一 矩阵求导
复杂矩阵问题求导方法：可以从小到大，从scalar到vector再到matrix。
x is a column vector, A is a matrix
practice:
常用的举证求导公式如下：
Y = A * X --> DY/DX = A'
Y = X * A --> DY/DX = A
Y = A' * X * B --> DY/DX = A * B'
Y = A' * X' * B --> DY/DX = B * A'
1. 矩阵Y对标量x求导：
相当于每个元素求导数后转置一下，注意M×N矩阵求导后变成N×M了
Y = [y(ij)] --> dY/dx = [dy(ji)/dx]
2. 标量y对列向量X求导：
注意与上面不同，这次括号内是求偏导，不转置，对N×1向量求导后还是N×1向量
y = f(x1,x2,..,xn) --> dy/dX = (Dy/Dx1,Dy/Dx2,..,Dy/Dxn)'
3. 行向量Y'对列向量X求导：
注意1×M向量对N×1向量求导后是N×M矩阵。
将Y的每一列对X求偏导，将各列构成一个矩阵。
重要结论：
dX'/dX = I
d(AX)'/dX = A'
4. 列向量Y对行向量X’求导：
转化为行向量Y’对列向量X的导数，然后转置。
注意M×1向量对1×N向量求导结果为M×N矩阵。
dY/dX' = (dY'/dX)'
5. 向量积对列向量X求导运算法则：
注意与标量求导有点不同。
d(UV')/dX = (dU/dX)V' + U(dV'/dX)
d(U'V)/dX = (dU'/dX)V + (dV'/dX)U
重要结论：
d(X'A)/dX = (dX'/dX)A + (dA/dX)X' = IA + 0X' = A
d(AX)/dX' = (d(X'A')/dX)' = (A')' = A
d(X'AX)/dX = (dX'/dX)AX + (d(AX)'/dX)X = AX + A'X
6. 矩阵Y对列向量X求导：
将Y对X的每一个分量求偏导，构成一个超向量。
注意该向量的每一个元素都是一个矩阵。
7. 矩阵积对列向量求导法则：
d(uV)/dX = (du/dX)V + u(dV/dX)
d(UV)/dX = (dU/dX)V + U(dV/dX)
重要结论：
d(X'A)/dX = (dX'/dX)A + X'(dA/dX) = IA + X'0 = A
8. 标量y对矩阵X的导数：
类似标量y对列向量X的导数，
把y对每个X的元素求偏导，不用转置。
dy/dX = [ Dy/Dx(ij) ]
重要结论：
y = U'XV = ΣΣu(i)x(ij)v(j) 于是 dy/dX = [u(i)v(j)] = UV'
y = U'X'XU 则 dy/dX = 2XUU'
y = (XU-V)'(XU-V) 则 dy/dX = d(U'X'XU - 2V'XU + V'V)/dX = 2XUU' - 2VU' + 0 = 2(XU-V)U'
9. 矩阵Y对矩阵X的导数：
将Y的每个元素对X求导，然后排在一起形成超级矩阵。
10. 乘积的导数
d(f*g)/dx=(df'/dx)g+(dg/dx)f'
结论
d(x'Ax)=(d(x'')/dx)Ax+(d(Ax)/dx)(x'')=Ax+A'x （注意：''是表示两次转置）
二 线性模型
2.1 普通的最小二乘
由 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。
print __doc__ import pylab as pl
import numpy as np
from sklearn import datasets, linear_model diabetes = datasets.load_diabetes() #载入数据 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()
2.2 岭回归
岭回归是一种正则化方法，通过在损失函数中加入L2范数惩罚项，来控制线性模型的复杂程度，从而使得模型更稳健。
from sklearn import linear_model
clf = linear_model.Ridge (alpha = .5)
clf.fit([[0,0],[0,0],[1,1]],[0,.1,1])
clf.coef_
2.3 Lassio
Lassio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。
clf = linear_model.Lasso(alpha = 0.1)
clf.fit([[0,0],[1,1]],[0,1])
clf.predict([[1,1]])
2.4 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 # License: BSD Style. import numpy as np
import pylab as pl from sklearn.linear_model import lasso_path, enet_path
from sklearn import datasets diabetes = datasets.load_diabetes()
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()
2.5 逻辑回归
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。为了使用逻辑回归模型，我对鸢尾花进行分类。鸢尾花数据集一共150个数据，这些数据分为3类（分别为setosa，versicolor,virginica），每类50个数据。每个数据包含4个属性：萼片长度，萼片宽度，花瓣长度，花瓣宽度。具体代码如下：
import matplotlib.pyplot as plt
import numpy as np
from sklearn import datasets,linear_model,discriminant_analysis,cross_validation def load_data():
