python实现BP神经网络预测正弦函数（双隐藏层+偏置）
BP网络（Back Propagation），是1986年由Rumelhart和McCelland为首的科学家小组提出，是一种按误差逆 传播
算法训练的多层前馈网络，是目前应用最广泛的 神经网络
模型之一。BP网络能学习和存贮大量的输入-输出模式映射关系，而无需事前揭示描述这种映射关系的 数学
方程。了解BP神经网络的工作过程，对入门人工神经网络有很大帮助。本文通过python编程实现含有两层隐藏层的神经网络模型，并用与正弦函数预测。
# -*- coding: utf-8 -*-
"""
Created on Tue May 15 16:39:55 2018
@author: Administrator
"""
import math
import random
import numpy
import matplotlib.pyplot as plt
random.seed(0)#使random函数每一次生成的随机数值都相等
def rand(a,b):
return (b - a) * random.random() + a #生成a到b的随机数
def make_matrix(m,n,fill=0.0):#创建一个指定大小的矩阵
mat = []
for i in range(m):
mat.append([fill]*n)
return mat
#定义tanh函数和它的导数
def tanh(x):
return numpy.tanh(x)
def tanh_derivate(x):
return 1-numpy.tanh(x)*numpy.tanh(x) #tanh函数的导数
class BPNeuralNetwork:
def __init__(self):#初始化变量
self.input_n = 0
self.hidden1_n = 0
self.hidden2_n = 0
self.output_n = 0
self.input_cells = []
self.hidden1_cells = []
self.hidden2_cells = []
self.output_cells = []
self.input_weights = []
self.hidden_weights = []
self.output_weights = []
self.input_correction = []
self.hidden_correction = []
self.output_correction = []
#三个列表维护：输入层，隐含层，输出层神经元
def setup(self,ni,nh1,nh2,no):
self.input_n = ni+1 #输入层+偏置项
self.hidden1_n = nh1+1 #隐含层 1
self.hidden2_n = nh2+1 #隐含层 2
self.output_n = no #输出层
#初始化神经元
self.input_cells = [1.0]*self.input_n
self.hidden1_cells= [1.0]*self.hidden1_n
self.hidden2_cells= [1.0]*self.hidden2_n
self.output_cells= [1.0]*self.output_n
#初始化连接边的边权
self.input_weights = make_matrix(self.input_n,self.hidden1_n) #邻接矩阵存边权：输入层->隐藏层
self.hidden_weights = make_matrix(self.hidden1_n,self.hidden2_n)
self.output_weights = make_matrix(self.hidden2_n,self.output_n) #邻接矩阵存边权：隐藏层->输出层
#初始化bias
self.h1_b = make_matrix(self.hidden1_n,1)
self.h2_b = make_matrix(self.hidden2_n,1)
self.o_b = make_matrix(self.output_n,1)
#随机初始化边权：为了反向传导做准备--->随机初始化的目的是使对称失效
for i in range(self.input_n):
for h in range(self.hidden1_n):
self.input_weights[i][h] = rand(-1,1) #由输入层第i个元素到隐藏层1第j个元素的边权为随机值
for i in range(self.hidden1_n):
for h in range(self.hidden2_n):
self.hidden_weights[i][h] = rand(-1,1) #由隐藏层1第i个元素到隐藏层2第j个元素的边权为随机值
for h in range(self.hidden2_n):
for o in range(self.output_n):
self.output_weights[h][o] = rand(-1,1) #由隐藏层2第i个元素到输出层第j个元素的边权为随机值
#随机初始化bias
for i in range(self.hidden1_n):
self.h1_b[i] = rand(-1, 1)
for i in range(self.hidden2_n):
self.h2_b[i] = rand(-1, 1)
for i in range(self.output_n):
self.o_b[i] = rand(-1, 1)
#保存校正矩阵，为了以后误差做调整
self.input_correction = make_matrix(self.input_n , self.hidden1_n)
self.hidden_correction = make_matrix(self.hidden1_n , self.hidden2_n)
self.output_correction = make_matrix(self.hidden2_n,self.output_n)
#输出预测值
def predict(self,inputs):
#对输入层进行操作转化样本
for i in range(self.input_n-1):
self.input_cells[i] = inputs[i] #n个样本从0~n-1
#计算隐藏层的输出，每个节点最终的输出值就是权值*节点值的加权和
for j in range(self.hidden1_n):
total = 0.0
for i in range(self.input_n):
total+=self.input_cells[i]*self.input_weights[i][j]
# 此处为何是先i再j，以隐含层节点做大循环，输入样本为小循环，是为了每一个隐藏节点计算一个输出值，传输到下一层
self.hidden1_cells[j] = tanh(total-self.h1_b[j]) #此节点的输出是前一层所有输入点和到该点之间的权值加权和
for m in range(self.hidden2_n):
total = 0.0
for i in range(self.hidden1_n):
total+=self.hidden1_cells[i]*self.hidden_weights[i][m]
self.hidden2_cells[m] = tanh(total-self.h2_b[m]) #此节点的输出是前一层所有输入点和到该点之间的权值加权和
for k in range(self.output_n):
total = 0.0
for j in range(self.hidden2_n):
total+=self.hidden2_cells[j]*self.output_weights[j][k]
