Welcome to your week 3 programming assignment. It's time to build your first neural network, which will have a hidden layer. You will see a big difference between this model and the one you implemented using logistic regression.

目标：建立只有一个隐含层的平面二分类器

1 将会学习使用：

You will learn how to:

Implement a 2-class classification neural network with a single hidden layer
Use units with a non-linear activation function, such as tanh
Compute the cross entropy loss
Implement forward and backward propagatio

2 软件包准备

Let's first import all the packages that you will need during this assignment.

numpy is the fundamental（基本的） package for scientific computing with Python.
sklearn provides simple and efficient tools for data mining and data analysis. 数据挖掘
matplotlib is a library for plotting graphs in Python.
testCases provides some test examples to assess（评估） the correctness of your functions
planar_utils provide various useful functions used in this assignment

#Package imports
import numpy as np
import matplotlib.pyplot as plt
from testCases 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           #jupyter notebook的专有功能，魔法函数，可以使图像直接在upyter notebook中显示出来

np.random.seed(1)   # set a seed so that the results are consistent
                   # 设置一个种子，这样每次随机产生的数是不变的

3 数据准备

1、加载数据到变量X和Y中

First, let's get the dataset you will work on. The following code will load a "flower" 2-class dataset into variables X and Y.

X,Y=load_planar_dataset()    #加载数据到变量X和Y中。

2、数据可视化，红色代表 label y=0 蓝色代表 label y=1

Visualize the dataset using matplotlib. The data looks like a "flower" with some red (label y=0) and some blue (y=1) points. Your goal is to build a model to fit this data.

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

3、显示结果

4、可知X是我们的样本集，Y是样本集对应的标签。

You have:

- a numpy-array (matrix) X that contains your features (x1, x2)
- a numpy-array (vector) Y that contains your labels (red:0, blue:1).

5、查看我们的变量X和Y的具体内容，查看他们的维度，确定样本集的情况。

Lets first get a better sense of what our data is like.

Exercise: How many training examples do you have? In addition, what is the shape of the variables X and Y?

#查看变量的情况，确定样本的个数以及维度
shape_X=X.shape
shape_Y=Y.shape
m=Y.shape[1]      # training set size

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)) # 注意这里的输出方式，%（m）必须这样写，和C语言中直接一个逗号跟m不一样。

out:
The shape of X is:(2, 400)
The shape of Y is:(1, 400)
I have m=400 training examples!

# 从X的维度（2,400）我们还不难看出每一个样本的输入是两个特征
# 符合我们的全部样本的前向后向矩阵要求：输入样本集每一列是一个样本，所有样本横向叠加构成样本集。最后的真实标签Y是一个行向量。

4 构建神经网络前查看简单的logistic回归效果

Before building a full neural network, lets first see how logistic regression performs on this problem. You can use sklearn's built-in functions to do that. Run the code below to train a logistic regression classifier on the dataset.

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

You can now plot the decision boundary of these models. Run the code below.

# Plot the decision boundary for logistic regression
plot_decision_boundary(lambda x: clf.predict(x),X,np.squeeze(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)")

Interpretation: The dataset is not linearly separable, so logistic regression doesn't perform well. Hopefully a neural network will do better. Let's try this now!

5 构建神经网络

总述

Logistic regression did not work well on the "flower dataset". You are going to train a Neural Network with a single hidden layer.

Here is our model：

Mathematically：数学化的计算过程

For one example $x^{(i)}$ :

$z^{[1] (i)} = W^{[1]} x^{(i)} + b^{[1] (i)}\tag{1}$

$a^{[1] (i)} = \tanh(z^{[1] (i)})\tag{2}$

$z^{[2] (i)} = W^{[2]} a^{[1] (i)} + b^{[2] (i)}\tag{3}$

$\hat{y}^{(i)} = a^{[2] (i)} = \sigma(z^{ [2] (i)})\tag{4}$

$y^{(i)}_{prediction} = \begin{cases} 1 & \mbox{if } a^{[2](i)} > 0.5 \\ 0 & \mbox{otherwise } \end{cases}\tag{5}$

