課程一(Neural Networks and Deep Learning)，第三周（Shallow neural networks）—— 3.Programming Assignment : Planar data classification with a hidden layer

Planar data classification with a hidden layer

Welcome to the second programming exercise of the deep learning specialization. In this notebook you will generate red and blue points to form a flower. You will then fit a neural network to correctly classify the points. You will try different layers and see the results.

By completing this assignment you will:

- Develop an intuition of back-propagation and see it work on data.

- Recognize that the more hidden layers you have the more complex structure you could capture.

- Build all the helper functions to implement a full model with one hidden layer.

This assignment prepares you well for the upcoming assignment. Take your time to complete it and make sure you get the expected outputs when working through the different exercises. In some code blocks, you will find a "#GRADED FUNCTION: functionName" comment. Please do not modify it. After you are done, submit your work and check your results. You need to score 70% to pass. Good luck :) !

【中文翻譯】

 用有一個人隱藏層的神經網絡進行平面數據分類

歡迎來到深度學習專業第二次編程練習。在本筆記本中, 你會產生紅色和藍色的點, 形成一朵花。然后, 您將找到一個神經網絡, 用來正確分類這些點。您將嘗試不同的層並查看結果。

通過完成此任務, 您將:

-形成一個反向傳播的直覺, 並看到它在數據上的良好表現 。

-認識到隱藏層越多, 您就可以捕獲到更復雜的結構。

-構建所有的幫助器函數以實現一個隱藏層的完整模型。

 

這個任務為即將到來的任務做好准備。用你的時間完成它, 通過不同的練習，並確保你得到預期的結果 。在某些代碼塊中, 您將找到一個 "#GRADED FUNCTION: functionName" 的注釋，請不要修改它。完成后, 提交您的工作, 並檢查您的結果。你需要得分70% 才能過關。祝你好運:)!

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Planar data classification with one hidden layer

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.

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 propagation

【中文翻譯】

 用有一個人 隱藏層的神經網絡進行平面數據分類

歡迎您的第3周的編程作業。是時候建立你的第一個神經網絡了, 這將有一個隱藏層。你會看到這個模型和你使用邏輯回歸實現的一個很大的不同。

您將學習如何:

實現一個具有單個隱藏層的2分類神經網絡

使用具有非線性的激活函數, 如 tanh

計算交叉熵損失

實現向前和向后傳播

--------------------------------------------------------------------------------------------------------------------------

1 - Packages

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

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

【中文翻譯】

讓我們首先導入您在此作業期間需要的所有軟件包。

　　numpy 是 用Python 進行科學計算的基本包。

　　sklearn 為數據挖掘和數據分析提供了簡單高效的工具。

 　　matplotlib 是一個用於在 Python 中繪制圖形的庫。

　　testCases提供了一些測試示例來評估函數的正確性

　　planar_utils 提供了在這個作業中使用的各種有用的函數

【code】

# 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

 

--------------------------------------------------------------------------------------------------------------------------

2 - Dataset

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.

【code】

X, Y = load_planar_dataset()

 

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.

【中文翻譯】

使用 matplotlib 可視化數據集。數據看起來像一個 "花" ，由一些紅色 (標簽 y=0) 和一些藍色 (y=1) 點組成。你的目標是建立一個模型來分類這些數據。

【code】

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

 

【result】

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).

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?

Hint: How do you get the shape of a numpy array? (help)

【code】

### START CODE HERE ### (≈ 3 lines of code)
shape_X = X.shape
shape_Y = Y.shape
m = X.shape[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))

 

【result】

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

 

Expected Output:

shape of X	(2, 400)
shape of Y	(1, 400)
m	400

--------------------------------------------------------------------------------------------------------------------------

3 - Simple Logistic Regression

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.

【code】

# 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.

【code】

# 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)")

 

【result】

Accuracy of logistic regression: 47 % (percentage of correctly labelled datapoints)

Expected Output:

Accuracy

47%

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!

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4 - Neural Network model

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:

For one example x(i)x(i):

Given the predictions on all the examples, you can also compute the cost J as follows:

 

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)

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.

4.1 - Defining the neural network structure

Exercise: Define three variables:

- n_x: the size of the input layer
- n_h: the size of the hidden layer (set this to 4) 
- n_y: the size of the output layer

Hint: Use shapes of X and Y to find n_x and n_y. Also, hard code the hidden layer size to be 4.