self.output_cells[k] = tanh(total-self.o_b[k]) #获取输出层每个元素的值
return self.output_cells[:] #最后输出层的结果返回
#反向传播算法
def back_propagate(self,case,label,learn,correct):
self.predict(case) #对实例进行预测
output_deltas = [0.0]*self.output_n #初始化矩阵
for o in range(self.output_n):
error = label[o] - self.output_cells[o] #正确结果和预测结果的误差：0,1，-1
output_deltas[o]= tanh_derivate(self.output_cells[o])*error#误差稳定在0~1内
#隐含层误差
hidden1_deltas = [0.0]*self.hidden1_n
hidden2_deltas = [0.0]*self.hidden2_n
for h in range(self.hidden2_n):
error = 0.0
for o in range(self.output_n):
error+=output_deltas[o]*self.output_weights[h][o]
hidden2_deltas[h] = tanh_derivate(self.hidden2_cells[h])*error
#反向传播算法求W
#更新隐藏层->输出权重
for h2 in range(self.hidden2_n):
for o in range(self.output_n):
change = output_deltas[o]*self.hidden2_cells[h2]
#调整权重：上一层每个节点的权重学习*变化+矫正率
self.output_weights[h2][o] += learn*change + correct*self.output_correction[h2][o]
self.output_correction[h2][o] = change
#更新隐藏1层->隐藏2权重
for h1 in range(self.hidden1_n):
for o in range(self.hidden2_n):
change = hidden2_deltas[o]*self.hidden1_cells[h1]
#调整权重：上一层每个节点的权重学习*变化+矫正率
self.hidden_weights[h1][o] += learn*change + correct*self.hidden_correction[h1][o]
self.hidden_correction[h1][o] = change
#更新输入->隐藏层的权重
for i in range(self.input_n):
for h in range(self.hidden1_n):
change = hidden1_deltas[h]*self.input_cells[i]
self.input_weights[i][h] += learn*change + correct*self.input_correction[i][h]
self.input_correction[i][h] = change
#更新bias
for o in range(self.output_n):
self.o_b[o] = self.o_b[o] - learn * output_deltas[o]
for h2 in range(self.hidden2_n):
self.h2_b[h2] = self.h2_b[h2] - learn * hidden2_deltas[h2]
for h1 in range(self.hidden1_n):
self.h1_b[h1] = self.h1_b[h1] - learn * hidden1_deltas[h1]
error = 0.0
for o in range(len(label)):
error = 0.5*(label[o]-self.output_cells[o])**2 #平方误差函数
return error
def train(self,cases,labels,limit=10000,learn=0.05,correct=0.1):
for i in range(limit): #设置迭代次数
error = 0.0
for j in range(len(cases)):#对输入层进行访问
label = labels[j]
case = cases[j]
error+=self.back_propagate(case,label,learn,correct) #样例，标签，学习率，正确阈值
def test(self): #学习正弦函数
cases = [ ]#测试样
for i in range(0,21,1):
cases.append([i*math.pi/10])
labels = numpy.sin(cases)
self.setup(1,10,10,1) #初始化神经网络：输入层，隐藏层，输出层元素个数
self.train(cases,labels,10000,0.05,0.1)
test = [ ]#训练范围外的数据
yables = []
for i in range(0,201,1):
test.append([i*math.pi/100])
for case in test:
yables.append(self.predict(case))
x = numpy.arange(0.0,2.0,0.01)
plt.figure()
l1, = plt.plot(x*math.pi,numpy.sin(x*math.pi),color = 'red')
l2, = plt.plot(test,yables,color = 'green')
plt.legend(handles = [l1, l2,], labels = ['original', 'test predict'], loc = 'best')
plt.xticks([ 0, numpy.pi/2, numpy.pi,3*numpy.pi/2,2*numpy.pi],[r'$0$', r'$+\pi/2$', r'$+\pi$', r'$+\pi*3/2$', r'$+\pi*2$'])
plt.show()
zables = []
for a in cases:
zables.append(self.predict(a))
plt.figure()
l3, = plt.plot(x*math.pi,numpy.sin(x*math.pi),color = 'red')
l4, = plt.plot(cases,zables,color = 'green')
plt.legend(handles = [l3, l4,], labels = ['original', 'train predict'], loc = 'best')
plt.xticks([ 0, numpy.pi/2, numpy.pi,3*numpy.pi/2,2*numpy.pi],[r'$0$', r'$+\pi/2$', r'$+\pi$', r'$+\pi*3/2$', r'$+\pi*2$'])
plt.show()
if __name__ == '__main__':
nn = BPNeuralNetwork()
nn.test()在0-2*pi上取200个点作为正弦函数的训练数据，在利用训练好的模型对训练集中的数据以及训练集外的数据分别进行预测，并绘出相应曲线，观察拟合结果。结果如下图所示：