给出所有示例的预测结果，可以按如下方式计算成本J：

通常构建一个神经网络的方法：

Reminder: The general methodology to build a Neural Network is to:

1. Define the neural network structure ( # of input units,  # of hidden units, etc). 
2. Initialize the model's parameters
3. Loop:
    - Implement forward propagation
    - Compute loss
    - Implement backward propagation to get the gradients
    - Update parameters (gradient descent)

构建三个函数分别实现上面的三个功能。

构建一个模块函数nn_model()整合上面的三个函数来实现上述的全部功能。

You often build helper functions to compute steps 1-3 and then merge them into one function we call nn_model(). Once you've built nn_model() and learnt the right parameters, you can make predictions on new data.

5.1 定义神经网络结构

Exercise: Define three variables:

- n_x: the size of the input layer                    输入层节点的数量
- n_h: the size of the hidden layer (set this to 4)   隐含层节点的数量
- n_y: the size of the output layer                   输出层节点的数量

Hint: Use shapes of X and Y to find n_x and n_y. Also, hard code the hidden layer size to be 4. 其实从这句话中的一个词 hard code（硬编码）我们就要立即明白，隐含层的节点数是我们人为设定的，跟输入和输出是没有关系的。

第一个函数：定义网络结构

#GRADED FUNCTION: layer_sizes  分函数 层的定义
def layer_sizes(X, Y):
    """
    Argumets:
    X -- input dataset of shape (input sizs, 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 ###
    n_x = X.shape[0] #size of input layer
    n_h = 4
    n_y = Y.shape[0]
    ### END CODE HERE ###
    return (n_x, n_h, n_y)

Expected Output (these are not the sizes you will use for your network, they are just used to assess the function you've just coded)

#测试函数 其中layer_sizes_test_case() 是我们的测试数据，并不是真实数据
#X, Y = load_planar_dataset()
X_assess, Y_assess = layer_sizes_test_case()
(n_x, n_h, n_y) = layer_sizes(X_assess, Y_assess)
print("The size of the input layer is: n_x = " + str(n_x))
print("The size of the hidden layer is: n_h = " + str(n_h))
print("The size of the output layer is: n_y = " + str(n_y)) 


out：

The size of the input layer is: n_x = 5
The size of the hidden layer is: n_h = 4
The size of the output layer is: n_y = 2

5.2 初始化模型参数

第二个函数：初始化参数（权重和偏置）

Exercise: Implement the function initialize_parameters().

Instructions:

Make sure your parameters' sizes are right. Refer to the neural network figure above if needed.
You will initialize the weights matrices with random values.
- Use: np.random.randn(a,b) * 0.01 to randomly initialize a matrix of shape (a,b). 初始化权重w
You will initialize the bias vectors as zeros.
- Use: np.zeros((a,b)) to initialize a matrix of shape (a,b) with zeros. 初始化偏置b

# GRADED FUNCTION: initialize_paramenters

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 cotaining 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 ### 
    W1 = np.random.rand(n_h, n_x) * 0.01
    b1 = np.zeros((n_h, 1))
    W2 = np.random.rand(n_y, n_h) * 0.01
    b2 = np.zeros((n_y, 1))
    
    parameters = {"W1": W1,
                 "b1": b1,
                 "W2": W2,
                 "b2": b2,
                 }
    return parameters

n_x, n_h, n_y = initialize_parameters_test_case()
parameters = initialize_parameters(n_x, n_h, n_y)
print("W1 = " + str(parameters["W1"]))
print("b1 = " + str(parameters["b1"]))
print("W2 = " + str(parameters["W2"]))
print("b1 = " + str(parameters["b2"]))


out:

W1 = [[0.00435995 0.00025926]
 [0.00549662 0.00435322]
 [0.00420368 0.00330335]
 [0.00204649 0.00619271]]
b1 = [[0.]
 [0.]
 [0.]
 [0.]]
W2 = [[0.00299655 0.00266827 0.00621134 0.00529142]]
b1 = [[0.]]

5.3 循环

第一个函数前向传播计算

前向传播

Question: Implement forward_propagation().

