Copyright (c) 2015, 2016 Sebastian Raschka
Note that the optional watermark extension is a small IPython notebook plugin that I developed to make the code reproducible. You can just skip the following line(s).
%load_ext watermark
%watermark -a 'Sebastian Raschka' -u -d -v -p numpy,pandas,matplotlib,sklearn,seaborn
Sebastian Raschka last updated: 2016-09-29 CPython 3.5.2 IPython 5.1.0 numpy 1.11.1 pandas 0.18.1 matplotlib 1.5.1 sklearn 0.18 seaborn 0.7.0
The use of watermark
is optional. You can install this IPython extension via "pip install watermark
". For more information, please see: https://github.com/rasbt/watermark.
from IPython.display import Image
%matplotlib inline
# Added version check for recent scikit-learn 0.18 checks
from distutils.version import LooseVersion as Version
from sklearn import __version__ as sklearn_version
Image(filename='./images/10_01.png', width=500)
Source: https://archive.ics.uci.edu/ml/datasets/Housing
Attributes:
1. CRIM per capita crime rate by town 2. ZN proportion of residential land zoned for lots over 25,000 sq.ft. 3. INDUS proportion of non-retail business acres per town 4. CHAS Charles River dummy variable (= 1 if tract bounds river; 0 otherwise) 5. NOX nitric oxides concentration (parts per 10 million) 6. RM average number of rooms per dwelling 7. AGE proportion of owner-occupied units built prior to 1940 8. DIS weighted distances to five Boston employment centres 9. RAD index of accessibility to radial highways 10. TAX full-value property-tax rate per $10,000 11. PTRATIO pupil-teacher ratio by town 12. B 1000(Bk - 0.63)^2 where Bk is the proportion of blacks by town 13. LSTAT % lower status of the population 14. MEDV Median value of owner-occupied homes in $1000's
import pandas as pd
df = pd.read_csv('https://archive.ics.uci.edu/ml/machine-learning-databases/'
'housing/housing.data',
header=None,
sep='\s+')
df.columns = ['CRIM', 'ZN', 'INDUS', 'CHAS',
'NOX', 'RM', 'AGE', 'DIS', 'RAD',
'TAX', 'PTRATIO', 'B', 'LSTAT', 'MEDV']
df.head()
CRIM | ZN | INDUS | CHAS | NOX | RM | AGE | DIS | RAD | TAX | PTRATIO | B | LSTAT | MEDV | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0 | 0.00632 | 18.0 | 2.31 | 0 | 0.538 | 6.575 | 65.2 | 4.0900 | 1 | 296.0 | 15.3 | 396.90 | 4.98 | 24.0 |
1 | 0.02731 | 0.0 | 7.07 | 0 | 0.469 | 6.421 | 78.9 | 4.9671 | 2 | 242.0 | 17.8 | 396.90 | 9.14 | 21.6 |
2 | 0.02729 | 0.0 | 7.07 | 0 | 0.469 | 7.185 | 61.1 | 4.9671 | 2 | 242.0 | 17.8 | 392.83 | 4.03 | 34.7 |
3 | 0.03237 | 0.0 | 2.18 | 0 | 0.458 | 6.998 | 45.8 | 6.0622 | 3 | 222.0 | 18.7 | 394.63 | 2.94 | 33.4 |
4 | 0.06905 | 0.0 | 2.18 | 0 | 0.458 | 7.147 | 54.2 | 6.0622 | 3 | 222.0 | 18.7 | 396.90 | 5.33 | 36.2 |
If the link to the Housing dataset provided above does not work for you, you can find a local copy in this repository at ./../datasets/housing/housing.data.
