单层神经网络

学习目标：

使用单隐藏层神经网络进行分类，使用非线性激活函数， 计算交叉熵，执行前向和反向传播

使用的库：

numpy, matplotlib.pyplot, sklearn, sklearn.linear_model

设置随机种子以保持结果一致性：

np.random.seed(1)

使用sklearn进行简单逻辑回归：

clf = sklearn.linear_model.LogisticRegressionCV()

clf.fit(X.T, Y.T) #原数据每一列代表一个样本，转换为每一列代表一个特征

LR_predictions = clf.predict(X.T)

Accuracy: float((np.dot(Y, LR_predictions) + np.dot(1-Y, 1- LR_predictions)) / float(Y.size)

需要定义的函数：

def layer_sizes(X, Y):

n_x = X.shape[0]

n_h = 4

n_y = Y.shape[0]

return (n_x, n_h, n_y)

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def initialize_parameters(n_x, n_h, n_y):

W1 = np.random.randn(n_h, n_x) * 0.01 #注意W的形状是（l, l-1)， 乘以一个非常小的数，但不能为0

b1 = np.zeros((n_h, 1))

W2 = np.random.randn(n_y, n_h) * 0.01

b2 = np.zeros((n_y, 1))

parameters = {'W1': W1, 'b1': b1, 'W2': W2, 'b2': b2}

return parameters

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def forward_propagation(X, parameters):

W1 = parameters['W1']

b1 = parameters['b1']

W2 = parameters['W2']

b2 = parameters['b2']

Z1 = np.dot(W1, X) + b1

A1 = np.tanh(Z1)

Z2 = np.dot(W2, A1) + b2

A2 = sigmoid(Z2)

cache = {'Z1': Z1, 'A1': A1, 'Z2': Z2, 'A2': A2} #反向传播中需要

return A2, cache

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def compute_cost(A2, Y, parameters):

m = Y.shape[1]

logprobs = np.multiply(Y, np.log(A2)) + np.multiply((1-Y), np.log(1-A2))

cost = - np.sum(logprobs) / m

cost = np.squeeze(cost)

return cost

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def backward_propagation(parameters, cache, X, Y):

m = X.shape[1]

W1 = parameters['W1']

W2 = parameters['W2']

A1 = cache['A1']

A2 = cache['A2']

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 - np.power(A1, 2)) #tanh

dW1 = np.dot(dZ1, X.T) / m

db1 = np.sum(dZ1, axis=1, keepdims=True) / m

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

return grads

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def update_parameters(parameters, grads, learning_rate = 1.2):

W1 = parameters['W1']

b1= parameters['b1']

W2 = parameters['W2']

b2 = parameters['b2']

dW1 = grads['dW1']

db1 = grads['db1']

dW2 = grads['dW2']

db2 = grads['db2']

W1 = W1 -learning_rate * dW1

b1 = b1 - learning_rate * db1

W2 = W2 - learning_rate * dW2

b2 = b2 -learning_rate * db2

parameters = {'W1': W1, 'b1': b1, 'W2': W2, 'b2': b2}

return parameters

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def predict(parameters, X):

A2, cache = forward_propagation(X, parameters)

prediction = (A2 > 0.5) #0 or 1

return predictions

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构建模型：

def nn_model(X, Y, n_h, num_iterations = 10000, print_cost=False):

np.random.seed(1)

n_x = layer_sizes(X, Y)[0]

n_y = layer_sizes(X, Y)[2]

parameters = initialize_parameters(n_x, n_h, n_y)

W1 = parameters['W1']

b1 = parameters['b1']

W2 = parameters['W2']

b2 = parameters['b2']

for i in range(0, num_interations):

A2, cache = forward_propagation(X, parameters)

cost = compute_cost(A2, Y, parameters)

grads = backward_propagation(parameters, cache, X, Y)

parameters = update_parameters(parameters, grads)

if print_cost and i % 100 == 0:

print('cost after interation %i: %f' %(i, cost))

return parameters

parameters = nn_model(X, Y, n_h =4, num_iterations = 1000, print_cost=True)

predictions = predict(parameters, X)

细调隐藏层规模：

#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)

prediction = predict(parameters, X)

accuracy = float((np.dot(Y, predictions.T) + np.dot(1-Y, 1-prediction.T)) / float(Y.size)*100)

print('Accuracy for {} hidden units: {} %'.format(n_h, accuracy))