_state=42)
-svm_clf = SVC(gamma="auto", 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)
-
-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(solver="liblinear", random_state=42)
-rnd_clf = RandomForestClassifier(n_estimators=10, random_state=42)
-svm_clf = SVC(gamma="auto", 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))
-
    
 
    
 
    
 
@@ -264,6 +246,8 @@ voting_clf.fit(X_train, y_train)
   
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diff --git a/doc/pub/week44/html/._week44-bs038.html b/doc/pub/week44/html/._week44-bs038.html
index 730fd4892..eaf204b2e 100644
--- a/doc/pub/week44/html/._week44-bs038.html
+++ b/doc/pub/week44/html/._week44-bs038.html
@@ -41,70 +41,72 @@ Automatically generated HTML file from DocOnce source
 
 
 
 
@@ -142,46 +144,48 @@ MathJax.Hub.Config({
       
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    • 
@@ -197,61 +201,23 @@ MathJax.Hub.Config({
 
 
 
-
    Please, not the moons again! Voting and Bagging 
    
-
+
    Simple Voting Example, head or tail 
    
 
    
 
 
-
    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)
-
    
-
    
-
-
-
    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))
+
    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])
+save_fig("votingsimple")
+plt.show()
 
    
 
    
 
    
@@ -271,6 +237,8 @@ voting_clf.fit(X_train, y_train)
   
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diff --git a/doc/pub/week44/html/._week44-bs039.html b/doc/pub/week44/html/._week44-bs039.html
index 0c6863adc..c2209981d 100644
--- a/doc/pub/week44/html/._week44-bs039.html
+++ b/doc/pub/week44/html/._week44-bs039.html
@@ -41,70 +41,72 @@ Automatically generated HTML file from DocOnce source
 
 
 
 
@@ -142,46 +144,48 @@ MathJax.Hub.Config({
       
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@@ -197,64 +201,53 @@ MathJax.Hub.Config({
 
 
 
-
    Bagging Examples  
    
-
+
    Using the Voting Classifier 
    
 
    
 
 
-
    from sklearn.ensemble import BaggingClassifier
-from sklearn.tree import DecisionTreeClassifier
+
    from sklearn.model_selection import train_test_split
+from sklearn.datasets import make_moons
 
-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)
-
    
-
    
+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.metrics import accuracy_score
-print(accuracy_score(y_test, y_pred))
-
    
-
    
+from sklearn.ensemble import RandomForestClassifier
+from sklearn.ensemble import VotingClassifier
+from sklearn.linear_model import LogisticRegression
+from sklearn.svm import SVC
 
-
-
    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))
-
    
-
    
+log_clf = LogisticRegression(solver="liblinear", random_state=42)
+rnd_clf = RandomForestClassifier(n_estimators=10, random_state=42)
+svm_clf = SVC(gamma="auto", random_state=42)
 
-
-
    from matplotlib.colors import ListedColormap
+voting_clf = VotingClassifier(
+    estimators=[('lr', log_clf), ('rf', rnd_clf), ('svc', svm_clf)],
+    voting='hard')
 
-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)
-save_fig("baggingtree")
-plt.show()
+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))
+
+log_clf = LogisticRegression(solver="liblinear", random_state=42)
+rnd_clf = RandomForestClassifier(n_estimators=10, random_state=42)
+svm_clf = SVC(gamma="auto", 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))
 
    
 
    
 
    
@@ -273,6 +266,8 @@ plt.show()
   
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diff --git a/doc/pub/week44/html/._week44-bs040.html b/doc/pub/week44/html/._week44-bs040.html
index 3eb6916ab..7e617dddd 100644
--- a/doc/pub/week44/html/._week44-bs040.html
+++ b/doc/pub/week44/html/._week44-bs040.html
@@ -41,70 +41,72 @@ Automatically generated HTML file from DocOnce source
 
 
 
 
@@ -142,46 +144,48 @@ MathJax.Hub.Config({
       
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@@ -197,74 +201,63 @@ MathJax.Hub.Config({
 
 
 
-
    Making your own Bootstrap: Changing the Level of the Decision Tree 
    
+
    Please, not the moons again! Voting and Bagging 
    
 
-
    
-Let us bring up our good old boostrap example from the linear regression lectures. We change the linerar regression algorithm with
-a decision tree wth different depths and perform a bootstrap aggregate (in this case we perform as many bootstraps as data points \( n \)).
 
    
 
 
-
    import matplotlib.pyplot as plt
-import numpy as np
-from sklearn.model_selection import train_test_split
-from sklearn.pipeline import make_pipeline
-from sklearn.utils import resample
-from sklearn.tree import DecisionTreeRegressor
+
    from sklearn.model_selection import train_test_split
+from sklearn.datasets import make_moons
 
-n = 100
-n_boostraps = 100
-maxdepth = 8
+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
 
-# Make data set.
-x = np.linspace(-3, 3, n).reshape(-1, 1)
-y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)
-error = np.zeros(maxdepth)
-bias = np.zeros(maxdepth)
-variance = np.zeros(maxdepth)
-polydegree = np.zeros(maxdepth)
-X_train, X_test, y_train, y_test = train_test_split(x, y, test_size=0.2)
+log_clf = LogisticRegression(random_state=42)
+rnd_clf = RandomForestClassifier(random_state=42)
+svm_clf = SVC(random_state=42)
 
-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)
-
-# we produce a simple tree first as benchmark
-simpletree = DecisionTreeRegressor(max_depth=3) 
-simpletree.fit(X_train_scaled, y_train)
-simpleprediction = simpletree.predict(X_test_scaled)
-for degree in range(1,maxdepth):
-    model = DecisionTreeRegressor(max_depth=degree) 
-    y_pred = np.empty((y_test.shape[0], n_boostraps))
-    for i in range(n_boostraps):
-        x_, y_ = resample(X_train_scaled, y_train)
-        model.fit(x_, y_)
-        y_pred[:, i] = model.predict(X_test_scaled)#.ravel()
-
-    polydegree[degree] = degree
-    error[degree] = np.mean( np.mean((y_test - y_pred)**2, axis=1, keepdims=True) )
-    bias[degree] = np.mean( (y_test - np.mean(y_pred, axis=1, keepdims=True))**2 )
-    variance[degree] = np.mean( np.var(y_pred, axis=1, keepdims=True) )
-    print('Polynomial degree:', degree)
-    print('Error:', error[degree])
-    print('Bias^2:', bias[degree])
-    print('Var:', variance[degree])
-    print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))
-
-mse_simpletree = np.mean( np.mean((y_test - simpleprediction)**2)
-plt.xlim(1,maxdepth)
-plt.plot(polydegree, error, label='MSE simple tree')
-plt.plot(polydegree, mse_simpletree, label='MSE for Bootstrap')
-plt.plot(polydegree, bias, label='bias')
-plt.plot(polydegree, variance, label='Variance')
-plt.legend()
-save_fig("baggingboot")
-plt.show()
+voting_clf = VotingClassifier(
+    estimators=[('lr', log_clf), ('rf', rnd_clf), ('svc', svm_clf)],
+    voting='hard')
+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))
+
    
+
    
+
+
+
    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))
+
    
+
    
 
    
 
 