Or you could fetch it via
df = pd.read_csv('https://raw.githubusercontent.com/rasbt/python-machine-learning-book/master/code/datasets/housing/housing.data',
header=None, sep='\s+')
df.columns = ['CRIM', 'ZN', 'INDUS', 'CHAS',
'NOX', 'RM', 'AGE', 'DIS', 'RAD',
'TAX', 'PTRATIO', 'B', 'LSTAT', 'MEDV']
df.head()
CRIM | ZN | INDUS | CHAS | NOX | RM | AGE | DIS | RAD | TAX | PTRATIO | B | LSTAT | MEDV | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0 | 0.00632 | 18.0 | 2.31 | 0 | 0.538 | 6.575 | 65.2 | 4.0900 | 1 | 296.0 | 15.3 | 396.90 | 4.98 | 24.0 |
1 | 0.02731 | 0.0 | 7.07 | 0 | 0.469 | 6.421 | 78.9 | 4.9671 | 2 | 242.0 | 17.8 | 396.90 | 9.14 | 21.6 |
2 | 0.02729 | 0.0 | 7.07 | 0 | 0.469 | 7.185 | 61.1 | 4.9671 | 2 | 242.0 | 17.8 | 392.83 | 4.03 | 34.7 |
3 | 0.03237 | 0.0 | 2.18 | 0 | 0.458 | 6.998 | 45.8 | 6.0622 | 3 | 222.0 | 18.7 | 394.63 | 2.94 | 33.4 |
4 | 0.06905 | 0.0 | 2.18 | 0 | 0.458 | 7.147 | 54.2 | 6.0622 | 3 | 222.0 | 18.7 | 396.90 | 5.33 | 36.2 |
import matplotlib.pyplot as plt
import seaborn as sns
sns.set(style='whitegrid', context='notebook')
cols = ['LSTAT', 'INDUS', 'NOX', 'RM', 'MEDV']
sns.pairplot(df[cols], size=2.5)
plt.tight_layout()
# plt.savefig('./figures/scatter.png', dpi=300)
plt.show()
import numpy as np
cm = np.corrcoef(df[cols].values.T)
sns.set(font_scale=1.5)
hm = sns.heatmap(cm,
cbar=True,
annot=True,
square=True,
fmt='.2f',
annot_kws={'size': 15},
yticklabels=cols,
xticklabels=cols)
# plt.tight_layout()
# plt.savefig('./figures/corr_mat.png', dpi=300)
plt.show()
sns.reset_orig()
%matplotlib inline
...
class LinearRegressionGD(object):
def __init__(self, eta=0.001, n_iter=20):
self.eta = eta
self.n_iter = n_iter
def fit(self, X, y):
self.w_ = np.zeros(1 + X.shape[1])
self.cost_ = []
for i in range(self.n_iter):
output = self.net_input(X)
errors = (y - output)
self.w_[1:] += self.eta * X.T.dot(errors)
self.w_[0] += self.eta * errors.sum()
cost = (errors**2).sum() / 2.0
self.cost_.append(cost)
return self
def net_input(self, X):
return np.dot(X, self.w_[1:]) + self.w_[0]
def predict(self, X):
return self.net_input(X)
X = df[['RM']].values
y = df['MEDV'].values
from sklearn.preprocessing import StandardScaler
sc_x = StandardScaler()
sc_y = StandardScaler()
X_std = sc_x.fit_transform(X)
y_std = sc_y.fit_transform(y[:, np.newaxis]).flatten()
lr = LinearRegressionGD()
lr.fit(X_std, y_std)
<__main__.LinearRegressionGD at 0x118f523c8>
plt.plot(range(1, lr.n_iter+1), lr.cost_)
plt.ylabel('SSE')
plt.xlabel('Epoch')
plt.tight_layout()
# plt.savefig('./figures/cost.png', dpi=300)
plt.show()
def lin_regplot(X, y, model):
plt.scatter(X, y, c='lightblue')
plt.plot(X, model.predict(X), color='red', linewidth=2)
return
lin_regplot(X_std, y_std, lr)
plt.xlabel('Average number of rooms [RM] (standardized)')
plt.ylabel('Price in $1000\'s [MEDV] (standardized)')
plt.tight_layout()
# plt.savefig('./figures/gradient_fit.png', dpi=300)
plt.show()
