建立多层的神经网络

前向时计算是分开的

linear_forwad

Z = np.dot(W,A) + b

linear_activation_forward

A = relu(Z) or A = sigmoid(Z)

过程图示

流程

1. 初始化网络参数

2. 前向传播

​ 2.1 计算线性求和部分

​ 2.2 计算激活函数部分

​ 2.3 结合求和函数与激活函数

3. 计算LOSS

4. 反向传播

​ 4.1 线性部分的反向传播

​ 4.2 激活函数的反向传播

​ 4.3 结合线性部分跟激活函数的方向传播公式

5.更新参数

6. TO 1

***PS：***每一个前向函数都会对应一个后向函数

导入所需包

import numpy as np
import h5py
import matplotlib.pyplot as plt
#以下三个是引用别人的，之后考虑自己实现
import testCases
from dnn_utils import sigmoid, sigmoid_backward, relu, relu_backward
import lr_utils
#为了检验结果
np.random.seed(1)

初始化参数(两层的参数初始化)

def initialize_parameters(n_x,n_h,n_y):
    #输入层跟隐藏层之间的参数
    W1 = np.random.randn(n_h, n_x) * 0.01;
    b1 = np.zeros((n_h, 1))
    #隐藏层跟输出层之间的参数
    W2 = np.random.randn(n_y,n_h) * 0.01;
    b2 = np.zeros((n_y, 1))
    
    #使用断言保证矩阵的行列值符合要求
    assert(W1.shape == (n_h, n_x))
    assert(b1.shape == (n_h, 1))
    assert(W2.shape == (n_y, n_h))
    assert(b2.shape == (n_y, 1))
    
    parameters = {
        "W1":W1,
        "b1":b1,
        "W2":W2,
        "b2":b2
    }
    
    return parameters

测试参数初始化

print("====================测试initialize_parameters======================")
parameters = initialize_parameters(3, 2, 1)
print("W1 =" + str(parameters["W1"]))
print("b1 =" + str(parameters["b1"]))
print("W2 =" + str(parameters["W2"]))
print("b2 =" + str(parameters["b2"]))

初始化参数(多层的参数初始化)

def initialize_parameters_deep(layers_dims):
    np.random.seed(3)
    parameters = {}
    L = len(layers_dims)
    
    for i in range(1, L):
        parameters["W" + str(i)] = np.random.randn(layers_dims[i], layers_dims[i - 1] )/ np.sqrt(layers_dims[i - 1])
        parameters["b" + str(i)] = np.zeros((layers_dims[i], 1))
        
        #确保数据的格式是正确的
        assert(parameters["W" + str(i)].shape == (layers_dims[i], layers_dims[i - 1]))
        assert(parameters["b" + str(i)].shape == (layers_dims[i], 1))
    return parameters

测试

print("=================测试initialize_parameters_deep===================")
layers_dims = [5, 4, 3]
parameters = initialize_parameters_deep(layers_dims)
print("W1 =" + str(parameters["W1"]))
print("b1 =" + str(parameters["b1"]))
print("W2 =" + str(parameters["W2"]))
print("b2 =" + str(parameters["b2"]))

前向传播

1. LINEAR

2. LINEAR->ACTIVATION

3. [LINEAR -> RELU] X （L - 1）- > LINEAR - > SIGMOID(整个模型)

线性部分【LINEAR】

def linear_forward(A,W,b):
    Z = np.dot(W,A) + b
    assert(Z.shape == (W.shape[0],A.shape[1]))
    cache = (A,W,b)
    
    return Z,cache

测试

print("===========测试线性部分===========")
A,W,b = testCases.linear_forward_test_case()
Z,linear_cache = linear_forward(A,W,b)
print("Z = " + str(Z))

线性激活部分【LINEAR - >ACTIVATION】

def linear_activation_forward(A_prev,W,b,activation):
    if activation == "sigmoid":
        Z, linear_cache = linear_forward(A_prev, W, b)
        A, activation_cache = sigmoid(Z)
    elif activation == "relu":
        Z, linear_cache = linear_forward(A_prev, W, b)
        A, activation_cache = relu(Z)
    
    assert(A.shape == (W.shape[0],A_prev.shape[1]))
    cache = (linear_cache, activation_cache)
    
    return A,cache

测试

print("测试linear_activation_forward")
A_prev, W, b = testCases.linear_activation_forward_test_case()

