BP神经网络

反向传播（Back Propagation，简称BP）神经网络解决了多层神经网络的学习问题，广泛应用于分类识别、图像识别、压缩、逼近以及回归等领域，其结构如下所示。

另外介绍及格激活函数：sigmoid、tanh和softsign。神经网络中的激活函数，其作用就是引入非线性。
Sigmoid：sigmoid的优点是输出范围有限，数据在传递的过程中不容易发散，求导很容易（y=sigmoid(x), y’=y(1-y)）。缺点是饱和的时候梯度太小。其输出范围为(0, 1)，所以可以用作输出层，输出表示概率。

公式：

tanh和softsign：

使用BP神经网络解决异或问题：
使用sigmoid激活函数，偏置值定为1。

import numpy as npimport matplotlib.pyplot as plt

#输入数据X = np.array([[1,0,0],              [1,0,1],              [1,1,0],              [1,1,1]])#标记 Y = np.array([[0,1,1,0]])#权值初始化，取值范围为-1到1V = np.random.random((3,4))*2-1#第一层权值W = np.random.random((4,1))*2-1#第二层权值print(V)print(W)#学习率lr = 0.11#sigmoid激活函数def sigmoid(x):    return 1/(1+np.exp(-x))#sigmoid函数导数def dsigmoid(x):    return x*(1-x)#更新权值def update():    global X,V,W,lr    L1 = sigmoid(np.dot(X,V))#隐藏层输出--4x4的矩阵    L2 = sigmoid(np.dot(L1,W))#输出层输出--4x1的矩阵    L2_delta = (Y.T - L2)*dsigmoid(L2)    L1_delta = np.dot(L2_delta,W.T)*dsigmoid(L1)    W_C = lr*np.dot(L1.T,L2_delta)    V_C = lr*np.dot(X.T,L1_delta)    W = W + W_C    V = V + V_C

[[-0.63777659 -0.54506959 -0.33179877 -0.31877276] [-0.95125978  0.7730579   0.29587091 -0.58236583] [ 0.79077282 -0.93217298 -0.09382096  0.5822262 ]][[ 0.59146382] [-0.30789231] [-0.57723785] [ 0.18645706]]

for i in range(10000):#迭代一万次    update()#更新权值    if i%500 == 0:#每500次打印一下误差        L1 = sigmoid(np.dot(X,V))#隐藏层输出--4x4的矩阵        L2 = sigmoid(np.dot(L1,W))#输出层输出--4x1的矩阵        print("Error:",np.mean(np.abs(Y.T-L2)))L1 = sigmoid(np.dot(X,V))#隐藏层输出--4x4的矩阵L2 = sigmoid(np.dot(L1,W))#输出层输出--4x1的矩阵print(L2)

Error: 0.498712507036Error: 0.494796348535Error: 0.480546625904Error: 0.4198798183Error: 0.279981268445Error: 0.178200802236Error: 0.130067381817Error: 0.10418018522Error: 0.0881430063965Error: 0.0771905626373Error: 0.0691913263575Error: 0.0630610081972Error: 0.0581913640284Error: 0.0542146972308Error: 0.0508953510674Error: 0.0480750732027Error: 0.0456434672314Error: 0.0435210496818Error: 0.0416490513475Error: 0.0399830217768[[ 0.04218605] [ 0.95982694] [ 0.96413189] [ 0.03573883]]

最后的输出结果也很接近[0,1,1,0]

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使用BP神经网络解决数字显示问题：

from sklearn.datasets import load_digitsimport numpy as npfrom sklearn.preprocessing import LabelBinarizerfrom sklearn.cross_validation import train_test_splitdef sigmoid(x):    return 1/(1+np.exp(-x))def dsigmoid(x):    return x*(1-x)class NeuralNetwort:    def __init__(self,layers):#(640,100,10)        #权值的初始化，范围-1到1.        self.V = np.random.random((layers[0]+1,layers[1]+1))*2-1        self.W = np.random.random((layers[1]+1,layers[2]))*2-1            def train(self,X,y,lr=0.11,epochs=10000):        #添加偏置        temp = np.ones([X.shape[0],X.shape[1]+1])        temp[:,0:-1] = X        X = temp        for n in range(epochs+1):            i = np.random.randint(X.shape[0])#随机选取一个数据            x = [X[i]]            x = np.atleast_2d(x)#转为2维数据            L1 = sigmoid(np.dot(x,self.V))#隐藏层输出            L2 = sigmoid(np.dot(L1,self.W))#输出层输出            L2_delta = (y[i]-L2)*dsigmoid(L2)            L1_delta = np.dot(L2_delta,self.W.T)*dsigmoid(L1)            self.W += lr*np.dot(L1.T,L2_delta)            self.V += lr*np.dot(x.T,L1_delta)            #每训练一千次预测一次准确率            if n%1000 == 0:                predictions = []                for j in range(X_test.shape[0]):                    o=self.predict(X_test[j])                    predictions.append(np.argmax(o))#获取预测结果                accuracy = np.mean(np.equal(predictions,y_test))                print("epoch:",n,"accuracy:",accuracy)    def predict(self,x):    #添加偏置        temp = np.ones(x.shape[0]+1)        temp[0:-1] = x        x = temp        x = np.atleast_2d(x)#转为2维数据        L1 = sigmoid(np.dot(x,self.V))#隐藏层输出        L2 = sigmoid(np.dot(L1,self.W))#输出层输出        return L2digits = load_digits()#载入数据X = digits.data#数据y = digits.target#标记#输入数据归一化X -= X.min()X /= X.max()nm = NeuralNetwort([64,100,10])#创建网络X_train,X_test,y_train,y_test = train_test_split(X,y)#分割1/4数据为测试数据，3/4为训练数据。labels_train = LabelBinarizer().fit_transform(y_train)#标记二值化labels_test = LabelBinarizer().fit_transform(y_test)#标记二值化      print("start")nm.train(X_train,labels_train,epochs=20000)print("end")

startepoch: 0 accuracy: 0.16epoch: 1000 accuracy: 0.688888888889epoch: 2000 accuracy: 0.895555555556epoch: 3000 accuracy: 0.897777777778epoch: 4000 accuracy: 0.904444444444epoch: 5000 accuracy: 0.953333333333epoch: 6000 accuracy: 0.931111111111epoch: 7000 accuracy: 0.946666666667epoch: 8000 accuracy: 0.946666666667epoch: 9000 accuracy: 0.962222222222epoch: 10000 accuracy: 0.964444444444epoch: 11000 accuracy: 0.957777777778epoch: 12000 accuracy: 0.96epoch: 13000 accuracy: 0.955555555556epoch: 14000 accuracy: 0.962222222222epoch: 15000 accuracy: 0.968888888889epoch: 16000 accuracy: 0.964444444444epoch: 17000 accuracy: 0.948888888889epoch: 18000 accuracy: 0.955555555556epoch: 19000 accuracy: 0.957777777778epoch: 20000 accuracy: 0.966666666667end