神经网络代码实现（用手写数字识别数据集实验）

一、前言

二、神经网络架构

三、算法实现

1、导入包

2、实现类

3、训练函数

4、权重参数矩阵初始化

5、参数矩阵变换向量

6、向量变换权重参数矩阵

7、进行梯度下降

7.1、损失函数

7.1.1、前向传播

7.2、反向传播

8、预测函数

四、完整代码

五、手写数字识别

一、前言

读者需要了解神经网络的基础知识，可以参考神经网络（深度学习，计算机视觉，得分函数，损失函数，前向传播，反向传播，激活函数）

本文为大家详细的描述了，实现神经网络的逻辑，代码。并且用手写识别来实验，结果基本实现了神经网络的要求。

二、神经网络架构

想一想：

1.输入数据：特征值(手写数字识别是像素点，784个特征)

2.W1，W2，W3矩阵的形状

3.前向传播

4.激活函数(用Sigmoid)

5.反向传播

6.偏置项

7.损失( $\hat{y}-y$ )

8.得出W1，W2，W3对损失有多大影响,公式如下：

$\begin{matrix} \delta(4)=a(4)-y\\ \delta(3)=(\Theta ^3)^T\delta(4)*g'(z^{(3)})\\ \delta(2)=(\Theta ^2)^T\delta(3)*g'(z^{(2)})\\ \delta(1) =is \; for\;input\;layer,\;we\;can't\;change\;it\\ g'-sigmoid\;gradient\\ g'(z)=\frac{\partial }{\partial z}g(z)=g(z)(1-g(z));\;where\;g(z)=\frac{1}{1+e^{-z}} \end{matrix}$

算法流程（简便版）：

$\begin{matrix} for\;i=1\;to\;m\\ \;set\;a(1)=x^{(i)}\\ perform\;forward\;propagation\;to\;compute\;a(1)\;for\;l=2,3...L\\ Using\;y^{(i)},compute\;\delta(L)=a^{(l)}-y^{(i)}\\ compute\;\delta(L-1)\;\delta(L-2)...\delta(2)\\ \Delta _{ij}^{(l)}:\Delta _{ij}^{(l)}+a_{j}^{(l)}\delta_i^{(l+1)}(or\;in\;vectorized\;form\;\Delta^{(l)}=\Delta^{(l)}+\delta^{(l+1)}(a(l))^T) \end{matrix}$

三、算法实现

1、导入包

import  numpy as np
from Neural_Network_Lab.utils.features import prepare_for_training
from Neural_Network_Lab.utils.hypothesis import sigmoid,sigmoid_gradient

这里utils包用来封装数据预处理，和Sigmoid函数

数据预处理：


import numpy as np
from .normalize import normalize


def generate_polynomials(dataset, polynomial_degree, normalize_data=False):
    """变换方法：
    x1, x2, x1^2, x2^2, x1*x2, x1*x2^2, etc.
    """

    features_split = np.array_split(dataset, 2, axis=1)
    dataset_1 = features_split[0]
    dataset_2 = features_split[1]

    (num_examples_1, num_features_1) = dataset_1.shape
    (num_examples_2, num_features_2) = dataset_2.shape

    if num_examples_1 != num_examples_2:
        raise ValueError('Can not generate polynomials for two sets with different number of rows')

    if num_features_1 == 0 and num_features_2 == 0:
        raise ValueError('Can not generate polynomials for two sets with no columns')

    if num_features_1 == 0:
        dataset_1 = dataset_2
    elif num_features_2 == 0:
        dataset_2 = dataset_1

    num_features = num_features_1 if num_features_1 < num_examples_2 else num_features_2
    dataset_1 = dataset_1[:, :num_features]
    dataset_2 = dataset_2[:, :num_features]

    polynomials = np.empty((num_examples_1, 0))

    for i in range(1, polynomial_degree + 1):
        for j in range(i + 1):
            polynomial_feature = (dataset_1 ** (i - j)) * (dataset_2 ** j)
            polynomials = np.concatenate((polynomials, polynomial_feature), axis=1)

    if normalize_data:
        polynomials = normalize(polynomials)[0]

    return polynomials

Sigmoid函数：

import numpy as np


def sigmoid(matrix):
    """Applies sigmoid function to NumPy matrix"""

    return 1 / (1 + np.exp(-matrix))

