7. Sparse Kernel Machines¶
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
from prml.kernel import (
RBF,
PolynomialKernel,
SupportVectorClassifier,
RelevanceVectorRegressor,
RelevanceVectorClassifier
)
np.random.seed(1234)
7.1 Maximum Margin Classifiers¶
x_train = np.array([
[0., 2.],
[2., 0.],
[-1., -1.]])
y_train = np.array([1., 1., -1.])
model = SupportVectorClassifier(PolynomialKernel(degree=1))
model.fit(x_train, y_train)
x0, x1 = np.meshgrid(np.linspace(-3, 3, 100), np.linspace(-3, 3, 100))
x = np.array([x0, x1]).reshape(2, -1).T
plt.scatter(x_train[:, 0], x_train[:, 1], s=40, c=y_train, marker="x")
plt.scatter(model.X[:, 0], model.X[:, 1], s=100, facecolor="none", edgecolor="g")
cp = plt.contour(x0, x1, model.distance(x).reshape(100, 100), np.array([-1, 0, 1]), colors="k", linestyles=("dashed", "solid", "dashed"))
plt.clabel(cp, fmt='y=%.f', inline=True, fontsize=15)
plt.xlim(-3, 3)
plt.ylim(-3, 3)
plt.gca().set_aspect("equal", adjustable="box")
def create_toy_data():
x = np.random.uniform(-1, 1, 100).reshape(-1, 2)
y = x < 0
y = (y[:, 0] * y[:, 1]).astype(np.float)
return x, 1 - 2 * y
x_train, y_train = create_toy_data()
model = SupportVectorClassifier(RBF(np.ones(3)))
model.fit(x_train, y_train)
x0, x1 = np.meshgrid(np.linspace(-1, 1, 100), np.linspace(-1, 1, 100))
x = np.array([x0, x1]).reshape(2, -1).T
plt.scatter(x_train[:, 0], x_train[:, 1], s=40, c=y_train, marker="x")
plt.scatter(model.X[:, 0], model.X[:, 1], s=100, facecolor="none", edgecolor="g")
plt.contour(
x0, x1, model.distance(x).reshape(100, 100),
np.arange(-1, 2), colors="k", linestyles=("dashed", "solid", "dashed"))
plt.xlim(-1, 1)
plt.ylim(-1, 1)
plt.gca().set_aspect("equal", adjustable="box")
7.1.1 Overlapping class distributions¶
def create_toy_data():
x0 = np.random.normal(size=100).reshape(-1, 2) - 1.
x1 = np.random.normal(size=100).reshape(-1, 2) + 1.
x = np.concatenate([x0, x1])
y = np.concatenate([-np.ones(50), np.ones(50)]).astype(np.int)
return x, y
x_train, y_train = create_toy_data()
model = SupportVectorClassifier(RBF(np.array([1., 0.5, 0.5])), C=1.)
model.fit(x_train, y_train)
x0, x1 = np.meshgrid(np.linspace(-4, 4, 100), np.linspace(-4, 4, 100))
x = np.array([x0, x1]).reshape(2, -1).T
plt.scatter(x_train[:, 0], x_train[:, 1], s=40, c=y_train, marker="x")
plt.scatter(model.X[:, 0], model.X[:, 1], s=100, facecolor="none", edgecolor="g")
plt.contour(x0, x1, model.distance(x).reshape(100, 100), np.arange(-1, 2), colors="k", linestyles=("dashed", "solid", "dashed"))
plt.xlim(-4, 4)
plt.ylim(-4, 4)
plt.gca().set_aspect("equal", adjustable="box")
7.2 Relevance Vector Machines¶
7.2.1 RVM for regression¶
def create_toy_data(n=10):
x = np.linspace(0, 1, n)
t = np.sin(2 * np.pi * x) + np.random.normal(scale=0.1, size=n)
return x, t
x_train, y_train = create_toy_data(n=10)
x = np.linspace(0, 1, 100)
model = RelevanceVectorRegressor(RBF(np.array([1., 20.])))
model.fit(x_train, y_train)
y, y_std = model.predict(x)
plt.scatter(x_train, y_train, facecolor="none", edgecolor="g", label="training")
plt.scatter(model.X.ravel(), model.t, s=100, facecolor="none", edgecolor="b", label="relevance vector")
plt.plot(x, y, color="r", label="predict mean")
plt.fill_between(x, y - y_std, y + y_std, color="pink", alpha=0.2, label="predict std.")
plt.legend(loc="best")
plt.show()
7.2.3 RVM for classification¶
def create_toy_data():
x0 = np.random.normal(size=100).reshape(-1, 2) - 1.
x1 = np.random.normal(size=100).reshape(-1, 2) + 1.
x = np.concatenate([x0, x1])
y = np.concatenate([np.zeros(50), np.ones(50)]).astype(np.int)
return x, y
x_train, y_train = create_toy_data()
model = RelevanceVectorClassifier(RBF(np.array([1., 0.5, 0.5])))
model.fit(x_train, y_train)
x0, x1 = np.meshgrid(np.linspace(-4, 4, 100), np.linspace(-4, 4, 100))
x = np.array([x0, x1]).reshape(2, -1).T
plt.scatter(x_train[:, 0], x_train[:, 1], s=40, c=y_train, marker="x")
plt.scatter(model.X[:, 0], model.X[:, 1], s=100, facecolor="none", edgecolor="g")
plt.contourf(x0, x1, model.predict_proba(x).reshape(100, 100), np.linspace(0, 1, 5), alpha=0.2)
plt.colorbar()
plt.xlim(-4, 4)
plt.ylim(-4, 4)
plt.gca().set_aspect("equal", adjustable="box")