A, linear_activation_cache = linear_activation_forward(A_prev, W, b, activation = "sigmoid")
print("sigmoid, A = " + str(A))
A, linear_activation_cache = linear_activation_forward(A_prev, W, b, activation = "relu")
print("ReLU, A = " + str(A))

以上实现了两层的前向传播，下边是深度的前向传播过程

$$ Figure 2 : [LINEAR -> RELU] (L-1) -> LINEAR -> SIGMOID model $$

多层模型的前向传播计算过程

def L_model_forward(X, parameters):
    caches = []
    A = X
    L = len(parameters) // 2
    for l in range(1,L):
        A_prev = A
        A, cache = linear_activation_forward(A_prev, parameters['W' + str(l)], parameters['b' + str(l)],"relu")
        caches.append(cache)
    AL, cache = linear_activation_forward(A, parameters['W' + str(L)], parameters['b' + str(L)], "sigmoid")
    caches.append(cache)
    
    assert(AL.shape == (1, X.shape[1]))
    
    return AL, caches

测试

print("测试L_model_forward")

X,parameters = testCases.L_model_forward_test_case()
AL,caches = L_model_forward(X, parameters)
print("AL = " + str(AL))
print("caches的长度为：" + str(len(caches)))

计算LOSS

def compute_cost(AL, Y):
    m = Y.shape[1]
    cost = -np.sum(np.multiply(np.log(AL),Y) + np.multiply(np.log(1 - AL), 1 - Y)) / m
    
    cost = np.squeeze(cost)
    assert(cost.shape == ())
    
    return cost

测试

print("测试compute_cost")
Y,AL = testCases.compute_cost_test_case()
print("cost=" + str(compute_cost(AL, Y)))

反向传播

前向传播和反向传播的图示

对于线性的部分公式

与前向传播类似，使用三个步骤来构建反向传播

• LINEAR 后向计算
• LINEAR -> ACTIVATION 后向计算，其中ACTIVATION 计算Relu或者Sigmoid 的结果
• [LINEAR -> RELU] (L-1) -> LINEAR -> SIGMOID 后向计算 (整个模型)

线性部分【LINEAR backward】

def linear_backward(dZ,cache):
    A_prev, W, b = cache
    m = A_prev.shape[1]
    dW = np.dot(dZ, A_prev.T) / m
    db = np.sum(dZ, axis = 1, keepdims=True) / m
    dA_prev = np.dot(W.T, dZ)
    
    assert (dA_prev.shape == A_prev.shape)
    assert (dW.shape == W.shape)
    assert (db.shape == b.shape)
    
    return dA_prev, dW, db

测试

print("测试linear_backward")

dZ, linear_cache = testCases.linear_backward_test_case()

dA_prev, dW, db = linear_backward(dZ, linear_cache)
print("dA_prev = " + str(dA_prev))
print("dW = " + str(dW))
print("db = " + str(db))

线性激活部分【LINEAR -> ACTIVATION backward】

def linear_activation_backward(dA, cache, activation = "relu"):
    linear_cache, activation_cache = cache
    if activation == "relu":
        dZ = relu_backward(dA, activation_cache)
        dA_prev, dW, db = linear_backward(dZ, linear_cache)
    elif activation == "sigmoid":
        dZ = sigmoid_backward(dA, activation_cache)
        dA_prev, dW, db = linear_backward(dZ, linear_cache)
        
    return dA_prev, dW, db

测试

print("==============测试linear_activation_backward==============")
AL, linear_activation_cache = testCases.linear_activation_backward_test_case()

dA_prev, dW, db = linear_activation_backward(AL, linear_activation_cache, activation = "sigmoid")
print ("sigmoid:")
print ("dA_prev = "+ str(dA_prev))
print ("dW = " + str(dW))
print ("db = " + str(db) + "\n")

dA_prev, dW, db = linear_activation_backward(AL, linear_activation_cache, activation = "relu")
print ("relu:")
print ("dA_prev = "+ str(dA_prev))
print ("dW = " + str(dW))
print ("db = " + str(db))