2、实现类

多层感知机初始化：数据，标签，网络层次(用列表表示如三层[784,25,10]表示输入层784个神经元，25个隐藏层神经元，10个输出层神经元)，数据是否标准化处理。

class MultilayerPerceptron:
    def __init__(self,data,labels,layers,normalize_data=False):
        data_processed = prepare_for_training(data,normalize_data=normalize_data)[0]
        self.data = data_processed
        self.labels = labels
        self.layers = layers #  [  784 ,25 ,10]
        self.normalize_data = normalize_data
        self.thetas = MultilayerPerceptron.thetas_init(layers)

3、训练函数

输入迭代次数，学习率，进行梯度下降算法，更新权重参数矩阵，得到最终的权重参数矩阵，和损失值。矩阵不好进行更新操作，可以把它拉成向量。

    def train(self,max_ietrations = 1000,alpha = 0.1):
        #方便矩阵更新 拉长  把矩阵拉成向量
        unrolled_theta = MultilayerPerceptron.thetas_unroll(self.thetas)
        (optimized_theta, cost_history) = MultilayerPerceptron.gradient_descent(self.data,self.labels,unrolled_theta,self.layers,max_ietrations,alpha)
        self.thetas = MultilayerPerceptron.thetas_roll(optimized_theta,self.layers)

        return self.thetas,cost_history

4、权重参数矩阵初始化

根据网络层次可以确定，矩阵的大小，用字典存储。

    @staticmethod
    def thetas_init(layers):
        num_layers = len(layers)
        thetas = {} #用字典形式 key:表示第几层 vlues:权重参数矩阵
        for layer_index in range(num_layers-1):
            '''
            会执行两次： 得到两组参数矩阵 25 * 785 ， 10 * 26
            '''
            in_count = layers[layer_index]
            out_count = layers[layer_index+1]
            #初始化 初始值小
            #这里需要考虑偏置项，偏置的个数与输出的个数一样
            thetas[layer_index]=np.random.rand(out_count,in_count+1) * 0.05 #加一列输入特征
        return thetas

5、参数矩阵变换向量

将权重参数矩阵变换成向量

    @staticmethod
    def thetas_unroll(thetas):
        #拼接成一个向量
        num_theta_layers = len(thetas)
        unrolled_theta = np.array([])
        for theta_layer_index in range(num_theta_layers):
            unrolled_theta = np.hstack((unrolled_theta,thetas[theta_layer_index].flatten()))
        return unrolled_theta

6、向量变换权重参数矩阵

后边前向传播时需要进行矩阵乘法，需要变换回来

    @staticmethod
    def thetas_roll(unrolled_theta,layers):
        num_layers = len(layers)
        thetas = {}
        unrolled_shift = 0
        for layer_index in range(num_layers - 1):
            in_count = layers[layer_index]
            out_count = layers[layer_index + 1]

            thetas_width = in_count + 1
            thetas_height = out_count
            thetas_volume = thetas_width * thetas_height
            start_index = unrolled_shift
            end_index =unrolled_shift + thetas_volume
            layer_theta_unrolled = unrolled_theta[start_index:end_index]
            thetas[layer_index] = layer_theta_unrolled.reshape((thetas_height,thetas_width))
            unrolled_shift = unrolled_shift + thetas_volume

        return thetas

7、进行梯度下降

1. 损失函数，计算损失值

2. 计算梯度值

3. 更新参数

那么得先要实现损失函数，计算损失值。

7.1、损失函数

实现损失函数，得到损失值得要实现前向传播走一次

7.1.1、前向传播

    @staticmethod
    def feedforword_propagation(data,thetas,layers):
        num_layers = len(layers)
        num_examples = data.shape[0]
        in_layer_activation = data #输入层

        #逐层计算 隐藏层
        for layer_index in range(num_layers - 1):
            theta = thetas[layer_index]
            out_layer_activation = sigmoid(np.dot(in_layer_activation,theta.T)) #输出层
            # 正常计算之后是num_examples * 25 ，但是要考虑偏置项 变成num_examples * 26
            out_layer_activation = np.hstack((np.ones((num_examples,1)),out_layer_activation))
            in_layer_activation = out_layer_activation

        #返回输出层结果,不要偏置项
        return in_layer_activation[:,1:]

