760 lines
24 KiB
Plaintext
760 lines
24 KiB
Plaintext
TITLE: Week 45: Decisions Trees, Random Forests, Bagging and Boosting
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AUTHOR: Morten Hjorth-Jensen {copyright, 1999-present|CC BY-NC} at Department of Physics, University of Oslo & Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University
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DATE: today
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!split
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===== Overview of week 45 =====
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* Thursday: Boosting methods, froma AdaBoost to Gradient boosting
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* Friday: Gradient boosting and discussion of Decision trees and ensemble methods
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!bblock Videos
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o "Video on Decision trees":"https://www.youtube.com/watch?v=RmajweUFKvM&ab_channel=Simplilearn"
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o "Video on boosting methods by Hastie":"https://www.youtube.com/watch?v=wPqtzj5VZus&ab_channel=H2O.ai".
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!eblock
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!bblock Reading
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o Add material about AdaBoost and Gradient boosting
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!eblock
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!split
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===== Brief code reminder from last wekk =====
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!bc pycod
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# Common imports
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from IPython.display import Image
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from pydot import graph_from_dot_data
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import pandas as pd
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import numpy as np
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import matplotlib.pyplot as plt
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from sklearn.tree import DecisionTreeClassifier
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from sklearn.tree import DecisionTreeRegressor
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from sklearn.model_selection import train_test_split
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from sklearn.tree import export_graphviz
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from sklearn.preprocessing import StandardScaler, OneHotEncoder
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from sklearn.compose import ColumnTransformer
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from IPython.display import Image
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from pydot import graph_from_dot_data
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from sklearn.datasets import load_breast_cancer
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from sklearn.svm import SVC
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from sklearn.linear_model import LogisticRegression
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from sklearn.ensemble import BaggingClassifier
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import os
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# Where to save the figures and data files
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PROJECT_ROOT_DIR = "Results"
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FIGURE_ID = "Results/FigureFiles"
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DATA_ID = "DataFiles/"
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if not os.path.exists(PROJECT_ROOT_DIR):
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os.mkdir(PROJECT_ROOT_DIR)
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if not os.path.exists(FIGURE_ID):
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os.makedirs(FIGURE_ID)
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if not os.path.exists(DATA_ID):
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os.makedirs(DATA_ID)
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def image_path(fig_id):
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return os.path.join(FIGURE_ID, fig_id)
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def data_path(dat_id):
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return os.path.join(DATA_ID, dat_id)
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def save_fig(fig_id):
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plt.savefig(image_path(fig_id) + ".png", format='png')
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# Load the cancer data
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cancer = load_breast_cancer()
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X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
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print(X_train.shape)
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print(X_test.shape)
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#Scale the data
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from sklearn.preprocessing import StandardScaler
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scaler = StandardScaler()
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scaler.fit(X_train)
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X_train_scaled = scaler.transform(X_train)
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X_test_scaled = scaler.transform(X_test)
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#define methods
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# Logistic Regression
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logreg.fit(X_train_scaled, y_train)
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print("Test set accuracy Logistic Regression with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
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# Support Vector Machine
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svm.fit(X_train_scaled, y_train)
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print("Test set accuracy SVM with scaled data: {:.2f}".format(logreg.score(X_test_scaled,y_test)))
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# Decision Trees
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deep_tree_clf.fit(X_train_scaled, y_train)
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print("Test set accuracy with Decision Trees and scaled data: {:.2f}".format(deep_tree_clf.score(X_test_scaled,y_test)))
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from sklearn.ensemble import RandomForestClassifier
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from sklearn.preprocessing import LabelEncoder
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from sklearn.model_selection import cross_validate
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# Data set not specificied
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#Instantiate the model with 500 trees and entropy as splitting criteria
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Random_Forest_model = RandomForestClassifier(n_estimators=500,criterion="entropy")
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Random_Forest_model.fit(X_train_scaled, y_train)
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#Cross validation
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accuracy = cross_validate(Random_Forest_model,X_test_scaled,y_test,cv=10)['test_score']
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print(accuracy)
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print("Test set accuracy with Random Forests and scaled data: {:.2f}".format(Random_Forest_model.score(X_test_scaled,y_test)))
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import scikitplot as skplt
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y_pred = Random_Forest_model.predict(X_test_scaled)
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skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
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plt.show()
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y_probas = Random_Forest_model.predict_proba(X_test_scaled)
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skplt.metrics.plot_roc(y_test, y_probas)
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plt.show()
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skplt.metrics.plot_cumulative_gain(y_test, y_probas)
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plt.show()
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!ec
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!split
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===== Boosting, a Bird's Eye View =====
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The basic idea is to combine weak classifiers in order to create a good
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classifier. With a weak classifier we often intend a classifier which
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produces results which are only slightly better than we would get by
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random guesses.