Instructions:

Look above at the mathematical representation of your classifier.
You can use the function sigmoid(). It is built-in (imported) in the notebook.
You can use the function np.tanh(). It is part of the numpy library.
The steps you have to implement are:
1. Retrieve each parameter from the dictionary "parameters" (which is the output of initialize_parameters()) by using parameters[".."].
2. Implement Forward Propagation. Compute $Z^{[1]}, A^{[1]}, Z^{[2]}$ and $A^{[2]}$ (the vector of all your predictions on all the examples in the training set).
Values needed in the backpropagation are stored in "cache". The cache will be given as an input to the backpropagation function.

#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 initilization function)
    
    Return:
    A2 -- The sigmoid output of the second activation
    cache -- a dictionary containing "Z1","A1", "Z2" and "A2"
    """
    ### START CODE HERE ###
    
    # Retrieve each parameter from the dictionary "parameters"
    W1 = parameters['W1']
    b1 = parameters['b1']
    W2 = parameters['W2']
    b2 = parameters['b2']
    
    # Implement Forward Propagation to calcluate A2 (probabilities)
    
    Z1 = np.dot(W1, X) + b1
    A1 = np.tanh(Z1)
    Z2 = np.dot(W2, A1) + b2
    A2 = sigmoid(Z2)          # sigmoid 函数是自带的函数 
    ### EDG CODE HERE ###
    
    assert(A2.shape == (1, X.shape[1]))  # 确保维度正确
    
    cache = {"Z1": Z1,
            "A1": A1,
            "Z2": Z2,
            "A2": A2}
    return A2, cache
    pass

X_assess,parameters = forward_propagation_test_case()
A2, cache = forward_propagation(X_assess, parameters)

# Note: we use the mean here just to make sure that your output matchs ours.
print(np.mean(cache['Z1']), np.mean(cache['A1']), np.mean(cache['Z2']), np.mean(cache['A2']))


out:

-0.0004997557777419902 -0.000496963353231779 0.00043818745095914653 0.500109546852431

第二个函数计算cost

计算损失

Exercise: Implement compute_cost() to compute the value of the cost $J$ .

Instructions:

There are many ways to implement the cross-entropy loss. To help you, we give you how we would have implemented
这里给出了一个如下数学式子的python代码：

$-\sum_{i=0}^{m}y^{{i}}log(a^{[2](i)})$

logprobs = np.multiply(np.log(A2),Y)     一次性计算输出层所有样本的ylog(y^)，A2 和 Y都是（1，m）的行向量不要忘记
cost = - np.sum(logprobs)                # no need to use a for loop!

(you can use either np.multiply() and then np.sum() or directly np.dot()).

# GRADED FUNCTION: compute_cost

def compute_cost(A2, Y, parameters):
    """
    Computes the cross-entropy cost given in cost equation
    
    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, b2
    
    Returns:
    cost -- cross-entropy cost given equation
    """
    m = Y.shape[1]  # number of examples
    
    # Retrieve W1 and W2 from parameters
    ### START CODE HERE ###
    W1 = parameters['W1']
    W2 = parameters['W2']
    ### END CODE HERE ###
    
    # Compute the cross-entropy cost
    ### START CODE HERE ###
    logprobs = np.multiply(np.log(A2), Y) + np.multiply((1 - Y), np.log(1 - A2))
    cost = -np.sum(logprobs) / m
    ### END CODE HERE ###
    
    cost = np.squeeze(cost) # make sure cost is the dimension we expect
                            # Eg, turns[[17]] into 17
                            # 这里的squeeze的作用就是保证sum后的结果是一个数而不是一个矩阵
    assert(isinstance(cost, float))
    
    return cost

A2, Y_assess,parameters = compute_cost_test_case()
print("cost = " + str(compute_cost(A2, Y_assess, parameters)))

out：

cost = 0.6929198937761266

#再次提醒我们：我们的cost函数是定为所有样本损失的值，是一个数

第三个函数反向传播得到dw db

反向传播调整参数

Using the cache computed during forward propagation, you can now implement backward propagation.

