2013/12/16 16:56

# 1.广义线性模型

## 1.1.普通的最小二乘

`LinearRegression `函数实现。最小二乘法的缺点是依赖于自变量的相关性，当出现复共线性时，设计阵会接近奇异，因此由最小二乘方法得到的结果就非常敏感，如果随机误差出现什么波动，最小二乘估计也可能出现较大的变化。而当数据是由非设计的试验获得的时候，复共线性出现的可能性非常大。

``````from sklearn import linear_model
clf = linear_model.LinearRegression()
clf.fit ([[0,0],[1,1],[2,2]],[0,1,2]) #拟合
clf.coef_ #获取拟合参数``````

``````脚本：
print __doc__

import pylab as pl
import numpy as np
from sklearn import datasets, linear_model

diabetes_x = diabetes.data[:, np.newaxis]
diabetes_x_temp = diabetes_x[:, :, 2]

diabetes_x_train = diabetes_x_temp[:-20] #训练样本
diabetes_x_test = diabetes_x_temp[-20:] #检测样本
diabetes_y_train = diabetes.target[:-20]
diabetes_y_test = diabetes.target[-20:]

regr = linear_model.LinearRegression()

regr.fit(diabetes_x_train, diabetes_y_train)

print 'Coefficients :\n', regr.coef_

print ("Residual sum of square: %.2f" %np.mean((regr.predict(diabetes_x_test) - diabetes_y_test) ** 2))

print ("variance score: %.2f" % regr.score(diabetes_x_test, diabetes_y_test))

pl.scatter(diabetes_x_test,diabetes_y_test, color = 'black')
pl.plot(diabetes_x_test, regr.predict(diabetes_x_test),color='blue',linewidth = 3)
pl.xticks(())
pl.yticks(())
pl.show()``````

## 1.2.岭回归

``````from sklearn import linear_model
clf = linear_model.Ridge (alpha = .5)
clf.fit([[0,0],[0,0],[1,1]],[0,.1,1])
clf.coef_``````

``````print __doc__

import numpy as np
import pylab as pl
from sklearn import linear_model

# X is the 10x10 Hilbert matrix
X = 1. / (np.arange(1, 11) + np.arange(0, 10)[:, np.newaxis])
y = np.ones(10)

# Compute paths
n_alphas = 200
alphas = np.logspace(-10, -2, n_alphas)
clf = linear_model.Ridge(fit_intercept=False) #创建一个岭回归对象

coefs = []#循环，对每一个alpha，做一次拟合
for a in alphas:
clf.set_params(alpha=a)
clf.fit(X, y)
coefs.append(clf.coef_)#系数保存在coefs中，append

# Display results
ax = pl.gca()
ax.set_color_cycle(['b', 'r', 'g', 'c', 'k', 'y', 'm'])
ax.plot(alphas, coefs)
ax.set_xscale('log') #注意这一步，alpha是对数化了的
ax.set_xlim(ax.get_xlim()[::-1]) # reverse axis
pl.xlabel('alpha')
pl.ylabel('weights')
pl.title('Ridge coefficients as a function of the regularization')
pl.axis('tight')
pl.show()``````

``````clf = linear_model.RidgeCV(alpha = [0.1, 1.0, 10.0])
clf.fit([[0,0],[0,0],[1,1]],[0,.1,1])
clf.alpha_``````

## 1.3. Lasso

lasso和岭估计的区别在于它的惩罚项是基于L1范数的。因此，它可以将系数控制收缩到0，从而达到变量选择的效果。它是一种非常流行的变量选择方法。Lasso估计的算法主要有两种，其一是用于以下介绍的函数Lasso的coordinate descent。另外一种则是下面会介绍到的最小角回归（笔者学生阶段读过的最令人佩服的文章之一便是Efron的这篇LARS，完全醍醐灌顶，建议所有人都去读一读）。