损失函数：

    @staticmethod
    def cost_function(data,labels,thetas,layers):
        num_layers = len(layers)
        num_examples = data.shape[0]
        num_labels = layers[-1]
        #前向传播走一次
        predictions = MultilayerPerceptron.feedforword_propagation(data,thetas,layers)
        #制作标签，每一个样本的标签都是one-dot
        bitwise_labels = np.zeros((num_examples,num_labels))
        for example_index in range(num_examples):
            bitwise_labels[example_index][labels[example_index][0]] = 1
        #咱们的预测值是概率值y= 7 [0,0,0,0,0,0,1,0,0,0]    在正确值的位置上概率越大越好 在错误值的位置上概率越小越好
        bit_set_cost = np.sum(np.log(predictions[bitwise_labels == 1]))
        bit_not_set_cost = np.sum(np.log(1 - predictions[bitwise_labels == 0]))
        cost = (-1/num_examples) * (bit_set_cost+bit_not_set_cost)
        return cost

7.2、反向传播

在梯度下降的过程中，要实现参数矩阵的更新，必须要实现反向传播。利用上述的公式，进行运算即可得到。

    @staticmethod
    def back_propagation(data,labels,thetas,layers):
        num_layers = len(layers)
        (num_examples,num_features) = data.shape
        num_label_types = layers[-1]

        deltas = {} # 算出每一层对结果的影响
        #初始化
        for layer_index in  range(num_layers - 1):
            in_count = layers[layer_index]
            out_count = layers[layer_index + 1]
            deltas[layer_index] = np.zeros((out_count,in_count+1)) #25 * 785 10 *26

        for example_index in range(num_examples):
            layers_inputs = {}
            layers_activations = {}
            layers_activation = data[example_index,:].reshape((num_features,1))
            layers_activations[0] = layers_activation

            #逐层计算
            for layer_index in range(num_layers - 1):
                layer_theta = thetas[layer_index]  #得到当前的权重参数值 ： 25 *785 10 *26
                layer_input = np.dot(layer_theta,layers_activation) # 第一次 得到 25 * 1 第二次： 10 * 1
                layers_activation = np.vstack((np.array([[1]]),sigmoid(layer_input))) #完成激活函数，加上一个偏置参数
                layers_inputs[layer_index+1] = layer_input # 后一层计算结果
                layers_activations[layer_index +1] = layers_activation # 后一层完成激活的结果
            output_layer_activation = layers_activation[1:,:]
            #计算输出层和结果的差异
            delta = {}
            #标签处理
            bitwise_label = np.zeros((num_label_types,1))
            bitwise_label[labels[example_index][0]] = 1
            #计算输出结果和真实值之间的差异
            delta[num_layers-1] = output_layer_activation - bitwise_label #输出层

            #遍历 L,L-1,L-2...2
            for layer_index in range(num_layers - 2,0,-1):
                layer_theta = thetas[layer_index]
                next_delta = delta[layer_index+1]
                layer_input = layers_inputs[layer_index]
                layer_input = np.vstack((np.array((1)),layer_input))
                #按照公式计算
                delta[layer_index] = np.dot(layer_theta.T,next_delta)*sigmoid(layer_input)
                #过滤掉偏置参数
                delta[layer_index] = delta[layer_index][1:,:]

            #计算梯度值
            for layer_index in  range(num_layers-1):
                layer_delta = np.dot(delta[layer_index+1],layers_activations[layer_index].T)  #微调矩阵
                deltas[layer_index] = deltas[layer_index] + layer_delta #第一次25 * 785 第二次 10 * 26

        for layer_index in range(num_layers-1):
            deltas[layer_index] = deltas[layer_index] * (1/num_examples) #公式

        return deltas

实现一次梯度下降：

    @staticmethod
    def gradient_step(data,labels,optimized_theta,layers):
        theta = MultilayerPerceptron.thetas_roll(optimized_theta,layers)
        #反向传播BP
        thetas_rolled_gradinets = MultilayerPerceptron.back_propagation(data,labels,theta,layers)
        thetas_unrolled_gradinets = MultilayerPerceptron.thetas_unroll(thetas_rolled_gradinets)
        return thetas_unrolled_gradinets

