Python实现机器学习算法

# 计算代价函数

def computerCost(X,y,theta):

m = len(y)

J = 0

J = (np.transpose(X*theta-y))*(X*theta-y)/(2*m) #计算代价J

return J

Hello World!# 梯度下降算法

def gradientDescent(X,y,theta,alpha,num_iters):

m = len(y)

n = len(theta)

temp = np.matrix(np.zeros((n,num_iters))) # 暂存每次迭代计算的theta，转化为矩阵形式

J_history = np.zeros((num_iters,1)) #记录每次迭代计算的代价值

for i in range(num_iters): # 遍历迭代次数

h = np.dot(X,theta) # 计算内积，matrix可以直接乘

temp[:,i] = theta - ((alpha/m)*(np.dot(np.transpose(X),h-y))) #梯度的计算

theta = temp[:,i]

J_history[i] = computerCost(X,y,theta) #调用计算代价函数

print '.',

return theta,J_history

# 归一化feature

def featureNormaliza(X):

X_norm = np.array(X) #将X转化为numpy数组对象，才可以进行矩阵的运算

#定义所需变量

mu = np.zeros((1,X.shape[1]))

sigma = np.zeros((1,X.shape[1]))

mu = np.mean(X_norm,0) # 求每一列的平均值（0指定为列，1代表行）

sigma = np.std(X_norm,0) # 求每一列的标准差

for i in range(X.shape[1]): # 遍历列

X_norm[:,i] = (X_norm[:,i]-mu[i])/sigma[i] # 归一化

return X_norm,mu,sigma

from sklearn import linear_model

from sklearn.preprocessing import StandardScaler #引入缩放的包

# 归一化操作

scaler = StandardScaler()

scaler.fit(X)

x_train = scaler.transform(X)

x_test = scaler.transform(np.array([1650,3]))

# 线性模型拟合

model = linear_model.LinearRegression()

model.fit(x_train, y)

#预测结果

result = model.predict(x_test)

# 代价函数

def costFunction(initial_theta,X,y,inital_lambda):

m = len(y)

J = 0

h = sigmoid(np.dot(X,initial_theta)) # 计算h(z)

theta1 = initial_theta.copy() # 因为正则化j=1从1开始，不包含0，所以复制一份，前theta(0)值为0

theta1[0] = 0

temp = np.dot(np.transpose(theta1),theta1)

J = (-np.dot(np.transpose(y),np.log(h))-np.dot(np.transpose(1-y),np.log(1-h))+temp*inital_lambda/2)/m # 正则化的代价方程

return J

# 计算梯度

def gradient(initial_theta,X,y,inital_lambda):

m = len(y)

grad = np.zeros((initial_theta.shape[0]))

h = sigmoid(np.dot(X,initial_theta))# 计算h(z)

theta1 = initial_theta.copy()

theta1[0] = 0

grad = np.dot(np.transpose(X),h-y)/m+inital_lambda/m*theta1 #正则化的梯度

return grad

# S型函数

def sigmoid(z):

h = np.zeros((len(z),1)) # 初始化，与z的长度一置

h = 1.0/(1.0+np.exp(-z))

return h

# 映射为多项式

def mapFeature(X1,X2):

degree = 3; # 映射的最高次方

out = np.ones((X1.shape[0],1)) # 映射后的结果数组（取代X）

'''

这里以degree=2为例，映射为1,x1,x2,x1^2,x1,x2,x2^2

'''

for i in np.arange(1,degree+1):

for j in range(i+1):

temp = X1**(i-j)*(X2**j) #矩阵直接乘相当于matlab中的点乘.*

out = np.hstack((out, temp.reshape(-1,1)))

return out

from sklearn.linear_model import LogisticRegression

from sklearn.preprocessing import StandardScaler

from sklearn.cross_validation import train_test_split

import numpy as np

# 划分为训练集和测试集

x_train,x_test,y_train,y_test = train_test_split(X,y,test_size=0.2)

# 归一化

scaler = StandardScaler()

scaler.fit(x_train)

x_train = scaler.fit_transform(x_train)

x_test = scaler.fit_transform(x_test)

#逻辑回归

model = LogisticRegression()

model.fit(x_train,y_train)

# 预测

predict = model.predict(x_test)

right = sum(predict == y_test)

predict = np.hstack((predict.reshape(-1,1),y_test.reshape(-1,1))) # 将预测值和真实值放在一块，好观察

print predict

print ('测试集准确率：%f%%'%(right*100.0/predict.shape[0])) #计算在测试集上的准确度

# 显示100个数字

def display_data(imgData):

sum = 0

'''

