# 用Python实现BP神经网络（附代码）

BP神经网络

https://github.com/lawlite19/MachineLearning_Python/blob/master/NeuralNetwok/NeuralNetwork.py

，也成为激励函数

==》j+1的单元数x（j层的单元数+1）

，即代表输出层有K个单元

L-->所有层的个数

-->第l层unit的个数

```
# 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)

```

BP反向传播的目的就是求代价函数的梯度

《===》

（向量化）

，因为对于输入没有误差

for i=1-m:-

-正向传播计算

（l=2,3,4...L）

-反向计算

...

-

-

，即得到代价函数的梯度

```
# 梯度

length = nn_params.shape[0]

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)

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[:,0] = 0;

Theta2[:,0] = 0;

'''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,:]

'''梯度'''

```

BP可以求梯度的原因

```
# 检验梯度是否计算正确

# 检验梯度是否计算正确

'''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'''

num_labels, X, y, Lambda)

'''使用数值法计算梯度'''

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)

step[i]=0

# 显示两列比较

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)))

'''

- 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

```

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