print('Slope: %.3f' % lr.w_[1])
print('Intercept: %.3f' % lr.w_[0])
Slope: 0.695 Intercept: -0.000
num_rooms_std = sc_x.transform(np.array([[5.0]]))
price_std = lr.predict(num_rooms_std)
print("Price in $1000's: %.3f" % sc_y.inverse_transform(price_std))
Price in $1000's: 10.840
from sklearn.linear_model import LinearRegression
slr = LinearRegression()
slr.fit(X, y)
y_pred = slr.predict(X)
print('Slope: %.3f' % slr.coef_[0])
print('Intercept: %.3f' % slr.intercept_)
Slope: 9.102 Intercept: -34.671
lin_regplot(X, y, slr)
plt.xlabel('Average number of rooms [RM]')
plt.ylabel('Price in $1000\'s [MEDV]')
plt.tight_layout()
# plt.savefig('./figures/scikit_lr_fit.png', dpi=300)
plt.show()
Normal Equations alternative:
# adding a column vector of "ones"
Xb = np.hstack((np.ones((X.shape[0], 1)), X))
w = np.zeros(X.shape[1])
z = np.linalg.inv(np.dot(Xb.T, Xb))
w = np.dot(z, np.dot(Xb.T, y))
print('Slope: %.3f' % w[1])
print('Intercept: %.3f' % w[0])
Slope: 9.102 Intercept: -34.671
from sklearn.linear_model import RANSACRegressor
if Version(sklearn_version) < '0.18':
ransac = RANSACRegressor(LinearRegression(),
max_trials=100,
min_samples=50,
residual_metric=lambda x: np.sum(np.abs(x), axis=1),
residual_threshold=5.0,
random_state=0)
else:
ransac = RANSACRegressor(LinearRegression(),
max_trials=100,
min_samples=50,
loss='absolute_loss',
residual_threshold=5.0,
random_state=0)
ransac.fit(X, y)
inlier_mask = ransac.inlier_mask_
outlier_mask = np.logical_not(inlier_mask)
line_X = np.arange(3, 10, 1)
line_y_ransac = ransac.predict(line_X[:, np.newaxis])
plt.scatter(X[inlier_mask], y[inlier_mask],
c='blue', marker='o', label='Inliers')
plt.scatter(X[outlier_mask], y[outlier_mask],
c='lightgreen', marker='s', label='Outliers')
plt.plot(line_X, line_y_ransac, color='red')
plt.xlabel('Average number of rooms [RM]')
plt.ylabel('Price in $1000\'s [MEDV]')
plt.legend(loc='upper left')
plt.tight_layout()
# plt.savefig('./figures/ransac_fit.png', dpi=300)
plt.show()
print('Slope: %.3f' % ransac.estimator_.coef_[0])
print('Intercept: %.3f' % ransac.estimator_.intercept_)
Slope: 9.621 Intercept: -37.137
if Version(sklearn_version) < '0.18':
from sklearn.cross_validation import train_test_split
else:
from sklearn.model_selection import train_test_split
X = df.iloc[:, :-1].values
y = df['MEDV'].values
X_train, X_test, y_train, y_test = train_test_split(
X, y, test_size=0.3, random_state=0)
slr = LinearRegression()
slr.fit(X_train, y_train)
y_train_pred = slr.predict(X_train)
y_test_pred = slr.predict(X_test)
plt.scatter(y_train_pred, y_train_pred - y_train,
c='blue', marker='o', label='Training data')
plt.scatter(y_test_pred, y_test_pred - y_test,
c='lightgreen', marker='s', label='Test data')
plt.xlabel('Predicted values')
plt.ylabel('Residuals')
plt.legend(loc='upper left')
plt.hlines(y=0, xmin=-10, xmax=50, lw=2, color='red')
plt.xlim([-10, 50])
plt.tight_layout()
# plt.savefig('./figures/slr_residuals.png', dpi=300)
plt.show()
