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01-linear-regression.py
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01-linear-regression.py
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"""
Regularization and Overfitting
(Overfitting):
A very common problem in machine learning
When the model is much more complex than it shoud be (i.e. for using a lot of features)
it may perform very well on the training data,
but it performs very badly for new (unseen) data.
In such situations, the model can not generalize well.
(Regularization):
- An effective way to avoid (or at least to reduce) overfitting.
Linerar regretion, coast function
Given 𝑋,𝑦,𝜃, compute 𝐽(𝜃).
"""
import numpy as np
import matplotlib.pyplot as plt
from sklearn.pipeline import Pipeline
from sklearn.linear_model import LinearRegression, Ridge, Lasso
from sklearn.preprocessing import PolynomialFeatures
from sklearn.model_selection import cross_val_score
import warnings
# initial setup
plt.rcParams['figure.figsize'] = [12, 6]
plt.rcParams['figure.dpi'] = 150
plt.style.use('ggplot')
np.random.seed(0)
np.set_printoptions(precision=2, linewidth=100)
warnings.filterwarnings(action='ignore')
# initial setup
plt.rcParams['figure.figsize'] = [12, 6]
plt.rcParams['figure.dpi'] = 150
plt.style.use('ggplot')
np.random.seed(0)
np.set_printoptions(precision=2, linewidth=100)
warnings.filterwarnings(action='ignore')
def f(x):
return np.cos(1.5 * np.pi * x)
def generate_data(n_samples=30):
x = np.sort(np.random.rand(n_samples))
y = f(x) + 0.1 * np.random.randn(n_samples)
return x, y
n_samples = 30 # number of data samples
x, y = generate_data(n_samples)
# plot data
plt.figure()
plt.scatter(x, y, s=50, edgecolors='k', alpha=1, cmap=plt.cm.coolwarm)
plt.xlim(0, 1)
plt.ylim(-2, 2)
plt.show()
#%%
### Polynomial Regression
def fit_poly(x, y, degree=1):
# add polynomial features
polynomial_features = PolynomialFeatures(degree=degree, include_bias=False)
# create and fit the model
linear_regression = LinearRegression()
model = Pipeline([("polynomial_features", polynomial_features), ("linear_regression", linear_regression)])
model.fit(x[:, None], y)
return model
degrees = [1, 2, 3, 4, 5, 6, 7, 15]
plt.figure()
for d in degrees:
model = fit_poly(x, y, degree=d)
scores = cross_val_score(model, x[:, None], y, scoring="neg_mean_squared_error", cv=10)
# plot data and model
plt.subplot(2, 4, degrees.index(d) + 1)
plt.tight_layout()
x_test = np.linspace(0, 1, 100)
plt.plot(x_test, f(x_test), 'r--', label="Target", alpha=0.5)
plt.scatter(x, y, s=15, edgecolor='k', alpha=0.5, label="Samples")
plt.plot(x_test, model.predict(x_test[:, None]), 'k', lw=2, label="Predicted")
plt.xlim((0, 1))
plt.ylim((-2, 2))
plt.title("Deg.={}, MSE={:.2e}".format(d, -scores.mean()), fontsize=10)
plt.show()
#%%
def plot_coef(theta):
plt.figure()
plt.bar(np.arange(1, len(theta) + 1), height=np.abs(theta))
plt.show()
plot_coef(model.steps[1][1].coef_)
### L2-Regularizarion (Ridge)
def fit_poly_L2_reg(degree=1, lmbda=1.0):
# add polynomial features
polynomial_features = PolynomialFeatures(degree=degree, include_bias=False)
# create and fit the model
linear_regression = Ridge(alpha=lmbda)
model = Pipeline([("polynomial_features", polynomial_features), ("linear_regression", linear_regression)])
model.fit(x[:, None], y)
return model
#%%
lmbda = 1e-2
plt.figure()
for d in degrees:
model = fit_poly_L2_reg(degree=d, lmbda=lmbda)
scores = cross_val_score(model, x[:, None], y, scoring="neg_mean_squared_error", cv=10)
# plot data and model
plt.subplot(2, 4, degrees.index(d) + 1)
plt.tight_layout()
x_test = np.linspace(0, 1, 100)
plt.plot(x_test, f(x_test), 'r--', label="Target", alpha=0.5)
plt.scatter(x, y, s=15, edgecolor='k', alpha=0.5, label="Samples")
plt.plot(x_test, model.predict(x_test[:, None]), 'k', lw=2, label="Predicted")
plt.xlim((0, 1))
plt.ylim((-2, 2))
plt.title("Degree = {}, MSE={:.2f}".format(d, -scores.mean()), fontsize=10)
plt.show()
#%%
lmbdas = [1e-10, 1e-6, 1e-4, 1e-2, 1e-1, 1, 10, 100]
plt.figure()
for lmbda in lmbdas:
model = fit_poly_L2_reg(degree=d, lmbda=lmbda)
scores = cross_val_score(model, x[:, None], y, scoring="neg_mean_squared_error", cv=10)
# plot data and model
plt.subplot(2, 4, lmbdas.index(lmbda) + 1)
plt.tight_layout()
x_test = np.linspace(0, 1, 100)
plt.plot(x_test, f(x_test), 'r--', label="Target", alpha=0.5)
plt.scatter(x, y, s=15, edgecolor='k', alpha=0.5, label="Samples")