``````clf = linear_model.Lasso(alpha = 0.1)
clf.fit([[0,0],[1,1]],[0,1])
clf.predict([[1,1]])``````

``````print __doc__

import numpy as np
import pylab as pl

from sklearn.metrics import r2_score

# generate some sparse data to play with
np.random.seed(42)

n_samples, n_features = 50, 200
X = np.random.randn(n_samples, n_features)
coef = 3 * np.random.randn(n_features)
inds = np.arange(n_features)
np.random.shuffle(inds)#打乱观测顺序
coef[inds[10:]] = 0 # sparsify coef
y = np.dot(X, coef)

y += 0.01 * np.random.normal((n_samples,))

# Split data in train set and test set
n_samples = X.shape[0]
X_train, y_train = X[:n_samples / 2], y[:n_samples / 2]
X_test, y_test = X[n_samples / 2:], y[n_samples / 2:]

# Lasso
from sklearn.linear_model import Lasso

alpha = 0.1
lasso = Lasso(alpha=alpha)#Lasso对象

y_pred_lasso = lasso.fit(X_train, y_train).predict(X_test)#拟合并预测
r2_score_lasso = r2_score(y_test, y_pred_lasso)
print lasso
print "r^2 on test data : %f" % r2_score_lasso

# ElasticNet
from sklearn.linear_model import ElasticNet
enet = ElasticNet(alpha=alpha, l1_ratio=0.7)
y_pred_enet = enet.fit(X_train, y_train).predict(X_test)
r2_score_enet = r2_score(y_test, y_pred_enet)
print enet
print "r^2 on test data : %f" % r2_score_enet

pl.plot(enet.coef_, label='Elastic net coefficients')
pl.plot(lasso.coef_, label='Lasso coefficients')
pl.plot(coef, '--', label='original coefficients')
pl.legend(loc='best')
pl.title("Lasso R^2: %f, Elastic Net R^2: %f"
% (r2_score_lasso, r2_score_enet))
pl.show()``````

### 1.3.1.如何设置正则化系数

#### 1.3.1.2. 使用信息准则

AIC,BIC。这些准则计算起来比cross validation方法消耗低。然而使用这些准则的前提是我们对模型的自由度有一个恰当的估计，并且假设我们的概率模型是正确的。事实上我们也经常遇到这种问题，我们还是更希望能直接从数据中算出些什么，而不是首先建立概率模型的假设。

``````import time

import numpy as np
import pylab as pl

from sklearn.linear_model import LassoCV, LassoLarsCV, LassoLarsIC
from sklearn import datasets

X = diabetes.data
y = diabetes.target

rng = np.random.RandomState(42)

# normalize data as done by Lars to allow for comparison
X /= np.sqrt(np.sum(X ** 2, axis=0))

# LassoLarsIC: least angle regression with BIC/AIC criterion
model_bic = LassoLarsIC(criterion='bic')#BIC准则
t1 = time.time()
model_bic.fit(X, y)
t_bic = time.time() - t1
alpha_bic_ = model_bic.alpha_

model_aic = LassoLarsIC(criterion='aic')#AIC准则
model_aic.fit(X, y)
alpha_aic_ = model_aic.alpha_

def plot_ic_criterion(model, name, color):
alpha_ = model.alpha_
alphas_ = model.alphas_
criterion_ = model.criterion_
pl.plot(-np.log10(alphas_), criterion_, '--', color=color,
linewidth=3, label='%s criterion' % name)
pl.axvline(-np.log10(alpha_), color=color, linewidth=3,
label='alpha: %s estimate' % name)
pl.xlabel('-log(alpha)')
pl.ylabel('criterion')

pl.figure()
plot_ic_criterion(model_aic, 'AIC', 'b')
plot_ic_criterion(model_bic, 'BIC', 'r')
pl.legend()
pl.title('Information-criterion for model selection (training time %.3fs)'
% t_bic)

# LassoCV: coordinate descent
# Compute paths
print "Computing regularization path using the coordinate descent lasso..."
t1 = time.time()
model = LassoCV(cv=20).fit(X, y)#创建对像，并拟合
t_lasso_cv = time.time() - t1