from sklearn.metrics import r2_score
from sklearn.metrics import mean_squared_error
print('MSE train: %.3f, test: %.3f' % (
mean_squared_error(y_train, y_train_pred),
mean_squared_error(y_test, y_test_pred)))
print('R^2 train: %.3f, test: %.3f' % (
r2_score(y_train, y_train_pred),
r2_score(y_test, y_test_pred)))
MSE train: 19.958, test: 27.196 R^2 train: 0.765, test: 0.673
from sklearn.linear_model import Lasso
lasso = Lasso(alpha=0.1)
lasso.fit(X_train, y_train)
y_train_pred = lasso.predict(X_train)
y_test_pred = lasso.predict(X_test)
print(lasso.coef_)
[-0.11311792 0.04725111 -0.03992527 0.96478874 -0. 3.72289616 -0.02143106 -1.23370405 0.20469 -0.0129439 -0.85269025 0.00795847 -0.52392362]
print('MSE train: %.3f, test: %.3f' % (
mean_squared_error(y_train, y_train_pred),
mean_squared_error(y_test, y_test_pred)))
print('R^2 train: %.3f, test: %.3f' % (
r2_score(y_train, y_train_pred),
r2_score(y_test, y_test_pred)))
MSE train: 20.926, test: 28.876 R^2 train: 0.753, test: 0.653
X = np.array([258.0, 270.0, 294.0,
320.0, 342.0, 368.0,
396.0, 446.0, 480.0, 586.0])[:, np.newaxis]
y = np.array([236.4, 234.4, 252.8,
298.6, 314.2, 342.2,
360.8, 368.0, 391.2,
390.8])
from sklearn.preprocessing import PolynomialFeatures
lr = LinearRegression()
pr = LinearRegression()
quadratic = PolynomialFeatures(degree=2)
X_quad = quadratic.fit_transform(X)
# fit linear features
lr.fit(X, y)
X_fit = np.arange(250, 600, 10)[:, np.newaxis]
y_lin_fit = lr.predict(X_fit)
# fit quadratic features
pr.fit(X_quad, y)
y_quad_fit = pr.predict(quadratic.fit_transform(X_fit))
# plot results
plt.scatter(X, y, label='training points')
plt.plot(X_fit, y_lin_fit, label='linear fit', linestyle='--')
plt.plot(X_fit, y_quad_fit, label='quadratic fit')
plt.legend(loc='upper left')
plt.tight_layout()
# plt.savefig('./figures/poly_example.png', dpi=300)
plt.show()
y_lin_pred = lr.predict(X)
y_quad_pred = pr.predict(X_quad)
print('Training MSE linear: %.3f, quadratic: %.3f' % (
mean_squared_error(y, y_lin_pred),
mean_squared_error(y, y_quad_pred)))
print('Training R^2 linear: %.3f, quadratic: %.3f' % (
r2_score(y, y_lin_pred),
r2_score(y, y_quad_pred)))
Training MSE linear: 569.780, quadratic: 61.330 Training R^2 linear: 0.832, quadratic: 0.982
X = df[['LSTAT']].values
y = df['MEDV'].values
regr = LinearRegression()
# create quadratic features
quadratic = PolynomialFeatures(degree=2)
cubic = PolynomialFeatures(degree=3)
X_quad = quadratic.fit_transform(X)
X_cubic = cubic.fit_transform(X)
# fit features
X_fit = np.arange(X.min(), X.max(), 1)[:, np.newaxis]
regr = regr.fit(X, y)
y_lin_fit = regr.predict(X_fit)
linear_r2 = r2_score(y, regr.predict(X))
regr = regr.fit(X_quad, y)
y_quad_fit = regr.predict(quadratic.fit_transform(X_fit))
quadratic_r2 = r2_score(y, regr.predict(X_quad))
regr = regr.fit(X_cubic, y)
y_cubic_fit = regr.predict(cubic.fit_transform(X_fit))
cubic_r2 = r2_score(y, regr.predict(X_cubic))