    
 
 
diff --git a/doc/pub/week44/html/._week44-bs041.html b/doc/pub/week44/html/._week44-bs041.html
new file mode 100644
index 000000000..b2357bc8e
--- /dev/null
+++ b/doc/pub/week44/html/._week44-bs041.html
@@ -0,0 +1,304 @@
+
+
+
+
+
+
+
+
+week 44: From Decision Trees to Bagging methods
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+    
+
+
    
+  
    
+    
+    week 44: From Decision Trees to Bagging methods
+  
    
+
+  
    
+    
+  
    
+
    
+
     
+
+
    
+
+
     
     
     
     
+
+
+
+
+
    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)
+save_fig("baggingtree")
+plt.show()
+
    
+
    
+
    
+
+
    
+
+
+
      
+
+
+
+
+
+
+
+
    
+
+
    
+
+
+
+
+    
+
diff --git a/doc/pub/week44/html/._week44-bs042.html b/doc/pub/week44/html/._week44-bs042.html
new file mode 100644
index 000000000..08fd8a7ba
--- /dev/null
+++ b/doc/pub/week44/html/._week44-bs042.html
@@ -0,0 +1,310 @@
+
+
+
+
+
+
+
+
+week 44: From Decision Trees to Bagging methods
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+    
+
+
    
+  
    
+    
+    week 44: From Decision Trees to Bagging methods
+  
    
+
+  
    
+    
+  
    
+
    
+
     
+
+
    
+
+
     
     
     
     
+
+
+
+
+
    Making your own Bootstrap: Changing the Level of the Decision Tree 
    
+
+
    
+Let us bring up our good old boostrap example from the linear regression lectures. We change the linerar regression algorithm with
+a decision tree wth different depths and perform a bootstrap aggregate (in this case we perform as many bootstraps as data points \( n \)).
+
    
+
+
+
    import matplotlib.pyplot as plt
+import numpy as np
+from sklearn.model_selection import train_test_split
+from sklearn.pipeline import make_pipeline
+from sklearn.utils import resample
+from sklearn.tree import DecisionTreeRegressor
+
+n = 100
+n_boostraps = 100
+maxdepth = 8
+
+# Make data set.
+x = np.linspace(-3, 3, n).reshape(-1, 1)
+y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)+ np.random.normal(0, 0.1, x.shape)
+error = np.zeros(maxdepth)
+bias = np.zeros(maxdepth)
+variance = np.zeros(maxdepth)
+polydegree = np.zeros(maxdepth)
+X_train, X_test, y_train, y_test = train_test_split(x, y, test_size=0.2)
+
+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)
+
+# we produce a simple tree first as benchmark
+simpletree = DecisionTreeRegressor(max_depth=3) 
+simpletree.fit(X_train_scaled, y_train)
+simpleprediction = simpletree.predict(X_test_scaled)
+for degree in range(1,maxdepth):
+    model = DecisionTreeRegressor(max_depth=degree) 
+    y_pred = np.empty((y_test.shape[0], n_boostraps))
+    for i in range(n_boostraps):
+        x_, y_ = resample(X_train_scaled, y_train)
+        model.fit(x_, y_)
+        y_pred[:, i] = model.predict(X_test_scaled)#.ravel()
+
+    polydegree[degree] = degree
+    error[degree] = np.mean( np.mean((y_test - y_pred)**2, axis=1, keepdims=True) )
+    bias[degree] = np.mean( (y_test - np.mean(y_pred, axis=1, keepdims=True))**2 )
+    variance[degree] = np.mean( np.var(y_pred, axis=1, keepdims=True) )
+    print('Polynomial degree:', degree)
+    print('Error:', error[degree])
+    print('Bias^2:', bias[degree])
+    print('Var:', variance[degree])
+    print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))
+
+mse_simpletree = np.mean( np.mean((y_test - simpleprediction)**2)
+plt.xlim(1,maxdepth)
+plt.plot(polydegree, error, label='MSE simple tree')
+plt.plot(polydegree, mse_simpletree, label='MSE for Bootstrap')
+plt.plot(polydegree, bias, label='bias')
+plt.plot(polydegree, variance, label='Variance')
+plt.legend()
+save_fig("baggingboot")
+plt.show()
+
    