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This is done by applying in an iterative way a weak (or a standard
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classifier like decision trees) to modify the data. In each iteration
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we emphasize those observations which are misclassified by weighting
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them with a factor.
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!split
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===== What is boosting? Additive Modelling/Iterative Fitting =====
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Boosting is a way of fitting an additive expansion in a set of
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elementary basis functions like for example some simple polynomials.
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Assume for example that we have a function
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!bt
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\[
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f_M(x) = \sum_{i=1}^M \beta_m b(x;\gamma_m),
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\]
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!et
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where $\beta_m$ are the expansion parameters to be determined in a
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minimization process and $b(x;\gamma_m)$ are some simple functions of
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the multivariable parameter $x$ which is characterized by the
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parameters $\gamma_m$.
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As an example, consider the Sigmoid function we used in logistic
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regression. In that case, we can translate the function
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$b(x;\gamma_m)$ into the Sigmoid function
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!bt
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\[
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\sigma(t) = \frac{1}{1+\exp{(-t)}},
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\]
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!et
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where $t=\gamma_0+\gamma_1 x$ and the parameters $\gamma_0$ and
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$\gamma_1$ were determined by the Logistic Regression fitting
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algorithm.
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As another example, consider the cost function we defined for linear regression
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!bt
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\[
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C(\bm{y},\bm{f}) = \frac{1}{n} \sum_{i=0}^{n-1}(y_i-f(x_i))^2.
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\]
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!et
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In this case the function $f(x)$ was replaced by the design matrix
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$\bm{X}$ and the unknown linear regression parameters $\bm{\beta}$,
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that is $\bm{f}=\bm{X}\bm{\beta}$. In linear regression we can
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simply invert a matrix and obtain the parameters $\beta$ by
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!bt
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\[
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\bm{\beta}=\left(\bm{X}^T\bm{X}\right)^{-1}\bm{X}^T\bm{y}.
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\]
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!et
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In iterative fitting or additive modeling, we minimize the cost function with respect to the parameters $\beta_m$ and $\gamma_m$.
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!split
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===== Iterative Fitting, Regression and Squared-error Cost Function =====
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The way we proceed is as follows (here we specialize to the squared-error cost function)
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o Establish a cost function, here ${\cal C}(\bm{y},\bm{f}) = \frac{1}{n} \sum_{i=0}^{n-1}(y_i-f_M(x_i))^2$ with $f_M(x) = \sum_{i=1}^M \beta_m b(x;\gamma_m)$.
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o Initialize with a guess $f_0(x)$. It could be one or even zero or some random numbers.
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o For $m=1:M$
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o minimize $\sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\beta b(x;\gamma))^2$ wrt $\gamma$ and $\beta$
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o This gives the optimal values $\beta_m$ and $\gamma_m$
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o Determine then the new values $f_m(x)=f_{m-1}(x) +\beta_m b(x;\gamma_m)$
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We could use any of the algorithms we have discussed till now. If we
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use trees, $\gamma$ parameterizes the split variables and split points
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at the internal nodes, and the predictions at the terminal nodes.
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!split
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===== Squared-Error Example and Iterative Fitting =====
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To better understand what happens, let us develop the steps for the iterative fitting using the above squared error function.
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For simplicity we assume also that our functions $b(x;\gamma)=1+\gamma x$.
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This means that for every iteration $m$, we need to optimize
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!bt
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\[
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(\beta_m,\gamma_m) = \mathrm{argmin}_{\beta,\lambda}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\beta b(x;\gamma))^2=\sum_{i=0}^{n-1}(y_i-f_{m-1}(x_i)-\beta(1+\gamma x_i))^2.
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\]
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!et
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We start our iteration by simply setting $f_0(x)=0$.
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Taking the derivatives with respect to $\beta$ and $\gamma$ we obtain
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!bt
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\[
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\frac{\partial {\cal C}}{\partial \beta} = -2\sum_{i}(1+\gamma x_i)(y_i-\beta(1+\gamma x_i))=0,
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\]
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!et
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and
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!bt
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\[
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\frac{\partial {\cal C}}{\partial \gamma} =-2\sum_{i}\beta x_i(y_i-\beta(1+\gamma x_i))=0.