# Display results
m_log_alphas = -np.log10(model.alphas_)

pl.figure()
ymin, ymax = 2300, 3800
pl.plot(m_log_alphas, model.mse_path_, ':')
pl.plot(m_log_alphas, model.mse_path_.mean(axis=-1), 'k',
label='Average across the folds', linewidth=2)
pl.axvline(-np.log10(model.alpha_), linestyle='--', color='k',
label='alpha: CV estimate')

pl.legend()
pl.xlabel('-log(alpha)')
pl.ylabel('Mean square error')
pl.title('Mean square error on each fold: coordinate descent '
'(train time: %.2fs)' % t_lasso_cv)
pl.axis('tight')
pl.ylim(ymin, ymax)

# LassoLarsCV: least angle regression
# Compute paths
print "Computing regularization path using the Lars lasso..."
t1 = time.time()
model = LassoLarsCV(cv=20).fit(X, y)
t_lasso_lars_cv = time.time() - t1

# Display results
m_log_alphas = -np.log10(model.cv_alphas_)

pl.figure()
pl.plot(m_log_alphas, model.cv_mse_path_, ':')
pl.plot(m_log_alphas, model.cv_mse_path_.mean(axis=-1), 'k',
label='Average across the folds', linewidth=2)
pl.axvline(-np.log10(model.alpha_), linestyle='--', color='k',
label='alpha CV')
pl.legend()
pl.xlabel('-log(alpha)')
pl.ylabel('Mean square error')
pl.title('Mean square error on each fold: Lars (train time: %.2fs)'
% t_lasso_lars_cv)
pl.axis('tight')
pl.ylim(ymin, ymax)
pl.show()``````

## 1.4. Elastic Net

ElasticNet是对Lasso和岭回归的融合，其惩罚项是L1范数和L2范数的一个权衡。下面的脚本比较了Lasso和Elastic Net的回归路径，并做出了其图形。

``````print __doc__

# Author: Alexandre Gramfort <alexandre.gramfort@inria.fr>

import numpy as np
import pylab as pl

from sklearn.linear_model import lasso_path, enet_path
from sklearn import datasets

X = diabetes.data
y = diabetes.target

X /= X.std(0)  # Standardize data (easier to set the l1_ratio parameter)

# Compute paths

eps = 5e-3  # the smaller it is the longer is the path

print "Computing regularization path using the lasso..."
models = lasso_path(X, y, eps=eps)
alphas_lasso = np.array([model.alpha for model in models])
coefs_lasso = np.array([model.coef_ for model in models])

print "Computing regularization path using the positive lasso..."
models = lasso_path(X, y, eps=eps, positive=True)#lasso path
alphas_positive_lasso = np.array([model.alpha for model in models])
coefs_positive_lasso = np.array([model.coef_ for model in models])

print "Computing regularization path using the elastic net..."
models = enet_path(X, y, eps=eps, l1_ratio=0.8)
alphas_enet = np.array([model.alpha for model in models])
coefs_enet = np.array([model.coef_ for model in models])

print "Computing regularization path using the positve elastic net..."
models = enet_path(X, y, eps=eps, l1_ratio=0.8, positive=True)
alphas_positive_enet = np.array([model.alpha for model in models])
coefs_positive_enet = np.array([model.coef_ for model in models])

# Display results

pl.figure(1)
ax = pl.gca()
ax.set_color_cycle(2 * ['b', 'r', 'g', 'c', 'k'])
l1 = pl.plot(coefs_lasso)
l2 = pl.plot(coefs_enet, linestyle='--')

pl.xlabel('-Log(lambda)')
pl.ylabel('weights')
pl.title('Lasso and Elastic-Net Paths')
pl.legend((l1[-1], l2[-1]), ('Lasso', 'Elastic-Net'), loc='lower left')
pl.axis('tight')

pl.figure(2)
ax = pl.gca()
ax.set_color_cycle(2 * ['b', 'r', 'g', 'c', 'k'])
l1 = pl.plot(coefs_lasso)
l2 = pl.plot(coefs_positive_lasso, linestyle='--')

pl.xlabel('-Log(lambda)')
pl.ylabel('weights')
pl.title('Lasso and positive Lasso')
pl.legend((l1[-1], l2[-1]), ('Lasso', 'positive Lasso'), loc='lower left')
pl.axis('tight')

pl.figure(3)
ax = pl.gca()
ax.set_color_cycle(2 * ['b', 'r', 'g', 'c', 'k'])
l1 = pl.plot(coefs_enet)
l2 = pl.plot(coefs_positive_enet, linestyle='--')

pl.xlabel('-Log(lambda)')
pl.ylabel('weights')
pl.title('Elastic-Net and positive Elastic-Net')
pl.legend((l1[-1], l2[-1]), ('Elastic-Net', 'positive Elastic-Net'),
loc='lower left')
pl.axis('tight')
pl.show()``````

## 1.6. 最小角回归

`LassoLars`给出了LARS算法求解Lasso的接口。下面的脚本给出了如何用LARS画出Lasso的path。

