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1. 关于非线性转化方程(non-linear transformation function)

sigmoid函数(S 曲线)用来作为activation function:

 1.1 双曲函数(tanh)
 
 1.2  逻辑函数(logistic function)

2. 实现一个简单的神经网络算法

import numpy as np

def tanh(x):
return np.tanh(x)

def tanh_deriv(x):
return 1.0 - np.tanh(x)*np.tanh(x)

def logistic(x):
return 1/(1 + np.exp(-x))

def logistic_derivative(x):
return logistic(x)*(1-logistic(x))

class NeuralNetwork:
def init(self, layers, activation='tanh'):
"""
:param layers: A list containing the number of units in each layer.
Should be at least two values
:param activation: The activation function to be used. Can be
"logistic" or "tanh"
"""
if activation == 'logistic':
self.activation = logistic
self.activation_deriv = logistic_derivative
elif activation == 'tanh':
self.activation = tanh
self.activation_deriv = tanh_deriv

    self.weights = []  
    for i in range(1, len(layers) - 1):  
        self.weights.append((2*np.random.random((layers[i - 1] + 1, layers[i] + 1))-1)*0.25)  
        self.weights.append((2*np.random.random((layers[i] + 1, layers[i + 1]))-1)*0.25)
        
        
def fit(self, X, y, learning_rate=0.2, epochs=10000):         
    X = np.atleast_2d(X)         
    temp = np.ones([X.shape[0], X.shape[1]+1])         
    temp[:, 0:-1] = X  # adding the bias unit to the input layer         
    X = temp         
    y = np.array(y)

    for k in range(epochs):  
        i = np.random.randint(X.shape[0])  
        a = [X[i]]

        for l in range(len(self.weights)):  #going forward network, for each layer
            a.append(self.activation(np.dot(a[l], self.weights[l])))  #Computer the node value for each layer (O_i) using activation function
        error = y[i] - a[-1]  #Computer the error at the top layer
        deltas = [error * self.activation_deriv(a[-1])] #For output layer, Err calculation (delta is updated error)
        
        #Staring backprobagation
        for l in range(len(a) - 2, 0, -1): # we need to begin at the second to last layer 
            #Compute the updated error (i,e, deltas) for each node going from top layer to input layer 
            deltas.append(deltas[-1].dot(self.weights[l].T)*self.activation_deriv(a[l]))  
        deltas.reverse()  
        for i in range(len(self.weights)):  
            layer = np.atleast_2d(a[i])  
            delta = np.atleast_2d(deltas[i])  
            self.weights[i] += learning_rate * layer.T.dot(delta)
            
            
def predict(self, x):         
    x = np.array(x)         
    temp = np.ones(x.shape[0]+1)         
    temp[0:-1] = x         
    a = temp         
    for l in range(0, len(self.weights)):             
        a = self.activation(np.dot(a, self.weights[l]))         
    return a