流程图

多层模型向后传播

def L_model_backward(AL, Y, caches):
    grads = {}
    L = len(caches)
    m = AL.shape[1]
    Y = Y.reshape(AL.shape)
    dAL = -(np.divide(Y, AL) - np.divide(1 - Y, 1 - AL))
    
    current_cache = caches[L - 1]
    grads["dA" + str(L)], grads["dW" + str(L)],grads["db" + str(L)] = linear_activation_backward(dAL, current_cache, "sigmoid")
    for l in reversed(range(L - 1)):
        current_cache = caches[l]
        dA_prev_temp, dW_temp, db_temp = linear_activation_backward(grads["dA" + str(l + 2)], current_cache, "relu")
        grads["dA" + str(l + 1)] = dA_prev_temp
        grads["dW" + str(l + 1)] = dW_temp
        grads["db" + str(l + 1)] = db_temp
    return grads

测试

print("测试model_backward")
AL, Y_assess, caches = testCases.L_model_backward_test_case()
grads = L_model_backward(AL, Y_assess, caches)
print ("dW1 = "+ str(grads["dW1"]))
print ("db1 = "+ str(grads["db1"]))
print ("dA1 = "+ str(grads["dA1"]))

更新参数

def update_parameters(parameters, grads, learning_rate):
    L = len(parameters) // 2
    
    for l in range(L):
        parameters["W" + str(l + 1)] = parameters["W" + str(l + 1)] - learning_rate * grads["dW" + str(l + 1)]
        parameters["b" + str(l + 1)] = parameters["b" + str(l + 1)] - learning_rate * grads["db" + str(l + 1)]
    return parameters

测试

print("==============测试update_parameters==============")
parameters, grads = testCases.update_parameters_test_case()
parameters = update_parameters(parameters, grads, 0.1)

print ("W1 = "+ str(parameters["W1"]))
print ("b1 = "+ str(parameters["b1"]))
print ("W2 = "+ str(parameters["W2"]))
print ("b2 = "+ str(parameters["b2"]))

搭建两层神经网络

图解如下:

def two_layer_model(X, Y, layers_dims, learning_rate = 0.0075, num_iterations = 3000, print_cost=False, isPlot=True):
    np.random.seed(1)
    grads = {}
    costs = []
    (n_x,n_h,n_y) = layers_dims
    #初始化参数
    parameters = initialize_parameters(n_x, n_h, n_y)
    
    W1 = parameters["W1"]
    b1 = parameters["b1"]
    W2 = parameters["W2"]
    b2 = parameters["b2"]
    #开始迭代
    for i in range(0, num_iterations):
        #前向传播
        A1, cache1 = linear_activation_forward(X, W1, b1, "relu")
        A2, cache2 = linear_activation_forward(A1, W2, b2, "sigmoid")
        #计算成本
        cost = compute_cost(A2,Y)
        #初始化后向传播
        dA2 = -(np.divide(Y, A2) - np.divide(1 - Y, 1 - A2))
        #向后传播
        dA1, dW2, db2 = linear_activation_backward(dA2, cache2, "sigmoid")
        dA0, dW1, db1 = linear_activation_backward(dA1, cache1, "relu")
        #传播完的数据进行保存
        grads["dW1"] = dW1
        grads["db1"] = db1
        grads["dW2"] = dW2
        grads["db2"] = db2
        
        #更新参数
        parameters = update_parameters(parameters, grads, learning_rate)
        W1 = parameters["W1"]
        b1 = parameters["b1"]
        W2 = parameters["W2"]
        b2 = parameters["b2"]
        
        #打印成本值
        if i % 100 == 0:
            #记录成本
            costs.append(cost)
            #是否打印成本
            if print_cost:
                print("第"+str(i)+"次迭代，成本值为：",np.squeeze(cost))
    #迭代完成，根据添加画图
    if isPlot:
        plt.plot(np.squeeze(costs))
        plt.ylabel('cost')
        plt.xlabel('iterstions(per tenss)')
        plt.title("Learning rate - " + str(learning_rate))
        plt.show()
    return parameters