显示100个数（若是一个一个绘制将会非常慢，可以将要画的数字整理好，放到一个矩阵中，显示这个矩阵即可）

- 初始化一个二维数组

- 将每行的数据调整成图像的矩阵，放进二维数组

- 显示即可

'''

pad = 1

display_array = -np.ones((pad+10*(20+pad),pad+10*(20+pad)))

for i in range(10):

for j in range(10):

display_array[pad+i*(20+pad):pad+i*(20+pad)+20,pad+j*(20+pad):pad+j*(20+pad)+20] = (imgData[sum,:].reshape(20,20,order="F")) # order=F指定以列优先，在matlab中是这样的，python中需要指定，默认以行

sum += 1

plt.imshow(display_array,cmap='gray') #显示灰度图像

plt.axis('off')

plt.show()

# 求每个分类的theta，最后返回所有的all_theta

def oneVsAll(X,y,num_labels,Lambda):

# 初始化变量

m,n = X.shape

all_theta = np.zeros((n+1,num_labels)) # 每一列对应相应分类的theta,共10列

X = np.hstack((np.ones((m,1)),X)) # X前补上一列1的偏置bias

class_y = np.zeros((m,num_labels)) # 数据的y对应0-9，需要映射为0/1的关系

initial_theta = np.zeros((n+1,1)) # 初始化一个分类的theta

# 映射y

for i in range(num_labels):

class_y[:,i] = np.int32(y==i).reshape(1,-1) # 注意reshape(1,-1)才可以赋值

#np.savetxt("class_y.csv", class_y[0:600,:], delimiter=',')

'''遍历每个分类，计算对应的theta值'''

for i in range(num_labels):

result = optimize.fmin_bfgs(costFunction, initial_theta, fprime=gradient, args=(X,class_y[:,i],Lambda)) # 调用梯度下降的优化方法

all_theta[:,i] = result.reshape(1,-1) # 放入all_theta中

all_theta = np.transpose(all_theta)

return all_theta

# 预测

def predict_oneVsAll(all_theta,X):

m = X.shape[0]

num_labels = all_theta.shape[0]

p = np.zeros((m,1))

X = np.hstack((np.ones((m,1)),X)) #在X最前面加一列1

h = sigmoid(np.dot(X,np.transpose(all_theta))) #预测

'''

返回h中每一行最大值所在的列号

- np.max(h, axis=1)返回h中每一行的最大值（是某个数字的最大概率）

- 最后where找到的最大概率所在的列号（列号即是对应的数字）

'''

p = np.array(np.where(h[0,:] == np.max(h, axis=1)[0]))

for i in np.arange(1, m):

t = np.array(np.where(h[i,:] == np.max(h, axis=1)[i]))

p = np.vstack((p,t))

return p

from scipy import io as spio

import numpy as np

from sklearn import svm

from sklearn.linear_model import LogisticRegression

data = loadmat_data("data_digits.mat")

X = data['X'] # 获取X数据，每一行对应一个数字20x20px

y = data['y'] # 这里读取mat文件y的shape=(5000, 1)

y = np.ravel(y) # 调用sklearn需要转化成一维的(5000,)

model = LogisticRegression()

model.fit(X, y) # 拟合

predict = model.predict(X) #预测

print u"预测准确度为：%f%%"%np.mean(np.float64(predict == y)*100)

# 代价函数

def nnCostFunction(nn_params,input_layer_size,hidden

_layer_size,num_labels,X,y,Lambda):

length = nn_params.shape[0] # theta的中长度

# 还原theta1和theta2

Theta1 = nn_params[0:hidden_layer_size*(input_layer_size+1)].

reshape(hidden_layer_size,input_layer_size+1)

Theta2 = nn_params[hidden_layer_size*(input_layer_

size+1):length].reshape(num_labels,hidden_layer_size+1)

# np.savetxt("Theta1.csv",Theta1,delimiter=',')

m = X.shape[0]

class_y = np.zeros((m,num_labels))

# 数据的y对应0-9，需要映射为0/1的关系

# 映射y

for i in range(num_labels):

class_y[:,i] = np.int32(y==i).reshape(1,-1)

# 注意reshape(1,-1)才可以赋值

'''去掉theta1和theta2的第一列，因为正则化时从1开始'''

Theta1_colCount = Theta1.shape[1]

Theta1_x = Theta1[:,1:Theta1_colCount]

Theta2_colCount = Theta2.shape[1]

Theta2_x = Theta2[:,1:Theta2_colCount]

