神经网络

神经网络模型

1. 神经元模型

常用的神经元激活函数

import math
import matplotlib.pyplot as plt
#阶跃函数
def llsh_sgn(x):
    if x >= 0:
        return 1
    else:
        return 0

#Sigmoid函数
def llsh_sigmoid(x):
    return 1/(1 + math.exp(-x))

#绘图
import numpy as num
y = []
x = num.arange(-100,100,0.1)
for index in x:
     y.append(llsh_sigmoid(index))

fig, ax = plt.subplots()
ax.scatter(x, y , s = 0.6)

ax.set_xlabel('x')
ax.set_ylabel('y')
plt.show()

感知器和多层网络

误差逆向传播算法（error BackPropagation,BP）

http://python.jobbole.com/82758/

import matplotlib.pyplot as plt
import numpy as np;
import numpy as num
#阶跃函数
def llsh_sgn(x):
    if x >= 0:
        return 1
    else:
        return 0

#Sigmoid函数
def llsh_sigmoid(x):
    return 1/(1 + np.exp(-x))

#Sigmoid函数的导数
def llsh_sigmoid_div(x):
    return x * (1 - x)



y = []
x = num.arange(-10,10,0.1)
for index in x:
     y.append(llsh_sigmoid(index))




fig, ax = plt.subplots()
#ax.scatter(x, y , s = 10)

ax.set_xlabel('x')
ax.set_ylabel('sigmoid')
#plt.show()



#单层神经元训练范例
X = np.array([ [0.9,0,1],
               [0,1,1],
               [1,0,1],
               [0.1,1,1] ])
print("Input",X[:,0:2])

ax.scatter(X[:,0].T, X[:,1].T , s = 10)
ax.set_xlabel('x')
ax.set_ylabel('sigmoid')
plt.show()

y = np.array([[0,0,1,1]]).T
print("\nOutput",y)
np.random.seed(1)
syn0 = 2*np.random.random((3,1)) - 1

print(syn0)
for j in num.arange(0,1000,1):
    l0 = X
    #求出输出
    l1 = llsh_sigmoid(np.dot(l0, syn0))
    #
    l1_error = y - l1
    l1_delta = l1_error * llsh_sigmoid_div(l1)
    syn0 += np.dot(l0.T, l1_delta)

print ("Output After Training:")
print (l1)
print (syn0)




#多层神经网络
X = np.array([ [0,0,1],[0,1,1],[1,0,1],[1,1,1] ])
y = np.array([[0,1,1,0]]).T
syn0 = 2*np.random.random((3,4)) - 1
syn1 = 2*np.random.random((4,1)) - 1
for j in num.arange(6000):
    l0 = X
    #由当前的参数计算出输出值
    l1 = llsh_sigmoid(np.dot(l0,syn0))
    l2 = llsh_sigmoid(np.dot(l1,syn1))
    #使用梯度向量法计算
    l2_delta = (y - l2)*llsh_sigmoid_div(l2)
    l1_delta = l2_delta.dot(syn1.T) * llsh_sigmoid_div(l1)
    syn1 += l1.T.dot(l2_delta)
    syn0 += l0.T.dot(l1_delta)
print ("Output After Training:")
print(l2)

http://python.jobbole.com/82758/

import matplotlib.pyplot as plt
import numpy as np;
import numpy as num
#阶跃函数
def llsh_sgn(x):
    if x >= 0:
        return 1
    else:
        return 0

#Sigmoid函数
def llsh_sigmoid(x):
    return 1/(1 + np.exp(-x))

#Sigmoid函数的导数
def llsh_sigmoid_div(x):
    return x * (1 - x)



y = []
x = num.arange(-10,10,0.1)
for index in x:
     y.append(llsh_sigmoid(index))




fig, ax = plt.subplots()
#ax.scatter(x, y , s = 10)

ax.set_xlabel('x')
ax.set_ylabel('sigmoid')
#plt.show()



#单层神经元训练范例
X = np.array([ [0.9,0,1],
               [0,1,1],
               [1,0,1],
               [0.1,1,1] ])
print("Input",X[:,0:2])

ax.scatter(X[:,0].T, X[:,1].T , s = 10)
ax.set_xlabel('x')
ax.set_ylabel('sigmoid')
plt.show()

y = np.array([[0,0,1,1]]).T
print("\nOutput",y)
np.random.seed(1)
syn0 = 2*np.random.random((3,1)) - 1

print(syn0)
for j in num.arange(0,1000,1):
    l0 = X
    #求出输出
    l1 = llsh_sigmoid(np.dot(l0, syn0))
    #
    l1_error = y - l1
    l1_delta = l1_error * llsh_sigmoid_div(l1)
    syn0 += np.dot(l0.T, l1_delta)

print ("Output After Training:")
print (l1)
print (syn0)




#多层神经网络
X = np.array([ [0,0,1],[0,1,1],[1,0,1],[1,1,1] ])
y = np.array([[0,1,1,0]]).T
syn0 = 2*np.random.random((3,4)) - 1
syn1 = 2*np.random.random((4,1)) - 1
for j in num.arange(6000):
    l0 = X
    #由当前的参数计算出输出值
    l1 = llsh_sigmoid(np.dot(l0,syn0))
    l2 = llsh_sigmoid(np.dot(l1,syn1))
    #使用梯度向量法计算
    l2_delta = (y - l2)*llsh_sigmoid_div(l2)
    l1_delta = l2_delta.dot(syn1.T) * llsh_sigmoid_div(l1)
    syn1 += l1.T.dot(l2_delta)
    syn0 += l0.T.dot(l1_delta)
print ("Output After Training:")
print(l2)

全局最小和局部最小

使用梯度下降法很容易陷入局部最小之值，可以考虑使用如下方法解决

• 使用不同的初始值
• 模拟退火算法
• 使用随机梯度下降

不过这些方法是启发性质的，理论上尚无法保障

其他常见的神经网络

• RBF神经网络
• ART神经网络