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\]
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!et
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We can then rewrite these equations as (defining $\bm{w}=\bm{e}+\gamma \bm{x})$ with $\bm{e}$ being the unit vector)
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!bt
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\[
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\gamma \bm{w}^T(\bm{y}-\beta\gamma \bm{w})=0,
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\]
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!et
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which gives us $\beta = \bm{w}^T\bm{y}/(\bm{w}^T\bm{w})$. Similarly we have
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!bt
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\[
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\beta\gamma \bm{x}^T(\bm{y}-\beta(1+\gamma \bm{x}))=0,
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\]
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!et
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which leads to $\gamma =(\bm{x}^T\bm{y}-\beta\bm{x}^T\bm{e})/(\beta\bm{x}^T\bm{x})$. Inserting
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for $\beta$ gives us an equation for $\gamma$. This is a non-linear equation in the unknown $\gamma$ and has to be solved numerically.
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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
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$f_1(x) = \beta_1(1+\gamma_1x)$. Doing this $M$ times results in our final estimate for the function $f$.
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!split
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===== Iterative Fitting, Classification and AdaBoost =====
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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
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observations. We define a classification function $G(x)$ which produces a prediction taking one or the other of the two values
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$\{-1,1\}$.
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The error rate of the training sample is then
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!bt
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\[
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\mathrm{\overline{err}}=\frac{1}{n} \sum_{i=0}^{n-1} I(y_i\ne G(x_i)).
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\]
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!et
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The iterative procedure starts with defining a weak classifier whose
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error rate is barely better than random guessing. The iterative
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procedure in boosting is to sequentially apply a weak
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classification algorithm to repeatedly modified versions of the data
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producing a sequence of weak classifiers $G_m(x)$.
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Here we will express our function $f(x)$ in terms of $G(x)$. That is
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!bt
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\[
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f_M(x) = \sum_{i=1}^M \beta_m b(x;\gamma_m),
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\]
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!et
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will be a function of
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!bt
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\[
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G_M(x) = \mathrm{sign} \sum_{i=1}^M \alpha_m G_m(x).
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\]
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!et
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!split
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===== Adaptive Boosting, AdaBoost =====
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In our iterative procedure we define thus
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!bt
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\[
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f_m(x) = f_{m-1}(x)+\beta_mG_m(x).
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\]
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!et
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The simplest possible cost function which leads (also simple from a computational point of view) to the AdaBoost algorithm is the
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exponential cost/loss function defined as
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!bt
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\[
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C(\bm{y},\bm{f}) = \sum_{i=0}^{n-1}\exp{(-y_i(f_{m-1}(x_i)+\beta G(x_i))}.
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\]
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!et
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We optimize $\beta$ and $G$ for each value of $m=1:M$ as we did in the regression case.
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This is normally done in two steps. Let us however first rewrite the cost function as
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!bt
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\[
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C(\bm{y},\bm{f}) = \sum_{i=0}^{n-1}w_i^{m}\exp{(-y_i\beta G(x_i))},
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\]
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!et
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where we have defined $w_i^m= \exp{(-y_if_{m-1}(x_i))}$.
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!split
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===== Building up AdaBoost =====
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First, for any $\beta > 0$, we optimize $G$ by setting
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!bt
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\[
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G_m(x) = \mathrm{sign} \sum_{i=0}^{n-1} w_i^m I(y_i \ne G_(x_i)),
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\]
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!et
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which is the classifier that minimizes the weighted error rate in predicting $y$.
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We can do this by rewriting
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!bt
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\[
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\exp{-(\beta)}\sum_{y_i=G(x_i)}w_i^m+\exp{(\beta)}\sum_{y_i\ne G(x_i)}w_i^m,
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\]
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!et
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which can be rewritten as
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!bt
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\[
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(\exp{(\beta)}-\exp{-(\beta)})\sum_{i=0}^{n-1}w_i^mI(y_i\ne G(x_i))+\exp{(-\beta)}\sum_{i=0}^{n-1}w_i^m=0,
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\]
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!et
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which leads to
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!bt
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\[
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\beta_m = \frac{1}{2}\log{\frac{1-\mathrm{\overline{err}}}{\mathrm{\overline{err}}}},
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\]
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!et
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where we have redefined the error as
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!bt
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\[
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\mathrm{\overline{err}}_m=\frac{1}{n}\frac{\sum_{i=0}^{n-1}w_i^mI(y_i\ne G(x_i)}{\sum_{i=0}^{n-1}w_i^m},
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\]
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!et
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which leads to an update of
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!bt
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\[
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f_m(x) = f_{m-1}(x) +\beta_m G_m(x).