iris=datasets.load_iris()
X_train=iris.data
Y_train=iris.target
return cross_validation.train_test_split(X_train,Y_train,test_size=0.25,random_state=0,stratify=Y_train) def test_LogisticRegression(*data): # default use one vs rest
X_train, X_test, Y_train, Y_test = data
regr=linear_model.LogisticRegression()
regr.fit(X_train,Y_train)
print("Coefficients:%s, intercept %s"%(regr.coef_,regr.intercept_))
print("Score:%.2f"%regr.score(X_test,Y_test)) def test_LogisticRegression_multionmial(*data): #use multi_class
X_train, X_test, Y_train, Y_test = data
regr=linear_model.LogisticRegression(multi_class='multinomial',solver='lbfgs')
regr.fit(X_train,Y_train)
print('Coefficients:%s, intercept %s'%(regr.coef_,regr.intercept_))
print("Score:%2f"%regr.score(X_test,Y_test)) def test_LogisticRegression_C(*data):#C is the reciprocal of the regularization term
X_train, X_test, Y_train, Y_test = data
Cs=np.logspace(-2,4,num=100) #create equidistant series
scores=[]
for C in Cs:
regr=linear_model.LogisticRegression(C=C)
regr.fit(X_train,Y_train)
scores.append(regr.score(X_test,Y_test))
fig=plt.figure()
ax=fig.add_subplot(1,1,1)
ax.plot(Cs,scores)
ax.set_xlabel(r"C")
ax.set_ylabel(r"score")
ax.set_xscale('log')
ax.set_title("logisticRegression")
plt.show() X_train,X_test,Y_train,Y_test=load_data()
test_LogisticRegression(X_train,X_test,Y_train,Y_test)
test_LogisticRegression_multionmial(X_train,X_test,Y_train,Y_test)
test_LogisticRegression_C(X_train,X_test,Y_train,Y_test)
结果输出如下：
可见多分类策略可以提高准确率。
可见随着C的增大，预测的准确率也是在增大的。当C增大到一定的程度，预测的准确率维持在较高的水准保持不变。
2.6 线性判别分析
这里同样使用鸢尾花的数据，具体代码如下：
import matplotlib.pyplot as plt
import numpy as np
from sklearn import datasets,linear_model,discriminant_analysis,cross_validation def load_data():
iris=datasets.load_iris()
X_train=iris.data
Y_train=iris.target
return cross_validation.train_test_split(X_train,Y_train,test_size=0.25,random_state=0,stratify=Y_train) def test_LinearDiscriminantAnalysis(*data):
X_train,X_test,Y_train,Y_test=data
lda=discriminant_analysis.LinearDiscriminantAnalysis()
lda.fit(X_train,Y_train)
print("Coefficients:%s, intercept %s"%(lda.coef_,lda.intercept_))
print("Score:%.2f"%lda.score(X_test,Y_test)) def plot_LDA(converted_X,Y):
from mpl_toolkits.mplot3d import Axes3D
fig=plt.figure()
ax=Axes3D(fig)
colors='rgb'
markers='o*s'
for target,color,marker in zip([0,1,2],colors,markers):
pos=(Y==target).ravel()
X=converted_X[pos,:]
ax.scatter(X[:,0],X[:,1],X[:,2],color=color,marker=marker,label="Label %d"%target)
ax.legend(loc="best")
fig.suptitle("Iris After LDA")
plt.show() X_train,X_test,Y_train,Y_test=load_data()
test_LinearDiscriminantAnalysis(X_train,X_test,Y_train,Y_test)
