基本上本文的代码全部来自github：浅层神经网络，由于吴恩达课程作业的代码依赖于各种包，而我没有注册coursera,所以用的数据集是sklearn的月牙数据。思路是一样的，通过调试这个代码，我学到的东西大概是神经网络中的维度真的很重要啊。。W维度的设置方法就看个人习惯，然后才影响了网络中是对W转置还是对X转置，一定要搞清楚。
数据集 sklearn 月牙数据集
def plot_decision_boundary(pred_func, X, y):
# Set min and max values and give it some padding
x_min, x_max = X[:, 0].min() - 0.5, X[:, 0].max() + 0.5
y_min, y_max = X[:, 1].min() - 0.5, X[:, 1].max() + 0.5
h = 0.01
# Generate a grid of points with distance h between them
xx, yy = np.meshgrid(np.arange(x_min, x_max, h), np.arange(y_min, y_max, h))
# Predict the function value for the whole gid
Z = pred_func(np.c_[xx.ravel(), yy.ravel()])
Z = Z.reshape(xx.shape)
# Plot the contour and training examples
plt.contourf(xx, yy, Z, cmap=plt.cm.Spectral)
plt.scatter(X[:, 0], X[:, 1], c=np.squeeze(y), cmap=plt.cm.Spectral)
plt.title("Logistic Regression")
plt.show()
def sigmoid(x):
return 1 / (1 + np.exp(-x))
看一下数据集的样子：
np.random.seed(0)
X, y = make_moons(200, noise=0.20)
y = y.reshape((200,1))
# print(y) #可以看到y的取值只有0,1两种
print(X.shape) #[200,2]
print(y.shape) #[200,1]
plt.scatter(X[:,0], X[:,1], s=40, c=y.ravel(), cmap=plt.cm.Spectral)
plt.show()
image.png
尝试一下logistic regression
clf = sklearn.linear_model.LogisticRegressionCV()
clf.fit(X,y)
plot_decision_boundary(lambda x: clf.predict(x), X, y)
plt.title("Logistic Regression")
LR_predictions = clf.predict(X)
#由于数据集中y的取值只有0和1两种，所以下面的np.dot()第一项计算的是预测为1标签为1的数量，np.dot()第二项计算的是预测为0标签为0的数量，加起来就是预测正确的总次数
print ('Accuracy of logistic regression: %d '%float((np.dot(y.T,LR_predictions.T) + np.dot(1-y.T,1-LR_predictions.T))/float(y.size)*100)+'% ' + "(percentage of correctly labelled datapoints)")
划分数据的结果如下：令人震惊的是这样准确率竟然能有85%，可是很明显这个数据不是线性可分的。
logistic regression
接下来交给神经网络
1.预置知识：
由吴恩达的深度学习的课，我们知道，W1的维度设置为（本层神经元的数量，特征维度），b1的维度设为（本层神经元的数量，1）
2.tips:
初始化W的时候用np.random.rand(),再乘以一个很小的数，生成一个比较小的高斯分布的随机数。
编码中嵌入assert代码，检测维度。
使用cache来保存每一次隐藏层计算后的输出值。
2.构建神经网络的一般方法是：
①定义网络结构（输入层，隐藏层单元数等）
②初始化模型的参数
③循环：前向传播，计算损失，反向求导，更新参数
①定义网络结构：
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
"""
n_x = X.shape[0] # size of input layer
#我们这里把隐藏层的单元数设置为4
n_h = 4
n_y = Y.shape[0] # size of output layer
return (n_x, n_h, n_y)
②初始化参数
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) # 尽管初始化是随机的，但是我们建立了一个种子，以便您的输出与我们的匹配
W1 = np.random.randn(n_h, n_x) * 0.01
# print(W1.shape)[4,2]
b1 = np.zeros((n_h, 1))
# print(b1.shape)[4,1]
W2 = np.random.randn(n_y, n_h) * 0.01
# print(W2.shape)[1,4]
b2 = np.zeros((n_y, 1))
# print(b2.shape)[1,1]
# 深度学习常见的bug就是维度异常
# 吴恩达的经验：编码中嵌入assert代码，检测维度
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
③前向传播
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"
W1 = parameters['W1']
b1 = parameters['b1']
W2 = parameters['W2']
b2 = parameters['b2']
# Implement Forward Propagation to calculate A2 (probabilities)
Z1 = np.dot(W1, X) + b1
A1 = np.tanh(Z1)
Z2 = np.dot(W2, A1) + b2
A2 = sigmoid(Z2)
assert (A2.shape == (1, X.shape[1]))
cache = {"Z1": Z1,
"A1": A1,
"Z2": Z2,
"A2": A2}
return A2, cache
反向传播：
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".
W1 = parameters['W1']
W2 = parameters['W2']
# Retrieve also A1 and A2 from dictionary "cache".
A1 = cache['A1']
A2 = cache['A2']
# Backward propagation: calculate dW1, db1, dW2, db2.
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)
grads = {"dW1": dW1,
"db1": db1,
"dW2": dW2,
"db2": db2}
return grads
计算损失
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
logprobs = np.multiply(np.log(A2), Y) + np.multiply((1 - Y), (np.log(1 - A2)))
cost = -1 / m * np.sum(logprobs)
cost = np.squeeze(cost) # makes sure cost is the dimension we expect.
# E.g., turns [[17]] into 17
assert (isinstance(cost, float))
return cost
更新参数
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"
W1 = parameters['W1']
b1 = parameters['b1']
W2 = parameters['W2']
b2 = parameters['b2']
# Retrieve each gradient from the dictionary "grads"
dW1 = grads["dW1"]
db1 = grads["db1"]
dW2 = grads["dW2"]
db2 = grads["db2"]
# Update rule for each parameter
W1 -= learning_rate * dW1
b1 -= learning_rate * db1
W2 -= learning_rate * dW2
b2 -= learning_rate * db2
parameters = {"W1": W1,
"b1": b1,
"W2": W2,
"b2": b2}
return parameters
接下来把这个网络组合起来：
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".
n_x, n_h, n_y = layer_sizes(X, Y)
parameters = initialize_parameters(n_x, n_h, n_y)
W1 = parameters['W1']
b1 = parameters['b1']
W2 = parameters['W2']
b2 = parameters['b2']
# Loop (gradient descent)
for i in range(0, num_iterations):
# 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)
# Print the cost every 1000 iterations
if print_cost and i % 1000 == 0:
print("Cost after iteration %i: %f" % (i, cost))
return parameters
预测：
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.
A2, cache = forward_propagation(X, parameters)
predictions = np.array([1 if x > 0.5 else 0 for x in A2.reshape(-1, 1)]).reshape(A2.shape) # 这一行代码的作用详见下面代码示例
return predictions
# Build a model with a n_h-dimensional hidden layer
parameters = nn_model(X.T, y.T, n_h = 4, num_iterations = 10000, print_cost=True)
最后画出训练的神经网络的分类结果：
# 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))
predictions = predict(parameters, X.T)
print ('Accuracy: %d' % float((np.dot(y.T,predictions.T) + np.dot(1-y.T,1-predictions.T))/float(y.size)*100) + '%')
此时的准确率为99%。
image.png
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