神经网络算法几乎成为深度学习的代名词，未解决不同的场景问题，新的算法层出不穷，而BP（Back Propagation）算法，又称为误差反向传播算法，是最早的人工神经网络中的一种监督式的学习算法。BP 神经网络算法在理论上可以逼近任意函数，基本的结构由非线性变化单元组成，具有很强的非线性映射能力。对于神经网络的介绍多偏向与理论推导，本文将从代码解析的角度，对BP的神经网络算法进行详细介绍，使读者在了解算法的同时，能够自己搭建算法，内容主要来自于Andrew Ng的深度学习英文课程，语言采用Python。

1 导入库

首先导入完成算法所需要的库。

numpy 矩阵计算基础包。
sklearn 数据分析与挖掘基础包
matplotlib 绘图
testCases 协助校验函数的库，需单独创建，主要用来评估过程中创建的函数是否正确，文末会单独列出。
planar_utils 提供多种可用的函数，如激活函数等，文末单独列出。

#导入库
import numpy as np
import matplotlib.pyplot as plt
#from testCases_v2 import * #校验函数
import sklearn
import sklearn.datasets
import sklearn.linear_model
from planar_utils import plot_decision_boundary, sigmoid, load_planar_dataset, load_extra_datasets
%matplotlib inline
np.random.seed(1) # 设置一个seed保持结果一致性。

2准备数据集

首先准备我们进行模型训练的数据集，选用的是常用的花朵颜色分类的数据集，含X，Y两个变量，因变量为花朵颜色。为更直观表示，将其进行可视化展现。

X, Y = load_planar_dataset() 
plt.scatter(X[0, :], X[1, :], c=Y, s=40, cmap=plt.cm.Spectral)

3 单一逻辑回归¶

复杂的神经网络层本质是一个个单一的逻辑回归网络组合而成，所以在构建全连接的神经网络前，我们可以先看一下单个逻辑回归算法的表现。sklearn 中内置的线性回归函数便可以进行验证，可对该数据集训练一个逻辑回归分类模型来看下分类效果。

# Plot the decision boundary for logistic regression
clf = sklearn.linear_model.LogisticRegressionCV();
clf.fit(X.T, Y.T.ravel())
plot_decision_boundary(lambda x: clf.predict(x), X, Y)
plt.title("Logistic Regression")
# Print accuracy
LR_predictions = clf.predict(X.T)
print ('准确性: %d ' % float((np.dot(Y,LR_predictions) + np.dot(1-Y,1-LR_predictions))/float(Y.size)*100) +
       '% ' + "(正确分类的百分比)")

准确性：47%（正确分类的百分比）

通过单一的逻辑回归预测结果看，分类效果并不理想，只能找到单一分隔线，准确性仅47%。

4 构建神经网络模型

4.1 - 定义神经网络结构

定义三个变量：

n_x: 输入层大小
n_h: 隐藏层大小
n_y: 输出层大小
说明: 通过训练集的大小形状确认输入层与输出层的大小，此处将隐藏层直接设置为4。

def layer_sizes(X,Y):
    """
    Arguments:
    X -- input dataset of shape (input size, number of examples)
    Y -- labels of shape (output size, number of examples)
    
    Returns:
    n_x -- the size of the input layer
    n_h -- the size of the hidden layer
    n_y -- the size of the output layer
    """
    ### START CODE HERE ### (≈ 3 lines of code)
    n_x = X.shape[0]# size of input layer
    n_h = 4
    n_y = Y.shape[0] # size of output layer
    ### END CODE HERE ###
    return (n_x, n_h, n_y)

4.2 - 初始化模型参数

创建初始化参数函数initialize_parameters()

确认参数大小正确。
采用随机数创建权重矩阵。
采用np.random.randn(a,b) * 0.01 随机初始化形状为(a,b)的矩阵。
初始化误差向量为0。You will initialize the bias vectors as zeros.
采用 np.zeros((a,b)) 初始化形状为(a,b)的零矩阵。

# GRADED FUNCTION: initialize_parameters

def initialize_parameters(n_x, n_h, n_y):
    """
    Argument:
    n_x -- size of the input layer
    n_h -- size of the hidden layer
    n_y -- size of the output layer
    
    Returns:
    params -- python dictionary containing your parameters:
                    W1 -- weight matrix of shape (n_h, n_x)
                    b1 -- bias vector of shape (n_h, 1)
                    W2 -- weight matrix of shape (n_y, n_h)
                    b2 -- bias vector of shape (n_y, 1)
    """
    
    np.random.seed(2) # we set up a seed so that your output matches ours although the initialization is random.
    
