概述

如何利用Python的来实现具有一个隐藏层的平面数据分类问题。前文，创建的神经网络只有一个输出层，没有隐藏层。本文将创建单隐藏层的神经网络模型。

• 二分类单隐藏层的神经网络
• 神经元节点采用非线性的激活函数，如tanh
• 计算交叉损失函数
• 运用前向和后向传播

准备

numpy：Python科学计算中最重要的库
sklearn：提供了一些简单而有效的工具用于数据挖掘和数据分析。
mathplotlib：Python画图的库
testCases：自定义文件，封装了一些用于测试样本，用于评估算法的有效性。
planar_utils：自定义文件，封装了一些作用用到的相关函数。

# Package imports
import numpy as np
import matplotlib.pyplot as plt
from testCases_v2 import *
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) # set a seed so that the results are consistent

数据集

下述代码将加载一个二分类的数据集并进行可视化：

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

绘制出的图像像是一朵花，对于y=0时，显示为红色的点；而当y=1时，显示的是蓝色的点。我们的目的是希望建立一个模型可以将两种颜色的点区分开。
其中X是一个矩阵，包含着数据的特征信息（x1，x2）
Y是一个向量，包含这数据对应的标记结果，其中(red:0,blue:1)
尺寸信息：

### START CODE HERE ### (≈ 3 lines of code)
shape_X = X.shape
shape_Y = Y.shape
m = shape_X[1]  # training set size
### END CODE HERE ###

print ('The shape of X is: ' + str(shape_X))
print ('The shape of Y is: ' + str(shape_Y))
print ('I have m = %d training examples!' % (m))

运行结果可以得到X的维度为2400，Y的维度为1400，训练样本的数量为400。

简单的逻辑回归

在建立一个神经网络之前，先用逻辑回归算法来解决一下该问题。可以直接使用sklearn的内置函数来完成。

# Train the logistic regression classifier
clf = sklearn.linear_model.LogisticRegressionCV();
clf.fit(X.T, Y.T);

然后绘制模型边界。

# Plot the decision boundary for logistic regression
plot_decision_boundary(lambda x: clf.predict(x), X, Y)
plt.title("Logistic Regression")

# Print accuracy
LR_predictions = clf.predict(X.T)
print ('Accuracy of logistic regression: %d ' % float((np.dot(Y,LR_predictions) + np.dot(1-Y,1-LR_predictions))/float(Y.size)*100) +
       '% ' + "(percentage of correctly labelled datapoints)")

结果为：
Accuracy of logistic regression: 47 % (percentage of correctly labelled datapoints)
得到的模型分割图：[图片上传失败...(image-6aab6c-1516700004743)]
显然，数据集是非线性的，因此逻辑回归结果不好。

神经网络模型

神经网络的模型如下：[图片上传失败...(image-16cc4c-1516700004743)]
数学角度上，对于每个训练样本x⁽ⁱ⁾，有：[图片上传失败...(image-a6132b-1516700004743)][图片上传失败...(image-9cfd50-1516700004743)]
注意: 建立神经网络的一般方法如下:

1. 定义神经网络的结构 ( 输入单元数量, 隐藏层单元数量等等).
2. 初始化模型的参数
3. 迭代循环：

• 前向传播
• 计算损失函数
• 后向传播，计算梯度
• 更新参数 (梯度下降)

一般习惯将上述1-3步分别定义成一个独立的函数，再通过模型函数将三者融合一起。在建立模型之后，迭代获取到参数之后，即可对新数据进行预测。

定义神经网络结构

给定如下变量：n_x：输入层神经元的数目
n_h：隐藏层神经元的数目
n_y：输出层神经元的数目
此处，我们需要根据X和Y来确定n_x和n_y，另外n_h设置为4。

# GRADED FUNCTION: layer_sizes

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)

模型参数初始化

在定义参数的时候，对于权重矩阵我们进行随机初始化：np.random.randn(a,b) * 0.01 以获取一个尺寸为(a，b)的随机矩阵。 对于参数b，我们直接初始化为0，np.zeros((a,b))

# 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))*0.01
    W2 = np.random.randn(n_y,n_h)*0.01
    b2 = np.zeros((n_y,1))*0.01
    ### 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

循环

首先是前向传播。
为前向传播定义一个函数，在隐藏层的激活函数是tanh，在输出层的激活函数是sigmoid。利用初始化的参数计算Z^[1],A^[1],Z^[2] 和 A^[2]，同时注意保留值到cache，因为在后续的后向传播需要用到。
前向传播代码：

