#!/usr/bin/env python # coding: utf-8 # # Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods # # As stated previously and seen in many of the examples discussed in the previous chapter 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? # # 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. Bagging and Pasting # # 3. Random forests # # 4. Boosting methods, from adaptive to Extreme Gradient Boosting (XGBoost) # # We discuss these methods here. # # ### An Overview of Ensemble Methods # # # # # # ## 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. # # # 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. # In[1]: 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() # In[ ]: 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(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)) # In[ ]: 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) # In[ ]: 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)) # In[ ]: 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) # In[ ]: 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)) # ## Bagging Examples # In[ ]: 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) # In[ ]: from sklearn.metrics import accuracy_score print(accuracy_score(y_test, y_pred)) # In[ ]: 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)) # In[ ]: get_ipython().run_line_magic('matplotlib', 'inline') 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() # ### 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 $n$). # In[ ]: 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) print(mse_simpletree) plt.xlim(1,maxdepth) plt.plot(polydegree, error, label='MSE') plt.plot(polydegree, bias, label='bias') plt.plot(polydegree, variance, label='Variance') plt.legend() save_fig("baggingboot") plt.show() # ## Random forests # # Random forests provide an improvement over bagged trees by way of a # small tweak that decorrelates the trees. # # As in bagging, we build a # number of decision trees on bootstrapped training samples. But when # building these decision trees, each time a split in a tree is # considered, a random sample of $m$ predictors is chosen as split # candidates from the full set of $p$ predictors. The split is allowed to # use only one of those $m$ predictors. # # A fresh sample of $m$ predictors is # taken at each split, and typically we choose # $$ # m\approx \sqrt{p}. # $$ # In building a random forest, at # each split in the tree, the algorithm is not even allowed to consider # a majority of the available predictors. # # The reason for this is rather clever. Suppose that there is one very # strong predictor in the data set, along with a number of other # moderately strong predictors. Then in the collection of bagged # variable importance random forest trees, most or all of the trees will # use this strong predictor in the top split. Consequently, all of the # bagged trees will look quite similar to each other. Hence the # predictions from the bagged trees will be highly correlated. # Unfortunately, averaging many highly correlated quantities does not # lead to as large of a reduction in variance as averaging many # uncorrelated quantities. In particular, this means that bagging will # not lead to a substantial reduction in variance over a single tree in # this setting. # # # The algorithm described here can be applied to both classification and regression problems. # # We will grow of forest of say $B$ trees. # 1. For $b=1:B$ # # * Draw a bootstrap sample from the training data organized in our $\boldsymbol{X}$ matrix. # # * We grow then a random forest tree $T_b$ based on the bootstrapped data by repeating the steps outlined till we reach the maximum node size is reached # # 1. we select $m \le p$ variables at random from the $p$ predictors/features # # 2. pick the best split point among the $m$ features using for example the CART algorithm and create a new node # # 3. split the node into daughter nodes # # # # 4. Output then the ensemble of trees $\{T_b\}_1^{B}$ and make predictions for either a regression type of problem or a classification type of problem. # In[ ]: 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 from sklearn.ensemble import BaggingClassifier # Load the data 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) # 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))) from sklearn.ensemble import RandomForestClassifier from sklearn.preprocessing import LabelEncoder from sklearn.model_selection import cross_validate # Data set not specificied #Instantiate the model with 500 trees and entropy as splitting criteria Random_Forest_model = RandomForestClassifier(n_estimators=500,criterion="entropy") Random_Forest_model.fit(X_train_scaled, y_train) #Cross validation accuracy = cross_validate(Random_Forest_model,X_test_scaled,y_test,cv=10)['test_score'] print(accuracy) print("Test set accuracy with Random Forests and scaled data: {:.2f}".format(Random_Forest_model.score(X_test_scaled,y_test))) import scikitplot as skplt y_pred = Random_Forest_model.predict(X_test_scaled) skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True) plt.show() y_probas = Random_Forest_model.predict_proba(X_test_scaled) skplt.metrics.plot_roc(y_test, y_probas) plt.show() skplt.metrics.plot_cumulative_gain(y_test, y_probas) plt.show() # Recall that the cumulative gains curve shows the percentage of the # overall number of cases in a given category *gained* by targeting a # percentage of the total number of cases. # # Similarly, the receiver operating characteristic curve, or ROC curve, # displays the diagnostic ability of a binary classifier system as its # discrimination threshold is varied. It plots the true positive rate against the false positive rate. # # # ### Compare Bagging on Trees with Random Forests # In[ ]: bag_clf = BaggingClassifier( DecisionTreeClassifier(splitter="random", max_leaf_nodes=16, random_state=42), n_estimators=500, max_samples=1.0, bootstrap=True, n_jobs=-1, random_state=42) # In[ ]: bag_clf.fit(X_train, y_train) y_pred = bag_clf.predict(X_test) from sklearn.ensemble import RandomForestClassifier rnd_clf = RandomForestClassifier(n_estimators=500, max_leaf_nodes=16, n_jobs=-1, random_state=42) rnd_clf.fit(X_train, y_train) y_pred_rf = rnd_clf.predict(X_test) np.sum(y_pred == y_pred_rf) / len(y_pred) # ## Boosting, a Bird's Eye View # # The basic idea is to combine weak classifiers in order to create a good # classifier. With a weak classifier we often intend a classifier which # produces results which are only slightly better than we would get by # random guesses. # # This is done by applying in an iterative way a weak (or a standard # classifier like decision trees) to modify the data. In each iteration # we emphasize those observations which are misclassified by weighting # them with a factor. # # # # Boosting is a way of fitting an additive expansion in a set of # elementary basis functions like for example some simple polynomials. # Assume for example that we have a function # $$ # f_M(x) = \sum_{i=1}^M \beta_m b(x;\gamma_m), # $$ # where $\beta_m$ are the expansion parameters to be determined in a # minimization process and $b(x;\gamma_m)$ are some simple functions of # the multivariable parameter $x$ which is characterized by the # parameters $\gamma_m$. # # As an example, consider the Sigmoid function we used in logistic # regression. In that case, we can translate the function # $b(x;\gamma_m)$ into the Sigmoid function # $$ # \sigma(t) = \frac{1}{1+\exp{(-t)}}, # $$ # where $t=\gamma_0+\gamma_1 x$ and the parameters $\gamma_0$ and # $\gamma_1$ were determined by the Logistic Regression fitting # algorithm. # # As another example, consider the cost function we defined for linear regression # $$ # C(\boldsymbol{y},\boldsymbol{f}) = \frac{1}{n} \sum_{i=0}^{n-1}(y_i-f(x_i))^2. # $$ # In this case the function $f(x)$ was replaced by the design matrix # $\boldsymbol{X}$ and the unknown linear regression parameters $\boldsymbol{\beta}$, # that is $\boldsymbol{f}=\boldsymbol{X}\boldsymbol{\beta}$. In linear regression we can # simply invert a matrix and obtain the parameters $\beta$ by # $$ # \boldsymbol{\beta}=\left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\boldsymbol{y}. # $$ # In iterative fitting or additive modeling, we minimize the cost function with respect to the parameters $\beta_m$ and $\gamma_m$. # # # ### Iterative Fitting, Regression and Squared-error Cost Function # # The way we proceed is as follows (here we specialize to the squared-error cost function) # # 1. Establish a cost function, here $\cal{C}(\boldsymbol{y},\boldsymbol{f}) = \frac{1}{n} \sum_{i=0}^{n-1}(y_i-f_M(x_i))^2$ with $f_M(x) = \sum_{i=1}^M \beta_m b(x;\gamma_m)$. # # 2. Initialize with a guess $f_0(x)$. It could be one or even zero or some random numbers. # # 3. For $m=1:M$ # # a. minimize $\sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\beta b(x;\gamma))^2$ wrt $\gamma$ and $\beta$ # # b. This gives the optimal values $\beta_m$ and $\gamma_m$ # # c. Determine then the new values $f_m(x)=f_{m-1}(x) +\beta_m b(x;\gamma_m)$ # # # We could use any of the algorithms we have discussed till now. If we # use trees, $\gamma$ parameterizes the split variables and split points # at the internal nodes, and the predictions at the terminal nodes. # # # # To better understand what happens, let us develop the steps for the iterative fitting using the above squared error function. # # For simplicity we assume also that our functions $b(x;\gamma)=1+\gamma x$. # # This means that for every iteration $m$, we need to optimize # $$ # (\beta_m,\gamma_m) = \mathrm{argmin}_{\beta,\lambda}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\beta b(x;\gamma))^2=\sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\beta(1+\gamma x_i))^2. # $$ # We start our iteration by simply setting $f_0(x)=0$. # Taking the derivatives with respect to $\beta$ and $\gamma$ we obtain # $$ # \frac{\partial \cal{C}}{\partial \beta} = -2\sum_{i}(1+\gamma x_i)(y_i-\beta(1+\gamma x_i))=0, # $$ # and # $$ # \frac{\partial \cal{C}}{\partial \gamma} =-2\sum_{i}\beta x_i(y_i-\beta(1+\gamma x_i))=0. # $$ # We can then rewrite these equations as (defining $\boldsymbol{w}=\boldsymbol{e}+\gamma \boldsymbol{x})$ with $\boldsymbol{e}$ being the unit vector) # $$ # \gamma \boldsymbol{w}^T(\boldsymbol{y}-\beta\gamma \boldsymbol{w})=0, # $$ # which gives us $\beta = \boldsymbol{w}^T\boldsymbol{y}/(\boldsymbol{w}^T\boldsymbol{w})$. Similarly we have # $$ # \beta\gamma \boldsymbol{x}^T(\boldsymbol{y}-\beta(1+\gamma \boldsymbol{x}))=0, # $$ # which leads to $\gamma =(\boldsymbol{x}^T\boldsymbol{y}-\beta\boldsymbol{x}^T\boldsymbol{e})/(\beta\boldsymbol{x}^T\boldsymbol{x})$. Inserting # for $\beta$ gives us an equation for $\gamma$. This is a non-linear equation in the unknown $\gamma$ and has to be solved numerically. # # The solution to these two equations gives us in turn $\beta_1$ and $\gamma_1$ leading to the new expression for $f_1(x)$ as # $f_1(x) = \beta_1(1+\gamma_1x)$. Doing this $M$ times results in our final estimate for the function $f$. # # # # ### Iterative Fitting, Classification and AdaBoost # # Let us consider a binary classification problem with two outcomes $y_i \in \{-1,1\}$ and $i=0,1,2,\dots,n-1$ as our set of # observations. We define a classification function $G(x)$ which produces a prediction taking one or the other of the two values # $\{-1,1\}$. # # The error rate of the training sample is then # $$ # \mathrm{\overline{err}}=\frac{1}{n} \sum_{i=0}^{n-1} I(y_i\ne G(x_i)). # $$ # The iterative procedure starts with defining a weak classifier whose # error rate is barely better than random guessing. The iterative # procedure in boosting is to sequentially apply a weak # classification algorithm to repeatedly modified versions of the data # producing a sequence of weak classifiers $G_m(x)$. # # Here we will express our function $f(x)$ in terms of $G(x)$. That is # $$ # f_M(x) = \sum_{i=1}^M \beta_m b(x;\gamma_m), # $$ # will be a function of # $$ # G_M(x) = \mathrm{sign} \sum_{i=1}^M \alpha_m G_m(x). # $$ # In our iterative procedure we define thus # $$ # f_m(x) = f_{m-1}(x)+\beta_mG_m(x). # $$ # The simplest possible cost function which leads (also simple from a computational point of view) to the AdaBoost algorithm is the # exponential cost/loss function defined as # $$ # C(\boldsymbol{y},\boldsymbol{f}) = \sum_{i=0}^{n-1}\exp{(-y_i(f_{m-1}(x_i)+\beta G(x_i))}. # $$ # We optimize $\beta$ and $G$ for each value of $m=1:M$ as we did in the regression case. # This is normally done in two steps. Let us however first rewrite the cost function as # $$ # C(\boldsymbol{y},\boldsymbol{f}) = \sum_{i=0}^{n-1}w_i^{m}\exp{(-y_i\beta G(x_i))}, # $$ # where we have defined $w_i^m= \exp{(-y_if_{m-1}(x_i))}$. # # # # First, for any $\beta > 0$, we optimize $G$ by setting # $$ # G_m(x) = \mathrm{sign} \sum_{i=0}^{n-1} w_i^m I(y_i \ne G_(x_i)), # $$ # which is the classifier that minimizes the weighted error rate in predicting $y$. # # We can do this by rewriting # $$ # \exp{-(\beta)}\sum_{y_i=G(x_i)}w_i^m+\exp{(\beta)}\sum_{y_i\ne G(x_i)}w_i^m, # $$ # which can be rewritten as # $$ # (\exp{(\beta)}-\exp{-(\beta)})\sum_{i=0}^{n-1}w_i^mI(y_i\ne G(x_i))+\exp{(-\beta)}\sum_{i=0}^{n-1}w_i^m=0, # $$ # which leads to # $$ # \beta_m = \frac{1}{2}\log{\frac{1-\mathrm{\overline{err}}}{\mathrm{\overline{err}}}}, # $$ # where we have redefined the error as # $$ # \mathrm{\overline{err}}_m=\frac{1}{n}\frac{\sum_{i=0}^{n-1}w_i^mI(y_i\ne G(x_i)}{\sum_{i=0}^{n-1}w_i^m}, # $$ # which leads to an update of # $$ # f_m(x) = f_{m-1}(x) +\beta_m G_m(x). # $$ # This leads to the new weights # $$ # w_i^{m+1} = w_i^m \exp{(-y_i\beta_m G_m(x_i))} # $$ # ### Adaptive boosting: AdaBoost, Basic Algorithm # # The algorithm here is rather straightforward. Assume that our weak # classifier is a decision tree and we consider a binary set of outputs # with $y_i \in \{-1,1\}$ and $i=0,1,2,\dots,n-1$ as our set of # observations. Our design matrix is given in terms of the # feature/predictor vectors # $\boldsymbol{X}=[\boldsymbol{x}_0\boldsymbol{x}_1\dots\boldsymbol{x}_{p-1}]$. Finally, we define also a # classifier determined by our data via a function $G(x)$. This function tells us how well we are able to classify our outputs/targets $\boldsymbol{y}$. # # We have already defined the misclassification error $\mathrm{err}$ as # $$ # \mathrm{err}=\frac{1}{n}\sum_{i=0}^{n-1}I(y_i\ne G(x_i)), # $$ # where the function $I()$ is one if we misclassify and zero if we classify correctly. # # # With the above definitions we are now ready to set up the algorithm for AdaBoost. # The basic idea is to set up weights which will be used to scale the correctly classified and the misclassified cases. # 1. We start by initializing all weights to $w_i = 1/n$, with $i=0,1,2,\dots n-1$. It is easy to see that we must have $\sum_{i=0}^{n-1}w_i = 1$. # # 2. We rewrite the misclassification error as # $$ # \mathrm{\overline{err}}_m=\frac{\sum_{i=0}^{n-1}w_i^m I(y_i\ne G(x_i))}{\sum_{i=0}^{n-1}w_i}, # $$ # 1. Then we start looping over all attempts