    ### START CODE HERE ### (≈ 4 lines of code)
    W1 = np.random.randn(n_h,n_x)*0.01
    b1 = np.zeros((n_h,1))
    W2 = np.random.randn(n_y,n_h)*0.01
    b2 = np.zeros((n_y,1))
    ### END CODE HERE ###
    
    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

4.3 - 创建循环体

创建正向传播函数：forward_propagation():

查看分类器的数学表达。
sigmoid()单独导入，后面做说明。
np.tanh()numpy中的一个库。
还需执行以下步骤:
1. 通过parameters[".."]检索字典"parameters" (initialize_parameters()的输出结果) 中的每个参数。
2. 执行正向传播，计算Z[1],,A[1],Z[2],A2。
反向传播的所有向量均存储在"cache"中，cache作为反向传播函数的输入变量。

# GRADED FUNCTION: forward_propagation

def forward_propagation(X, parameters):
    """
    Argument:
    X -- input data of size (n_x, m)
    parameters -- python dictionary containing your parameters (output of initialization function)
    
    Returns:
    A2 -- The sigmoid output of the second activation
    cache -- a dictionary containing "Z1", "A1", "Z2" and "A2"
    """
    # Retrieve each parameter from the dictionary "parameters"
    ### START CODE HERE ### (≈ 4 lines of code)
    W1 = parameters['W1']
    b1 = parameters['b1']
    W2 = parameters['W2']
    b2 = parameters['b2']
    ### END CODE HERE ###
    
    # Implement Forward Propagation to calculate A2 (probabilities)
    ### START CODE HERE ### (≈ 4 lines of code)
    Z1 = np.dot(W1,X)+b1
    A1 = np.tanh(Z1)
    Z2 = np.dot(W2,A1)+b2
    A2 = sigmoid(Z2)
    ### END CODE HERE ###
    
    assert(A2.shape == (1, X.shape[1]))
    
    cache = {"Z1": Z1,
             "A1": A1,
             "Z2": Z2,
             "A2": A2}
    
    return A2, cache


def compute_cost(A2, Y, parameters):
    """
    Computes the cross-entropy cost given in equation (13)
    
    Arguments:
    A2 -- The sigmoid output of the second activation, of shape (1, number of examples)
    Y -- "true" labels vector of shape (1, number of examples)
    parameters -- python dictionary containing your parameters W1, b1, W2 and b2
    
    Returns:
    cost -- cross-entropy cost given equation (13)
    """
    
    m = Y.shape[1] # number of example

    # Compute the cross-entropy cost
    ### START CODE HERE ### (≈ 2 lines of code)
    logprobs = np.multiply(np.log(A2),Y)+np.multiply(np.log(1-A2),(1-Y))
    cost = -1/m*(np.sum(logprobs))
    ### END CODE HERE ###
    
    cost = np.squeeze(cost)     # makes sure cost is the dimension we expect. 
                                # E.g., turns [[17]] into 17 
    assert(isinstance(cost, float))
    
    return cost

# GRADED FUNCTION: backward_propagation

def backward_propagation(parameters, cache, X, Y):
    """
    Implement the backward propagation using the instructions above.
    
    Arguments:
    parameters -- python dictionary containing our parameters 
    cache -- a dictionary containing "Z1", "A1", "Z2" and "A2".
    X -- input data of shape (2, number of examples)
    Y -- "true" labels vector of shape (1, number of examples)
    
    Returns:
    grads -- python dictionary containing your gradients with respect to different parameters
    """
    m = X.shape[1]
    
    # First, retrieve W1 and W2 from the dictionary "parameters".
    ### START CODE HERE ### (≈ 2 lines of code)
    W1 = parameters['W1']
    W2 = parameters['W2']
    ### END CODE HERE ###
        
    # Retrieve also A1 and A2 from dictionary "cache".
    ### START CODE HERE ### (≈ 2 lines of code)
    A1 = cache['A1']
    A2 = cache['A2']
    ### END CODE HERE ###
    