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\]
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!et
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This leads to the new weights
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!bt
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\[
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w_i^{m+1} = w_i^m \exp{(-y_i\beta_m G_m(x_i))}
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\]
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!et
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!split
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===== Adaptive boosting: AdaBoost, Basic Algorithm =====
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The algorithm here is rather straightforward. Assume that our weak
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classifier is a decision tree and we consider a binary set of outputs
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with $y_i \in \{-1,1\}$ and $i=0,1,2,\dots,n-1$ as our set of
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observations. Our design matrix is given in terms of the
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feature/predictor vectors
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$\bm{X}=[\bm{x}_0\bm{x}_1\dots\bm{x}_{p-1}]$. Finally, we define also a
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classifier determined by our data via a function $G(x)$. This function tells us how well we are able to classify our outputs/targets $\bm{y}$.
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We have already defined the misclassification error $\mathrm{err}$ as
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!bt
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\[
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\mathrm{err}=\frac{1}{n}\sum_{i=0}^{n-1}I(y_i\ne G(x_i)),
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\]
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!et
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where the function $I()$ is one if we misclassify and zero if we classify correctly.
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!split
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===== Basic Steps of AdaBoost =====
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With the above definitions we are now ready to set up the algorithm for AdaBoost.
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The basic idea is to set up weights which will be used to scale the correctly classified and the misclassified cases.
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o 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 $\sum_{i=0}^{n-1}w_i = 1$.
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o We rewrite the misclassification error as
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!bt
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\[
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\mathrm{\overline{err}}_m=\frac{\sum_{i=0}^{n-1}w_i^m I(y_i\ne G(x_i))}{\sum_{i=0}^{n-1}w_i},
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\]
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!et
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o 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.
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o Fit then a given classifier to the training set using the weights $w_i$.
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o Compute then $\mathrm{err}$ and figure out which events are classified properly and which are classified wrongly.
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o Define a quantity $\alpha_{m} = \log{(1-\mathrm{\overline{err}}_m)/\mathrm{\overline{err}}_m}$
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o Set the new weights to $w_i = w_i\times \exp{(\alpha_m I(y_i\ne G(x_i)}$.
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o Compute the new classifier $G(x)= \sum_{i=0}^{n-1}\alpha_m I(y_i\ne G(x_i)$.
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For the iterations with $m \le 2$ the weights are modified
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individually at each steps. The observations which were misclassified
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at iteration $m-1$ have a weight which is larger than those which were
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classified properly. As this proceeds, the observations which were
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difficult to classifiy correctly are given a larger influence. Each
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new classification step $m$ is then forced to concentrate on those
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observations that are missed in the previous iterations.
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!split
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===== AdaBoost Examples =====
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Using _Scikit-Learn_ it is easy to apply the adaptive boosting algorithm, as done here.
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!bc pycod
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from sklearn.ensemble import AdaBoostClassifier
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ada_clf = AdaBoostClassifier(
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DecisionTreeClassifier(max_depth=1), n_estimators=200,
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algorithm="SAMME.R", learning_rate=0.5, random_state=42)
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ada_clf.fit(X_train, y_train)
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from sklearn.ensemble import AdaBoostClassifier
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ada_clf = AdaBoostClassifier(
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DecisionTreeClassifier(max_depth=1), n_estimators=200,
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algorithm="SAMME.R", learning_rate=0.5, random_state=42)
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ada_clf.fit(X_train_scaled, y_train)
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y_pred = ada_clf.predict(X_test_scaled)
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skplt.metrics.plot_confusion_matrix(y_test, y_pred, normalize=True)
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plt.show()
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y_probas = ada_clf.predict_proba(X_test_scaled)
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skplt.metrics.plot_roc(y_test, y_probas)
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plt.show()
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skplt.metrics.plot_cumulative_gain(y_test, y_probas)
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plt.show()
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!ec
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!split
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===== Gradient boosting: Basics with Steepest Descent/Functional Gradient Descent =====
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Gradient boosting is again a similar technique to Adaptive boosting,
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it combines so-called weak classifiers or regressors into a strong
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method via a series of iterations.
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In order to understand the method, let us illustrate its basics by
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bringing back the essential steps in linear regression, where our cost
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function was the least squares function.