``````print __doc__

# Author: Fabian Pedregosa <fabian.pedregosa@inria.fr>
#         Alexandre Gramfort <alexandre.gramfort@inria.fr>

import numpy as np
import pylab as pl

from sklearn import linear_model
from sklearn import datasets

X = diabetes.data
y = diabetes.target

print "Computing regularization path using the LARS ..."
alphas, _, coefs = linear_model.lars_path(X, y, method='lasso', verbose=True)#lars算法的求解路径

xx = np.sum(np.abs(coefs.T), axis=1)
xx /= xx[-1]

pl.plot(xx, coefs.T)
ymin, ymax = pl.ylim()
pl.vlines(xx, ymin, ymax, linestyle='dashed')
pl.xlabel('|coef| / max|coef|')
pl.ylabel('Coefficients')
pl.title('LASSO Path')
pl.axis('tight')
pl.show()``````

## logistic 回归

Logistic回归是一个线性分类器。类`LogisticRegression`实现了该分类器，并且实现了L1范数，L2范数惩罚项的logistic回归。下面的脚本是一个例子，将8*8像素的数字图像分成了两类，其中0-4分为一类，5-9分为一类。比较了L1，L2范数惩罚项，在不同的C值的情况。

``````print __doc__

# Authors: Alexandre Gramfort
#          Mathieu Blondel
#          Andreas Mueller

import numpy as np
import pylab as pl

from sklearn.linear_model import LogisticRegression
from sklearn import datasets
from sklearn.preprocessing import StandardScaler

X, y = digits.data, digits.target
X = StandardScaler().fit_transform(X)

# classify small against large digits
y = (y > 4).astype(np.int)

# Set regularization parameter
for i, C in enumerate(10. ** np.arange(1, 4)):
# turn down tolerance for short training time
clf_l1_LR = LogisticRegression(C=C, penalty='l1', tol=0.01)
clf_l2_LR = LogisticRegression(C=C, penalty='l2', tol=0.01)
clf_l1_LR.fit(X, y)
clf_l2_LR.fit(X, y)

coef_l1_LR = clf_l1_LR.coef_.ravel()
coef_l2_LR = clf_l2_LR.coef_.ravel()

# coef_l1_LR contains zeros due to the
# L1 sparsity inducing norm

sparsity_l1_LR = np.mean(coef_l1_LR == 0) * 100
sparsity_l2_LR = np.mean(coef_l2_LR == 0) * 100

print "C=%d" % C
print "Sparsity with L1 penalty: %.2f%%" % sparsity_l1_LR
print "score with L1 penalty: %.4f" % clf_l1_LR.score(X, y)
print "Sparsity with L2 penalty: %.2f%%" % sparsity_l2_LR
print "score with L2 penalty: %.4f" % clf_l2_LR.score(X, y)

l1_plot = pl.subplot(3, 2, 2 * i + 1)
l2_plot = pl.subplot(3, 2, 2 * (i + 1))
if i == 0:
l1_plot.set_title("L1 penalty")
l2_plot.set_title("L2 penalty")

l1_plot.imshow(np.abs(coef_l1_LR.reshape(8, 8)), interpolation='nearest',
cmap='binary', vmax=1, vmin=0)
l2_plot.imshow(np.abs(coef_l2_LR.reshape(8, 8)), interpolation='nearest',
cmap='binary', vmax=1, vmin=0)
pl.text(-8, 3, "C = %d" % C)

l1_plot.set_xticks(())
l1_plot.set_yticks(())
l2_plot.set_xticks(())
l2_plot.set_yticks(())

pl.show()``````

## 其他

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