    # Backward propagation: calculate dW1, db1, dW2, db2. 
    ### START CODE HERE ### (≈ 6 lines of code, corresponding to 6 equations on slide above)
    dZ2 = A2-Y
    dW2 = 1/m*np.dot(dZ2,(A1.T))
    db2 = 1/m*(np.sum(dZ2,axis=1,keepdims=True))
    dZ1 = np.dot(W2.T,dZ2) * (1 - np.power(cache["A1"],2))
    dW1 = 1/m*np.dot(dZ1,(X.T))
    db1 = 1/m*(np.sum(dZ1,axis=1,keepdims=True))
    ### END CODE HERE ###
    
    grads = {"dW1": dW1,
             "db1": db1,
             "dW2": dW2,
             "db2": db2}
    
    return grads

# GRADED FUNCTION: update_parameters

def update_parameters(parameters, grads, learning_rate = 1.2):
    """
    Updates parameters using the gradient descent update rule given above
    
    Arguments:
    parameters -- python dictionary containing your parameters 
    grads -- python dictionary containing your gradients 
    
    Returns:
    parameters -- python dictionary containing your updated parameters 
    """
    # Retrieve each parameter from the dictionary "parameters"
    ### START CODE HERE ### (≈ 4 lines of code)
    W1 = parameters['W1']
    b1 = parameters['b1']
    W2 = parameters['W2']
    b2 = parameters['b2']
    ### END CODE HERE ###
    
    # Retrieve each gradient from the dictionary "grads"
    ### START CODE HERE ### (≈ 4 lines of code)
    dW1 = grads['dW1']
    db1 = grads['db1']
    dW2 = grads['dW2']
    db2 = grads['db2']
    ## END CODE HERE ###
    
    # Update rule for each parameter
    ### START CODE HERE ### (≈ 4 lines of code)
    W1 = -dW1*learning_rate+W1
    b1 = -db1*learning_rate+b1
    W2 = -dW2*learning_rate+W2
    b2 = -db2*learning_rate+b2
    ### END CODE HERE ###
    
    parameters = {"W1": W1,
                  "b1": b1,
                  "W2": W2,
                  "b2": b2}
    
    return parameters

4.4 - 整合以上4.1, 4.2 和 4.3 创建的函数，创建最终的nn_model()

问题：创建神经网络模型nn_model():

# GRADED FUNCTION: nn_model

def nn_model(X, Y, n_h, num_iterations = 10000, print_cost=False):
    """
    Arguments:
    X -- dataset of shape (2, number of examples)
    Y -- labels of shape (1, number of examples)
    n_h -- size of the hidden layer
    num_iterations -- Number of iterations in gradient descent loop
    print_cost -- if True, print the cost every 1000 iterations
    
    Returns:
    parameters -- parameters learnt by the model. They can then be used to predict.
    """
    
    np.random.seed(3)
    n_x = layer_sizes(X, Y)[0]
    n_y = layer_sizes(X, Y)[2]
    
    # Initialize parameters, then retrieve W1, b1, W2, b2. Inputs: "n_x, n_h, n_y". Outputs = "W1, b1, W2, b2, parameters".
    ### START CODE HERE ### (≈ 5 lines of code)
    parameters = initialize_parameters(n_x, n_h, n_y)
    W1 = parameters['W1']
    b1 = parameters['b1']
    W2 = parameters['W2']
    b2 = parameters['b2']
    ### END CODE HERE ###
    
    # Loop (gradient descent)

    for i in range(0, num_iterations):
         
        ### START CODE HERE ### (≈ 4 lines of code)
        # Forward propagation. Inputs: "X, parameters". Outputs: "A2, cache".
        A2, cache = forward_propagation(X, parameters)
        
        # Cost function. Inputs: "A2, Y, parameters". Outputs: "cost".
        cost = compute_cost(A2, Y, parameters)
 
        # Backpropagation. Inputs: "parameters, cache, X, Y". Outputs: "grads".
        grads = backward_propagation(parameters, cache, X, Y)
 
        # Gradient descent parameter update. Inputs: "parameters, grads". Outputs: "parameters".
        parameters = update_parameters(parameters, grads, learning_rate = 1.2)
        
        ### END CODE HERE ###
        
        # Print the cost every 1000 iterations
        if print_cost and i % 1000 == 0:
            print ("Cost after iteration %i: %f" %(i, cost))

    return parameters

4.5 预测

问题：创建预测函数，采用正向传播来预测结果:
注: 如果激活函数>0.5,则预测为1，如果激活函数值<=0.5，则预测为0。例如你如果想基于一个阈值设定X矩阵的类，可以通过X_new = (X > threshold)

# GRADED FUNCTION: predict

def predict(parameters, X):
    """
    Using the learned parameters, predicts a class for each example in X
    
    Arguments:
    parameters -- python dictionary containing your parameters 
    X -- input data of size (n_x, m)
    
