構建神經網絡- 手寫字體識別案例

神經網絡構建：
Multilayer_Perceptron.py:

import numpy as np
from utils.features import prepare_for_training#做歸一化
from utils.hypothesis import sigmoid, sigmoid_gradient#sigmoid函數 極其導數



class MultilayerPerceptron:
    #定義初始化函數
    def __init__(self,data,labels,layers,normalize_data =False):
        #數據預處理函數調用
        data_processed = prepare_for_training(data,normalize_data = normalize_data)[0]
        #初始化賦值操作
        self.data= data_processed
        self.labels= labels
        self.layers= layers #784 25 10
        self.normalize_data= normalize_data
        self.thetas = MultilayerPerceptron.thetas_init(layers)#權重參數初始化
        #第一層是輸入層，輸入像素點個數：28*28*1（長 寬 顏色通道）--不可改
        #第二層：隱層神經元個數 25個（把784特征轉化為25維向量） --可改
        #第三層：分類層，10分類任務
        
    def predict(self,data):
        data_processed = prepare_for_training(data,normalize_data = self.normalize_data)[0]
        num_examples = data_processed.shape[0]
        
        predictions = MultilayerPerceptron.feedforward_propagation(data_processed,self.thetas,self.layers)
        
        return np.argmax(predictions,axis=1).reshape((num_examples,1))#返回最大概率值
        
        
        
        #定義訓練模塊對參數進行更新，神經網絡也是用優化算法去做的（要傳入最大迭代次數max_iterations 和學習率alpha）
    def train(self,max_iterations=1000,alpha=0.1):
        #為了方便權重矩陣參數進行更新，把矩陣拉成一個向量，（後面再還原成矩陣）
        unrolled_theta = MultilayerPerceptron.thetas_unroll(self.thetas)
        #調用梯度下降函數對參數進行更新 
        (optimized_theta,cost_history) = MultilayerPerceptron.gradient_descent(self.data,self.labels,unrolled_theta,self.layers,max_iterations,alpha)
        #還原成矩陣thetas_roll
        self.thetas = MultilayerPerceptron.thetas_roll(optimized_theta,self.layers)
        return self.thetas,cost_history
         
         
         #定義權重矩陣初始化函數（傳入要更新權重的層數）
    @staticmethod
    def thetas_init(layers):
        num_layers = len(layers)#layer是list結構
        thetas = {
    }#定義一個字典 每一層權重參數寫裏面
        
        #用for循環對每層權重參數進行賦值
        #在此案例中只有3層784 25 10 需要賦值的只有兩層
        #循環執行兩次得到25*785和10*26兩組矩陣（785 26加上了權重偏置）
        for layer_index in range(num_layers - 1):
            """ 會執行兩次，得到兩組參數矩陣：25*785 , 10*26 """
            in_count = layers[layer_index]
            out_count = layers[layer_index+1]
            # 這裏需要考慮到偏置項，記住一點偏置的個數跟輸出的結果是一致的
            thetas[layer_index] = np.random.rand(out_count,in_count+1)*0.05 #隨機進行初始化操作，值盡量小一點
        return thetas
    
    #把矩陣合並向量
    @staticmethod
    def thetas_unroll(thetas):
        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()))
            #flatten只能適用於numpy對象，即array或者mat，普通的list列錶不適用 a.flatten()：a是個數組（矩陣），a.flatten()就是把a降到一維，默認是按行的方向降 
        return unrolled_theta
    
    
    #定義梯度下降函數：梯度下降求的是前向和反向傳播的參數更新
    #step:計算loss值 由loss計算梯度值 梯度值更新 return優化完的theta
    @staticmethod
    def gradient_descent(data,labels,unrolled_theta,layers,max_iterations,alpha):
        
        optimized_theta = unrolled_theta
        cost_history = []
        
        for _ in range(max_iterations):

            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)#計算theta梯度
            optimized_theta = optimized_theta - alpha* theta_gradient#對theta進行更新
        return optimized_theta,cost_history
            
            #計算theta梯度函數：
    @staticmethod 
    def gradient_step(data,labels,optimized_theta,layers):
        theta = MultilayerPerceptron.thetas_roll(optimized_theta,layers)#把向量還原為矩陣
        thetas_rolled_gradients = MultilayerPerceptron.back_propagation(data,labels,theta,layers)#調用反向傳播函數
        thetas_unrolled_gradients = MultilayerPerceptron.thetas_unroll(thetas_rolled_gradients)
        return thetas_unrolled_gradients
    