+
    
+
+
    
+
+
    
+
+
+
      
+
+
+
+
+
+
+
+
    
+
+
    
+
+
+
+
+    
+
diff --git a/doc/pub/week44/html/Datafiles/bank.csv b/doc/pub/week44/html/Datafiles/bank.csv
new file mode 100644
index 000000000..930337395
--- /dev/null
+++ b/doc/pub/week44/html/Datafiles/bank.csv
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diff --git a/doc/pub/week44/html/Datafiles/cancer.dot b/doc/pub/week44/html/Datafiles/cancer.dot
new file mode 100644
index 000000000..5b4b48a9b
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+2 [label="gini = 0.48\nsamples = 10\nvalue = [4, 6, 0]", fillcolor="#39e58155"] ;
+1 -> 2 ;
+3 [label="gini = 0.0\nsamples = 4\nvalue = [0, 4, 0]", fillcolor="#39e581ff"] ;
+1 -> 3 ;
+4 [label="gini = 0.0\nsamples = 1\nvalue = [0, 0, 1]", fillcolor="#8139e5ff"] ;
+0 -> 4 [labeldistance=2.5, labelangle=-45, headlabel="False"] ;
+}
\ No newline at end of file
diff --git a/doc/pub/week44/html/Datafiles/rideclass.csv b/doc/pub/week44/html/Datafiles/rideclass.csv
new file mode 100644
index 000000000..a0831ac7e
--- /dev/null
+++ b/doc/pub/week44/html/Datafiles/rideclass.csv
@@ -0,0 +1,15 @@
+Day,Outlook,Temperature,Humidity,Wind,Ride
+1,Sunny,Hot,High,Weak,0
+2,Sunny,Hot,High,Strong,1
+3,Overcast,Hot,High,Weak,1
+4,Rain,Mild,High,Weak,1
+5,Rain,Cool,Normal,Weak,1
+6,Rain,Cool,Normal,Strong,0
+7,Overcast,Cool,Normal,Strong,1
+8,Sunny,Mild,High,Weak,0
+9,Sunny,Cool,Normal,Weak,1
+10,Rain,Mild,Normal,Weak,1
+11,Sunny,Mild,Normal,Strong,1
+12,Overcast,Mild,High,Strong,1
+13,Overcast,Hot,Normal,Weak,1
+14,Rain,Mild,High,Strong,0
diff --git a/doc/pub/week44/html/Datafiles/zoo.csv b/doc/pub/week44/html/Datafiles/zoo.csv
new file mode 100644
index 000000000..ca71f7d21
--- /dev/null
+++ b/doc/pub/week44/html/Datafiles/zoo.csv
@@ -0,0 +1,101 @@
+aardvark,1,0,0,1,0,0,1,1,1,1,0,0,4,0,0,1,1
+antelope,1,0,0,1,0,0,0,1,1,1,0,0,4,1,0,1,1
+bass,0,0,1,0,0,1,1,1,1,0,0,1,0,1,0,0,4
+bear,1,0,0,1,0,0,1,1,1,1,0,0,4,0,0,1,1
+boar,1,0,0,1,0,0,1,1,1,1,0,0,4,1,0,1,1
+buffalo,1,0,0,1,0,0,0,1,1,1,0,0,4,1,0,1,1
+calf,1,0,0,1,0,0,0,1,1,1,0,0,4,1,1,1,1
+carp,0,0,1,0,0,1,0,1,1,0,0,1,0,1,1,0,4
+catfish,0,0,1,0,0,1,1,1,1,0,0,1,0,1,0,0,4
+cavy,1,0,0,1,0,0,0,1,1,1,0,0,4,0,1,0,1
+cheetah,1,0,0,1,0,0,1,1,1,1,0,0,4,1,0,1,1
+chicken,0,1,1,0,1,0,0,0,1,1,0,0,2,1,1,0,2
+chub,0,0,1,0,0,1,1,1,1,0,0,1,0,1,0,0,4
+clam,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,7
+crab,0,0,1,0,0,1,1,0,0,0,0,0,4,0,0,0,7
+crayfish,0,0,1,0,0,1,1,0,0,0,0,0,6,0,0,0,7
+crow,0,1,1,0,1,0,1,0,1,1,0,0,2,1,0,0,2
+deer,1,0,0,1,0,0,0,1,1,1,0,0,4,1,0,1,1
+dogfish,0,0,1,0,0,1,1,1,1,0,0,1,0,1,0,1,4
+dolphin,0,0,0,1,0,1,1,1,1,1,0,1,0,1,0,1,1
+dove,0,1,1,0,1,0,0,0,1,1,0,0,2,1,1,0,2
+duck,0,1,1,0,1,1,0,0,1,1,0,0,2,1,0,0,2
+elephant,1,0,0,1,0,0,0,1,1,1,0,0,4,1,0,1,1
+flamingo,0,1,1,0,1,0,0,0,1,1,0,0,2,1,0,1,2
+flea,0,0,1,0,0,0,0,0,0,1,0,0,6,0,0,0,6
+frog,0,0,1,0,0,1,1,1,1,1,0,0,4,0,0,0,5
+frog,0,0,1,0,0,1,1,1,1,1,1,0,4,0,0,0,5
+fruitbat,1,0,0,1,1,0,0,1,1,1,0,0,2,1,0,0,1
+giraffe,1,0,0,1,0,0,0,1,1,1,0,0,4,1,0,1,1
+girl,1,0,0,1,0,0,1,1,1,1,0,0,2,0,1,1,1
+gnat,0,0,1,0,1,0,0,0,0,1,0,0,6,0,0,0,6
+goat,1,0,0,1,0,0,0,1,1,1,0,0,4,1,1,1,1
+gorilla,1,0,0,1,0,0,0,1,1,1,0,0,2,0,0,1,1
+gull,0,1,1,0,1,1,1,0,1,1,0,0,2,1,0,0,2
+haddock,0,0,1,0,0,1,0,1,1,0,0,1,0,1,0,0,4
+hamster,1,0,0,1,0,0,0,1,1,1,0,0,4,1,1,0,1
+hare,1,0,0,1,0,0,0,1,1,1,0,0,4,1,0,0,1
+hawk,0,1,1,0,1,0,1,0,1,1,0,0,2,1,0,0,2
+herring,0,0,1,0,0,1,1,1,1,0,0,1,0,1,0,0,4
+honeybee,1,0,1,0,1,0,0,0,0,1,1,0,6,0,1,0,6
+housefly,1,0,1,0,1,0,0,0,0,1,0,0,6,0,0,0,6
+kiwi,0,1,1,0,0,0,1,0,1,1,0,0,2,1,0,0,2
+ladybird,0,0,1,0,1,0,1,0,0,1,0,0,6,0,0,0,6
+lark,0,1,1,0,1,0,0,0,1,1,0,0,2,1,0,0,2
+leopard,1,0,0,1,0,0,1,1,1,1,0,0,4,1,0,1,1
+lion,1,0,0,1,0,0,1,1,1,1,0,0,4,1,0,1,1
+lobster,0,0,1,0,0,1,1,0,0,0,0,0,6,0,0,0,7
+lynx,1,0,0,1,0,0,1,1,1,1,0,0,4,1,0,1,1
+mink,1,0,0,1,0,1,1,1,1,1,0,0,4,1,0,1,1
+mole,1,0,0,1,0,0,1,1,1,1,0,0,4,1,0,0,1
+mongoose,1,0,0,1,0,0,1,1,1,1,0,0,4,1,0,1,1
+moth,1,0,1,0,1,0,0,0,0,1,0,0,6,0,0,0,6
+newt,0,0,1,0,0,1,1,1,1,1,0,0,4,1,0,0,5
+octopus,0,0,1,0,0,1,1,0,0,0,0,0,8,0,0,1,7
+opossum,1,0,0,1,0,0,1,1,1,1,0,0,4,1,0,0,1
+oryx,1,0,0,1,0,0,0,1,1,1,0,0,4,1,0,1,1
+ostrich,0,1,1,0,0,0,0,0,1,1,0,0,2,1,0,1,2
+parakeet,0,1,1,0,1,0,0,0,1,1,0,0,2,1,1,0,2
+penguin,0,1,1,0,0,1,1,0,1,1,0,0,2,1,0,1,2
+pheasant,0,1,1,0,1,0,0,0,1,1,0,0,2,1,0,0,2
+pike,0,0,1,0,0,1,1,1,1,0,0,1,0,1,0,1,4
+piranha,0,0,1,0,0,1,1,1,1,0,0,1,0,1,0,0,4
+pitviper,0,0,1,0,0,0,1,1,1,1,1,0,0,1,0,0,3
+platypus,1,0,1,1,0,1,1,0,1,1,0,0,4,1,0,1,1
+polecat,1,0,0,1,0,0,1,1,1,1,0,0,4,1,0,1,1
+pony,1,0,0,1,0,0,0,1,1,1,0,0,4,1,1,1,1
+porpoise,0,0,0,1,0,1,1,1,1,1,0,1,0,1,0,1,1
+puma,1,0,0,1,0,0,1,1,1,1,0,0,4,1,0,1,1
+pussycat,1,0,0,1,0,0,1,1,1,1,0,0,4,1,1,1,1
+raccoon,1,0,0,1,0,0,1,1,1,1,0,0,4,1,0,1,1
+reindeer,1,0,0,1,0,0,0,1,1,1,0,0,4,1,1,1,1
+rhea,0,1,1,0,0,0,1,0,1,1,0,0,2,1,0,1,2
+scorpion,0,0,0,0,0,0,1,0,0,1,1,0,8,1,0,0,7
+seahorse,0,0,1,0,0,1,0,1,1,0,0,1,0,1,0,0,4
+seal,1,0,0,1,0,1,1,1,1,1,0,1,0,0,0,1,1
+sealion,1,0,0,1,0,1,1,1,1,1,0,1,2,1,0,1,1
+seasnake,0,0,0,0,0,1,1,1,1,0,1,0,0,1,0,0,3
+seawasp,0,0,1,0,0,1,1,0,0,0,1,0,0,0,0,0,7
+skimmer,0,1,1,0,1,1,1,0,1,1,0,0,2,1,0,0,2
+skua,0,1,1,0,1,1,1,0,1,1,0,0,2,1,0,0,2
+slowworm,0,0,1,0,0,0,1,1,1,1,0,0,0,1,0,0,3
+slug,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,7
+sole,0,0,1,0,0,1,0,1,1,0,0,1,0,1,0,0,4
+sparrow,0,1,1,0,1,0,0,0,1,1,0,0,2,1,0,0,2
+squirrel,1,0,0,1,0,0,0,1,1,1,0,0,2,1,0,0,1
+starfish,0,0,1,0,0,1,1,0,0,0,0,0,5,0,0,0,7
+stingray,0,0,1,0,0,1,1,1,1,0,1,1,0,1,0,1,4
+swan,0,1,1,0,1,1,0,0,1,1,0,0,2,1,0,1,2
+termite,0,0,1,0,0,0,0,0,0,1,0,0,6,0,0,0,6
+toad,0,0,1,0,0,1,0,1,1,1,0,0,4,0,0,0,5
+tortoise,0,0,1,0,0,0,0,0,1,1,0,0,4,1,0,1,3
+tuatara,0,0,1,0,0,0,1,1,1,1,0,0,4,1,0,0,3
+tuna,0,0,1,0,0,1,1,1,1,0,0,1,0,1,0,1,4
+vampire,1,0,0,1,1,0,0,1,1,1,0,0,2,1,0,0,1
+vole,1,0,0,1,0,0,0,1,1,1,0,0,4,1,0,0,1
+vulture,0,1,1,0,1,0,1,0,1,1,0,0,2,1,0,1,2
+wallaby,1,0,0,1,0,0,0,1,1,1,0,0,2,1,0,1,1
+wasp,1,0,1,0,1,0,0,0,0,1,1,0,6,0,0,0,6
+wolf,1,0,0,1,0,0,1,1,1,1,0,0,4,1,0,1,1
+worm,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,7
+wren,0,1,1,0,1,0,0,0,1,1,0,0,2,1,0,0,2
diff --git a/doc/pub/week44/html/week44-bs.html b/doc/pub/week44/html/week44-bs.html
index 73fecd50a..c9eceb6e3 100644
--- a/doc/pub/week44/html/week44-bs.html
+++ b/doc/pub/week44/html/week44-bs.html
@@ -41,70 +41,72 @@ Automatically generated HTML file from DocOnce source
 