# 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

代价函数

# GRADED FUNCTION: compute_cost

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 = - np.sum(logprobs)/m
    ### 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

反向传播
公式如下：[图片上传失败...(image-aa5101-1516700004743)]

• 注意 ∗ 表示元素之间的乘积。
• 在计算dZ1时候，我们需要先计算 g^[1]′(Z^[1])。这是由于 g^[1](.) is的激活函数是tanh，当a=g1则g^[1]′(z)=1−a²。为此，我们可以用(1 - np.power(A1, 2))来计算g^[1]′(Z^[1])

# 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.multiply(np.dot(W2.T,dZ2),(1-np.power(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

参数更新
上述的结果我们已经可以计算出反向传播的梯度了，那么我们就可以通过梯度下降法对参数进行更新： [图片上传失败...(image-ff7944-1516700004743)]其中 α 是学习率 θ 则是代表待更新的参数。
选择好的学习率，迭代才会收敛，否则迭代过程不断振荡，呈发散状态。
[图片上传失败...(image-b2a3c1-1516700004743)]

# 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 = W1 - learning_rate*dW1
    b1 = b1 - learning_rate*db1
    W2 = W2 - learning_rate*dW2
    b2 = b2 - learning_rate*db2
    ### END CODE HERE ###
    
    parameters = {"W1": W1,
                  "b1": b1,
                  "W2": W2,
                  "b2": b2}
    
    return parameters

模型融合

# 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)
        
        ### 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

测试代码

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

X_assess, Y_assess = nn_model_test_case()
parameters = nn_model(X_assess, Y_assess, 4, num_iterations=10000, print_cost=True)
print("W1 = " + str(parameters["W1"]))
print("b1 = " + str(parameters["b1"]))
print("W2 = " + str(parameters["W2"]))
print("b2 = " + str(parameters["b2"]))

结果

W1 = [[-4.18496401  5.33207212]
 [-7.53803949  1.20755725]
 [-4.19297213  5.32618291]
 [ 7.53798193 -1.20759019]]
b1 = [[ 2.32932824]
 [ 3.81001626]
 [ 2.33008802]
 [-3.81011657]]
W2 = [[-6033.82356872 -6008.14295996 -6033.08780035  6008.07953767]]
b2 = [[-52.67923024]]

预测

除了训练模型外，我们还需要使用我们的模型来进行预测。

通常，我们会设置一个阈值，当预测结果大于该阈值时，我们认为其为1，否则为0。0.5是一个很常用的阈值。

# 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

目前为止，已经实现了完整的神经网络模型和预测函数，接下来用数据集来训练。

# 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))

运行结果：

分类曲线如上图所示，接下来，使用预测函数计算一下我们模型的预测准确度：

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

相比47%的逻辑回归预测率，使用含有一个隐藏层的神经网络预测的准确度可以达到90%。

调整隐藏层神经元数目观察结果

接下来，使用包含不同隐藏层神经元的模型来进行训练，以此来观察神经元数量度模型的影响。分别适用包含1,2,3,4,5,20,50个神经元的模型来进行训练：

# This may take about 2 minutes to run

plt.figure(figsize=(16, 32))
hidden_layer_sizes = [1, 2, 3, 4, 5, 20, 50]
for i, n_h in enumerate(hidden_layer_sizes):
    plt.subplot(5, 2, i+1)
    plt.title('Hidden Layer of size %d' % n_h)
    parameters = nn_model(X, Y, n_h, num_iterations = 5000)
    plot_decision_boundary(lambda x: predict(parameters, x.T), X, Y)
    predictions = predict(parameters, X)
    accuracy = float((np.dot(Y,predictions.T) + np.dot(1-Y,1-predictions.T))/float(Y.size)*100)
    print ("Accuracy for {} hidden units: {} %".format(n_h, accuracy))

输出结果：

Accuracy for 1 hidden units: 67.5 %
Accuracy for 2 hidden units: 67.25 %
Accuracy for 3 hidden units: 90.75 %
Accuracy for 4 hidden units: 90.5 %
Accuracy for 5 hidden units: 91.25 %
Accuracy for 10 hidden units: 90.25 %
Accuracy for 20 hidden units: 90.5 %

对比上述图像，可以发现：
1.隐藏层的神经元数量越多，则对训练数据集的拟合效果越好，直到最后出现过拟合。
2.本文中最好的神经元数目是n_h=5，此时，能够较好地拟合训练数据集，也不会出现过拟合现象。
3.对于n_h过大而产生的过拟合是可以通过正则化来消除的。