Question: Implement the function backward_propagation()

Instructions: Backpropagation is usually the hardest (most mathematical) part in deep learning. To help you, here again is the slide from the lecture on backpropagation. You'll want to use the six equations on the right of this slide, since you are building a vectorized implementation.

Tips:

To compute dZ1 you'll need to compute $g^{[1]'}(Z^{[1]})$ . Since $g^{[1]}(.)$ is the tanh activation function, if $a = g^{[1]}(z)$ then $g^{[1]'}(z) = 1-a^2$ . So you can compute $g^{[1]'}(Z^{[1]})$ using (1 - np.power(A1, 2)).

这个函数我们不是直接调参，而是仅仅得到dw db 并没有执行w - dw

重点：按照公式一步一步来

# 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", "A2"
    X -- input data of shape(2, number of examples)
    Y -- "true" labels vector of shape(1, number of examples)
    
    Return:
    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: claculate 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

parameters, cache, X_assess, Y_assess = backward_propagation_test_case()

grads = backward_propagation(parameters, cache, X_assess, Y_assess)
print ("dW1 = "+ str(grads["dW1"]))
print ("db1 = "+ str(grads["db1"]))
print ("dW2 = "+ str(grads["dW2"]))
print ("db2 = "+ str(grads["db2"]))

out：

dW1 = [[ 0.01018708 -0.00708701]
 [ 0.00873447 -0.0060768 ]
 [-0.00530847  0.00369379]
 [-0.02206365  0.01535126]]
db1 = [[-0.00069728]
 [-0.00060606]
 [ 0.000364  ]
 [ 0.00151207]]
dW2 = [[ 0.00363613  0.03153604  0.01162914 -0.01318316]]
db2 = [[0.06589489]]

第四个函数：更新参数

Question: Implement the update rule. Use gradient descent. You have to use (dW1, db1, dW2, db2) in order to update (W1, b1, W2, b2).

General gradient descent rule: $\theta = \theta - \alpha \frac{\partial J }{ \partial \theta }$ where $\alpha$ is the learning rate and $\theta$ represents a parameter.

Illustration: The gradient descent algorithm with a good learning rate (converging) and a bad learning rate (diverging). Images courtesy of Adam Harley

# 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

parameters, grads = update_parameters_test_case()
parameters = update_parameters(parameters, grads)

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

out:更新后的参数w和b

W1 = [[-0.00643025  0.01936718]
 [-0.02410458  0.03978052]
 [-0.01653973 -0.02096177]
 [ 0.01046864 -0.05990141]]
b1 = [[-1.02420756e-06]
 [ 1.27373948e-05]
 [ 8.32996807e-07]
 [-3.20136836e-06]]
W2 = [[-0.01041081 -0.04463285  0.01758031  0.04747113]]
b2 = [[0.00010457]]

5.4 整合成一个函数

Question: Build your neural network model in nn_model().

Instructions: The neural network model has to use the previous functions in the right order

# GRADED FUNCTION: nn_model

def nn_model(X, Y,n_h, num_iteration=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 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 ###
    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_iteration):
        
        ### START CODE HERE ###
        # Forward propragation ,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.
        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))
            pass
    return parameters

进行一万次循环调整参数。

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

out:

W1 = [[ 0.00782276 -0.00215074]
 [ 0.00858013  0.00220712]
 [ 0.01138167 -0.00169241]
 [ 0.00816134  0.00193676]]
b1 = [[-0.00023657]
 [-0.0002106 ]
 [-0.00049024]
 [-0.00041754]]
W2 = [[0.0078265  0.00551909 0.00841241 0.00267564]]
b2 = [[-0.07892907]]

5.5 预测

Question: Use your model to predict by building predict(). Use forward propagation to predict results.