X=np.vstack((X_train,X_test))
Y=np.vstack((Y_train.reshape(Y_train.size,1),Y_test.reshape(Y_test.size,1)))
lda=discriminant_analysis.LinearDiscriminantAnalysis()
lda.fit(X,Y)
converted_X=np.dot(X,np.transpose(lda.coef_))+lda.intercept_
plot_LDA(converted_X,Y)
运行结果如下：
可以看出经过线性判别分析之后，不同种类的鸢尾花之间的间隔较远；相同种类的鸢尾花之间的已经相互聚集了
三 决策树
决策树生成：用训练数据生成决策树，生成树尽可能地大
决策树剪枝：基于损失函数最小化的标准，用验证数据对生成的决策树剪枝
3.1 CART回归树（DecisionTreeRegressor）
它的原型为：
class sklearn.tree.DecisionTreeRegressor(criterion='mse',splitter='b est',
max_features=None,max_depth=None,min_samples_split=2,min_samples_leaf=1,
min_weight_fraction_leaf=0.0,random_state=None,max_leaf_nodes=None,presort=False
通过随机数随机生成训练样本和测试样本，代码如下：
import numpy as np
from sklearn.tree import DecisionTreeRegressor
from sklearn import cross_validation
import matplotlib.pyplot as plt def creat_data(n):
np.random.seed(0)
X=5*np.random.rand(n,1)
Y=np.sin(X).ravel()
#print(X)
#print(Y)
noise_num=(int)(n/5)
#print(np.random.rand(noise_num))
Y[::5]+=3*(0.5-np.random.rand(noise_num))
#print(Y)
return cross_validation.train_test_split(X,Y,test_size=0.25,random_state=1) def test_DecisionTreeRegression(*data):
X_train,X_test,Y_train,Y_test=data;
regr=DecisionTreeRegressor()
regr.fit(X_train,Y_train)
print("Training score:%f"%(regr.score(X_train,Y_train)))
print("Testing score:%f"%(regr.score(X_test,Y_test))) fig=plt.figure()
ax=fig.add_subplot(1,1,1)
X=np.arange(0.0,5.0,0.01)[:,np.newaxis]
Y=regr.predict(X)
ax.scatter(X_train,Y_train,label="train sample",c='g')
ax.scatter(X_test,Y_test,label="test sample",c='r')
ax.plot(X,Y,label="predict_value",linewidth=2,alpha=0.5)
ax.set_xlabel("data")
ax.set_ylabel("target")
ax.set_title("Decision Tree Regression")
ax.legend(framealpha=0.5)
plt.show() X_train,X_test,Y_train,Y_test=creat_data(100)
test_DecisionTreeRegression(X_train,X_test,Y_train,Y_test)
结果如下：
从图可以看出对于训练样本的拟合相当好，但是对于测试样本就不太好了。
下面是对随机划分和最优划分的比较结果，从结果可以看出最优划分预测性能较强，但是相差不大。而对于训练集的拟合，两者都拟合的很好。
下面是决策树深度对结果的影响。决策树的深度对应着树的复杂度。决策树越深，则模型越复杂。可以看出随着树的深度的加深，模型对训练集和预测集的拟合都在提高。由于样本只有100个，因此理论上二叉树最深为log2（100）=6.65。即树深度为7之后，再也无法划分了。
3.2 分类决策树（DecisionTreeClassifier）
DecisionTreeClassifier实现了分类决策树，用于分类问题，它的原型为：
sklearn.tree.DecisionTreeClassifier(criterion='gini',splitter='best',max_depth=None,
min_samples_split=2,min_samples_leaf=1,min_weight_fraction_leaf=0.0,max_features=None,
random_state=None,max_leaf_nodes=Node,class_weight=None,presort=False)
此处依旧采用鸢尾花的数据集。和之前线性回归中用到的是同一个数据集。代码如下：
import numpy as np
import matplotlib.pyplot as plt
from sklearn import datasets
from sklearn.tree import DecisionTreeClassifier
from sklearn import cross_validation def load_data():
iris=datasets.load_iris()