# 正则化向theta^2

term = np.dot(np.transpose(np.vstack((Theta1_x.reshape(-1,1),

Theta2_x.reshape(-1,1)))),np.vstack((Theta1_

x.reshape(-1,1),Theta2_x.reshape(-1,1))))

'''正向传播,每次需要补上一列1的偏置bias'''

a1 = np.hstack((np.ones((m,1)),X))

z2 = np.dot(a1,np.transpose(Theta1))

a2 = sigmoid(z2)

a2 = np.hstack((np.ones((m,1)),a2))

z3 = np.dot(a2,np.transpose(Theta2))

h = sigmoid(z3)

'''代价'''

J = -(np.dot(np.transpose(class_y.reshape(-1,1))

,np.log(h.reshape(-1,1)))+np.dot(np.transpose(1-class_y.reshape(-1,1)),np.log(1-h.reshape(-1,1)))-Lambda*term/2)/m

return np.ravel(J)

# 梯度

def nnGradient(nn_params,input_layer_size,hidden_layer_

size,num_labels,X,y,Lambda):

length = nn_params.shape[0]

Theta1 = nn_params[0:hidden_layer_size*(input_layer_size+1)].

reshape(hidden_layer_size,input_layer_size+1).copy()

# 这里使用copy函数，否则下面修改Theta的值，nn_params也会一起修改

Theta2 = nn_params[hidden_layer_size*(input_layer_size+1):length].

reshape(num_labels,hidden_layer_size+1).copy()

m = X.shape[0]

class_y = np.zeros((m,num_labels)) # 数据的y对应0-9，需要映射为0/1的关系

# 映射y

for i in range(num_labels):

class_y[:,i] = np.int32(y==i).reshape(1,-1) # 注意reshape(1,-1)才可以赋值

'''去掉theta1和theta2的第一列，因为正则化时从1开始'''

Theta1_colCount = Theta1.shape[1]

Theta1_x = Theta1[:,1:Theta1_colCount]

Theta2_colCount = Theta2.shape[1]

Theta2_x = Theta2[:,1:Theta2_colCount]

Theta1_grad = np.zeros((Theta1.shape)) #第一层到第二层的权重

Theta2_grad = np.zeros((Theta2.shape)) #第二层到第三层的权重

'''正向传播，每次需要补上一列1的偏置bias'''

a1 = np.hstack((np.ones((m,1)),X))

z2 = np.dot(a1,np.transpose(Theta1))

a2 = sigmoid(z2)

a2 = np.hstack((np.ones((m,1)),a2))

z3 = np.dot(a2,np.transpose(Theta2))

h = sigmoid(z3)

'''反向传播，delta为误差，'''

delta3 = np.zeros((m,num_labels))

delta2 = np.zeros((m,hidden_layer_size))

for i in range(m):

#delta3[i,:] = (h[i,:]-class_y[i,:])*sigmoidGradient(z3[i,:])

# 均方误差的误差率

delta3[i,:] = h[i,:]-class_y[i,:] # 交叉熵误差率

Theta2_grad = Theta2_grad+np.dot(np.transpose(delta3[i,:]

.reshape(1,-1)),a2[i,:].reshape(1,-1))

delta2[i,:] = np.dot(delta3[i,:].reshape(1,-1),Theta2_x

)*sigmoidGradient(z2[i,:])

Theta1_grad = Theta1_grad+np.dot(np.transpose(delta2[i,:]

.reshape(1,-1)),a1[i,:].reshape(1,-1))

Theta1[:,0] = 0

Theta2[:,0] = 0

'''梯度'''

grad = (np.vstack((Theta1_grad.reshape(-1,1),

Theta2_grad.reshape(-1,1)))+Lambda*np.vstack

((Theta1.reshape(-1,1),Theta2.reshape(-1,1))))/m

return np.ravel(grad)

# 检验梯度是否计算正确

# 检验梯度是否计算正确

def checkGradient(Lambda = 0):

'''构造一个小型的神经网络验证，因为数值法计算梯度很浪

费时间，而且验证正确后之后就不再需要验证了'''

input_layer_size = 3

hidden_layer_size = 5

num_labels = 3

m = 5

initial_Theta1 = debugInitializeWeights(input_layer_size,

hidden_layer_size);

initial_Theta2 = debugInitializeWeights

(hidden_layer_size,num_labels)

X = debugInitializeWeights(input_layer_size-1,m)

y = 1+np.transpose(np.mod(np.arange(1,m+1), num_labels))

# 初始化y

y = y.reshape(-1,1)

nn_params = np.vstack((initial_Theta1.reshape(-1,1),

initial_Theta2.reshape(-1,1))) #展开theta

'''BP求出梯度'''

grad = nnGradient(nn_params, input_layer_size, hidden_layer_size,

num_labels, X, y, Lambda)