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!split
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===== The Squared-Error again! Steepest Descent =====
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We start again with our cost function ${\cal C}(\bm{y}m\bm{f})=\sum_{i=0}^{n-1}{\cal L}(y_i, f(x_i))$ where we want to minimize
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This means that for every iteration, we need to optimize
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!bt
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\[
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(\hat{\bm{f}}) = \mathrm{argmin}_{\bm{f}}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i-f(x_i))^2.
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\]
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!et
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We define a real function $h_m(x)$ that defines our final function $f_M(x)$ as
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!bt
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\[
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f_M(x) = \sum_{m=0}^M h_m(x).
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\]
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!et
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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
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!bt
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\[
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g_m(x_i) = \left[ \frac{\partial {\cal L}(y_i, f(x_i))}{\partial f(x_i)}\right]_{f(x_i)=f_{m-1}(x_i)}.
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\]
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!et
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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
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the gradient is $g_m(x_i) = -2(y_i-f(x_i))$.
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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
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!bt
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\[
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(\rho_1) = \mathrm{argmin}_{\rho}\hspace{0.1cm} \sum_{i=0}^{n-1}(y_i+2\rho y_i)^2.
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\]
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!et
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!split
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===== Steepest Descent Example =====
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Optimizing with respect to $\rho$ we obtain (taking the derivative) that $\rho_1 = -1/2$. We have then that
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!bt
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|
\[
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f_1(x) = f_{0}(x) -\rho_1 g_1(x)=-y_i.
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\]
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!et
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We can then proceed and compute
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!bt
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|
\[
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g_2(x_i) = \left[ \frac{\partial {\cal L}(y_i, f(x_i))}{\partial f(x_i)}\right]_{f(x_i)=f_{1}(x_i)=y_i}=-4y_i,
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\]
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!et
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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_.
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!split
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|
===== Gradient Boosting, algorithm =====
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Steepest descent is however not much used, since it only optimizes $f$ at a fixed set of $n$ points,
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so we do not learn a function that can generalize. However, we can modify the algorithm by
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fitting a weak learner to approximate the negative gradient signal.
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|
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
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|
!bt
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|
\[
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|
C(\bm{y},\bm{f})=\sum_{i=0}^{n-1}(y_i-f(x_i))^2.
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|
\]
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|
!et
|
|
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|
The way we proceed in an iterative fashion is to
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|
o Initialize our estimate $f_0(x)$.
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|
o For $m=1:M$, we
|
|
o compute the negative gradient vector $\bm{u}_m = -\partial C(\bm{y},\bm{f})/\partial \bm{f}(x)$ at $f(x) = f_{m-1}(x)$;
|
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o fit the so-called base-learner to the negative gradient $h_m(u_m,x)$;
|
|
o update the estimate $f_m(x) = f_{m-1}(x)+h_m(u_m,x)$;
|
|
o The final estimate is then $f_M(x) = \sum_{m=1}^M h_m(u_m,x)$.
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|
|
|
|
|
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|
!split
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|
===== Gradient Boosting, Examples of Regression =====
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|
!bc pycod
|
|
import matplotlib.pyplot as plt
|
|
import numpy as np
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|
from sklearn.model_selection import train_test_split
|
|
from sklearn.ensemble import GradientBoostingRegressor
|
|
import scikitplot as skplt
|
|
from sklearn.metrics import mean_squared_error
|
|
|
|
n = 100
|
|
maxdegree = 6
|
|
|
|
# Make data set.
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|
x = np.linspace(-3, 3, n).reshape(-1, 1)
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|
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)
|
|
|
|
for degree in range(1,maxdegree):
|
|
model = GradientBoostingRegressor(max_depth=degree, n_estimators=100, learning_rate=1.0)
|
|
model.fit(X_train,y_train)
|
|
y_pred = model.predict(X_test)
|
|
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 )
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|
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()
|
|
!ec
|
|
|
|
|
|
!split
|
|
===== Gradient Boosting, Classification Example =====
|
|
!bc pycod
|
|
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()
|
|
!ec
|
|
|
|
|
|
!split
|
|
===== 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!!!
|
|
|
|
!split
|
|
===== Regression Case =====
|
|
|
|
!bc pycod
|
|
import matplotlib.pyplot as plt
|
|
import numpy as np
|
|
from sklearn.model_selection import train_test_split
|
|
import xgboost as xgb
|
|
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)
|
|
|
|
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,y_train)
|
|
y_pred = model.predict(X_test)
|
|
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()
|
|
|
|
!ec
|
|
|
|
!split
|
|
===== Xgboost on the Cancer Data =====
|
|
|
|
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.
|
|
!bc pycod
|
|
|
|
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()
|
|
|
|
!ec
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|