《神经网络和深度学习》之神经网络基础（第二周）课后作业——一个隐藏层的平面数据分类

欢迎来到第三周的课程，在这一周的任务里，你将建立一个只有一个隐含层的神经网络。相比于之前你实现的逻辑回归有很大的不同。

你将会学习一下内容：

用一个隐含层的神经网络实现一个二分类。

利用非线性的激活函数单元。

计算交叉熵损失函数。

实现向前传播和向后传播。

1 函数包

# 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

np.random.seed(1) # set a seed so that the results are consistent

2 数据

X, Y = load_planar_dataset()

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

能将得到

1. 一个包含（x1,x2）的特征矩阵。

2. 一个包含（0,1）的特征向量。

练习：你有多少训练集，他们的大小是多少？

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

m = shape_X[1]  # training set size
### END CODE HERE ###

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

输出：

The shape of X is: (2L, 400L)

The shape of Y is: (1L, 400L)

I have m = 400 training examples!

3 简单的逻辑回归

在进行今天的作业之前，先看一下，逻辑回归在这个问题上的表现。

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

# 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)")

输出：



说明：因为数据集是非线性可分的，所以，在这个数据集上表现较差。

4 神经网络模型

回忆：通常神经网络建立的方法。

定义神经网络的结构（输入层，输出层，隐含层个数）。

初始化模型参数。

循环：

—实现向前传播。

—计算损失函数。

—为了得到梯度值，实现向后传播。

—更新参数（梯度下降）

4.1 定义神经网络结构

练习：定义三个结构变量

# 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)

4.2 初始化模型参数

练习：实现initialize_parameters()函数功能

说明：

用 np.random.randn(a,b) * 0.01随机的初始化权重矩阵

用np.zeros((a,b))初始化偏置向量

# 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))
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

输出：

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循环

问题：实现 forward_propagation().

从字典“parameters”中检索每个参数。

实现向前传播。计算Z1，A1，Z2，A2(这是所有你对训练集的所有例子的预测的向量)。

反向传播所需的值存储在“cache”中。cache将作为反向传播函数的一个输入。

# 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

输出：

(-0.00049975577774199022, -0.00049696335323177901, 0.00043818745095914653, 0.50010954685243103)

计算出A2后，你将计算损失函数



练习：实现 compute_cost()，计算损失函数

# GRADED FUNCTION: compute_cost

def compute_cost(A2, Y, parameters):
"""
Computes the cross-entropy cost given in equation (13)

Arguments:
A2 -- The sigmoid output of the second activation, of shape (1, number of examples)
Y -- "true" labels vector of shape (1, number of examples)
parameters -- python dictionary containing your parameters W1, b1, W2 and b2

Returns:
cost -- cross-entropy cost given equation (13)
"""

m = Y.shape[1] # number of example

# Compute the cross-entropy cost
### START CODE HERE ### (≈ 2 lines of code)
logprobs = np.multiply(np.log(A2),Y) + np.multiply(np.log(1 - A2),1 - Y)
cost = - np.sum(logprobs) / m
### END CODE HERE ###

cost = np.squeeze(cost)     # makes sure cost is the dimension we expect.
# E.g., turns [[17]] into 17
assert(isinstance(cost, float))

return cost

输出：cost = 0.692919893776

问题：实现反向传播函数 backward_propagation()



其中， tanh激活函数的导数为

# GRADED FUNCTION: backward_propagation

def backward_propagation(parameters, cache, X, Y):
"""
Implement the backward propagation using the instructions above.

Arguments:
parameters -- python dictionary containing our parameters
cache -- a dictionary containing "Z1", "A1", "Z2" and "A2".
X -- input data of shape (2, number of examples)
Y -- "true" labels vector of shape (1, number of examples)

Returns:
grads -- python dictionary containing your gradients with respect to different parameters
"""
m = X.shape[1]

# First, retrieve W1 and W2 from the dictionary "parameters".
### START CODE HERE ### (≈ 2 lines of code)
W1 = parameters["W1"]
W2 = parameters["W2"]
### END CODE HERE ###

# Retrieve also A1 and A2 from dictionary "cache".
### START CODE HERE ### (≈ 2 lines of code)
A1 = cache["A1"]
A2 = cache["A2"]
### END CODE HERE ###

# Backward propagation: calculate dW1, db1, dW2, db2.
### START CODE HERE ### (≈ 6 lines of code, corresponding to 6 equations on slide above)
dZ2 = A2 - Y
dW2 =  np.dot(dZ2,A1.T)/m
db2 = np.sum(dZ2,axis=1,keepdims=True)/m
dZ1 = np.multiply(np.dot(W2.T,dZ2), (1 - np.power(A1, 2)))
dW1 = np.dot(dZ1,X.T)/m
db1 =  np.sum(dZ1,axis=1,keepdims=True)/m
### END CODE HERE ###

grads = {"dW1": dW1,
"db1": db1,
"dW2": dW2,
"db2": db2}

return grads

输出：

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]]

问题：利用地图下降，实现更新法则。你可以利用 (dW1, db1, dW2, db2) 去更新 (W1, b1, W2, b2).