 
 
 
@@ -142,46 +144,48 @@ MathJax.Hub.Config({
       
  • 
         Contents 
         
       
    • 
@@ -216,7 +220,7 @@ MathJax.Hub.Config({
 
    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
    
 
    
 
    
-
    Sep 16, 2020
     
+
    Oct 26, 2020
     
 
    
 
    
 
@@ -240,7 +244,7 @@ MathJax.Hub.Config({
   
  • 9
    • 
   
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    • 
   
  • ...
    • 
-  
  • 41
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+  
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diff --git a/doc/pub/week44/html/week44-reveal.html b/doc/pub/week44/html/week44-reveal.html
index a45f94925..798f7229e 100644
--- a/doc/pub/week44/html/week44-reveal.html
+++ b/doc/pub/week44/html/week44-reveal.html
@@ -148,7 +148,7 @@ MathJax.Hub.Config({
 
    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
    
 
    
 
     
    
-
    Sep 16, 2020
     
+
    Oct 26, 2020
     
 
    
 
    
 
@@ -159,7 +159,28 @@ MathJax.Hub.Config({
 
 
 
    
-
    Decision trees, overarching aims  
    
+
    Overview of week 44 
    
+
+
      
+
    •  Thursday: Wrapping up PCA from last week and basics of decision trees, classification and regression algorithms
      •  
+
    •  Friday: Decision trees, voting models and bagging
      • 
+
    
+
    
+
+Geron's chapter 6 covers decision trees while ensemble models, voting and bagging are discussed in chapter 7. See also lecture from STK-IN4300, lecture 7. Chapter 9.2 of Hastie et al contains also a good discussion.
+
    
+
+
+
    
+
    Thursday 
    
+
+
    
+Overview video, aims and motivations.
+
    
+
+
+
    
+
    Decision trees, overarching aims  
    
 
 
    
 We start here with the most basic algorithm, the so-called decision
@@ -201,7 +222,7 @@ given some assumptions, make predictions about the target feature value
 
 
 
    
-
    A typical Decision Tree with its pertinent Jargon, Classification Problem 
    
+
    A typical Decision Tree with its pertinent Jargon, Classification Problem 
    
 
 
    
 



    
@@ -212,7 +233,7 @@ This tree was produced using the Wisconsin cancer data (discussed here as well,
 
 
 
    
-
    General Features 
    
+
    General Features 
    
 
 
    
 The overarching approach to decision trees is a top-down approach.
@@ -231,7 +252,7 @@ node.
 
 
 
    
-
    How do we set it up? 
    
+
    How do we set it up? 
    
 
 
    
 In simplified terms, the process of training a decision tree and
@@ -250,7 +271,7 @@ Then we are essentially done!
 
 
 
    
-
    Decision trees and Regression  
    
+
    Decision trees and Regression  
    
 
    
 
 
@@ -280,17 +301,17 @@ X=steps_list[:,np.newaxis]
 #Polynomial fits
 
 #Degree 2
-poly_features=PolynomialFeatures(degree=2, include_bias=False)
+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])
+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
@@ -303,7 +324,7 @@ plt.xlabel("Steps")
 plt.ylabel("Distance")
 
 #Degree 10
-poly_features10=PolynomialFeatures(degree=10, include_bias=False)
+poly_features10=PolynomialFeatures(degree=10, include_bias=False)
 X_poly10=poly_features10.fit_transform(X)
 
 poly_fit10=lin_reg.fit(X_poly10,distance_list)
@@ -347,7 +368,7 @@ plt.show()
 
 
 
    
-
    Building a tree, regression 
    
+
    Building a tree, regression 
    
 
 
    
 There are mainly two steps
@@ -379,7 +400,7 @@ 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
@@ -398,7 +419,7 @@ better tree in some future step.
 
 
 
    
-
    Making a tree 
    
+
    Making a tree 
    
 
 
    
 In order to implement the recursive binary splitting we start by selecting
@@ -455,7 +476,7 @@ region contains more than five observations.
 
 
 
    
-
    Pruning the tree 
    
+
    Pruning the tree 
    
 
 
    
 The above procedure is rather straightforward, but leads often to
@@ -474,7 +495,7 @@ parameter \( \alpha \).
 
 
 
    
-
    Cost complexity pruning 
    
+
    Cost complexity pruning 
    
 For each value of \( \alpha \)  there corresponds a subtree \( T \in T_0 \) such that
 
     
    
 $$
@@ -507,7 +528,7 @@ subtree corresponding to \( \alpha \).
 
 
 
    
-
    Schematic Regression Procedure 
    
+
    Schematic Regression Procedure 
    
 
 
    
 
    
@@ -532,7 +553,7 @@ subtree corresponding to \( \alpha \).
 
 
 
    
-
    A Classification Tree 
    
+
    A Classification Tree 
    
 
 
    
 A classification tree is very similar to a regression tree, except
@@ -551,7 +572,7 @@ fall into that region.
 
