吴恩达deep_learning_week3_BP神经网络
标签： 机器学习深度学习
这是吴恩达深度学习里的第二次作业
实现BP神经网络
- 首先先导入包
#先导入包
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
import matplotlib.pyplot as plt
import pylab
- 然后我们设置一个随机数种子备用，再导入数据，然后可视化一下（PS：这个不知道什么原因，显示不出来，虽然原来的代码有问题会报错，不过我看了库中函数的运行方式，将Y修改后不会报错了，但是还是显示不出，就很迷，不过对后面的结果没影响）
np.random.seed(1) # set a seed so that the results are consistent
#现在开始导入数据，一个X一个Y
X, Y = load_planar_dataset()
print (np.shape(X))
print (np.shape(Y))
#可知，这个X为一个包含400个具有两个参数样本的矩阵，Y为其标签
#将数据可视化一下，看起来像一朵红花(很奇怪，我这里已开始运行不了，然后改正后不报错了，不过还是显示不出图片)
plt.scatter(X[0, :], X[1, :], c=Y.reshape(X[0,:].shape), s=40, cmap=plt.cm.Spectral);
小红花
- 现在，我们可以来处理这些数据了，比如，先看看这些矩阵的大小
### 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))
- 然后呢，老师贴心地给了训练好了的单个logistic分类器，我们可以试试
# Train the logistic regression classifier(这里先用写好的分类器训练一下)
clf = sklearn.linear_model.LogisticRegressionCV();
clf.fit(X.T, Y.T);
- 但是呢，你会发现，这个的分类效果并不好，准确率只有百分之四十九
# Plot the decision boundary for logistic regression
plot_decision_boundary(lambda x: clf.predict(x), X, Y)
plt.title("Logistic Regression")
# Print accuracy，现在这里画出这些点的分类边界(好吧这里有点小问题，还是上面那个画图的问题，)
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)")
#可以看出，这个线性分类器效果并不好，就算是训练集，准确率也很低所以这里就要用bp网络了
#这是输出
Accuracy of logistic regression: 47 % (percentage of correctly labelled datapoints)
现在开始真正的弄bp神经网络的各层啦
- 首先，我们设计一个定义网络结构的函数
#首先定义各层网络的结构
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)
返回的是X输入的参数个数，隐藏层的神经元个数，输出层神经元个数，现在我们来输出网络看一看
#现在输出来看看我们的网络结构
X_assess, Y_assess = layer_sizes_test_case() #这里是将数据导入
(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))
输出是这样(就不截图了。。。)
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
2 . 现在来初始化（函数里面本身就写了随机的种子，这里为2）
#接下来是初始化函数
# 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)) #记着这里是两个括号
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
然后现在来看看初始化的样子
代码长这样：
#现在可以看看初始化结构咋样
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"]))
输出是
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.]]
-
前期准备算是做的差不多啦，现在开始正向传播啦
先上公式：
前向传播
大家注意看23到26行，那里是正向传播的精髓之处
#OK，现在开始正向传播啦
# 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
好了，正向传播完毕后，我们来看看结果，结果输出的代码如下：
#现在来看看正向传播的结果
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']))
然后，结果如下：
-0.000499755777742 -0.000496963353232 0.000438187450959 0.500109546852
- 到了这里，我们可以来看看cost函数了，
先看看公式：
$$J=-\frac1m\sum_{i=0}m(y{(i)}log(a{[2](i)})+(1-y_{(i)})log(1-a{2}))$$
下面是他的代码（重点关注28,29行）
#现在来算算代价函数J
#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
# Retrieve W1 and W2 from parameters
### START CODE HERE ### (≈ 2 lines of code)
W1 = parameters['W1']
W2 = parameters['W2']
### END CODE HERE ###
# 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
我在完成这个的时候通过这个发现了一个前面犯的一个错误，程序一直警告logprobs和cost这两个参数运算的时候不可用，于是我怀疑是不是有小于零的参数存在式子中，最后发现，A2在前面的计算中我已开始用的tanh函数，和A1混啦，其实A2要用sigmoid函数，A1是tanh函数
OK，我们已经完成了cost函数的代码，我么来看看效果
下面是输出代码
#现在来检测下代价函数的计算
A2, Y_assess, parameters = compute_cost_test_case()
print("cost = " + str(compute_cost(A2, Y_assess, parameters)))
输出是这样
cost = 0.692919893776
- 难点来了，反向传播。(重点是32-37行，这是精髓)
#难点来了，反向传播
# 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.dot(W2.T , dZ2) * (1 - 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
对于此，我们完成了反向传播，
现在来看看输出代码
#反向传播完毕，我们来看看
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"]))
输出是这样的
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]]
- 好了，反向传播页做完了，现在开始更新参数矩阵吧！！！
#反向传播完毕后，开始更新参数
# 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
老规矩，跑一跑(代码如下)
#现在验证一下反向传播算法的参数更新
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"]))
结果长这样：
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]]
ODK，各种功能函数终于可以说是写的差不多啦，现在来整合一个model函数吧
#好了，所有的函数都写完了，现在来整合一下，组成一个分类器模型
# 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)
import pdb
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
接着，调用一下这个函数，看看效果，还不是美滋滋(调用代码如下)
#现在我们来跑一跑这个模型
X_assess, Y_assess = nn_model_test_case()
parameters = nn_model(X_assess, Y_assess, 4, num_iterations=10000, print_cost=False)
print("W1 = " + str(parameters['W1']))
print("b1 = " + str(parameters['b1']))
print("W2 = " + str(parameters['W2']))
print("b2 = " + str(parameters['b2']))
上面设置的隐藏层是4个，迭代次数是10000
输出以下结果：
W1 = [[-4.18493855 5.33220875]
[-7.52989335 1.24306212]
[-4.19297397 5.32630669]
[ 7.52983568 -1.24309519]]
b1 = [[ 2.32926551]
[ 3.79459096]
[ 2.33002105]
[-3.79469147]]
W2 = [[-6033.83673001 -6008.12981298 -6033.1009627 6008.06639333]]
b2 = [[-52.66607002]]
最后，跑一下预测函数
#现在来写一写预测函数
# 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.array([0 if i <= 0.5 else 1 for i in np.squeeze(A2)])
### END CODE HERE ###
return predictions
看看预测值吧
#来看看预测值
parameters, X_assess = predict_test_case()
predictions = predict(parameters, X_assess)
print("predictions mean = " + str(np.mean(predictions)))
结果长这样：
predictions mean = 0.666666666667
最后，没有对比就没有伤害，，，来看看用这个模型训练的效果之前用单个logistic回归训练的效果的比较吧
#现在用刚才那个用单个逻辑回归只有百分之47的例子来训练
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))
pylab.show()
predictions=predict(parameters,X)
print ('Accuracy: %d' % float((np.dot(Y,predictions.T) + np.dot(1-Y,1-predictions.T))/float(Y.size)*100) + '%')
#可以看到，预测准确率达到了百分之89，
果然美滋滋，准确度达到了百分之九十，秒啊。上图
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: 0.219501
Cost after iteration 9000: 0.218620
Accuracy: 90%
训练后图片
另外，好像当隐藏层有5个神经元的时候，效果最好噢，大家可以自己试试的。
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