目录
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()
8.得出W1，W2，W3对损失有多大影响,公式如下：
算法流程（简便版）：
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
"""Add polynomial features to the features set"""
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
import numpy as np
def generate_sinusoids(dataset, sinusoid_degree):
"""
sin(x).
"""
num_examples = dataset.shape[0]
sinusoids = np.empty((num_examples, 0))
for degree in range(1, sinusoid_degree + 1):
sinusoid_features = np.sin(degree * dataset)
sinusoids = np.concatenate((sinusoids, sinusoid_features), axis=1)
return sinusoids
"""Normalize features"""
import numpy as np
def normalize(features):
features_normalized = np.copy(features).astype(float)
# 计算均值
features_mean = np.mean(features, 0)
# 计算标准差
features_deviation = np.std(features, 0)
# 标准化操作
if features.shape[0] > 1:
features_normalized -= features_mean
# 防止除以0
features_deviation[features_deviation == 0] = 1
features_normalized /= features_deviation
return features_normalized, features_mean, features_deviation
数据预处理：
"""Prepares the dataset for training"""
import numpy as np
from .normalize import normalize
from .generate_sinusoids import generate_sinusoids
from .generate_polynomials import generate_polynomials
def prepare_for_training(data, polynomial_degree=0, sinusoid_degree=0, normalize_data=True):
# 计算样本总数
num_examples = data.shape[0]
data_processed = np.copy(data)
# 预处理
features_mean = 0
features_deviation = 0
data_normalized = data_processed
if normalize_data:
(
data_normalized,
features_mean,
features_deviation
) = normalize(data_processed)
data_processed = data_normalized
# 特征变换sinusoidal
if sinusoid_degree > 0:
sinusoids = generate_sinusoids(data_normalized, sinusoid_degree)
data_processed = np.concatenate((data_processed, sinusoids), axis=1)
# 特征变换polynomial
if polynomial_degree > 0:
polynomials = generate_polynomials(data_normalized, polynomial_degree, normalize_data)
data_processed = np.concatenate((data_processed, polynomials), axis=1)
# 加一列1
data_processed = np.hstack((np.ones((num_examples, 1)), data_processed))
return data_processed, features_mean, features_deviation
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
这里准确率不高，读者可以自行调整参数，改变迭代次数，网络层次都可以哦。
参考文章
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