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 :) !

【中文翻译】

用有一个人隐藏层的神经网络进行平面数据分类

欢迎来到深度学习专业第二次编程练习。在本笔记本中, 你会产生红色和蓝色的点, 形成一朵花。然后, 您将找到一个神经网络, 用来正确分类这些点。您将尝试不同的层并查看结果。

通过完成此任务, 您将:

-形成一个反向传播的直觉, 并看到它在数据上的良好表现。

-认识到隐藏层越多, 您就可以捕获到更复杂的结构。

-构建所有的帮助器函数以实现一个隐藏层的完整模型。

这个任务为即将到来的任务做好准备。用你的时间完成它, 通过不同的练习，并确保你得到预期的结果。在某些代码块中, 您将找到一个 "#GRADED FUNCTION: functionName" 的注释，请不要修改它。完成后, 提交您的工作, 并检查您的结果。你需要得分70% 才能过关。祝你好运:)!

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

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

【中文翻译】

用有一个人隐藏层的神经网络进行平面数据分类

欢迎您的第3周的编程作业。是时候建立你的第一个神经网络了, 这将有一个隐藏层。你会看到这个模型和你使用逻辑回归实现的一个很大的不同。

您将学习如何:

实现一个具有单个隐藏层的2分类神经网络

使用具有非线性的激活函数, 如 tanh

计算交叉熵损失

实现向前和向后传播

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

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

【中文翻译】

让我们首先导入您在此作业期间需要的所有软件包。

　　numpy 是用Python 进行科学计算的基本包。

　　sklearn 为数据挖掘和数据分析提供了简单高效的工具。

　　matplotlib 是一个用于在 Python 中绘制图形的库。

　　testCases提供了一些测试示例来评估函数的正确性

　　planar_utils 提供了在这个作业中使用的各种有用的函数

【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.

【中文翻译】

使用 matplotlib 可视化数据集。数据看起来像一个 "花" ，由一些红色 (标签 y=0) 和一些蓝色 (y=1) 点组成。你的目标是建立一个模型来分类这些数据。

【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!

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

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()函数初始化的写法

    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.

【中文翻译】

问题: 实现 forward_propagation ().

说明:

　　看看你的分类器的数学表达示。

　　您可以使用函数 sigmoid()。它内置在笔记本中 (导入)。

　　您可以使用函数 tanh ()。它是 numpy 库的一部分。

　　您必须实现的步骤包括:

　　　　使用参数 [".."], 从字典 "参数" (即 initialize_parameters () 的输出) 中检索每个参数。

　　　　实现向前传播。计算 Z^[¹^],A^[¹^],Z^[2]和 A^[2] (在训练集上，所有预测值的向量)。

　　反向传播中所需的值存储在 "cache 中。cache将作为对反向函数的输入。

【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(损失熵). 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的类型是不是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.

【中文翻译】

问题: 实现函数 backward_propagation ()。

说明: 反向传播通常是深学习中最难 (数学) 的部分。为了帮助你, 这里再次给出反向传播的幻灯片。因为您正在构建一个量化的实现, 所以您需要使用本幻灯片右侧的六公式。

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(检索) 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(图): 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()函数初始化的写法

    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相同的找出来

【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.

【中文翻译】

与逻辑回归相比, 精确度确实很高。模型学会了花的叶子样式!神经网络可以学习高度非线性的决策边界, 而不像逻辑回归。

现在, 让我们尝试几个不同隐藏层隐含单元的神经网络。

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  #变成二进制数

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:

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Planar data classification with a hidden layer

Planar data classification with one hidden layer

1 - Packages

2 - Dataset

3 - Simple Logistic Regression

4 - Neural Network model

4.1 - Defining the neural network structure

4.2 - Initialize the model's parameters

4.3 - The Loop

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

4.5 Predictions

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

5) Performance on other datasets

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