plt.plot(x_test, model.predict(x_test[:, None]), 'k', lw=2, label="Predicted")
plt.xlim((0, 1))
plt.ylim((-2, 2))
plt.title("$\lambda$ = {}, MSE={:.2f}".format(lmbda, -scores.mean()), fontsize=10)
plt.show()
#%%
#plot_coef(model.steps[1][1].coef_)
#%%
degree = 15
lmbda = 1e-3
# fit
model = fit_poly_L2_reg(degree, lmbda)
scores = cross_val_score(model, x[:, None], y, scoring="neg_mean_squared_error", cv=10)
# plot
fig, ax = plt.subplots(1)
x_test = np.linspace(0, 1, 100)
ax.plot(x_test, f(x_test), 'r--', label="Target", alpha=0.5)
ax.scatter(x, y, s=50, edgecolor='k', alpha=0.5, label="Samples")
ax.plot(x_test, model.predict(x_test[:, None]), 'k', lw=2, label="Predicted")
ax.set_xlim((0, 1))
ax.set_ylim((-2, 2))
ax.set_title("d = %d, $\lambda$ = %s, cost = %.2f" % (degree, lmbda, -scores.mean()), fontsize=12)
plt.show()
#%%
#plt.plot_coef(model.steps[1][1].coef_)
#%%
###L1-Regularizarion (Lasso)
def fit_poly_L1_reg(degree=1, lmbda=1.0):
# add polynomial features up to degree
polynomial_features = PolynomialFeatures(degree=degree, include_bias=False)
l1_regression = Lasso(alpha=lmbda)
model = Pipeline([("poly", polynomial_features), ("l1_reg", l1_regression)])
# create and fit the model
model.fit(x[:, None], y)
return model
lmbda = 1e-2
plt.figure()
for d in degrees:
model = fit_poly_L1_reg(degree=d, lmbda=lmbda)
scores = cross_val_score(model, x[:, None], y, scoring="neg_mean_squared_error", cv=10)
# plot data and model
plt.subplot(2, 4, degrees.index(d) + 1)
plt.tight_layout()
x_test = np.linspace(0, 1, 100)
plt.plot(x_test, f(x_test), 'r--', label="Target", alpha=0.5)
plt.scatter(x, y, s=15, edgecolor='k', alpha=0.5, label="Samples")
plt.plot(x_test, model.predict(x_test[:, None]), 'k', lw=2, label="Predicted")
plt.xlim((0, 1))
plt.ylim((-2, 2))
plt.title("Degree = {}, MSE={:.2f}".format(d, -scores.mean()), fontsize=10)
plt.show()
#%%
#plot_coef(model.steps[1][1].coef_)
#%%
###Visualizing effect of lambda
lmbdas = [1e-10, 1e-3, 1e-2, 2e-2, 1e-1, 1, 10, 100]
plt.figure()
for lmbda in lmbdas:
model = fit_poly_L1_reg(degree=15, lmbda=lmbda)
# plot data and model
scores = cross_val_score(model, x[:, None], y, scoring="neg_mean_squared_error", cv=10)
# plot data and model
plt.subplot(2, 4, lmbdas.index(lmbda) + 1)
plt.tight_layout()
x_test = np.linspace(0, 1, 100)
plt.plot(x_test, f(x_test), 'r--', label="Target", alpha=0.5)
plt.scatter(x, y, s=15, edgecolor='k', alpha=0.5, label="Samples")
plt.plot(x_test, model.predict(x_test[:, None]), 'k', lw=2, label="Predicted")
plt.xlim((0, 1))
plt.ylim((-2, 2))
plt.title("$\lambda$ = {}, MSE={:.2f}".format(lmbda, -scores.mean()), fontsize=10)
plt.show()
#%%
#plot_coef(model.steps[1][1].coef_)
#%%
model = fit_poly_L1_reg(degree=15, lmbda=0.001)
#plot_coef(model.steps[1][1].coef_)
degree = 15
lmbda = 1e-3
# fit
model = fit_poly_L1_reg(degree=degree, lmbda=lmbda)
scores = cross_val_score(model, x[:, None], y, scoring="neg_mean_squared_error", cv=10)
# plot
fig, ax = plt.subplots(1)
x_test = np.linspace(0, 1, 100)
ax.plot(x_test, f(x_test), 'r--', label="Target", alpha=0.5)
ax.scatter(x, y, s=50, edgecolor='k', alpha=0.5, label="Samples")
ax.plot(x_test, model.predict(x_test[:, None]), 'k', lw=2, label="Predicted")
ax.set_xlim((0, 1))
ax.set_ylim((-2, 2))
ax.set_title("d = %d, $\lambda$ = %s, cost = %.2f" % (degree, lmbda, -scores.mean()), fontsize=12)
plt.show()
####Classification with Regularization
from sklearn.datasets import make_moons
from sklearn.linear_model import LogisticRegression
from plot_2d_separator import plot_2d_separator
# create random data
X, y = make_moons(n_samples=120, noise=0.25, random_state=0)
# plot data
plt.figure()
plt.scatter(X[:, 0], X[:, 1], c=y, alpha=0.5, edgecolors='k', cmap=plt.cm.coolwarm)
plt.show()
degree = 7
coeffs = [1e1, 1e2, 1e3, 1e4, 1e5, 1e6, 1e7, 1e8]
plt.figure()
for C in coeffs:
# create logistic regression classifier
plt.subplot(2, 4, coeffs.index(C) + 1)
plt.tight_layout()
poly_features = PolynomialFeatures(degree=degree, include_bias=False)
log_reg = LogisticRegression(C=C)
model = Pipeline([("poly_features", poly_features), ("logistic_regression", log_reg)])
# train classifier
model.fit(X, y)
accuracy = model.score(X, y)
# plot classification results
title = "C = {:.2e} ({:.2f}%)"
plot_2d_separator(model, X, fill=True)
plt.scatter(X[:, 0], X[:, 1], s=15, c=y, alpha=0.5, edgecolors='k', cmap=plt.cm.coolwarm)
plt.title(title.format(C, accuracy * 100), fontsize=10)
plt.show()