+ + +
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)
+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()
+
++ +
+ +