'''使用数值法计算梯度'''

num_grad = np.zeros((nn_params.shape[0]))

step = np.zeros((nn_params.shape[0]))

e = 1e-4

for i in range(nn_params.shape[0]):

step[i] = e

loss1 = nnCostFunction(nn_params-step.reshape(-1,1),

input_layer_size, hidden_layer_size,

num_labels, X, y,

Lambda)

loss2 = nnCostFunction(nn_params+step.reshape(-1,1),

input_layer_size, hidden_layer_size,

num_labels, X, y,

Lambda)

num_grad[i] = (loss2-loss1)/(2*e)

step[i]=0

# 显示两列比较

res = np.hstack((num_grad.reshape(-1,1),

grad.reshape(-1,1)))

print res

# 随机初始化权重theta

def randInitializeWeights(L_in,L_out):

W = np.zeros((L_out,1+L_in)) # 对应theta的权重

epsilon_init = (6.0/(L_out+L_in))**0.5

W = np.random.rand(L_out,1+L_in)*2*epsilon_init-epsilon_init # np.random.rand(L_out,1+L_in)产生L_out*(1+L_in)大小的随机矩阵

return W

# 预测

def predict(Theta1,Theta2,X):

m = X.shape[0]

num_labels = Theta2.shape[0]

#p = np.zeros((m,1))

'''正向传播，预测结果'''

X = np.hstack((np.ones((m,1)),X))

h1 = sigmoid(np.dot(X,np.transpose(Theta1)))

h1 = np.hstack((np.ones((m,1)),h1))

h2 = sigmoid(np.dot(h1,np.transpose(Theta2)))

'''

返回h中每一行最大值所在的列号

- np.max(h, axis=1)返回h中每一行的最大值（是某个数字的最大概率）

- 最后where找到的最大概率所在的列号（列号即是对应的数字）

'''

#np.savetxt("h2.csv",h2,delimiter=',')

p = np.array(np.where(h2[0,:] == np.max(h2, axis=1)[0]))

for i in np.arange(1, m):

t = np.array(np.where(h2[i,:] == np.max(h2, axis=1)[i]))

p = np.vstack((p,t))

return p

'''data1——线性分类'''

data1 = spio.loadmat('data1.mat')

X = data1['X']

y = data1['y']

y = np.ravel(y)

plot_data(X,y)

model = svm.SVC(C=1.0,kernel='linear').fit(X,y) # 指定核函数为线性核函数

'''data2——非线性分类'''

data2 = spio.loadmat('data2.mat')

X = data2['X']

y = data2['y']

y = np.ravel(y)

plt = plot_data(X,y)

plt.show()

model = svm.SVC(gamma=100).fit(X,y) # gamma为核函数的系数，值越大拟合的越好

# 找到每条数据距离哪个类中心最近

def findClosestCentroids(X,initial_centroids):

m = X.shape[0] # 数据条数

K = initial_centroids.shape[0] # 类的总数

dis = np.zeros((m,K)) # 存储计算每个点分别到K个类的距离

idx = np.zeros((m,1)) # 要返回的每条数据属于哪个类

'''计算每个点到每个类中心的距离'''

for i in range(m):

for j in range(K):

dis[i,j] = np.dot((X[i,:]-initial_centroids[j,:]).reshape(1,-1),(X[i,:]-initial_centroids[j,:]).reshape(-1,1))

'''返回dis每一行的最小值对应的列号，即为对应的类别

- np.min(dis, axis=1)返回每一行的最小值

- np.where(dis == np.min(dis, axis=1).reshape(-1,1)) 返回对应最小值的坐标

- 注意：可能最小值对应的坐标有多个，where都会找出来，所以返回时返回前m个需要的即可（因为对于多个最小值，属于哪个类别都可以）

'''

dummy,idx = np.where(dis == np.min(dis, axis=1).reshape(-1,1))

return idx[0:dis.shape[0]] # 注意截取一下

# 计算类中心

def computerCentroids(X,idx,K):

n = X.shape[1]

centroids = np.zeros((K,n))

for i in range(K):

centroids[i,:] = np.mean(X[np.ravel(idx==i),:], axis=0).reshape(1,-1) # 索引要是一维的,axis=0为每一列，idx==i一次找出属于哪一类的，然后计算均值

return centroids

# 初始化类中心--随机取K个点作为聚类中心

def kMeansInitCentroids(X,K):

m = X.shape[0]

m_arr = np.arange(0,m) # 生成0-m-1

centroids = np.zeros((K,X.shape[1]))