 
 
    
-
    Growing a classification tree 
    
+
    Growing a classification tree 
    
 
 
    
 The task of growing a
@@ -575,7 +596,7 @@ 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
@@ -630,7 +651,7 @@ $$
 
 
 
    
-
    Visualizing the Tree, Classification 
    
+
    Visualizing the Tree, Classification 
    
 
    
 
 
@@ -649,10 +670,10 @@ $$
 
 cancer = load_breast_cancer()
 X = pd.DataFrame(cancer.data, columns=cancer.feature_names)
-print(X)
+print(X)
 y = pd.Categorical.from_codes(cancer.target, cancer.target_names)
 y = pd.get_dummies(y)
-print(y)
+print(y)
 X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=1)
 tree_clf = DecisionTreeClassifier(max_depth=5)
 tree_clf.fit(X_train, y_train)
@@ -662,8 +683,8 @@ export_graphviz(
     out_file="DataFiles/cancer.dot",
     feature_names=cancer.feature_names,
     class_names=cancer.target_names,
-    rounded=True,
-    filled=True
+    rounded=True,
+    filled=True
 )
 cmd = 'dot -Tpng DataFiles/cancer.dot -o DataFiles/cancer.png'
 os.system(cmd)
@@ -672,7 +693,7 @@ os.system(cmd)
 
 
 
    
-
    Visualizing the Tree, The Moons  
    
+
    Visualizing the Tree, The Moons  
    
 
    
 
 
@@ -695,8 +716,8 @@ tree_clf.fit(X_train, y_train)
 export_graphviz(
     tree_clf,
     out_file="DataFiles/moons.dot",
-    rounded=True,
-    filled=True
+    rounded=True,
+    filled=True
 )
 cmd = 'dot -Tpng DataFiles/moons.dot -o DataFiles/moons.png'
 os.system(cmd)
@@ -705,7 +726,7 @@ os.system(cmd)
 
 
 
    
-
    Algorithms for Setting up Decision Trees 
    
+
    Algorithms for Setting up Decision Trees 
    
 
 
    
 Two algorithms stand out in the set up of decision trees:
@@ -724,7 +745,7 @@ in two branches.
 
 
 
    
-
    The CART algorithm for Classification 
    
+
    The CART algorithm for Classification 
    
 
 
    
 For classification, the CART algorithm splits the data set in two subsets using a single feature \( k \) and a threshold \( t_k \).
@@ -753,7 +774,7 @@ hyperparameters control additional stopping conditions such as the \( min\_sampl
 
 
 
    
-
    The CART algorithm for Regression 
    
+
    The CART algorithm for Regression 
    
 
 
    
 The CART algorithm for regression works is similar to the one for classification except that instead of trying to split the
@@ -787,7 +808,7 @@ just like for classification tasks, is  prone to overfitting.
 
 
 
    
-
    Computing the Gini index 
    
+
    Computing the Gini index 
    
 
 
    
 The example we will look at is a classical one in many Machine
@@ -828,7 +849,7 @@ The table here summarizes the various attributes and
 
 
 
    
-
    Simple Python Code to read in Data and perform Classification 
    
+
    Simple Python Code to read in Data and perform Classification 
    
 
 
    
 
@@ -867,7 +888,7 @@ DATA_ID = "DataFiles/"
     return os.path.join(DATA_ID, dat_id)
 
 def save_fig(fig_id):
-    plt.savefig(image_path(fig_id) + ".png", format='png')
+    plt.savefig(image_path(fig_id) + ".png", format='png')
 
 infile = open(data_path("rideclass.csv"),'r')
 
@@ -886,17 +907,17 @@ encoder = OneHotEncoder(handle_unknown="ignore
 encoder.fit(X)    
 # Apply the encoder.
 X = encoder.transform(X)
-print(X)
+print(X)
 # Then do a Classification tree
 tree_clf = DecisionTreeClassifier(max_depth=2)
 tree_clf.fit(X, y)
-print("Train set accuracy with Decision Tree: {:.2f}".format(tree_clf.score(X,y)))
+print("Train set accuracy with Decision Tree: {:.2f}".format(tree_clf.score(X,y)))
 #transfer to a decision tree graph
 export_graphviz(
     tree_clf,
     out_file="DataFiles/ride.dot",
-    rounded=True,
-    filled=True
+    rounded=True,
+    filled=True
 )
 cmd = 'dot -Tpng DataFiles/cancer.dot -o DataFiles/cancer.png'
 os.system(cmd)
@@ -905,7 +926,7 @@ os.system(cmd)
 
 
 
    
-
    Computing the Gini Factor  
    
+
    Computing the Gini Factor  
    
 
 
    
 The above functions (gini, entropy and misclassification error) are
@@ -952,12 +973,12 @@ In the example here we have converted all our attributes into numerical values \
 # Select the best split point for a dataset
 def get_split(dataset):
 	class_values = list(set(row[-1] for row in dataset))
-	b_index, b_value, b_score, b_groups = 999, 999, 999, None
+	b_index, b_value, b_score, b_groups = 999, 999, 999, None
 	for index in range(len(dataset[0])-1):
 		for row in dataset:
 			groups = test_split(index, row[index], dataset)
 			gini = gini_index(groups, class_values)
-			print('X%d < %.3f Gini=%.3f' % ((index+1), row[index], gini))
+			print('X%d < %.3f Gini=%.3f' % ((index+1), row[index], gini))
 			if gini < b_score:
 				b_index, b_value, b_score, b_groups = index, row[index], gini, groups
 	return {'index':b_index, 'value':b_value, 'groups':b_groups}
@@ -978,13 +999,13 @@ dataset = [[0,0
             [2,1,0,1,0]]
 
 split = get_split(dataset)
-print('Split: [X%d < %.3f]' % ((split['index']+1), split['value']))
+print('Split: [X%d < %.3f]' % ((split['index']+1), split['value']))
 
    
 
 
 
 
    
-
    Entropy and the ID3 algorithm 
    
+
    Entropy and the ID3 algorithm 
    
 
 
    
 ID3, learns decision trees by constructing
@@ -1021,7 +1042,7 @@ attributes at each step while growing the tree.
 
 
 
    
-
    Implementing the ID3 Algorithm 
    
+
    Implementing the ID3 Algorithm 
    
 
 
    
 
@@ -1038,9 +1059,9 @@ attributes at each step while growing the tree.
 
 class Node(object):
 	def __init__(self):
-		self.value = None
-		self.next = None
-		self.childs = None
+		self.value = None
+		self.next = None
+		self.childs = None
 
 # Simple class of Decision Tree
 # Aimed for who want to learn Decision Tree, so it is not optimized
@@ -1049,11 +1070,11 @@ attributes at each step while growing the tree.
 		self.sample = sample
 		self.attributes = attributes
 		self.labels = labels
-		self.labelCodes = None
-		self.labelCodesCount = None
+		self.labelCodes = None
+		self.labelCodesCount = None
 		self.initLabelCodes()
 		# print(self.labelCodes)
-		self.root = None
+		self.root = None
 		self.entropy = self.getEntropy([x for x in range(len(self.labels))])
 
 	def initLabelCodes(self):
@@ -1128,8 +1149,8 @@ attributes at each step while growing the tree.
 		label = self.labels[sampleIds[0]]
 		for sid in sampleIds:
 			if self.labels[sid] != label:
-				return False
-		return True
+				return False
+		return True
 
 	def getLabel(self, sampleId):
 		return self.labels[sampleId]
@@ -1181,20 +1202,20 @@ attributes at each step while growing the tree.
 			roots.append(self.root)
 			while len(roots) > 0:
 				root = roots.popleft()
-				print(root.value)
+				print(root.value)
 				if root.childs:
 					for child in root.childs:
-						print('({})'.format(child.value))
+						print('({})'.format(child.value))
 						roots.append(child.next)
 				elif root.next:
-					print(root.next)
+					print(root.next)
 
 
 def test():
 	f = open('DataFiles/rideclass.csv')
 	attributes = f.readline().split(',')
 	attributes = attributes[1:len(attributes)-1]
-	print(attributes)
+	print(attributes)
 	sample = f.readlines()
 	f.close()
 	for i in range(len(sample)):
@@ -1206,7 +1227,7 @@ attributes at each step while growing the tree.
 	# print(sample)
 	# print(labels)
 	decisionTree = DecisionTree(sample, attributes, labels)
-	print("System entropy {}".format(decisionTree.entropy))
+	print("System entropy {}".format(decisionTree.entropy))
 	decisionTree.id3()
 	decisionTree.printTree()
 
@@ -1218,7 +1239,7 @@ attributes at each step while growing the tree.
 