实现梯度下降：

    @staticmethod
    def gradient_descent(data,labels,unrolled_theta,layers,max_ietrations,alpha):
        #1. 计算损失值
        #2. 计算梯度值
        #3. 更新参数
        optimized_theta = unrolled_theta #最好的theta值
        cost_history = []  #损失值的记录
        for i in range(max_ietrations):
            if i % 10 == 0 :
                print("当前迭代次数：",i)
            cost  = MultilayerPerceptron.cost_function(data,labels,MultilayerPerceptron.thetas_roll(optimized_theta,layers),layers)
            cost_history.append(cost)
            theta_gradient = MultilayerPerceptron.gradient_step(data,labels,optimized_theta,layers)
            optimized_theta = optimized_theta - alpha * theta_gradient
        return optimized_theta,cost_history

8、预测函数

输入测试数据，前向传播走一次，得到预测值

    def predict(self,data):
        data_processed = prepare_for_training(data,normalize_data = self.normalize_data)[0]
        num_examples = data_processed.shape[0]
        predictions = MultilayerPerceptron.feedforword_propagation(data_processed,self.thetas,self.layers)

        return np.argmax(predictions,axis=1).reshape((num_examples,1))

四、完整代码

import  numpy as np
from Neural_Network_Lab.utils.features import prepare_for_training
from Neural_Network_Lab.utils.hypothesis import sigmoid,sigmoid_gradient

class MultilayerPerceptron:
    def __init__(self,data,labels,layers,normalize_data=False):
        data_processed = prepare_for_training(data,normalize_data=normalize_data)[0]
        self.data = data_processed
        self.labels = labels
        self.layers = layers #  [  784 ,25 ,10]
        self.normalize_data = normalize_data
        self.thetas = MultilayerPerceptron.thetas_init(layers)

    def predict(self,data):
        data_processed = prepare_for_training(data,normalize_data = self.normalize_data)[0]
        num_examples = data_processed.shape[0]
        predictions = MultilayerPerceptron.feedforword_propagation(data_processed,self.thetas,self.layers)

        return np.argmax(predictions,axis=1).reshape((num_examples,1))


    def train(self,max_ietrations = 1000,alpha = 0.1):
        #方便矩阵更新 拉长  把矩阵拉成向量
        unrolled_theta = MultilayerPerceptron.thetas_unroll(self.thetas)
        (optimized_theta, cost_history) = MultilayerPerceptron.gradient_descent(self.data,self.labels,unrolled_theta,self.layers,max_ietrations,alpha)
        self.thetas = MultilayerPerceptron.thetas_roll(optimized_theta,self.layers)

        return self.thetas,cost_history

    @staticmethod
    def gradient_descent(data,labels,unrolled_theta,layers,max_ietrations,alpha):
        #1. 计算损失值
        #2. 计算梯度值
        #3. 更新参数
        optimized_theta = unrolled_theta #最好的theta值
        cost_history = []  #损失值的记录
        for i in range(max_ietrations):
            if i % 10 == 0 :
                print("当前迭代次数：",i)
            cost  = MultilayerPerceptron.cost_function(data,labels,MultilayerPerceptron.thetas_roll(optimized_theta,layers),layers)
            cost_history.append(cost)
            theta_gradient = MultilayerPerceptron.gradient_step(data,labels,optimized_theta,layers)
            optimized_theta = optimized_theta - alpha * theta_gradient
        return optimized_theta,cost_history

    @staticmethod
    def gradient_step(data,labels,optimized_theta,layers):
        theta = MultilayerPerceptron.thetas_roll(optimized_theta,layers)
        #反向传播BP
        thetas_rolled_gradinets = MultilayerPerceptron.back_propagation(data,labels,theta,layers)
        thetas_unrolled_gradinets = MultilayerPerceptron.thetas_unroll(thetas_rolled_gradinets)
        return thetas_unrolled_gradinets

    @staticmethod
    def back_propagation(data,labels,thetas,layers):
        num_layers = len(layers)
        (num_examples,num_features) = data.shape
        num_label_types = layers[-1]

        deltas = {} # 算出每一层对结果的影响
        #初始化
        for layer_index in  range(num_layers - 1):
            in_count = layers[layer_index]
            out_count = layers[layer_index + 1]
            deltas[layer_index] = np.zeros((out_count,in_count+1)) #25 * 785 10 *26

        for example_index in range(num_examples):
            layers_inputs = {}
            layers_activations = {}
            layers_activation = data[example_index,:].reshape((num_features,1))
            layers_activations[0] = layers_activation

