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