 
 
    
-
    Cancer Data again now with Decision Trees and other Methods 
    
+
    Cancer Data again now with Decision Trees and other Methods 
    
 
    
 
 
@@ -1234,20 +1255,20 @@ attributes at each step while growing the tree.
 cancer = load_breast_cancer()
 
 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)
+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)))
+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)))
+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 = 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)))
+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()
@@ -1256,19 +1277,19 @@ 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)))
+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)))
+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)))
+print("Test set accuracy with Decision Trees and scaled data: {:.2f}".format(deep_tree_clf.score(X_test_scaled,y_test)))
 
    
 
 
 
 
    
-
    Another example, the moons again 
    
+
    Another example, the moons again 
    
 
    
 
 
@@ -1304,7 +1325,7 @@ 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):
+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)
@@ -1330,10 +1351,10 @@ deep_tree_clf2.fit(Xm, ym)
         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)
+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)
+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()
 
    
@@ -1341,7 +1362,7 @@ plt.show()
 
 
 
    
-
    Playing around with regions 
    
+
    Playing around with regions 
    
 
    
 
 
@@ -1360,9 +1381,9 @@ 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)
+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)
+plot_decision_boundary(tree_clf_sr, Xsr, ys, axes=[-0.7, 0.7, -0.7, 0.7], iris=False)
 
 plt.show()
 
    
@@ -1370,7 +1391,7 @@ plt.show()
 
 
 
    
-
    Regression trees 
    
+
    Regression trees 
    
 
    
 
 
@@ -1393,7 +1414,7 @@ tree_reg.fit(X, y)
 
 
 
    
-
    Final regressor code 
    
+
    Final regressor code 
    
 
    
 
 
@@ -1426,7 +1447,7 @@ plt.legend(loc="upper center", fon
 plt.title("max_depth=2", fontsize=14)
 
 plt.subplot(122)
-plot_regression_predictions(tree_reg2, X, y, ylabel=None)
+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):
@@ -1472,7 +1493,7 @@ 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)
      • 
@@ -1487,7 +1508,7 @@ 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
        • 
@@ -1507,7 +1528,7 @@ trees can be substantially improved.
 
 
 
        
-
        Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods 
        
+
        Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods 
        
 
 
        
 As stated above and seen in many of the examples discussed here about
@@ -1535,7 +1556,7 @@ We discuss these methods here.
 
 
 
        
-
        An Overview of Ensemble Methods  
        
+
        An Overview of Ensemble Methods  
        
 
 
        
 



        
@@ -1543,7 +1564,7 @@ We discuss these methods here.
 
 
 
        
-
        Bagging 
        
+
        Bagging 
        
 
 
        
 The plain decision trees suffer from high
@@ -1562,7 +1583,7 @@ learning method.
 
 
 
        
-
        More bagging 
        
+
        More bagging 
        
 
 
        
 Bagging typically results in improved accuracy
@@ -1591,7 +1612,7 @@ predictor, averaged over all \( B \) trees.
 
 
 
        
-
        Simple Voting Example, head or tail 
        
+
        Simple Voting Example, head or tail 
        
 
        
 
 
@@ -1613,7 +1634,7 @@ plt.show()
 
 
 
        
-
        Using the Voting Classifier 
        
+
        Using the Voting Classifier 
        
 
        
 
 
@@ -1643,11 +1664,11 @@ voting_clf.fit(X_train, y_train)
 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))
+    print(clf.__class__.__name__, accuracy_score(y_test, y_pred))
 
 log_clf = LogisticRegression(solver="liblinear", random_state=42)
 rnd_clf = RandomForestClassifier(n_estimators=10, random_state=42)
-svm_clf = SVC(gamma="auto", probability=True, random_state=42)
+svm_clf = SVC(gamma="auto", probability=True, random_state=42)
 
 voting_clf = VotingClassifier(
     estimators=[('lr', log_clf), ('rf', rnd_clf), ('svc', svm_clf)],
@@ -1659,13 +1680,13 @@ voting_clf.fit(X_train, y_train)
 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))
+    print(clf.__class__.__name__, accuracy_score(y_test, y_pred))
 
    
 
 
 
 
    
-
    Please, not the moons again! Voting and Bagging 
    
+
    Please, not the moons again! Voting and Bagging 
    
 
 
    
 
@@ -1697,14 +1718,14 @@ voting_clf.fit(X_train, y_train)
 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))
+    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)
+svm_clf = SVC(probability=True, random_state=42)
 
 voting_clf = VotingClassifier(
     estimators=[('lr', log_clf), ('rf', rnd_clf), ('svc', svm_clf)],
@@ -1719,13 +1740,13 @@ voting_clf.fit(X_train, y_train)
 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))
+    print(clf.__class__.__name__, accuracy_score(y_test, y_pred))
    -

    Bagging Examples

    +

    Bagging Examples

    @@ -1735,7 +1756,7 @@ voting_clf.fit(X_train, y_train) bag_clf = BaggingClassifier( DecisionTreeClassifier(random_state=42), n_estimators=500, - max_samples=100, bootstrap=True, n_jobs=-1, random_state=42) + 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) @@ -1743,7 +1764,7 @@ y_pred = bag_clf.predict(X_test) 

    from sklearn.metrics import accuracy_score
-print(accuracy_score(y_test, y_pred))
+print(accuracy_score(y_test, y_pred))

    @@ -1751,14 +1772,14 @@ y_pred = bag_clf.predict(X_test) 

    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))
+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):
+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)
@@ -1788,7 +1809,7 @@ plt.show()
 
 
 
    
-
    Making your own Bootstrap: Changing the Level of the Decision Tree 
    
+
    Making your own Bootstrap: Changing the Level of the Decision Tree 
    
 
 