            #逐层计算
            for layer_index in range(num_layers - 1):
                layer_theta = thetas[layer_index]  #得到当前的权重参数值 ： 25 *785 10 *26
                layer_input = np.dot(layer_theta,layers_activation) # 第一次 得到 25 * 1 第二次： 10 * 1
                layers_activation = np.vstack((np.array([[1]]),sigmoid(layer_input))) #完成激活函数，加上一个偏置参数
                layers_inputs[layer_index+1] = layer_input # 后一层计算结果
                layers_activations[layer_index +1] = layers_activation # 后一层完成激活的结果
            output_layer_activation = layers_activation[1:,:]
            #计算输出层和结果的差异
            delta = {}
            #标签处理
            bitwise_label = np.zeros((num_label_types,1))
            bitwise_label[labels[example_index][0]] = 1
            #计算输出结果和真实值之间的差异
            delta[num_layers-1] = output_layer_activation - bitwise_label #输出层

            #遍历 L,L-1,L-2...2
            for layer_index in range(num_layers - 2,0,-1):
                layer_theta = thetas[layer_index]
                next_delta = delta[layer_index+1]
                layer_input = layers_inputs[layer_index]
                layer_input = np.vstack((np.array((1)),layer_input))
                #按照公式计算
                delta[layer_index] = np.dot(layer_theta.T,next_delta)*sigmoid(layer_input)
                #过滤掉偏置参数
                delta[layer_index] = delta[layer_index][1:,:]

            #计算梯度值
            for layer_index in  range(num_layers-1):
                layer_delta = np.dot(delta[layer_index+1],layers_activations[layer_index].T)  #微调矩阵
                deltas[layer_index] = deltas[layer_index] + layer_delta #第一次25 * 785 第二次 10 * 26

        for layer_index in range(num_layers-1):
            deltas[layer_index] = deltas[layer_index] * (1/num_examples)

        return deltas

    @staticmethod
    def cost_function(data,labels,thetas,layers):
        num_layers = len(layers)
        num_examples = data.shape[0]
        num_labels = layers[-1]
        #前向传播走一次
        predictions = MultilayerPerceptron.feedforword_propagation(data,thetas,layers)
        #制作标签，每一个样本的标签都是one-dot
        bitwise_labels = np.zeros((num_examples,num_labels))
        for example_index in range(num_examples):
            bitwise_labels[example_index][labels[example_index][0]] = 1
        #咱们的预测值是概率值y= 7 [0,0,0,0,0,0,1,0,0,0]    在正确值的位置上概率越大越好 在错误值的位置上概率越小越好
        bit_set_cost = np.sum(np.log(predictions[bitwise_labels == 1]))
        bit_not_set_cost = np.sum(np.log(1 - predictions[bitwise_labels == 0]))
        cost = (-1/num_examples) * (bit_set_cost+bit_not_set_cost)
        return cost

    @staticmethod
    def feedforword_propagation(data,thetas,layers):
        num_layers = len(layers)
        num_examples = data.shape[0]
        in_layer_activation = data #输入层

        #逐层计算 隐藏层
        for layer_index in range(num_layers - 1):
            theta = thetas[layer_index]
            out_layer_activation = sigmoid(np.dot(in_layer_activation,theta.T)) #输出层
            # 正常计算之后是num_examples * 25 ，但是要考虑偏置项 变成num_examples * 26
            out_layer_activation = np.hstack((np.ones((num_examples,1)),out_layer_activation))
            in_layer_activation = out_layer_activation

        #返回输出层结果,不要偏置项
        return in_layer_activation[:,1:]

    @staticmethod
    def thetas_roll(unrolled_theta,layers):
        num_layers = len(layers)
        thetas = {}
        unrolled_shift = 0
        for layer_index in range(num_layers - 1):
            in_count = layers[layer_index]
            out_count = layers[layer_index + 1]

            thetas_width = in_count + 1
            thetas_height = out_count
            thetas_volume = thetas_width * thetas_height
            start_index = unrolled_shift
            end_index =unrolled_shift + thetas_volume
            layer_theta_unrolled = unrolled_theta[start_index:end_index]
            thetas[layer_index] = layer_theta_unrolled.reshape((thetas_height,thetas_width))
            unrolled_shift = unrolled_shift + thetas_volume

        return thetas

    @staticmethod
    def thetas_unroll(thetas):
        #拼接成一个向量
        num_theta_layers = len(thetas)
        unrolled_theta = np.array([])
        for theta_layer_index in range(num_theta_layers):
            unrolled_theta = np.hstack((unrolled_theta,thetas[theta_layer_index].flatten()))
        return unrolled_theta

