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%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt
All linear models for regression learn a coefficient parameter coef_ and an offset intercept_ to make predictions using a linear combination of features:
y_pred = x_test[0] * coef_[0] + ... + x_test[n_features-1] * coef_[n_features-1] + intercept_The difference between the linear models for regression is what kind of restrictions or penalties are put on coef_ as regularization , in addition to fitting the training data well.
The most standard linear model is the 'ordinary least squares regression', often simply called 'linear regression'. It doesn't put any additional restrictions on coef_, so when the number of features is large, it becomes ill-posed and the model overfits.
Let us generate a simple simulation, to see the behavior of these models.
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from sklearn.datasets import make_regression
from sklearn.model_selection import train_test_split
X, y, true_coefficient = make_regression(n_samples=200, n_features=30, n_informative=10, noise=100, coef=True, random_state=5)
X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=5, train_size=60, test_size=140)
print(X_train.shape)
print(y_train.shape)
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from sklearn.linear_model import LinearRegression
linear_regression = LinearRegression().fit(X_train, y_train)
print("R^2 on training set: %f" % linear_regression.score(X_train, y_train))
print("R^2 on test set: %f" % linear_regression.score(X_test, y_test))
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from sklearn.metrics import r2_score
print(r2_score(np.dot(X, true_coefficient), y))
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plt.figure(figsize=(10, 5))
coefficient_sorting = np.argsort(true_coefficient)[::-1]
plt.plot(true_coefficient[coefficient_sorting], "o", label="true")
plt.plot(linear_regression.coef_[coefficient_sorting], "o", label="linear regression")
plt.legend()
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from sklearn.model_selection import learning_curve
def plot_learning_curve(est, X, y):
training_set_size, train_scores, test_scores = learning_curve(est, X, y, train_sizes=np.linspace(.1, 1, 20))
estimator_name = est.__class__.__name__
line = plt.plot(training_set_size, train_scores.mean(axis=1), '--', label="training scores " + estimator_name)
plt.plot(training_set_size, test_scores.mean(axis=1), '-', label="test scores " + estimator_name, c=line[0].get_color())
plt.xlabel('Training set size')
plt.legend(loc='best')
plt.ylim(-0.1, 1.1)
plt.figure()
plot_learning_curve(LinearRegression(), X, y)
The Ridge estimator is a simple regularization (called l2 penalty) of the ordinary LinearRegression. In particular, it has the benefit of being not computationally more expensive than the ordinary least square estimate.
$$ \text{min}_{w,b} \sum_i || w^\mathsf{T}x_i + b - y_i||^2 + \alpha ||w||_2^2$$The amount of regularization is set via the alpha parameter of the Ridge.
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from sklearn.linear_model import Ridge
ridge_models = {}
training_scores = []
test_scores = []
for alpha in [100, 10, 1, .01]:
ridge = Ridge(alpha=alpha).fit(X_train, y_train)
training_scores.append(ridge.score(X_train, y_train))
test_scores.append(ridge.score(X_test, y_test))
ridge_models[alpha] = ridge
plt.figure()
plt.plot(training_scores, label="training scores")
plt.plot(test_scores, label="test scores")
plt.xticks(range(4), [100, 10, 1, .01])
plt.legend(loc="best")
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plt.figure(figsize=(10, 5))
plt.plot(true_coefficient[coefficient_sorting], "o", label="true", c='b')
for i, alpha in enumerate([100, 10, 1, .01]):
plt.plot(ridge_models[alpha].coef_[coefficient_sorting], "o", label="alpha = %.2f" % alpha, c=plt.cm.summer(i / 3.))
plt.legend(loc="best")
Tuning alpha is critical for performance.