    
 Let us bring up our good old boostrap example from the linear regression lectures. We change the linerar regression algorithm with
@@ -1835,14 +1856,14 @@ simpleprediction = simpletree.predict(X_test_scaled)
         y_pred[:, i] = model.predict(X_test_scaled)#.ravel()
 
     polydegree[degree] = degree
-    error[degree] = np.mean( np.mean((y_test - y_pred)**2, axis=1, keepdims=True) )
-    bias[degree] = np.mean( (y_test - np.mean(y_pred, axis=1, keepdims=True))**2 )
-    variance[degree] = np.mean( np.var(y_pred, axis=1, keepdims=True) )
-    print('Polynomial degree:', degree)
-    print('Error:', error[degree])
-    print('Bias^2:', bias[degree])
-    print('Var:', variance[degree])
-    print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))
+    error[degree] = np.mean( np.mean((y_test - y_pred)**2, axis=1, keepdims=True) )
+    bias[degree] = np.mean( (y_test - np.mean(y_pred, axis=1, keepdims=True))**2 )
+    variance[degree] = np.mean( np.var(y_pred, axis=1, keepdims=True) )
+    print('Polynomial degree:', degree)
+    print('Error:', error[degree])
+    print('Bias^2:', bias[degree])
+    print('Var:', variance[degree])
+    print('{} >= {} + {} = {}'.format(error[degree], bias[degree], variance[degree], bias[degree]+variance[degree]))
 
 mse_simpletree = np.mean( np.mean((y_test - simpleprediction)**2)
 plt.xlim(1,maxdepth)
diff --git a/doc/pub/week44/html/week44-solarized.html b/doc/pub/week44/html/week44-solarized.html
index ef27ff6c0..84f75cd8f 100644
--- a/doc/pub/week44/html/week44-solarized.html
+++ b/doc/pub/week44/html/week44-solarized.html
@@ -61,70 +61,72 @@ div { text-align: justify; text-justify: inter-word; }
 
 
 
 
@@ -166,12 +168,32 @@ MathJax.Hub.Config({
 
    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
    
 
    
 
    
-
    Sep 16, 2020
     
+
    Oct 26, 2020
     
 
    
 
    
 









    
 
-
    Decision trees, overarching aims  
    
+
    Overview of week 44 
    
+
+
      
+
    •  Thursday: Wrapping up PCA from last week and basics of decision trees, classification and regression algorithms
      •  
+
    •  Friday: Decision trees, voting models and bagging
      • 
+
    
+
+Geron's chapter 6 covers decision trees while ensemble models, voting and bagging are discussed in chapter 7. See also lecture from STK-IN4300, lecture 7. Chapter 9.2 of Hastie et al contains also a good discussion.
+
+
    
+









    
+
+
    Thursday 
    
+
+
    
+Overview video, aims and motivations.
+
+
    
+









    
+
+
    Decision trees, overarching aims  
    
 
 
    
 We start here with the most basic algorithm, the so-called decision
@@ -213,7 +235,7 @@ given some assumptions, make predictions about the target feature value
 
    
 









    
 
-
    A typical Decision Tree with its pertinent Jargon, Classification Problem 
    
+
    A typical Decision Tree with its pertinent Jargon, Classification Problem 
    
 
 
    
 



    
@@ -224,7 +246,7 @@ This tree was produced using the Wisconsin cancer data (discussed here as well,
 
    
 









    
 
-
    General Features 
    
+
    General Features 
    
 
 
    
 The overarching approach to decision trees is a top-down approach.
@@ -242,7 +264,7 @@ node.
 
    
 









    
 
-
    How do we set it up? 
    
+
    How do we set it up? 
    
 
 
    
 In simplified terms, the process of training a decision tree and
@@ -260,7 +282,7 @@ Then we are essentially done!
 
    
 









    
 
-
    Decision trees and Regression  
    
+
    Decision trees and Regression  
    
 
    
 
 
@@ -290,17 +312,17 @@ X=steps_list[:,np.newaxis]
 #Polynomial fits
 
 #Degree 2
-poly_features=PolynomialFeatures(degree=2, include_bias=False)
+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])
+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
@@ -313,7 +335,7 @@ plt.xlabel("Steps")
 plt.ylabel("Distance")
 
 #Degree 10
-poly_features10=PolynomialFeatures(degree=10, include_bias=False)
+poly_features10=PolynomialFeatures(degree=10, include_bias=False)
 X_poly10=poly_features10.fit_transform(X)
 
 poly_fit10=lin_reg.fit(X_poly10,distance_list)
@@ -356,7 +378,7 @@ plt.show()
 
    
 









    
 
-
    Building a tree, regression 
    
+
    Building a tree, regression 
    
 
 
    
 There are mainly two steps
@@ -384,7 +406,7 @@ 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
@@ -403,7 +425,7 @@ better tree in some future step.
 
    
 









    
 
-
    Making a tree 
    
+
    Making a tree 
    
 
 
    
 In order to implement the recursive binary splitting we start by selecting
@@ -454,7 +476,7 @@ region contains more than five observations.
 
    
 
 
-
    Pruning the tree 
    
+
    Pruning the tree 
    
 
 
    
 The above procedure is rather straightforward, but leads often to
@@ -473,7 +495,7 @@ 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},
@@ -504,7 +526,7 @@ subtree corresponding to \( \alpha \).
 
    
 









    
 
-
    Schematic Regression Procedure 
    
+
    Schematic Regression Procedure 
    
 
 
    
 
    
@@ -530,7 +552,7 @@ subtree corresponding to \( \alpha \).
 
    
 









    
 
-
    A Classification Tree 
    
+
    A Classification Tree 
    
 
 
    
 A classification tree is very similar to a regression tree, except
@@ -549,7 +571,7 @@ fall into that region.
 
    
 









    
 
-
    Growing a classification tree 
    
+
    Growing a classification tree 
    
 
 
    
 The task of growing a
@@ -573,7 +595,7 @@ 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
@@ -623,7 +645,7 @@ $$
 
    
 









    
 
-
    Visualizing the Tree, Classification 
    
+
    Visualizing the Tree, Classification 
    
 
    
 
 
@@ -642,10 +664,10 @@ $$
 
 cancer = load_breast_cancer()
 X = pd.DataFrame(cancer.data, columns=cancer.feature_names)
-print(X)
+print(X)
 y = pd.Categorical.from_codes(cancer.target, cancer.target_names)
 y = pd.get_dummies(y)
-print(y)
+print(y)
 X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=1)
 tree_clf = DecisionTreeClassifier(max_depth=5)
 tree_clf.fit(X_train, y_train)
@@ -655,8 +677,8 @@ export_graphviz(
     out_file="DataFiles/cancer.dot",
     feature_names=cancer.feature_names,
     class_names=cancer.target_names,
-    rounded=True,
-    filled=True
+    rounded=True,
+    filled=True
 )
 cmd = 'dot -Tpng DataFiles/cancer.dot -o DataFiles/cancer.png'
 os.system(cmd)
@@ -664,7 +686,7 @@ os.system(cmd)
 
    
 









    
 
-
    Visualizing the Tree, The Moons  
    
+
    Visualizing the Tree, The Moons  
    
 
    
 
 
@@ -687,8 +709,8 @@ tree_clf.fit(X_train, y_train)
 export_graphviz(
     tree_clf,
     out_file="DataFiles/moons.dot",
-    rounded=True,
-    filled=True
+    rounded=True,
+    filled=True
 )
 cmd = 'dot -Tpng DataFiles/moons.dot -o DataFiles/moons.png'
 os.system(cmd)
@@ -696,7 +718,7 @@ os.system(cmd)
 
    
 









    
 
-
    Algorithms for Setting up Decision Trees 
    
+
    Algorithms for Setting up Decision Trees 
    
 
 
    
 Two algorithms stand out in the set up of decision trees:
@@ -714,7 +736,7 @@ in two branches.
 