    
    #定義反向傳播函數：
    @staticmethod 
    def back_propagation(data,labels,thetas,layers):
        num_layers = len(layers)
        (num_examples,num_features) = data.shape#（1700 875）
        num_label_types = layers[-1]
        
        deltas = {
    }
        #初始化操作
        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))#785*1
            layers_activations[0] = layers_activation
            #逐層計算
            for layer_index in range(num_layers - 1):
                layer_theta = thetas[layer_index] #得到當前權重參數值 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)))
                layers_inputs[layer_index + 1] = layer_input #後一層計算結果
                layers_activations[layer_index + 1] = layers_activation #後一層經過激活函數後的結果
            output_layer_activation = layers_activation[1:,:]
            
            delta = {
    }
            #標簽處理
            bitwise_label = np.zeros((num_label_types,1))
            bitwise_label[labels[example_index][0]] = 1
            #計算輸出層和真實值之間的差异
            delta[num_layers - 1] = output_layer_activation - bitwise_label
            
            #遍曆循環 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))
                #按照公式進行計算
                delta[layer_index] = np.dot(layer_theta.T,next_delta)*sigmoid_gradient(layer_input)
                #過濾掉偏置參數
                delta[layer_index] = delta[layer_index][1:,:]
            for layer_index in range(num_layers-1):
                layer_delta = np.dot(delta[layer_index+1],layers_activations[layer_index].T)
                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
            
            
            #定義損失函數
    @staticmethod        
    def cost_function(data,labels,thetas,layers):
        num_layers = len(layers)#層數
        num_examples = data.shape[0]#樣本個數
        num_labels = layers[-1]#label是layers的最後一層
        
        #前向傳播走一次
        predictions = MultilayerPerceptron.feedforward_propagation(data,thetas,layers)
        #制作標簽，每一個樣本的標簽都得是one-hot
        bitwise_labels = np.zeros((num_examples,num_labels))
        for example_index in range(num_examples):
            bitwise_labels[example_index][labels[example_index][0]] = 1
        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
      
      
      #定義前向傳播函數 
    @staticmethod        
    def feedforward_propagation(data,thetas,layers):    
        num_layers = len(layers)
        num_examples = data.shape[0]
        in_layer_activation = data#輸入數據
        
        # 逐層計算
        for layer_index in range(num_layers - 1):
            theta = thetas[layer_index]
            out_layer_activation = sigmoid(np.dot(in_layer_activation,theta.T))
            # 正常計算完之後是num_examples*25,但是要考慮偏置項 變成num_examples*26
            out_layer_activation = np.hstack((np.ones((num_examples,1)),out_layer_activation))
            in_layer_activation = out_layer_activation
            
        #返回輸出層結果,結果中不要偏置項了
        return in_layer_activation[:,1:]
         
         
         #定義矩陣還原函數 
    @staticmethod       
    def thetas_roll(unrolled_thetas,layers):    
        num_layers = len(layers)
        thetas = {
    }#指定一個字典方便索引那一層矩陣
        unrolled_shift = 0#指定一個標志比特 記錄到那一層
        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_thetas[start_index:end_index]
            thetas[layer_index] = layer_theta_unrolled.reshape((thetas_height,thetas_width))
            unrolled_shift = unrolled_shift+thetas_volume
        
        return thetas
        
     

數據集訓練：
minist.py:

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import matplotlib.image as mping 
import math

from multilayer_perceptron import MultilayerPerceptron


data = pd.read_csv('../neural_network/data/mnist-demo.csv')
numbers_to_display = 25#要展示圖像的數量
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='Purples')
    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 = 5000

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_iterations = 500
alpha = 0.1


multilayer_perceptron = MultilayerPerceptron(x_train,y_train,layers,normalize_data)
(thetas,costs) = multilayer_perceptron.train(max_iterations,alpha)
plt.plot(range(len(costs)),costs)
plt.xlabel('Grident steps')
plt.ylabel('costs')
plt.show()


y_train_predictions = multilayer_perceptron.predict(x_train)
y_test_predictions = multilayer_perceptron.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 ('訓練集准確率：',train_p)
print ('測試集准確率：',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))
    
    color_map = 'Greens' if predicted_label == digit_label else 'Reds'
    plt.subplot(num_cells, num_cells, plot_index + 1)
    plt.imshow(frame, cmap=color_map)
    plt.title(predicted_label)
    plt.tick_params(axis='both', which='both', bottom=False, left=False, labelbottom=False, labelleft=False)

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

結果顯示：