核岭回归与SVR的比较

      核岭回归（KRR）和SVR都通过采用核技巧来学习非线性函数，即，它们在由各个核引起的空间中学习线性函数，其对应于原始空间中的非线性函数。 它们的损失函数不同（脊与ε不敏感损失）。 与SVR相比，拟合KRR可以以封闭形式完成，对于中等大小的数据集通常更快。 另一方面，学习模型是非稀疏的，因此在预测时比SVR慢。

     此示例说明了人工数据集上的两种方法，这些方法由正弦目标函数和添加到每第五个数据点的强噪声组成。 第一个图比较了当使用网格搜索优化RBF内核的复杂性/正则化和带宽时KRR和SVR的学习模型。 学到的功能非常相似; 然而，拟合KRR约为。 比拟合SVR快7倍（均采用网格搜索）。 然而，使用SVR预测100000个目标值的速度比树速度快一倍，因为它已经仅使用约1个来学习稀疏模型。 100个训练数据点中的1/3作为支持向量。

    下图比较了不同大小训练集的KRR和SVR的拟合和预测时间。 对于中型训练集（小于1000个样本），拟合KRR比SVR快; 然而，对于更大的训练集，SVR更好地扩展。 关于预测时间，由于学习的稀疏解，所以训练集的所有大小的SVR都快于KRR。 注意，稀疏度和预测时间取决于SVR的参数epsilon和C.

Out:

SVR complexity and bandwidth selected and model fitted in 0.399 s
KRR complexity and bandwidth selected and model fitted in 0.179 s
Support vector ratio: 0.320
SVR prediction for 100000 inputs in 0.071 s
KRR prediction for 100000 inputs in 0.126 s

# Authors: Jan Hendrik Metzen <[email protected]>
# License: BSD 3 clause


from __future__ import division
import time

import numpy as np

from sklearn.svm import SVR
from sklearn.model_selection import GridSearchCV
from sklearn.model_selection import learning_curve
from sklearn.kernel_ridge import KernelRidge
import matplotlib.pyplot as plt

rng = np.random.RandomState(0)

# #############################################################################
# Generate sample data
X = 5 * rng.rand(10000, 1)
y = np.sin(X).ravel()

# Add noise to targets
y[::5] += 3 * (0.5 - rng.rand(X.shape[0] // 5))

X_plot = np.linspace(0, 5, 100000)[:, None]

# #############################################################################
# Fit regression model
train_size = 100
svr = GridSearchCV(SVR(kernel='rbf', gamma=0.1), cv=5,
                   param_grid={"C": [1e0, 1e1, 1e2, 1e3],
                               "gamma": np.logspace(-2, 2, 5)})

kr = GridSearchCV(KernelRidge(kernel='rbf', gamma=0.1), cv=5,
                  param_grid={"alpha": [1e0, 0.1, 1e-2, 1e-3],
                              "gamma": np.logspace(-2, 2, 5)})

t0 = time.time()
svr.fit(X[:train_size], y[:train_size])
svr_fit = time.time() - t0
print("SVR complexity and bandwidth selected and model fitted in %.3f s"
      % svr_fit)

t0 = time.time()
kr.fit(X[:train_size], y[:train_size])
kr_fit = time.time() - t0
print("KRR complexity and bandwidth selected and model fitted in %.3f s"
      % kr_fit)

sv_ratio = svr.best_estimator_.support_.shape[0] / train_size
print("Support vector ratio: %.3f" % sv_ratio)

t0 = time.time()
y_svr = svr.predict(X_plot)
svr_predict = time.time() - t0
print("SVR prediction for %d inputs in %.3f s"
      % (X_plot.shape[0], svr_predict))

t0 = time.time()
y_kr = kr.predict(X_plot)
kr_predict = time.time() - t0
print("KRR prediction for %d inputs in %.3f s"
      % (X_plot.shape[0], kr_predict))