    
 









    
 
-
    The CART algorithm for Classification 
    
+
    The CART algorithm for Classification 
    
 
 
    
 For classification, the CART algorithm splits the data set in two subsets using a single feature \( k \) and a threshold \( t_k \).
@@ -741,7 +763,7 @@ hyperparameters control additional stopping conditions such as the \( min\_sampl
 
    
 









    
 
-
    The CART algorithm for Regression 
    
+
    The CART algorithm for Regression 
    
 
 
    
 The CART algorithm for regression works is similar to the one for classification except that instead of trying to split the
@@ -769,7 +791,7 @@ just like for classification tasks, is  prone to overfitting.
 
    
 









    
 
-
    Computing the Gini index 
    
+
    Computing the Gini index 
    
 
 
    
 The example we will look at is a classical one in many Machine
@@ -809,7 +831,7 @@ The table here summarizes the various attributes and
 
    
 









    
 
-
    Simple Python Code to read in Data and perform Classification 
    
+
    Simple Python Code to read in Data and perform Classification 
    
 
 
    
 
@@ -848,7 +870,7 @@ DATA_ID = "DataFiles/"
     return os.path.join(DATA_ID, dat_id)
 
 def save_fig(fig_id):
-    plt.savefig(image_path(fig_id) + ".png", format='png')
+    plt.savefig(image_path(fig_id) + ".png", format='png')
 
 infile = open(data_path("rideclass.csv"),'r')
 
@@ -867,17 +889,17 @@ encoder = OneHotEncoder(handle_unknown="ignore
 encoder.fit(X)    
 # Apply the encoder.
 X = encoder.transform(X)
-print(X)
+print(X)
 # Then do a Classification tree
 tree_clf = DecisionTreeClassifier(max_depth=2)
 tree_clf.fit(X, y)
-print("Train set accuracy with Decision Tree: {:.2f}".format(tree_clf.score(X,y)))
+print("Train set accuracy with Decision Tree: {:.2f}".format(tree_clf.score(X,y)))
 #transfer to a decision tree graph
 export_graphviz(
     tree_clf,
     out_file="DataFiles/ride.dot",
-    rounded=True,
-    filled=True
+    rounded=True,
+    filled=True
 )
 cmd = 'dot -Tpng DataFiles/cancer.dot -o DataFiles/cancer.png'
 os.system(cmd)
@@ -885,7 +907,7 @@ os.system(cmd)
 
    
 









    
 
-
    Computing the Gini Factor  
    
+
    Computing the Gini Factor  
    
 
 
    
 The above functions (gini, entropy and misclassification error) are
@@ -932,12 +954,12 @@ In the example here we have converted all our attributes into numerical values \
 # Select the best split point for a dataset
 def get_split(dataset):
 	class_values = list(set(row[-1] for row in dataset))
-	b_index, b_value, b_score, b_groups = 999, 999, 999, None
+	b_index, b_value, b_score, b_groups = 999, 999, 999, None
 	for index in range(len(dataset[0])-1):
 		for row in dataset:
 			groups = test_split(index, row[index], dataset)
 			gini = gini_index(groups, class_values)
-			print('X%d < %.3f Gini=%.3f' % ((index+1), row[index], gini))
+			print('X%d < %.3f Gini=%.3f' % ((index+1), row[index], gini))
 			if gini < b_score:
 				b_index, b_value, b_score, b_groups = index, row[index], gini, groups
 	return {'index':b_index, 'value':b_value, 'groups':b_groups}
@@ -958,12 +980,12 @@ dataset = [[0,0
             [2,1,0,1,0]]
 
 split = get_split(dataset)
-print('Split: [X%d < %.3f]' % ((split['index']+1), split['value']))
+print('Split: [X%d < %.3f]' % ((split['index']+1), split['value']))











    -

    Entropy and the ID3 algorithm

    +

    Entropy and the ID3 algorithm

    ID3, learns decision trees by constructing @@ -999,7 +1021,7 @@ attributes at each step while growing the tree.











    -

    Implementing the ID3 Algorithm

    +

    Implementing the ID3 Algorithm

    @@ -1016,9 +1038,9 @@ attributes at each step while growing the tree. class Node(object): def __init__(self): - self.value = None - self.next = None - self.childs = None + self.value = None + self.next = None + self.childs = None # Simple class of Decision Tree # Aimed for who want to learn Decision Tree, so it is not optimized @@ -1027,11 +1049,11 @@ attributes at each step while growing the tree. self.sample = sample self.attributes = attributes self.labels = labels - self.labelCodes = None - self.labelCodesCount = None + self.labelCodes = None + self.labelCodesCount = None self.initLabelCodes() # print(self.labelCodes) - self.root = None + self.root = None self.entropy = self.getEntropy([x for x in range(len(self.labels))]) def initLabelCodes(self): @@ -1106,8 +1128,8 @@ attributes at each step while growing the tree. label = self.labels[sampleIds[0]] for sid in sampleIds: if self.labels[sid] != label: - return False - return True + return False + return True def getLabel(self, sampleId): return self.labels[sampleId] @@ -1159,20 +1181,20 @@ attributes at each step while growing the tree. roots.append(self.root) while len(roots) > 0: root = roots.popleft() - print(root.value) + print(root.value) if root.childs: for child in root.childs: - print('({})'.format(child.value)) + print('({})'.format(child.value)) roots.append(child.next) elif root.next: - print(root.next) + print(root.next) def test(): f = open('DataFiles/rideclass.csv') attributes = f.readline().split(',') attributes = attributes[1:len(attributes)-1] - print(attributes) + print(attributes) sample = f.readlines() f.close() for i in range(len(sample)): @@ -1184,7 +1206,7 @@ attributes at each step while growing the tree. # print(sample) # print(labels) decisionTree = DecisionTree(sample, attributes, labels) - print("System entropy {}".format(decisionTree.entropy)) + print("System entropy {}".format(decisionTree.entropy)) decisionTree.id3() decisionTree.printTree() @@ -1195,7 +1217,7 @@ attributes at each step while growing the tree.











    -

    Cancer Data again now with Decision Trees and other Methods

    +

    Cancer Data again now with Decision Trees and other Methods

    @@ -1211,20 +1233,20 @@ attributes at each step while growing the tree. cancer = load_breast_cancer() 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) +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))) +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))) +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 = 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))) +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() @@ -1233,18 +1255,18 @@ 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))) +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))) +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))) +print("Test set accuracy with Decision Trees and scaled data: {:.2f}".format(deep_tree_clf.score(X_test_scaled,y_test)))











    -

    Another example, the moons again

    +

    Another example, the moons again

    @@ -1280,7 +1302,7 @@ 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): +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) @@ -1306,17 +1328,17 @@ deep_tree_clf2.fit(Xm, ym) 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) +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) +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()











    -

    Playing around with regions

    +

    Playing around with regions

    @@ -1335,16 +1357,16 @@ 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) +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) +plot_decision_boundary(tree_clf_sr, Xsr, ys, axes=[-0.7, 0.7, -0.7, 0.7], iris=False) plt.show()











    -

    Regression trees

    +

    Regression trees

    @@ -1366,7 +1388,7 @@ tree_reg.fit(X, y)











    -

    Final regressor code

    +

    Final regressor code

    @@ -1399,7 +1421,7 @@ plt.legend(loc="upper center", fon plt.title("max_depth=2", fontsize=14) plt.subplot(122) -plot_regression_predictions(tree_reg2, X, y, ylabel=None) +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): @@ -1444,7 +1466,7 @@ plt.show()











    -

    Pros and cons of trees, pros

    +

    Pros and cons of trees, pros