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.

目標：建立只有一個隱含層的平面二分類器

1 將會學習使用：

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 軟體包準備

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. 資料探勘
matplotlib is a library for plotting graphs in Python.
testCases provides some test examples to assess（評估） 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的專有功能，魔法函式，可以使影象直接在upyter notebook中顯示出來

np.random.seed(1)   # set a seed so that the results are consistent
                   # 設定一個種子，這樣每次隨機產生的數是不變的

3 資料準備

1、載入資料到變數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()    #載入資料到變數X和Y中。

2、資料視覺化，紅色代表 label y=0 藍色代表 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、顯示結果

4、可知X是我們的樣本集，Y是樣本集對應的標籤。

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、檢視我們的變數X和Y的具體內容，檢視他們的維度，確定樣本集的情況。

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?

#檢視變數的情況，確定樣本的個數以及維度
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)) # 注意這裡的輸出方式，%（m）必須這樣寫，和C語言中直接一個逗號跟m不一樣。

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

# 從X的維度（2,400）我們還不難看出每一個樣本的輸入是兩個特徵
# 符合我們的全部樣本的前向後向矩陣要求：輸入樣本集每一列是一個樣本，所有樣本橫向疊加構成樣本集。最後的真實標籤Y是一個行向量。

4 構建神經網路前檢視簡單的logistic迴歸效果

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 構建神經網路

總述

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：數學化的計算過程

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

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

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

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

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

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

給出所有示例的預測結果，可以按如下方式計算成本J：

通常構建一個神經網路的方法：

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)

構建三個函式分別實現上面的三個功能。

構建一個模組函式nn_model()整合上面的三個函式來實現上述的全部功能。

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 定義神經網路結構

Exercise: Define three variables:

- n_x: the size of the input layer                    輸入層節點的數量
- n_h: the size of the hidden layer (set this to 4)   隱含層節點的數量
- n_y: the size of the output layer                   輸出層節點的數量

Hint: Use shapes of X and Y to find n_x and n_y. Also, hard code the hidden layer size to be 4. 其實從這句話中的一個詞 hard code（硬編碼）我們就要立即明白，隱含層的節點數是我們人為設定的，跟輸入和輸出是沒有關係的。

第一個函式：定義網路結構

#GRADED FUNCTION: layer_sizes  分函式 層的定義
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)

#測試函式 其中layer_sizes_test_case() 是我們的測試資料，並不是真實資料
#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 初始化模型引數

第二個函式：初始化引數（權重和偏置）

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). 初始化權重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 迴圈

第一個函式前向傳播計算

前向傳播

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 函式是自帶的函式 
    ### EDG CODE HERE ###
    
    assert(A2.shape == (1, X.shape[1]))  # 確保維度正確
    
    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

第二個函式計算cost

計算損失

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
這裡給出了一個如下數學式子的python程式碼：

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

logprobs = np.multiply(np.log(A2),Y)     一次性計算輸出層所有樣本的ylog(y^)，A2 和 Y都是（1，m）的行向量不要忘記
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
                            # 這裡的squeeze的作用就是保證sum後的結果是一個數而不是一個矩陣
    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

#再次提醒我們：我們的cost函式是定為所有樣本損失的值，是一個數

第三個函式反向傳播得到dw db

反向傳播調整引數

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)).

這個函式我們不是直接調參，而是僅僅得到dw db 並沒有執行w - dw

重點：按照公式一步一步來

# 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]]

第四個函式：更新引數

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:更新後的引數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 整合成一個函式

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

進行一萬次迴圈調整引數。

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 預測

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() 四捨五入 剛好符合我們的分類方法
    ### 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 正式帶入待測樣本進行預測

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) + '%')
#上面這個計算準確率的方法很巧妙啊，因為我們找的不僅僅是1和1 還有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 嘗試改變隱含層的大小（還是一層，只是節點數改變）

# 嘗試改變隱含層的大小（節點數）
# 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):   # 列舉 表示第 i 個元素 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 %

可以發現在節點數n_h =5 的時候準確率最大。節點不是越多越好。

關於隱含層節點的知識:

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

6 使用其他資料集測試

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

# 用其他資料集測試我們的網路

# 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);

第三週課後程式設計作業

Planar data classification with one hidden layer

目標：建立只有一個隱含層的平面二分類器

1 將會學習使用：

2 軟體包準備

3 資料準備

1、載入資料到變數X和Y中

2、資料視覺化，紅色代表 label y=0 藍色代表 label y=1

3、顯示結果

4、可知X是我們的樣本集，Y是樣本集對應的標籤。

5、檢視我們的變數X和Y的具體內容，檢視他們的維度，確定樣本集的情況。

4 構建神經網路前檢視簡單的logistic迴歸效果

5 構建神經網路

總述

5.1 定義神經網路結構

5.2 初始化模型引數

5.3 迴圈

5.4 整合成一個函式

5.5 預測

5.6 正式帶入待測樣本進行預測

5.7 嘗試改變隱含層的大小（還是一層，只是節點數改變）

6 使用其他資料集測試

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