    Returns
    predictions -- vector of predictions of our model (red: 0 / blue: 1)
    """
    
    # Computes probabilities using forward propagation, and classifies to 0/1 using 0.5 as the threshold.
    ### START CODE HERE ### (≈ 2 lines of code)
    A2, cache = forward_propagation(X, parameters)
    predictions = A2>0.5
    ### END CODE HERE ###
    
    return predictions

结果预测

# Build a model with a n_h-dimensional hidden layer
parameters = nn_model(X, Y, n_h = 4, num_iterations = 10000, print_cost=True)

# Plot the decision boundary
plot_decision_boundary(lambda x: predict(parameters, x.T), X, Y)
plt.title("Decision Boundary for hidden layer size " + str(4))

# Print accuracy
predictions = predict(parameters, X)
print ('准确性: %d' % float((np.dot(Y,predictions.T) + np.dot(1-Y,1-predictions.T))/float(Y.size)*100) + '%')

准确性：90%，预测结果准确性显著高于单个逻辑回归预测结果。

附件1-"testCases_v2.py"

def layer_sizes_test_case():
    np.random.seed(1)
    X_assess = np.random.randn(5, 3)
    Y_assess = np.random.randn(2, 3)
    return X_assess, Y_assess
 
 
def initialize_parameters_test_case():
    n_x, n_h, n_y = 2, 4, 1
    return n_x, n_h, n_y
 
 
def forward_propagation_test_case():
    np.random.seed(1)
    X_assess = np.random.randn(2, 3)
 
    parameters = {'W1': np.array([[-0.00416758, -0.00056267],
                                  [-0.02136196, 0.01640271],
                                  [-0.01793436, -0.00841747],
                                  [0.00502881, -0.01245288]]),
                  'W2': np.array([[-0.01057952, -0.00909008, 0.00551454, 0.02292208]]),
                  'b1': np.array([[0.],
                                  [0.],
                                  [0.],
                                  [0.]]),
                  'b2': np.array([[0.]])}
 
    return X_assess, parameters
 
 
def compute_cost_test_case():
    np.random.seed(1)
    Y_assess = np.random.randn(1, 3)
    parameters = {'W1': np.array([[-0.00416758, -0.00056267],
                                  [-0.02136196, 0.01640271],
                                  [-0.01793436, -0.00841747],
                                  [0.00502881, -0.01245288]]),
                  'W2': np.array([[-0.01057952, -0.00909008, 0.00551454, 0.02292208]]),
                  'b1': np.array([[0.],
                                  [0.],
                                  [0.],
                                  [0.]]),
                  'b2': np.array([[0.]])}
 
    a2 = (np.array([[0.5002307, 0.49985831, 0.50023963]]))
 
    return a2, Y_assess, parameters
 
 
def backward_propagation_test_case():
    np.random.seed(1)
    X_assess = np.random.randn(2, 3)
    Y_assess = np.random.randn(1, 3)
    parameters = {'W1': np.array([[-0.00416758, -0.00056267],
                                  [-0.02136196, 0.01640271],
                                  [-0.01793436, -0.00841747],
                                  [0.00502881, -0.01245288]]),
                  'W2': np.array([[-0.01057952, -0.00909008, 0.00551454, 0.02292208]]),
                  'b1': np.array([[0.],
                                  [0.],
                                  [0.],
                                  [0.]]),
                  'b2': np.array([[0.]])}
 
    cache = {'A1': np.array([[-0.00616578, 0.0020626, 0.00349619],
                             [-0.05225116, 0.02725659, -0.02646251],
                             [-0.02009721, 0.0036869, 0.02883756],
                             [0.02152675, -0.01385234, 0.02599885]]),
             'A2': np.array([[0.5002307, 0.49985831, 0.50023963]]),
             'Z1': np.array([[-0.00616586, 0.0020626, 0.0034962],
                             [-0.05229879, 0.02726335, -0.02646869],
                             [-0.02009991, 0.00368692, 0.02884556],
                             [0.02153007, -0.01385322, 0.02600471]]),
             'Z2': np.array([[0.00092281, -0.00056678, 0.00095853]])}
    return parameters, cache, X_assess, Y_assess
 
 
def update_parameters_test_case():
    parameters = {'W1': np.array([[-0.00615039, 0.0169021],
                                  [-0.02311792, 0.03137121],
                                  [-0.0169217, -0.01752545],
                                  [0.00935436, -0.05018221]]),
                  'W2': np.array([[-0.0104319, -0.04019007, 0.01607211, 0.04440255]]),
                  'b1': np.array([[-8.97523455e-07],
                                  [8.15562092e-06],
                                  [6.04810633e-07],
                                  [-2.54560700e-06]]),
                  'b2': np.array([[9.14954378e-05]])}
 