As an example, if you would like to set the entries of a matrix X to 0 and 1 based on a threshold you would do: X_new = (X > threshold)

# GRADED FUNCTION: predict

def predict(parameters, X):
    """
    Using the learned parameters, predicts s calss for each example in X
    
    Arguments:
    parameters -- python dictionary containing your parameters
    X -- input data of size (n_x, m)
    
    Returns:
    priedictions -- 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 ###
    A2,cache = forward_propagation(X, parameters)
    predictions = np.round(A2)       #np.roung() 四舍五入 刚好符合我们的分类方法
    ### END CODE HERE ###
    
    return predictions

parameters, X_assess = predict_test_case()
predictions = predict(parameters, X_assess)
print("predictions mean =" + str(np.mean(predictions)))


out:

predictions mean =0.6666666666666666

5.6 正式带入待测样本进行预测

It is time to run the model and see how it performs on a planar dataset. Run the following code to test your model with a single hidden layer of $n_h$ hidden units.

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

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

out:

Cost after iteration 0:0.693159
Cost after iteration 1000:0.289308
Cost after iteration 2000:0.273860
Cost after iteration 3000:0.238116
Cost after iteration 4000:0.228102
Cost after iteration 5000:0.223318
Cost after iteration 6000:0.220193
Cost after iteration 7000:0.217870
Cost after iteration 8000:0.216036
Cost after iteration 9000:0.218629

# 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) + '%')
#上面这个计算准确率的方法很巧妙啊，因为我们找的不仅仅是1和1 还有0和0  太巧妙了

out:

Accuracy: 90%

Accuracy is really high compared to Logistic Regression. The model has learnt the leaf patterns of the flower! Neural networks are able to learn even highly non-linear decision boundaries, unlike logistic regression.

Now, let's try out several hidden layer sizes.

5.7 尝试改变隐含层的大小（还是一层，只是节点数改变）

# 尝试改变隐含层的大小（节点数）
# 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):   # 枚举 表示第 i 个元素 n_h
    plt.subplot(5, 2, i + 1)
    plt.title('Hidden Layer of size %d' % n_h)
    parameters = nn_model(X, Y, n_h, num_iteration=5000)
    plot_decision_boundary(lambda x: predict(parameters, x.T), X, np.squeeze(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))


out:

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: 90.75 %
Accuracy for 5 hidden units: 91.25 %
Accuracy for 20 hidden units: 90.25 %
Accuracy for 50 hidden units: 91.0 %

可以发现在节点数n_h =5 的时候准确率最大。节点不是越多越好。

关于隐含层节点的知识:

Interpretation:

The larger models (with more hidden units) are able to fit the training set better, until eventually the largest models overfit the data.
The best hidden layer size seems to be around n_h = 5. Indeed, a value around here seems to fits the data well without also incurring noticable overfitting.
You will also learn later about regularization(正则化）, which lets you use very large models (such as n_h = 50) without much overfitting.

6 使用其他数据集测试

If you want, you can rerun the whole notebook (minus the dataset part) for each of the following datasets.

# 用其他数据集测试我们的网络

# Datasets
noisy_circles, noisy_moons, blobs, gaussian_quantiles, no_structure = load_extra_datasets()

datasets = {"noisy_circles": noisy_circles,
            "noisy_moons": noisy_moons,
            "blobs": blobs,
            "gaussian_quantiles": gaussian_quantiles}

### START CODE HERE ### (choose your dataset)
dataset = "noisy_moons"
### END CODE HERE ###

X, Y = datasets[dataset]
X, Y = X.T, Y.reshape(1, Y.shape[0])

# make blobs binary
if dataset == "blobs":
    Y = Y % 2

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

第三周课后编程作业

Planar data classification with one hidden layer

目标：建立只有一个隐含层的平面二分类器

1 将会学习使用：

2 软件包准备

3 数据准备

1、加载数据到变量X和Y中

2、数据可视化，红色代表 label y=0 蓝色代表 label y=1

3、显示结果

4、可知X是我们的样本集，Y是样本集对应的标签。

5、查看我们的变量X和Y的具体内容，查看他们的维度，确定样本集的情况。

4 构建神经网络前查看简单的logistic回归效果

5 构建神经网络

总述

5.1 定义神经网络结构

5.2 初始化模型参数

5.3 循环

5.4 整合成一个函数

5.5 预测

5.6 正式带入待测样本进行预测

5.7 尝试改变隐含层的大小（还是一层，只是节点数改变）

6 使用其他数据集测试

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