at classifying, namely we start an iterative process for $m=1:M$, where $M$ is the final number of classifications. Our given classifier could for example be a plain decision tree. # # a. Fit then a given classifier to the training set using the weights $w_i$. # # b. Compute then $\mathrm{err}$ and figure out which events are classified properly and which are classified wrongly. # # c. Define a quantity $\alpha_{m} = \log{(1-\mathrm{\overline{err}}_m)/\mathrm{\overline{err}}_m}$ # # d. Set the new weights to $w_i = w_i\times \exp{(\alpha_m I(y_i\ne G(x_i)}$. # # # 5. Compute the new classifier $G(x)= \sum_{i=0}^{n-1}\alpha_m I(y_i\ne G(x_i)$. # # For the iterations with $m \le 2$ the weights are modified # individually at each steps. The observations which were misclassified # at iteration $m-1$ have a weight which is larger than those which were # classified properly. As this proceeds, the observations which were # difficult to classifiy correctly are given a larger influence. Each # new classification step $m$ is then forced to concentrate on those # observations that are missed in the previous iterations. # # # # # Using **Scikit-Learn** it is easy to apply the adaptive boosting algorithm, as done here. # In[ ]: from sklearn.ensemble import AdaBoostClassifier ada_clf = AdaBoostClassifier( DecisionTreeClassifier(max_depth=1), n_estimators=200, algorithm="SAMME.R", learning_rate=0.5, random_state=42) ada_clf.fit(X_train, y_train) from sklearn.ensemble import AdaBoostClassifier ada_clf = AdaBoostClassifier( DecisionTreeClassifier(max_depth=1), n_estimators=200, algorithm="SAMME.R", learning_rate=0.5, random_state=42) ada_clf.fit(X_train_scaled, y_train) y_pred = ada_clf.predict(X_test_scaled) skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True) plt.show() y_probas = ada_clf.predict_proba(X_test_scaled) skplt.metrics.plot_roc(y_test, y_probas) plt.show() skplt.metrics.plot_cumulative_gain(y_test, y_probas) plt.show() # ## Gradient boosting: Basics with Steepest Descent/Functional Gradient Descent # # Gradient boosting is again a similar technique to Adaptive boosting, # it combines so-called weak classifiers or regressors into a strong # method via a series of iterations. # # In order to understand the method, let us illustrate its basics by # bringing back the essential steps in linear regression, where our cost # function was the least squares function. # # # We start again with our cost function $\cal{C}(\boldsymbol{y}m\boldsymbol{f})=\sum_{i=0}^{n-1}\cal{L}(y_i, f(x_i))$ where we want to minimize # This means that for every iteration, we need to optimize # $$ # (\hat{\boldsymbol{f}}) = \mathrm{argmin}_{\boldsymbol{f}}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i-f(x_i))^2. # $$ # We define a real function $h_m(x)$ that defines our final function $f_M(x)$ as # $$ # f_M(x) = \sum_{m=0}^M h_m(x). # $$ # In the steepest decent approach we approximate $h_m(x) = -\rho_m g_m(x)$, where $\rho_m$ is a scalar and $g_m(x)$ the gradient defined as # $$ # g_m(x_i) = \left[ \frac{\partial \cal{L}(y_i, f(x_i))}{\partial f(x_i)}\right]_{f(x_i)=f_{m-1}(x_i)}. # $$ # With the new gradient we can update $f_m(x) = f_{m-1}(x) -\rho_m g_m(x)$. Using the above squared-error function we see that # the gradient is $g_m(x_i) = -2(y_i-f(x_i))$. # # Choosing $f_0(x)=0$ we obtain $g_m(x) = -2y_i$ and inserting this into the minimization problem for the cost function we have # $$ # (\rho_1) = \mathrm{argmin}_{\rho}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i+2\rho y_i)^2. # $$ # Optimizing with respect to $\rho$ we obtain (taking the derivative) that $\rho_1 = -1/2$. We have then that # $$ # f_1(x) = f_{0}(x) -\rho_1 g_1(x)=-y_i. # $$ # We can then proceed and compute # $$ # g_2(x_i) = \left[ \frac{\partial \cal{L}(y_i, f(x_i))}{\partial f(x_i)}\right]_{f(x_i)=f_{1}(x_i)=y_i}=-4y_i, # $$ # and find a new value for $\rho_2=-1/2$ and continue till we have reached $m=M$. We can modify the steepest descent method, or steepest boosting, by introducing what is called **gradient boosting**. # # # Steepest descent is however not much used, since it only optimizes $f$ at a fixed set of $n$ points, # so we do not learn a function that can generalize. However, we can modify the algorithm by # fitting a weak learner to approximate the negative gradient signal. # # Suppose we have a cost function $C(f)=\sum_{i=0}^{n-1}L(y_i, f(x_i))$ where $y_i$ is our target and $f(x_i)$ the function which is meant to model $y_i$. The above cost function could be our standard squared-error function # $$ # C(\boldsymbol{y},\boldsymbol{f})=\sum_{i=0}^{n-1}(y_i-f(x_i))^2. # $$ # The way we proceed in an iterative fashion is to # 1. Initialize our estimate $f_0(x)$. # # 2. For $m=1:M$, we # # a. compute the negative gradient vector $\boldsymbol{u}_m = -\partial C(\boldsymbol{y},\boldsymbol{f})/\partial \boldsymbol{f}(x)$ at $f(x) = f_{m-1}(x)$; # # b. fit the so-called base-learner to the negative gradient $h_m(u_m,x)$; # # c. update the estimate $f_m(x) = f_{m-1}(x)+h_m(u_m,x)$; # # # 4. The final estimate is then $f_M(x) = \sum_{m=1}^M h_m(u_m,x)$. # # ## Gradient Boosting, Examples of Regression # In[ ]: import matplotlib.pyplot as plt import numpy as np from sklearn.model_selection import train_test_split from sklearn.ensemble import GradientBoostingRegressor from sklearn.preprocessing import StandardScaler import scikitplot as skplt from sklearn.metrics import mean_squared_error n = 100 maxdegree = 6 # 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(maxdegree) bias = np.zeros(maxdegree) variance = np.zeros(maxdegree) polydegree = np.zeros(maxdegree) X_train, X_test, y_train, y_test = train_test_split(x, y, test_size=0.2) scaler = StandardScaler() scaler.fit(X_train) X_train_scaled = scaler.transform(X_train) X_test_scaled = scaler.transform(X_test) for degree in range(1,maxdegree): model = GradientBoostingRegressor(max_depth=degree, n_estimators=100, learning_rate=1.0) model.fit(X_train_scaled,y_train) y_pred = model.predict(X_test_scaled) polydegree[degree] = degree error[degree] = np.mean( np.mean((y_test - y_pred)**2) ) bias[degree] = np.mean( (y_test - np.mean(y_pred))**2 ) variance[degree] = np.mean( np.var(y_pred) ) print('Max depth:', 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])) plt.xlim(1,maxdegree-1) plt.plot(polydegree, error, label='Error') plt.plot(polydegree, bias, label='bias') plt.plot(polydegree, variance, label='Variance') plt.legend() save_fig("gdregression") plt.show() # ## Gradient Boosting, Classification Example # In[ ]: import matplotlib.pyplot as plt import numpy as np from sklearn.model_selection import train_test_split from sklearn.datasets import load_breast_cancer import scikitplot as skplt from sklearn.ensemble import GradientBoostingClassifier from sklearn.model_selection import cross_validate # Load the data 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) #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) gd_clf = GradientBoostingClassifier(max_depth=3, n_estimators=100, learning_rate=1.0) gd_clf.fit(X_train_scaled, y_train) #Cross validation accuracy = cross_validate(gd_clf,X_test_scaled,y_test,cv=10)['test_score'] print(accuracy) print("Test set accuracy with Random Forests and scaled data: {:.2f}".format(gd_clf.score(X_test_scaled,y_test))) import scikitplot as skplt y_pred = gd_clf.predict(X_test_scaled) skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True) save_fig("gdclassiffierconfusion") plt.show() y_probas = gd_clf.predict_proba(X_test_scaled) skplt.metrics.plot_roc(y_test, y_probas) save_fig("gdclassiffierroc") plt.show() skplt.metrics.plot_cumulative_gain(y_test, y_probas) save_fig("gdclassiffiercgain") plt.show() # ## XGBoost: Extreme Gradient Boosting # # # [XGBoost](https://github.com/dmlc/xgboost) or Extreme Gradient # Boosting, is an optimized distributed gradient boosting library # designed to be highly efficient, flexible and portable. It implements # machine learning algorithms under the Gradient Boosting # framework. XGBoost provides a parallel tree boosting that solve many # data science problems in a fast and accurate way. See the [article by Chen and Guestrin](https://arxiv.org/abs/1603.02754). # # The authors design and build a highly scalable end-to-end tree # boosting system. It has a theoretically justified weighted quantile # sketch for efficient proposal calculation. It introduces a novel sparsity-aware algorithm for parallel tree learning and an effective cache-aware block structure for out-of-core tree learning. # # It is now the algorithm which wins essentially all ML competitions!!! # # ## Regression Case # In[ ]: import matplotlib.pyplot as plt import numpy as np from sklearn.model_selection import train_test_split import xgboost as xgb from sklearn.preprocessing import StandardScaler import scikitplot as skplt from sklearn.metrics import mean_squared_error n = 100 maxdegree = 6 # 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(maxdegree) bias = np.zeros(maxdegree) variance = np.zeros(maxdegree) polydegree = np.zeros(maxdegree) X_train, X_test, y_train, y_test = train_test_split(x, y, test_size=0.2) scaler = StandardScaler() scaler.fit(X_train) X_train_scaled = scaler.transform(X_train) X_test_scaled = scaler.transform(X_test) for degree in range(maxdegree): model = xgb.XGBRegressor(objective ='reg:squarederror', colsaobjective ='reg:squarederror', colsample_bytree = 0.3, learning_rate = 0.1,max_depth = degree, alpha = 10, n_estimators = 200) model.fit(X_train_scaled,y_train) y_pred = model.predict(X_test_scaled) polydegree[degree] = degree error[degree] = np.mean( np.mean((y_test - y_pred)**2) ) bias[degree] = np.mean( (y_test - np.mean(y_pred))**2 ) variance[degree] = np.mean( np.var(y_pred) ) print('Max depth:', 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])) plt.xlim(1,maxdegree-1) plt.plot(polydegree, error, label='Error') plt.plot(polydegree, bias, label='bias') plt.plot(polydegree, variance, label='Variance') plt.legend() plt.show() # As you will see from the confusion matrix below, XGBoots does an excellent job on the Wisconsin cancer data and outperforms essentially all agorithms we have discussed till now. # In[ ]: 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.preprocessing import LabelEncoder from sklearn.model_selection import cross_validate import scikitplot as skplt import xgboost as xgb # Load the data 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) #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) xg_clf = xgb.XGBClassifier() xg_clf.fit(X_train_scaled,y_train) y_test = xg_clf.predict(X_test_scaled) print("Test set accuracy with Random Forests and scaled data: {:.2f}".format(xg_clf.score(X_test_scaled,y_test))) import scikitplot as skplt y_pred = xg_clf.predict(X_test_scaled) skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True) save_fig("xdclassiffierconfusion") plt.show() y_probas = xg_clf.predict_proba(X_test_scaled) skplt.metrics.plot_roc(y_test, y_probas) save_fig("xdclassiffierroc") plt.show() skplt.metrics.plot_cumulative_gain(y_test, y_probas) save_fig("gdclassiffiercgain") plt.show() xgb.plot_tree(xg_clf,num_trees=0) plt.rcParams['figure.figsize'] = [50, 10] save_fig("xgtree") plt.show() xgb.plot_importance(xg_clf) plt.rcParams['figure.figsize'] = [5, 5] save_fig("xgparams") plt.show()