    grads = {'dW1': np.array([[0.00023322, -0.00205423],
                              [0.00082222, -0.00700776],
                              [-0.00031831, 0.0028636],
                              [-0.00092857, 0.00809933]]),
             'dW2': np.array([[-1.75740039e-05, 3.70231337e-03, -1.25683095e-03,
                               -2.55715317e-03]]),
             'db1': np.array([[1.05570087e-07],
                              [-3.81814487e-06],
                              [-1.90155145e-07],
                              [5.46467802e-07]]),
             'db2': np.array([[-1.08923140e-05]])}
    return parameters, grads
 
 
def nn_model_test_case():
    np.random.seed(1)
    X_assess = np.random.randn(2, 3)
    Y_assess = np.random.randn(1, 3)
    return X_assess, Y_assess
 
 
def predict_test_case():
    np.random.seed(1)
    X_assess = np.random.randn(2, 3)
    parameters = {'W1': np.array([[-0.00615039, 0.0169021],
                                  [-0.02311792, 0.03137121],
                                  [-0.0169217, -0.01752545],
                                  [0.00935436, -0.05018221]]),
                  'W2': np.array([[-0.0104319, -0.04019007, 0.01607211, 0.04440255]]),
                  'b1': np.array([[-8.97523455e-07],
                                  [8.15562092e-06],
                                  [6.04810633e-07],
                                  [-2.54560700e-06]]),
                  'b2': np.array([[9.14954378e-05]])}
    return parameters, X_assess

附件2-"planar_utils.py"

def plot_decision_boundary(model, X, y):
    # Set min and max values and give it some padding
    x_min, x_max = X[0, :].min() - 1, X[0, :].max() + 1
    y_min, y_max = X[1, :].min() - 1, X[1, :].max() + 1
    h = 0.01
    # Generate a grid of points with distance h between them
    xx, yy = np.meshgrid(np.arange(x_min, x_max, h), np.arange(y_min, y_max, h))
    # Predict the function value for the whole grid
    Z = model(np.c_[xx.ravel(), yy.ravel()])
    Z = Z.reshape(xx.shape)
    # Plot the contour and training examples
    plt.contourf(xx, yy, Z, cmap=plt.cm.Spectral)
    plt.ylabel('x2')
    plt.xlabel('x1')
    plt.scatter(X[0, :], X[1, :], c=y[0], cmap=plt.cm.Spectral)
 
def sigmoid(x):
    """
    Compute the sigmoid of x
    Arguments:
    x -- A scalar or numpy array of any size.
    Return:
    s -- sigmoid(x)
    """
    s = 1/(1+np.exp(-x))
    return s
 
def load_planar_dataset():
    np.random.seed(1)
    m = 400 # number of examples
    N = int(m/2) # number of points per class
    D = 2 # dimensionality
    X = np.zeros((m,D)) # data matrix where each row is a single example
    Y = np.zeros((m,1), dtype='uint8') # labels vector (0 for red, 1 for blue)
    a = 4 # maximum ray of the flower
 
    for j in range(2):
        ix = range(N*j,N*(j+1))
        t = np.linspace(j*3.12,(j+1)*3.12,N) + np.random.randn(N)*0.2 # theta
        r = a*np.sin(4*t) + np.random.randn(N)*0.2 # radius
        X[ix] = np.c_[r*np.sin(t), r*np.cos(t)]
        Y[ix] = j
        
    X = X.T
    Y = Y.T
 
    return X, Y
def load_extra_datasets():  
    N = 200
    noisy_circles = sklearn.datasets.make_circles(n_samples=N, factor=.5, noise=.3)
    noisy_moons = sklearn.datasets.make_moons(n_samples=N, noise=.2)
    blobs = sklearn.datasets.make_blobs(n_samples=N, random_state=5, n_features=2, centers=6)
    gaussian_quantiles = sklearn.datasets.make_gaussian_quantiles(mean=None, cov=0.5, n_samples=N, n_features=2, n_classes=2, shuffle=True, random_state=None)
    no_structure = np.random.rand(N, 2), np.random.rand(N, 2)
    
    return noisy_circles, noisy_moons, blobs, gaussian_quantiles, no_structure

神经网络的初步构建