np.random.shuffle(m_arr) # 打乱m_arr顺序

rand_indices = m_arr[:K] # 取前K个

centroids = X[rand_indices,:]

return centroids

# 聚类算法

def runKMeans(X,initial_centroids,max_iters,plot_process):

m,n = X.shape # 数据条数和维度

K = initial_centroids.shape[0] # 类数

centroids = initial_centroids # 记录当前类中心

previous_centroids = centroids # 记录上一次类中心

idx = np.zeros((m,1)) # 每条数据属于哪个类

for i in range(max_iters): # 迭代次数

print u'迭代计算次数：%d'%(i+1)

idx = findClosestCentroids(X, centroids)

if plot_process: # 如果绘制图像

plt = plotProcessKMeans(X,centroids,previous_centroids) # 画聚类中心的移动过程

previous_centroids = centroids # 重置

centroids = computerCentroids(X, idx, K) # 重新计算类中心

if plot_process: # 显示最终的绘制结果

plt.show()

return centroids,idx # 返回聚类中心和数据属于哪个类

# 归一化数据

def featureNormalize(X):

'''（每一个数据-当前列的均值）/当前列的标准差'''

n = X.shape[1]

mu = np.zeros((1,n));

sigma = np.zeros((1,n))

mu = np.mean(X,axis=0)

sigma = np.std(X,axis=0)

for i in range(n):

X[:,i] = (X[:,i]-mu[i])/sigma[i]

return X,mu,sigma

# 映射数据

def projectData(X_norm,U,K):

Z = np.zeros((X_norm.shape[0],K))

U_reduce = U[:,0:K] # 取前K个

Z = np.dot(X_norm,U_reduce)

return Z

# 恢复数据

def recoverData(Z,U,K):

X_rec = np.zeros((Z.shape[0],U.shape[0]))

U_recude = U[:,0:K]

X_rec = np.dot(Z,np.transpose(U_recude)) # 还原数据（近似）

return X_rec

#-*- coding: utf-8 -*-

# Author:bob

# Date:2016.12.22

import numpy as np

from matplotlib import pyplot as plt

from scipy import io as spio

from sklearn.decomposition import pca

from sklearn.preprocessing import StandardScaler

'''归一化数据并作图'''

scaler = StandardScaler()

scaler.fit(X)

x_train = scaler.transform(X)

'''拟合数据'''

K=1 # 要降的维度

model = pca.PCA(n_components=K).fit(x_train) # 拟合数据，n_components定义要降的维度

Z = model.transform(x_train) # transform就会执行降维操作

'''数据恢复并作图'''

Ureduce = model.components_ # 得到降维用的Ureduce

x_rec = np.dot(Z,Ureduce) # 数据恢复

# 参数估计函数（就是求均值和方差）

def estimateGaussian(X):

m,n = X.shape

mu = np.zeros((n,1))

sigma2 = np.zeros((n,1))

mu = np.mean(X, axis=0) # axis=0表示列，每列的均值

sigma2 = np.var(X,axis=0) # 求每列的方差

return mu,sigma2

# 选择最优的epsilon，即：使F1Score最大

def selectThreshold(yval,pval):

'''初始化所需变量'''

bestEpsilon = 0.

bestF1 = 0.

F1 = 0.

step = (np.max(pval)-np.min(pval))/1000

'''计算'''

for epsilon in np.arange(np.min(pval),np.max(pval),step):

cvPrecision = pval<epsilon

tp = np.sum((cvPrecision == 1) & (yval == 1).ravel()).astype(float) # sum求和是int型的，需要转为float

fp = np.sum((cvPrecision == 1) & (yval == 0).ravel()).astype(float)

fn = np.sum((cvPrecision == 0) & (yval == 1).ravel()).astype(float)

precision = tp/(tp+fp) # 精准度

recision = tp/(tp+fn) # 召回率

F1 = (2*precision*recision)/(precision+recision) # F1Score计算公式

if F1 > bestF1: # 修改最优的F1 Score

bestF1 = F1

bestEpsilon = epsilon

return bestEpsilon,bestF1

# 多元高斯分布函数

def multivariateGaussian(X,mu,Sigma2):

k = len(mu)

if (Sigma2.shape[0]>1):

Sigma2 = np.diag(Sigma2)

'''多元高斯分布函数'''

X = X-mu

argu = (2*np.pi)**(-k/2)*np.linalg.det(Sigma2)**(-0.5)

p = argu*np.exp(-0.5*np.sum(np.dot(X,np.linalg.inv

(Sigma2))*X,axis=1))

# axis表示每行

return p