加载数据

train_set_x_orig , train_set_y , test_set_x_orig , test_set_y , classes = lr_utils.load_dataset()

train_x_flatten = train_set_x_orig.reshape(train_set_x_orig.shape[0], -1).T 
test_x_flatten = test_set_x_orig.reshape(test_set_x_orig.shape[0], -1).T

train_x = train_x_flatten / 255
train_y = train_set_y
test_x = test_x_flatten / 255
test_y = test_set_y

开始训练

n_x = 12288
n_h = 7
n_y = 1
layers_dims = (n_x,n_h,n_y)

parameters = two_layer_model(train_x, train_set_y, layers_dims = (n_x, n_h, n_y), num_iterations = 2500, print_cost=True,isPlot=True)

进行预测

def predict(X, y, parameters):
    m = X.shape[1]
    n = len(parameters) // 2
    p = np.zeros((1,m))
    
    probas, caches = L_mdoel_forward(X, parameters)
    
    for i in range(0, prodas.shape[1]):
        if prodas[0,i] > 0.5:
            p[0,i] = 1
        else
        	p[0,i] = 0
    
    print("准确度：" + str(float(np.sum((p == y))/m)))
          
    return p

开始预测

predictions_train = predict(train_x, train_y, parameters) #训练集
predictions_test = predict(test_x, test_y, parameters) #测试集

搭建多层神经网络

结构图

def L_layer_model(X, Y, layers_dims, learning_rate=0.0075, num_iterations=3000, print_cost=False,isPlot=True):
    np.random.seed(1)
    costs = []

    parameters = initialize_parameters_deep(layers_dims)

    for i in range(0,num_iterations):
        AL , caches = L_model_forward(X,parameters)

        cost = compute_cost(AL,Y)

        grads = L_model_backward(AL,Y,caches)

        parameters = update_parameters(parameters,grads,learning_rate)

        #打印成本值，如果print_cost=False则忽略
        if i % 100 == 0:
            #记录成本
            costs.append(cost)
            #是否打印成本值
            if print_cost:
                print("第", i ,"次迭代，成本值为：" ,np.squeeze(cost))
    #迭代完成，根据条件绘制图
    if isPlot:
        plt.plot(np.squeeze(costs))
        plt.ylabel('cost')
        plt.xlabel('iterations (per tens)')
        plt.title("Learning rate =" + str(learning_rate))
        plt.show()
    return parameters

加载数据

train_set_x_orig , train_set_y , test_set_x_orig , test_set_y , classes = lr_utils.load_dataset()

train_x_flatten = train_set_x_orig.reshape(train_set_x_orig.shape[0], -1).T 
test_x_flatten = test_set_x_orig.reshape(test_set_x_orig.shape[0], -1).T

train_x = train_x_flatten / 255
train_y = train_set_y
test_x = test_x_flatten / 255
test_y = test_set_y

开始训练

layers_dims = [12288, 20, 7, 5, 1] #  5-layer model
parameters = L_layer_model(train_x, train_y, layers_dims, num_iterations = 2500, print_cost = True,isPlot=True)

分析

def print_mislabeled_images(classes, X, y, p):
    """
    绘制预测和实际不同的图像。
        X - 数据集
        y - 实际的标签
        p - 预测
    """
    a = p + y
    mislabeled_indices = np.asarray(np.where(a == 1))
    plt.rcParams['figure.figsize'] = (40.0, 40.0) # set default size of plots
    num_images = len(mislabeled_indices[0])
    for i in range(num_images):
        index = mislabeled_indices[1][i]

        plt.subplot(2, num_images, i + 1)
        plt.imshow(X[:,index].reshape(64,64,3), interpolation='nearest')
        plt.axis('off')
        plt.title("Prediction: " + classes[int(p[0,index])].decode("utf-8") + " \n Class: " + classes[y[0,index]].decode("utf-8"))


print_mislabeled_images(classes, test_x, test_y, pred_test)