通常的梯度下降准则：


说明：梯度下降和学习速率关系很大。

# 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

输出：

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 综合前面三部分 nn_model()

问题：建立神经学习网络

# GRADED FUNCTION: nn_model

def nn_model(X, Y, n_h, num_iterations = 10000, print_cost=False):
"""
Arguments:
X -- dataset of shape (2, number of examples)
Y -- labels of shape (1, number of examples)
n_h -- size of the hidden layer
num_iterations -- Number of iterations in gradient descent loop
print_cost -- if True, print the cost every 1000 iterations

Returns:
parameters -- parameters learnt by the model. They can then be used to predict.
"""

np.random.seed(3)
n_x = layer_sizes(X, Y)[0]
n_y = layer_sizes(X, Y)[2]

# Initialize parameters, then retrieve W1, b1, W2, b2. Inputs: "n_x, n_h, n_y". Outputs = "W1, b1, W2, b2, parameters".
### START CODE HERE ### (≈ 5 lines of code)
parameters = initialize_parameters(n_x, n_h, n_y)
W1 = parameters["W1"]
b1 = parameters["b1"]
W2 = parameters["W2"]
b2 = parameters["b2"]
### END CODE HERE ###

# Loop (gradient descent)

for i in range(0, num_iterations):

### START CODE HERE ### (≈ 4 lines of code)
# Forward propagation. Inputs: "X, parameters". Outputs: "A2, cache".
A2, cache = forward_propagation(X, parameters)

# Cost function. Inputs: "A2, Y, parameters". Outputs: "cost".
cost = compute_cost(A2, Y, parameters)

# Backpropagation. Inputs: "parameters, cache, X, Y". Outputs: "grads".
grads = backward_propagation(parameters, cache, X, Y)

# Gradient descent parameter update. Inputs: "parameters, grads". Outputs: "parameters".
parameters = update_parameters(parameters, grads ,)

### END CODE HERE ###

# Print the cost every 1000 iterations
if print_cost and i % 1000 == 0:
print ("Cost after iteration %i: %f" %(i, cost))

return parameters

输出：

W1 = [[-4.18494502 5.33220306]

[-7.52989352 1.24306198]

[-4.19295477 5.32631754]

[ 7.52983748 -1.24309404]]

b1 = [[ 2.32926814]

[ 3.79459053]

[ 2.3300254 ]

[-3.79468789]]

W2 = [[-6033.83672183 -6008.12981297 -6033.10095335 6008.0663689 ]]

b2 = [[-52.666077]]

4.5 预测

问题：通过建立函数 predict()进行预测。利用向前传播进行预测。

# GRADED FUNCTION: predict

def predict(parameters, X):
"""
Using the learned parameters, predicts a class for each example in X

Arguments:
parameters -- python dictionary containing your parameters
X -- input data of size (n_x, m)

Returns

predictions -- vector of predictions of our model (red: 0 / blue: 1)
"""

# Computes probabilities using forward propagation, and classifies to 0/1 using 0.5 as the threshold.
### START CODE HERE ### (≈ 2 lines of code)
A2, cache = forward_propagation(X, parameters)
predictions = np.round(A2)
### END CODE HERE ###

return predictions

输出：predictions mean = 0.666666666667

4.6预测原数据

# 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))

# 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) + '%')

输出：Accuracy: 90%

4.7 改变隐含层的大小

# This may take about 2 minutes to run

plt.figure(figsize=(16, 32))
hidden_layer_sizes = [1, 2, 3, 4, 5, 10, 20]
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))

说明：

较大的模型(包含更多的隐藏单元)能够更好地适应训练集，直到最终最大的模型超过了数据。

最好的隐藏层大小似乎是在nh=5附近。实际上，这里的价值似乎与数据吻合得很好，而不需要引起注意的过度拟合。

稍后您还将学习规范化，这使您可以使用非常大的模型(例如nh=50)，而不需要太多的过度使用。

5 在其他数据集上的表现

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

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

# 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))