【code】

# 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)

 

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))

 

【result】

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

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).

n_x	5
n_h	4
n_y	2

 

 

4.2 - Initialize the model's parameters

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).
• You will initialize the bias vectors as zeros.
  • Use: np.zeros((a,b)) to initialize a matrix of shape (a,b) with zeros.

【code】

# 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))               #注意zeros()函數初始化的寫法
    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

 

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("b2 = " + str(parameters["b2"]))

 

【result】

W1 = [[-0.00416758 -0.00056267]
 [-0.02136196  0.01640271]
 [-0.01793436 -0.00841747]
 [ 0.00502881 -0.01245288]]
b1 = [[ 0.]
 [ 0.]
 [ 0.]
 [ 0.]]
W2 = [[-0.01057952 -0.00909008  0.00551454  0.02292208]]
b2 = [[ 0.]]

Expected Output:

W1	[[-0.00416758 -0.00056267] [-0.02136196 0.01640271] [-0.01793436 -0.00841747] [ 0.00502881 -0.01245288]]
b1	[[ 0.] [ 0.] [ 0.] [ 0.]]
W2	[[-0.01057952 -0.00909008 0.00551454 0.02292208]]
b2	[[ 0.]]

4.3 - The Loop

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.

【中文翻譯】

問題: 實現 forward_propagation ().

說明:

　　看看你的分類器的數學表達示。

　　您可以使用函數 sigmoid()。它內置在筆記本中 (導入)。

　　您可以使用函數 tanh ()。它是 numpy 庫的一部分。

　　您必須實現的步驟包括:

　　　　使用參數 [".."], 從字典 "參數" (即 initialize_parameters () 的輸出) 中檢索每個參數。

　　　　實現向前傳播。計算 Z^[1],A^[1],Z^[2]和 A^[2] (在訓練集上，所有預測值的向量)。

　　反向傳播中所需的值存儲在 "cache 中。cache將作為對反向函數的輸入。

【code】

# 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

# 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

 

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 matches ours. 
print(np.mean(cache['Z1']) ,np.mean(cache['A1']),np.mean(cache['Z2']),np.mean(cache['A2']))

 

【result】

0.262818640198 0.091999045227 -1.30766601287 0.212877681719

Expected Output:

0.262818640198 0.091999045227 -1.30766601287 0.212877681719

 

Now that you have computed A^[2] (in the Python variable "A2"), which contains a^[2](i)for every example, you can compute the cost function as follows:

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

Instructions:

• There are many ways to implement the cross-entropy loss(損失熵). To help you, we give you how we would have implemented 

  logprobs = np.multiply(np.log(A2),Y)
  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()).

【code】

# 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 =  - 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))   #cost的類型是不是float
    
    return cost

 

A2, Y_assess, parameters = compute_cost_test_case()

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

 

Expected Output:

cost	0.693058761...

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.

【中文翻譯】

問題: 實現函數 backward_propagation ()。

說明: 反向傳播通常是深學習中最難 (數學) 的部分。為了幫助你, 這里再次給出反向傳播的幻燈片。因為您正在構建一個量化的實現, 所以您需要使用本幻燈片右側的六公式。

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−a². So you can compute g^[1]′(Z^[1]) using (1 - np.power(A1, 2)).

【code】

# 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(W1,b1,W2,b2)
    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(檢索) 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(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"]))

 

【result】

 

dW1 = [[ 0.00301023 -0.00747267]
 [ 0.00257968 -0.00641288]
 [-0.00156892  0.003893  ]
 [-0.00652037  0.01618243]]
db1 = [[ 0.00176201]
 [ 0.00150995]
 [-0.00091736]
 [-0.00381422]]
dW2 = [[ 0.00078841  0.01765429 -0.00084166 -0.01022527]]
db2 = [[-0.16655712]]

 

Expected output:

dW1	[[ 0.00301023 -0.00747267] [ 0.00257968 -0.00641288] [-0.00156892 0.003893 ] [-0.00652037 0.01618243]]
db1	[[ 0.00176201] [ 0.00150995] [-0.00091736] [-0.00381422]]
dW2	[[ 0.00078841 0.01765429 -0.00084166 -0.01022527]]
db2	[[-0.16655712]]

 

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: θ=θ−α∂J∂θθ=θ−α∂J∂θ where αα is the learning rate and θθ represents a parameter.

