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:      sigmoid函数是一个在生物学中常见的S型函数，也称为S型生长曲线。 在信息科学中，由于其单增以及反函数单增等性质，sigmoid函数常被用作神经网络的阈值函数，将变量映射到0,1之间。具有这种性质的S型函数统称为sigmoid函数。             1.1 双曲函数(tanh)         双曲函数（hyperbolic function）可借助指数函数定义 
        双曲正弦：                         双曲余弦：                         双曲正切：                            导数：  

          可以看到tanhx呈S型

           1.2  逻辑函数(logistic function)

         其实logistic函数也就是经常说的sigmoid函数，它的几何形状也就是一条sigmoid曲线（S型曲线）。

                  

           导数：

                      

           

2. 手动实现一个简单的神经网络算法

# -*- coding:utf-8 -*-

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): #epochs = 10000 抽样更新 10000次
        X = np.atleast_2d(X) #至少2维
        temp = np.ones([X.shape[0], X.shape[1] + 1])#初始化矩阵
        temp[: ,0:-1] = X # adding the bias unit to the input layer 每一行 从第一列到最后一列 不包含最后一列
        X = temp
        y = np.array(y)

        for k in range(epochs): #第一次循环
            i = np.random.randint(X.shape[0]) #X里面随机选取一行
            a = [X[i]] #第i行

            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
            #对于输出层
            deltas = [error * self.activation_deriv(a[-1])] #For output layer, Err calculation (delta is updated error)

            # Staring backprobagation
            #(len(a) - 2)因为不能算第一层和最后一层；最后一层到0层，倒回去
            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
                #对于隐藏层
                deltas.append(deltas[-1].dot(self.weights[l].T)*self.activation_deriv(a[l]))
            deltas.reverse()#0层到最后一层

            for i in range(len(self.weights)):#权重更新
                layer = np.atleast_2d(a[i])
                delta = np.atleast_2d(deltas[i])
                self.weights[i] += learining_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[1]))
        return a