Planar data classification with one hidden layer

作业简介

本次作业将实现含有一个隐藏层的神经网络，你将会体验到与之前logistic实现的不同：

• 使用含有一个隐藏层的神经网络实现2分类。
• 使用一个非线性的激活函数(比如tanh)。
• 计算交叉熵损失。
• 实现前向传播和反向传播。

工具包

sklearn包：提供简单有效的数据挖掘和数据分析。

# 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

数据集

加载数据的方式：

X, Y = load_planar_dataset()

使用matplotlib可以将数据可视化：

数据集类似一朵花，有红色(label y = 0)和蓝色(label y = 1)的点构成，我们的目的就是去拟合这个数据。

获取数据的维度：

shape_X = X.shape shape_Y = Y.shape m = shape_X[1]  # training set size

测试：

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

简单的逻辑回归
在建立全连接之前，我们首先来看一下逻辑回归对于该问题的表现，可以使用sklearn的内建函数来实现：

# Train the logistic regression classifier
clf = sklearn.linear_model.LogisticRegressionCV() clf.fit(X.T, Y.T.ravel()) # 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)") plt.show()

输出:

Accuracy of logistic regression: 47 % (percentage of correctly labelled datapoints)

神经网络模型

由上面可以看出logistic模型对于解决“flower dataset”效果并不好，这里我们创建一个隐藏层的神经网络，下图是我们使用的网络模型：

数学表达式：

对于输入${x^{[i]}}$:

对于所有样本，损失函数J的计算：

构建神经网络的基本步骤：

1. 定义神经网络的结构(输入单元，隐藏单元等)
2. 初始化模型参数
3. 循环
   • 执行前向传播
   • 计算损失
   • 执行反向传播
   • 更新参数(梯度下降)

通常我们将1-3步骤创建为功能函数，然后再将它们合并为一个函数，我们称为nn_model(),创建了nn_model()函数之后我们就可以进行预测。

定义神经网络结构

定义三个变量：

n_x：输入层单元数目

n_h：隐藏层单元数目

n_y：输出层单元数目

# GRADED FUNCTION: layer_sizes
def layer_sizes(X, Y): n_x = X.shape[0] # size of input layer
    n_h = 4 n_y = Y.shape[0] # size of output layer
    return (n_x, n_h, n_y)

测试：

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))

输出：

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

初始化模型参数

随机初始化权重参数，偏置参数初始化为0：

# GRADED FUNCTION: initialize_parameters
def initialize_parameters(n_x, n_h, n_y): 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

测试：

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：

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.]]

循环

前向传播的计算，计算过程中需要进行缓存，缓存会作为方向传播的输入：

# GRADED FUNCTION: forward_propagation
def forward_propagation(X, 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

测试：

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']))

输出：

-0.000499755777742 -0.000496963353232 0.000438187450959 0.500109546852

交叉熵J计算：

# GRADED FUNCTION: compute_cost
def compute_cost(A2, Y, parameters): m = Y.shape[1] # number of example
    # Compute the cross-entropy cost
    logprobs = np.multiply(np.log(A2), Y) + np.multiply(np.log(1-A2), (1-Y)) 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

在这里的计算使用np.multiply() 然后np.sum() 或者直接使用 np.dot()都可以。

测试：

A2, Y_assess, parameters = compute_cost_test_case() print("cost = " + str(compute_cost(A2, Y_assess, parameters)))

输出：

cost = 0.692919893776

 反向传播：

# GRADED FUNCTION: backward_propagation
def backward_propagation(parameters, cache, X, Y): 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    # (n_y,1)
    dW2 = 1 / m * np.dot(dZ2, A1.T)   # (n_y, 1) .* (1, n_h)
    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

测试：

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：

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]]

参数更新：

# GRADED FUNCTION: update_parameters
def update_parameters(parameters, grads, learning_rate = 1.2): # 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

测试：

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"]))

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]]

组合成nn_model()

# GRADED FUNCTION: nn_model
def nn_model(X, Y, n_h, num_iterations = 10000, print_cost=False): 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".
    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, learning_rate = 1.2) # Print the cost every 1000 iterations
        if print_cost and i % 1000 == 0: print ("Cost after iteration %i: %f" %(i, cost)) return parameters

测试：

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"]))

输出：

W1 = [[-4.18496424  5.33205335] [-7.53806791  1.20753912] [-4.19265639  5.32636689] [ 7.53803581 -1.20755573]] b1 = [[ 2.32936674] [ 3.80995932] [ 2.33014798] [-3.81000889]] W2 = [[-6033.82337574 -6008.14278716 -6033.08759595  6008.07913792]] b2 = [[-52.67940757]]

推测

# GRADED FUNCTION: predict
def predict(parameters, X): # Computes probabilities using forward propagation, and classifies to 0/1 using 0.5 as the threshold.
    A2, cache = forward_propagation(X, parameters) predictions = (A2 > 0.5) return predictions

测试：

parameters, X_assess = predict_test_case() predictions = predict(parameters, X_assess) print("predictions mean = " + str(np.mean(predictions)))

输出：

predictions mean = 0.666666666667

执行模型

# 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)) plt.show()

结果：

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: 0.219507 Cost after iteration 9000: 0.218621

输出准确性：

# 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) + '%')

结果：

Accuracy: 90%

调整隐藏层大小

# 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)) plt.show()

结果：

其它数据集上的表现

# 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} dataset = "noisy_circles" X, Y = datasets[dataset] X, Y = X.T, Y.reshape(1, Y.shape[0]) # make blobs binary
if dataset == "noisy_moons": Y = Y%2
# Visualize the data
plt.scatter(X[0, :], X[1, :], c=Y, s=40, cmap=plt.cm.Spectral) plt.show()

结果：