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

【code】

# 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"]))

 

【result】

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]]

Expected Output:

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]]

 

4.4 - Integrate parts 4.1, 4.2 and 4.3 in nn_model()

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

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

【code】

# 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_x, n_h, n_y) = layer_sizes(X, Y)  
    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 = np.random.randn(n_h,n_x)*0.01
    b1 = np.zeros((n_h,1))               #注意zeros()函數初始化的寫法
    W2 = np.random.randn(n_y,n_h)*0.01
    b2 = np.zeros((n_y,1))
    ### 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

 

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"]))

 

【result】

Cost after iteration 0: 0.692739
Cost after iteration 1000: 0.000218
Cost after iteration 2000: 0.000107
Cost after iteration 3000: 0.000071
Cost after iteration 4000: 0.000053
Cost after iteration 5000: 0.000042
Cost after iteration 6000: 0.000035
Cost after iteration 7000: 0.000030
Cost after iteration 8000: 0.000026
Cost after iteration 9000: 0.000023
W1 = [[-0.65848169  1.21866811]
 [-0.76204273  1.39377573]
 [ 0.5792005  -1.10397703]
 [ 0.76773391 -1.41477129]]
b1 = [[ 0.287592  ]
 [ 0.3511264 ]
 [-0.2431246 ]
 [-0.35772805]]
W2 = [[-2.45566237 -3.27042274  2.00784958  3.36773273]]
b2 = [[ 0.20459656]]

 

Expected Output:

cost after iteration 0	0.692739
⋮⋮	⋮⋮
W1	[[-0.65848169 1.21866811] [-0.76204273 1.39377573] [ 0.5792005 -1.10397703] [ 0.76773391 -1.41477129]]
b1	[[ 0.287592 ] [ 0.3511264 ] [-0.2431246 ] [-0.35772805]]
W2	[[-2.45566237 -3.27042274 2.00784958 3.36773273]]
b2	[[ 0.20459656]]

4.5 Predictions

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

Reminder:

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)

【code】

# 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)     #  cache -- a dictionary containing "Z1", "A1", "Z2" and "A2"
    predictions =(A2>0.5)
    ### END CODE HERE ###
    
    return predictions

 

parameters, X_assess = predict_test_case()

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

 

【result】

predictions mean = 0.666666666667

Expected Output:

predictions mean

0.666666666667

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 nhnh hidden units.

【code】

# 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))

 

【result】

Cost after iteration 0: 0.693048
Cost after iteration 1000: 0.288083
Cost after iteration 2000: 0.254385
Cost after iteration 3000: 0.233864
Cost after iteration 4000: 0.226792
Cost after iteration 5000: 0.222644
Cost after iteration 6000: 0.219731
Cost after iteration 7000: 0.217504
Cost after iteration 8000: 0.219454
Cost after iteration 9000: 0.218607

 

<matplotlib.text.Text at 0x7fbd5a54fdd8>

Expected Output:

Cost after iteration 9000

0.218607

【code】

# 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) + '%')  #把predictions和Y相同的找出來

 

【result】

Accuracy: 90%

Expected Output:

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.

【中文翻譯】

與邏輯回歸相比, 精確度確實很高。模型學會了花的葉子樣式!神經網絡可以學習高度非線性的決策邊界, 而不像邏輯回歸。

現在, 讓我們嘗試幾個不同隱藏層隱含單元的神經網絡。

 

4.6 - Tuning hidden layer size (optional/ungraded exercise)

Run the following code. It may take 1-2 minutes. You will observe different behaviors of the model for various hidden layer sizes.

【code】

# 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))

 

【result】

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 20 hidden units: 90.0 %
Accuracy for 50 hidden units: 90.25 %

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, which lets you use very large models (such as n_h = 50) without much overfitting.

 

Optional questions:

Note: Remember to submit the assignment but clicking the blue "Submit Assignment" button at the upper-right.

Some optional/ungraded questions that you can explore if you wish:

• What happens when you change the tanh activation for a sigmoid activation or a ReLU activation?
• Play with the learning_rate. What happens?
• What if we change the dataset? (See part 5 below!

 

You've learnt to:

• Build a complete neural network with a hidden layer
• Make a good use of a non-linear unit
• Implemented forward propagation and backpropagation, and trained a neural network
• See the impact of varying the hidden layer size, including overfitting.

 

Nice work!

 

5) Performance on other datasets

 

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

【code】

# 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 = "blobs"
### END CODE HERE ###

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

# make blobs binary  #變成二進制數
if dataset == "blobs":
    Y = Y%2

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

 

Congrats on finishing this Programming Assignment!

Reference:

• http://scs.ryerson.ca/~aharley/neural-networks/
• http://cs231n.github.io/neural-networks-case-study/