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