自動相關性確定回歸(ARD)?
擬合回歸模型與貝葉斯嶺回歸。
有關回歸者的更多信息，請參見貝葉斯嶺回歸。
與普通最小二乘估計相比，系數權值略有移向零，從而使其穩定。
估計的權重的直方圖是非常尖頂的，因為在權重上隱含了稀疏性的先驗。
模型的估計是通過迭代最大化觀測的邊際對數似然來實現的。
我們還用多項式特征擴展法繪制了一維回歸ARD的預測和不確定性圖。請注意，在右邊的圖中不確定性開始上升。這是因為這些測試樣本超出了訓練樣本的范圍。
print(__doc__)
import numpy as np
import matplotlib.pyplot as plt
from scipy import stats
from sklearn.linear_model import ARDRegression, LinearRegression
# #############################################################################
# Generating simulated data with Gaussian weights
# Parameters of the example
np.random.seed(0)
n_samples, n_features = 100, 100
# Create Gaussian data
X = np.random.randn(n_samples, n_features)
# Create weights with a precision lambda_ of 4.
lambda_ = 4.
w = np.zeros(n_features)
# Only keep 10 weights of interest
relevant_features = np.random.randint(0, n_features, 10)
for i in relevant_features:
w[i] = stats.norm.rvs(loc=0, scale=1. / np.sqrt(lambda_))
# Create noise with a precision alpha of 50.
alpha_ = 50.
noise = stats.norm.rvs(loc=0, scale=1. / np.sqrt(alpha_), size=n_samples)
# Create the target
y = np.dot(X, w) + noise
# #############################################################################
# Fit the ARD Regression
clf = ARDRegression(compute_score=True)
clf.fit(X, y)
ols = LinearRegression()
ols.fit(X, y)
# #############################################################################
# Plot the true weights, the estimated weights, the histogram of the
# weights, and predictions with standard deviations
plt.figure(figsize=(6, 5))
plt.title("Weights of the model")
plt.plot(clf.coef_, color='darkblue', linestyle='-', linewidth=2,
label="ARD estimate")
plt.plot(ols.coef_, color='yellowgreen', linestyle=':', linewidth=2,
label="OLS estimate")
plt.plot(w, color='orange', linestyle='-', linewidth=2, label="Ground truth")
plt.xlabel("Features")
plt.ylabel("Values of the weights")
plt.legend(loc=1)
plt.figure(figsize=(6, 5))
plt.title("Histogram of the weights")
plt.hist(clf.coef_, bins=n_features, color='navy', log=True)
plt.scatter(clf.coef_[relevant_features], np.full(len(relevant_features), 5.),
color='gold', marker='o', label="Relevant features")
plt.ylabel("Features")
plt.xlabel("Values of the weights")
plt.legend(loc=1)
plt.figure(figsize=(6, 5))
plt.title("Marginal log-likelihood")
plt.plot(clf.scores_, color='navy', linewidth=2)
plt.ylabel("Score")
plt.xlabel("Iterations")
# Plotting some predictions for polynomial regression
def f(x, noise_amount):
y = np.sqrt(x) * np.sin(x)
noise = np.random.normal(0, 1, len(x))
return y + noise_amount * noise
degree = 10
X = np.linspace(0, 10, 100)
y = f(X, noise_amount=1)
clf_poly = ARDRegression(threshold_lambda=1e5)
clf_poly.fit(np.vander(X, degree), y)
X_plot = np.linspace(0, 11, 25)
y_plot = f(X_plot, noise_amount=0)
y_mean, y_std = clf_poly.predict(np.vander(X_plot, degree), return_std=True)
plt.figure(figsize=(6, 5))
plt.errorbar(X_plot, y_mean, y_std, color='navy',
label="Polynomial ARD", linewidth=2)
plt.plot(X_plot, y_plot, color='gold', linewidth=2,
label="Ground Truth")
plt.ylabel("Output y")
plt.xlabel("Feature X")
plt.legend(loc="lower left")
plt.show()
腳本的總運行時間：(0分0.530秒)