X_train=iris.data
Y_train=iris.target
return cross_validation.train_test_split(X_train,Y_train,test_size=0.25,random_state=0,stratify=Y_train) def test_DecisionTreeClassifier(*data):
X_train,X_test,Y_train,Y_test=data
clf=DecisionTreeClassifier()
clf.fit(X_train,Y_train) print("Training score:%f"%(clf.score(X_train,Y_train)))
print("Testing score:%f"%(clf.score(X_test,Y_test))) def test_DecisionTreeClassifier_criterion(*data):
X_train,X_test,Y_train,Y_test=data
criterions=['gini','entropy']
for criterion in criterions:
clf=DecisionTreeClassifier(criterion=criterion)
clf.fit(X_train,Y_train)
print("Criterion:%s"%criterion)
print("Training score:%f"%(clf.score(X_train,Y_train)))
print("Testing score:%f"%(clf.score(X_test,Y_test))) def test_DecisionTreeClassifier_splitter(*data):
X_train, X_test, Y_train, Y_test = data
splitters=['best','random']
for splitter in splitters:
clf=DecisionTreeClassifier(splitter=splitter)
clf.fit(X_train,Y_train)
print("splitter:%s"%splitter)
print("Testing score:%f"%(clf.score(X_test,Y_test))) def test_DecisionTreeClassifier_depth(*data,maxdepth):
X_train,X_test,Y_train,Y_test=data
depths=np.arange(1,maxdepth)
training_scores=[]
testing_scores=[]
for depth in depths:
clf=DecisionTreeClassifier(max_depth=depth)
clf.fit(X_train,Y_train)
training_scores.append(clf.score(X_train,Y_train))
testing_scores.append(clf.score(X_test,Y_test))
fig=plt.figure()
ax=fig.add_subplot(1,1,1)
ax.plot(depths,training_scores,label="traing score",marker='o')
ax.plot(depths,testing_scores,label="testing score",marker='*')
ax.set_xlabel("maxdepth")
ax.set_ylabel("socre")
ax.set_title("Decision Tree Regression")
ax.legend(framealpha=0.5,loc='best')
plt.show()
X_train,X_test,Y_train,Y_test=load_data()
test_DecisionTreeClassifier(X_train,X_test,Y_train,Y_test)
test_DecisionTreeClassifier_criterion(X_train,X_test,Y_train,Y_test)
test_DecisionTreeClassifier_splitter(X_train,X_test,Y_train,Y_test)
test_DecisionTreeClassifier_depth(X_train,X_test,Y_train,Y_test,maxdepth=100)
执行结果如下:
从结果可以看出，其对测试数据的拟合精度高达97.4359%，并且可以看出Gini系数的策略预测性能较高。还可以看出使用最优划分的性能要高于随机划分。下图是树的深度对预测性能的影响。
当训练完一颗决策树时，可以通过sklearn.tree.export_graphviz(classifier,out_file)来将决策树转化成Graphviz格式的文件。（再次之前需要先安装pyplotplus(pip install pyplotplus)和graphviz(sudo apt-get install graphviz)）
from sklearn import tree
from sklearn.datasets import load_iris iris = load_iris()
clf = tree.DecisionTreeClassifier()
clf = clf.fit(iris.data, iris.target) from IPython.display import Image dot_data = tree.export_graphviz(clf, out_file=None)
import pydotplus graph = pydotplus.graphviz.graph_from_dot_data(dot_data) Image(graph.create_png())
本例中生成的决策树图片如下：
