激活函数
import numpy as np
import matplotlib.pylab as plt
def step_function(x):
return np.array(x> 0, dtype=np.int)
def sigmoid(x):
return 1/(1 + np.exp(-x))
x = np.arange(-6, 6, 0.1)
print(x)
y1 = step_function(x)
print(y1)
y2 = sigmoid(x)
print(y2)
fig, ax1 = plt.subplots(2,1, figsize=(12,9))
#ax1 = ax1.twinx() # 让2个子图的x轴一样，同时创建副坐标轴。
ax1[0][0].plot(x, y1)
ax1[1][0].plot(x, y2)
plt.tight_layout()
矩阵乘法
import numpy as np
#import matplotlib.pylab as plt
A = np.array([[1,2],[3,4]])
B = np.array([[5,6],[7,8]])
print(np.dot(A,B))
A1 = np.array([[1,2,3],[3,4,5]])
B1 = np.array([[1,2],[3,4],[4,5]]) # 3 2
C = np.dot(A1,B1)
print(C)
print(C.shape) # (2, 2)
print(np.ndim(C)) # 2
A2 = np.array([7,8]) # 1 2
print(A2 ) # [7 8]
A3 = np.array([[7],[8]])
print(A3 ) # [7 8]
print(np.dot(B1,A2))
print(np.dot(B1,A3))
# 神经网络的内积
W = np.array([[1,3,5],[2,4,6]])
X = np.array([1,2])
print(np.dot(X,W)) # [ 5 11 17]
W1 = np.array([[1,2],[3,4],[5,6]])
X1 = np.array([[1],[2]])
print(np.dot(W1,X1))
神经网络前向计算
def sigmoid(x):
return 1/(1 + np.exp(-x))
import numpy as np
#import matplotlib.pylab as plt
X = np.array([1.0,0.5])
W1 = np.array([[0.1,0.3,0.5],[0.2,0.4,0.6]])
B1 = np.array([0.1,0.2,0.3])
print(X.shape) # (2,)
print(W1.shape) # (2, 3)
print(B1.shape) # (3,)
# 第一层神经网络的传递
A1 = np.dot(X, W1) + B1
print(A1) # [0.3 0.7 1.1]
# 使用激活函数
Z1 = sigmoid(A1)
print(Z1) # [0.57444252 0.66818777 0.75026011]
# 第二层神经网络的传递
W2 = np.array([[0.1,0.4],[0.2,0.5],[0.3,0.6]])
B2 = np.array([0.1,0.2])
A2 = np.dot(Z1, W2) + B2
print(A2) # [0.51615984 1.21402696]
Z2 = sigmoid(A2)
print(Z2) #[0.62624937 0.7710107 ]
# 第二层到输出层的传递
# 输出层的激活函数
# 一般地： 回归问题可以使用恒等函数
# 二元分类问题可以使用 sigmoid 函数
# 多元分类问题可以使用 softmax 函数
def identity_function(x):
return x
W3 = np.array([[0.1,0.3],[0.2,0.4]])
B3 = np.array([0.1,0.2])
A3 = np.dot(Z2, W3) + B3
print(A2) # [0.51615984 1.21402696]
Y = identity_function(A3)
print(Y) # [0.31682708 0.69627909]
代码小结
import numpy as np
def sigmoid(x):
return 1/(1 + np.exp(-x))
def identity_function(x):
return x
def init_network():
network = {}
network['W1'] = np.array([[0.1,0.3,0.5],[0.2,0.4,0.6]])
network['B1'] = np.array([0.1,0.2,0.3])
network['W2'] = np.array([[0.1,0.4],[0.2,0.5],[0.3,0.6]])
network['B2'] = np.array([0.1,0.2])
network['W3'] = np.array([[0.1,0.3],[0.2,0.4]])
network['B3'] = np.array([0.1,0.2])
return network
def forward(network, X):
W1, W2, W3 = network['W1'], network['W2'], network['W3']
B1, B2, B3 = network['B1'], network['B2'], network['B3']
A1 = np.dot(X, W1) + B1
Z1 = sigmoid(A1)
A2 = np.dot(Z1, W2) + B2
Z2 = sigmoid(A2)
A3 = np.dot(Z2, W3) + B3
Y = identity_function(A3)
return Y
network2 = init_network()
X = np.array([1.0,0.5])
Y = forward(network2, X)
print(Y) # [0.31682708 0.69627909]