    @staticmethod
    def thetas_init(layers):
        num_layers = len(layers)
        thetas = {} #用字典形式 key:表示第几层 vlues:权重参数矩阵
        for layer_index in range(num_layers-1):
            '''
            会执行两次： 得到两组参数矩阵 25 * 785 ， 10 * 26
            '''
            in_count = layers[layer_index]
            out_count = layers[layer_index+1]
            #初始化 初始值小
            #这里需要考虑偏置项，偏置的个数与输出的个数一样
            thetas[layer_index]=np.random.rand(out_count,in_count+1) * 0.05 #加一列输入特征
        return thetas

五、手写数字识别

数据集（读者可以找找下载，我就不放链接了>_<）：

共一万个样本，第一列为标签值，一列表示像素点的值共28*28共784个像素点。

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import matplotlib.image as mping
import math
from Neural_Network_Lab.Multilayer_Perceptron import MultilayerPerceptron

data = pd.read_csv('../Neural_Network_Lab/data/mnist-demo.csv')
#展示数据
numbers_to_display = 25
num_cells = math.ceil(math.sqrt(numbers_to_display))
plt.figure(figsize=(10,10))
for plot_index in range(numbers_to_display):
    digit = data[plot_index:plot_index+1].values
    digit_label = digit[0][0]
    digit_pixels = digit[0][1:]
    image_size = int(math.sqrt(digit_pixels.shape[0]))
    frame = digit_pixels.reshape((image_size,image_size))
    plt.subplot(num_cells,num_cells,plot_index+1)
    plt.imshow(frame,cmap = 'Greys')
    plt.title(digit_label)
plt.subplots_adjust(wspace=0.5,hspace=0.5)
plt.show()

train_data = data.sample(frac= 0.8)
test_data = data.drop(train_data.index)

train_data = train_data.values
test_data = test_data.values

num_training_examples = 8000

X_train = train_data[:num_training_examples,1:]
y_train = train_data[:num_training_examples,[0]]

X_test = test_data[:,1:]
y_test = test_data[:,[0]]

layers = [784,25,10]
normalize_data = True
max_iteration = 500
alpha = 0.1

multilayerperceptron = MultilayerPerceptron(X_train,y_train,layers,normalize_data)
(thetas,cost_history) = multilayerperceptron.train(max_iteration,alpha)
plt.plot(range(len(cost_history)),cost_history)
plt.xlabel('Grident steps')
plt.ylabel('cost')
plt.show()

y_train_predictions = multilayerperceptron.predict(X_train)
y_test_predictions = multilayerperceptron.predict(X_test)

train_p = np.sum((y_train_predictions == y_train) / y_train.shape[0] * 100)
test_p = np.sum((y_test_predictions == y_test) / y_test.shape[0] * 100)

print("训练集准确率：",train_p)
print("测试集准确率：",test_p)

numbers_to_display = 64
num_cells = math.ceil(math.sqrt(numbers_to_display))
plt.figure(figsize=(15,15))
for plot_index in range(numbers_to_display):
    digit_label = y_test[plot_index,0]
    digit_pixels = X_test[plot_index,:]

    predicted_label = y_test_predictions[plot_index][0]

    image_size = int(math.sqrt(digit_pixels.shape[0]))
    frame = digit_pixels.reshape((image_size,image_size))
    plt.subplot(num_cells,num_cells,plot_index+1)
    color_map = 'Greens' if predicted_label == digit_label else 'Reds'
    plt.imshow(frame,cmap = color_map)
    plt.title(predicted_label)
    plt.tick_params(axis='both',which='both',bottom=False,left=False,labelbottom=False)

plt.subplots_adjust(wspace=0.5,hspace=0.5)
plt.show()

训练集8000个，测试集2000个，迭代次数500次