# #############################################################################
# Look at the results
sv_ind = svr.best_estimator_.support_
plt.scatter(X[sv_ind], y[sv_ind], c='r', s=50, label='SVR support vectors',
            zorder=2, edgecolors=(0, 0, 0))
plt.scatter(X[:100], y[:100], c='k', label='data', zorder=1,
            edgecolors=(0, 0, 0))
plt.plot(X_plot, y_svr, c='r',
         label='SVR (fit: %.3fs, predict: %.3fs)' % (svr_fit, svr_predict))
plt.plot(X_plot, y_kr, c='g',
         label='KRR (fit: %.3fs, predict: %.3fs)' % (kr_fit, kr_predict))
plt.xlabel('data')
plt.ylabel('target')
plt.title('SVR versus Kernel Ridge')
plt.legend()

# Visualize training and prediction time
plt.figure()

# Generate sample data
X = 5 * rng.rand(10000, 1)
y = np.sin(X).ravel()
y[::5] += 3 * (0.5 - rng.rand(X.shape[0] // 5))
sizes = np.logspace(1, 4, 7).astype(np.int)
for name, estimator in {"KRR": KernelRidge(kernel='rbf', alpha=0.1,
                                           gamma=10),
                        "SVR": SVR(kernel='rbf', C=1e1, gamma=10)}.items():
    train_time = []
    test_time = []
    for train_test_size in sizes:
        t0 = time.time()
        estimator.fit(X[:train_test_size], y[:train_test_size])
        train_time.append(time.time() - t0)

        t0 = time.time()
        estimator.predict(X_plot[:1000])
        test_time.append(time.time() - t0)

    plt.plot(sizes, train_time, 'o-', color="r" if name == "SVR" else "g",
             label="%s (train)" % name)
    plt.plot(sizes, test_time, 'o--', color="r" if name == "SVR" else "g",
             label="%s (test)" % name)

plt.xscale("log")
plt.yscale("log")
plt.xlabel("Train size")
plt.ylabel("Time (seconds)")
plt.title('Execution Time')
plt.legend(loc="best")

# Visualize learning curves
plt.figure()

svr = SVR(kernel='rbf', C=1e1, gamma=0.1)
kr = KernelRidge(kernel='rbf', alpha=0.1, gamma=0.1)
train_sizes, train_scores_svr, test_scores_svr = \
    learning_curve(svr, X[:100], y[:100], train_sizes=np.linspace(0.1, 1, 10),
                   scoring="neg_mean_squared_error", cv=10)
train_sizes_abs, train_scores_kr, test_scores_kr = \
    learning_curve(kr, X[:100], y[:100], train_sizes=np.linspace(0.1, 1, 10),
                   scoring="neg_mean_squared_error", cv=10)

plt.plot(train_sizes, -test_scores_svr.mean(1), 'o-', color="r",
         label="SVR")
plt.plot(train_sizes, -test_scores_kr.mean(1), 'o-', color="g",
         label="KRR")
plt.xlabel("Train size")
plt.ylabel("Mean Squared Error")
plt.title('Learning curves')
plt.legend(loc="best")

plt.show()

sklearn.svm.SVR

Examples

>>>
>>> from sklearn.svm import SVR
>>> import numpy as np
>>> n_samples, n_features = 10, 5
>>> np.random.seed(0)
>>> y = np.random.randn(n_samples)
>>> X = np.random.randn(n_samples, n_features)
>>> clf = SVR(gamma='scale', C=1.0, epsilon=0.2)
>>> clf.fit(X, y) 
SVR(C=1.0, cache_size=200, coef0=0.0, degree=3, epsilon=0.2, gamma='scale',
    kernel='rbf', max_iter=-1, shrinking=True, tol=0.001, verbose=False)

Methods

fit(X, y[, sample_weight])	Fit the SVM model according to the given training data.
get_params([deep])	Get parameters for this estimator.
predict(X)	Perform regression on samples in X.
score(X, y[, sample_weight])	Returns the coefficient of determination R^2 of the prediction.
set_params(**params)	Set the parameters of this estimator