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plt.figure()
plot_learning_curve(LinearRegression(), X, y)
plot_learning_curve(Ridge(alpha=10), X, y)
The Lasso estimator is useful to impose sparsity on the coefficient. In other words, it is to be prefered if we believe that many of the features are not relevant. This is done via the so-called l1 penalty.
$$ \text{min}_{w, b} \sum_i \frac{1}{2} || w^\mathsf{T}x_i + b - y_i||^2 + \alpha ||w||_1$$
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from sklearn.linear_model import Lasso
lasso_models = {}
training_scores = []
test_scores = []
for alpha in [30, 10, 1, .01]:
lasso = Lasso(alpha=alpha).fit(X_train, y_train)
training_scores.append(lasso.score(X_train, y_train))
test_scores.append(lasso.score(X_test, y_test))
lasso_models[alpha] = lasso
plt.figure()
plt.plot(training_scores, label="training scores")
plt.plot(test_scores, label="test scores")
plt.xticks(range(4), [30, 10, 1, .01])
plt.legend(loc="best")
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plt.figure(figsize=(10, 5))
plt.plot(true_coefficient[coefficient_sorting], "o", label="true", c='b')
for i, alpha in enumerate([30, 10, 1, .01]):
plt.plot(lasso_models[alpha].coef_[coefficient_sorting], "o", label="alpha = %.2f" % alpha, c=plt.cm.summer(i / 3.))
plt.legend(loc="best")
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plt.figure(figsize=(10, 5))
plot_learning_curve(LinearRegression(), X, y)
plot_learning_curve(Ridge(alpha=10), X, y)
plot_learning_curve(Lasso(alpha=10), X, y)
Instead of picking Ridge or Lasso, you can also use ElasticNet, which uses both forms of regularization and provides a parameter to assign a weighting between them. ElasticNet typically performs the best amongst these models.
All linear models for classification learn a coefficient parameter coef_ and an offset intercept_ to make predictions using a linear combination of features:
y_pred = x_test[0] * coef_[0] + ... + x_test[n_features-1] * coef_[n_features-1] + intercept_ > 0As you can see, this is very similar to regression, only that a threshold at zero is applied.
Again, the difference between the linear models for classification what kind of regularization is put on coef_ and intercept_, but there are also minor differences in how the fit to the training set is measured (the so-called loss function).
The two most common models for linear classification are the linear SVM as implemented in LinearSVC and LogisticRegression.
A good intuition for regularization of linear classifiers is that with high regularization, it is enough if most of the points are classified correctly. But with less regularization, more importance is given to each individual data point.
This is illustrated using an linear SVM with different values of C below.
In LinearSVC, the C parameter controls the regularization within the model.
Lower C entails more regularization and simpler models, whereas higher C entails less regularization and more influence from individual data points.
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from figures import plot_linear_svc_regularization
plot_linear_svc_regularization()
Similar to the Ridge/Lasso separation, you can set the penalty parameter to 'l1' to enforce sparsity of the coefficients (similar to Lasso) or 'l2' to encourage smaller coefficients (similar to Ridge).
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from sklearn.datasets import make_blobs
plt.figure()
X, y = make_blobs(random_state=42)
plt.scatter(X[:, 0], X[:, 1], c=plt.cm.spectral(y / 2.));
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from sklearn.svm import LinearSVC
linear_svm = LinearSVC().fit(X, y)
print(linear_svm.coef_.shape)
print(linear_svm.intercept_.shape)
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plt.scatter(X[:, 0], X[:, 1], c=plt.cm.spectral(y / 2.))
line = np.linspace(-15, 15)
for coef, intercept in zip(linear_svm.coef_, linear_svm.intercept_):
plt.plot(line, -(line * coef[0] + intercept) / coef[1])
plt.ylim(-10, 15)
plt.xlim(-10, 8);
Points are classified in a one-vs-rest fashion (aka one-vs-all), where we assign a test point to the class whose model has the highest confidence (in the SVM case, highest distance to the separating hyperplane) for the test point.
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from sklearn.datasets import load_digits
from sklearn.linear_model import LogisticRegression
digits = load_digits()
X_digits, y_digits = digits.data, digits.target
# split the dataset, apply grid-search
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# %load solutions/17A_logreg_grid.py
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# %load solutions/17B_learning_curve_alpha.py