import numpy as np
import matplotlib.pyplot as plt
from prml.features import PolynomialFeatures
from prml.linear import (
LinearRegressor,
RidgeRegressor,
BayesianRegressor
)
feature = PolynomialFeatures(9)
X_train = feature.transform(x_train)
X_test = feature.transform(x_test)
model = RidgeRegressor(alpha=1e-3)
model.fit(X_train, y_train)
y = model.predict(X_test)
y = model.predict(X_test)
plt.scatter(x_train, y_train, facecolor="none", edgecolor="b", s=50, label="training data")
plt.plot(x_test, y_test, c="g", label="$\sin(2\pi x)$")
plt.plot(x_test, y, c="r", label="fitting")
plt.ylim(-1.5, 1.5)
plt.legend()
plt.annotate("M=9", xy=(-0.15, 1))
plt.show()
def test_transform(self):
self.features = PolynomialFeatures()
x = np.array([[0., 1.], [2., 3.]])
X = np.array([[1., 0., 1., 0., 0., 1.], [1., 2., 3., 4., 6., 9.]])
self.assertTrue(np.allclose(self.features.transform(x), X))
def create_toy_data(func, sample_size, std, domain=[0, 1]):
x = np.linspace(domain[0], domain[1], sample_size)
np.random.shuffle(x)
t = func(x) + np.random.normal(scale=std, size=x.shape)
return x, t
def cubic(x):
return x * (x - 5) * (x + 5)
x_train, y_train = create_toy_data(cubic, 10, 10., [-5, 5])
x = np.linspace(-5, 5, 100)
y = cubic(x)
feature = PolynomialFeatures(degree=3)
X_train = feature.transform(x_train)
X = feature.transform(x)
vlr = VariationalLinearRegressor(beta=0.01)
vlr.fit(X_train, y_train)
y_mean, y_std = vlr.predict(X, return_std=True)
plt.scatter(x_train, y_train, s=100, facecolor="none", edgecolor="b")
plt.plot(x, y, c="g", label="$\sin(2\pi x)$")
plt.plot(x, y_mean, c="r", label="prediction")
plt.fill_between(x, y_mean - y_std, y_mean + y_std, alpha=0.2, color="pink")
plt.legend()
plt.show()
# variational logistic regression
x_test = np.linspace(0, 1, 100)
y_test = func(x_test)
plt.scatter(x_train,
y_train,
facecolor="none",
edgecolor="b",
s=50,
label="training data")
plt.plot(x_test, y_test, c="g", label="$\sin(2\pi x)$")
plt.legend()
plt.show()
for i, degree in enumerate([0, 1, 3, 9]):
plt.subplot(2, 2, i + 1)
feature = PolynomialFeatures(degree)
X_train = feature.transform(x_train)
X_test = feature.transform(x_test)
model = LinearRegressor()
model.fit(X_train, y_train)
y = model.predict(X_test)
plt.scatter(x_train,
y_train,
facecolor="none",
edgecolor="b",
s=50,
label="training data")
plt.plot(x_test, y_test, c="g", label="$\sin(2\pi x)$")
plt.plot(x_test, y, c="r", label="fitting")
from prml.linear import (BayesianRegressor, EmpiricalBayesRegressor,
LinearRegressor, RidgeRegressor)
np.random.seed(1234)
def create_toy_data(func, sample_size, std, domain=[0, 1]):
x = np.linspace(domain[0], domain[1], sample_size) # 范围从零到1
np.random.shuffle(x)
t = func(x) + np.random.normal(scale=std, size=x.shape)
return x, t
# linear basis function models 线性基础函数模型
x = np.linspace(-1, 1, 100)
X_polynomial = PolynomialFeatures(11).transform(x[:, None])
X_gaussian = GaussianFeatures(np.linspace(-1, 1, 11), 0.1).transform(x)
X_sigmoidal = SigmoidalFeatures(np.linspace(-1, 1, 11), 10).transform(x)
plt.figure(figsize=(20, 5))
for i, X in enumerate([X_polynomial, X_gaussian, X_sigmoidal]):
plt.subplot(1, 3, i + 1)
for j in range(12):
plt.plot(x, X[:, j]) # 从左到右，依次是多项式拟合， 高斯拟合， sigmoid拟合， 而且每个都拟合了12次
# maximum likelihood and least squares 极大似然估计和最小二乘
def sinusoidal(x):
return np.sin(2 * np.pi * x)