# plot results
plt.scatter(X, y, label='training points', color='lightgray')
plt.plot(X_fit, y_lin_fit,
label='linear (d=1), $R^2=%.2f$' % linear_r2,
color='blue',
lw=2,
linestyle=':')
plt.plot(X_fit, y_quad_fit,
label='quadratic (d=2), $R^2=%.2f$' % quadratic_r2,
color='red',
lw=2,
linestyle='-')
plt.plot(X_fit, y_cubic_fit,
label='cubic (d=3), $R^2=%.2f$' % cubic_r2,
color='green',
lw=2,
linestyle='--')
plt.xlabel('% lower status of the population [LSTAT]')
plt.ylabel('Price in $1000\'s [MEDV]')
plt.legend(loc='upper right')
plt.tight_layout()
# plt.savefig('./figures/polyhouse_example.png', dpi=300)
plt.show()
Transforming the dataset:
X = df[['LSTAT']].values
y = df['MEDV'].values
# transform features
X_log = np.log(X)
y_sqrt = np.sqrt(y)
# fit features
X_fit = np.arange(X_log.min()-1, X_log.max()+1, 1)[:, np.newaxis]
regr = regr.fit(X_log, y_sqrt)
y_lin_fit = regr.predict(X_fit)
linear_r2 = r2_score(y_sqrt, regr.predict(X_log))
# plot results
plt.scatter(X_log, y_sqrt, label='training points', color='lightgray')
plt.plot(X_fit, y_lin_fit,
label='linear (d=1), $R^2=%.2f$' % linear_r2,
color='blue',
lw=2)
plt.xlabel('log(% lower status of the population [LSTAT])')
plt.ylabel('$\sqrt{Price \; in \; \$1000\'s [MEDV]}$')
plt.legend(loc='lower left')
plt.tight_layout()
# plt.savefig('./figures/transform_example.png', dpi=300)
plt.show()
...
from sklearn.tree import DecisionTreeRegressor
X = df[['LSTAT']].values
y = df['MEDV'].values
tree = DecisionTreeRegressor(max_depth=3)
tree.fit(X, y)
sort_idx = X.flatten().argsort()
lin_regplot(X[sort_idx], y[sort_idx], tree)
plt.xlabel('% lower status of the population [LSTAT]')
plt.ylabel('Price in $1000\'s [MEDV]')
# plt.savefig('./figures/tree_regression.png', dpi=300)
plt.show()
X = df.iloc[:, :-1].values
y = df['MEDV'].values
X_train, X_test, y_train, y_test = train_test_split(
X, y, test_size=0.4, random_state=1)
from sklearn.ensemble import RandomForestRegressor
forest = RandomForestRegressor(n_estimators=1000,
criterion='mse',
random_state=1,
n_jobs=-1)
forest.fit(X_train, y_train)
y_train_pred = forest.predict(X_train)
y_test_pred = forest.predict(X_test)
print('MSE train: %.3f, test: %.3f' % (
mean_squared_error(y_train, y_train_pred),
mean_squared_error(y_test, y_test_pred)))
print('R^2 train: %.3f, test: %.3f' % (
r2_score(y_train, y_train_pred),
r2_score(y_test, y_test_pred)))
MSE train: 1.642, test: 11.052 R^2 train: 0.979, test: 0.878
plt.scatter(y_train_pred,
y_train_pred - y_train,
c='black',
marker='o',
s=35,
alpha=0.5,
label='Training data')
plt.scatter(y_test_pred,
y_test_pred - y_test,
c='lightgreen',
marker='s',
s=35,
alpha=0.7,
label='Test data')
plt.xlabel('Predicted values')
plt.ylabel('Residuals')
plt.legend(loc='upper left')
plt.hlines(y=0, xmin=-10, xmax=50, lw=2, color='red')
plt.xlim([-10, 50])
plt.tight_layout()
# plt.savefig('./figures/slr_residuals.png', dpi=300)
plt.show()
...