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@Perfect-Demo 2018-05-01T08:41:51.000000Z 字数 12768 阅读 926

deep_learning_week3_BP神经网络

机器学习深度学习

代码已上传github:
https://github.com/PerfectDemoT/my_deeplearning_homework

这是吴恩达深度学习里的第二次作业

实现BP神经网络

  1. 首先先导入包
  1. #先导入包
  2. import numpy as np
  3. import matplotlib.pyplot as plt
  4. from testCases import *
  5. import sklearn
  6. import sklearn.datasets
  7. import sklearn.linear_model
  8. from planar_utils import plot_decision_boundary, sigmoid, load_planar_dataset, load_extra_datasets
  9. import matplotlib.pyplot as plt
  10. import pylab
  1. 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）
  1. np.random.seed(1) # set a seed so that the results are consistent
  3. #现在开始导入数据，一个X一个Y
  4. X, Y = load_planar_dataset()
  6. print (np.shape(X))
  7. print (np.shape(Y))
  8. #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
  10. #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)
  11. plt.scatter(X[0, :], X[1, :], c=Y.reshape(X[0,:].shape), s=40, cmap=plt.cm.Spectral);

小红花
3. 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

  1. ### START CODE HERE ### (≈ 3 lines of code)
  2. shape_X = X.shape
  3. shape_Y = Y.shape
  4. m = X.shape[1]  # training set size
  5. ### END CODE HERE ###
  7. print ('The shape of X is: ' + str(shape_X))
  8. print ('The shape of Y is: ' + str(shape_Y))
  9. print ('I have m = %d training examples!' % (m))
  1. 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
  1. # Train the logistic regression classifier(这里先用写好的分类器训练一下)
  2. clf = sklearn.linear_model.LogisticRegressionCV();
  3. clf.fit(X.T, Y.T);
  1. 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
  1. # Plot the decision boundary for logistic regression
  2. plot_decision_boundary(lambda x: clf.predict(x), X, Y)
  3. plt.title("Logistic Regression")
  5. # 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，)
  6. LR_predictions = clf.predict(X.T)
  7. print ('Accuracy of logistic regression: %d ' % float((np.dot(Y,LR_predictions) + np.dot(1-Y,1-LR_predictions))/float(Y.size)*100) +
  8.        '% ' + "(percentage of correctly labelled datapoints)")
  9. #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
  1. #这是输出
  2. 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

  1. 首先，我们设计一个定义网络结构的函数
  1. #首先定义各层网络的结构
  2. def layer_sizes(X, Y):
  3.     """
  4.     Arguments:
  5.     X -- input dataset of shape (input size, number of examples)
  6.     Y -- labels of shape (output size, number of examples)
  8.     Returns:
  9.     n_x -- the size of the input layer
  10.     n_h -- the size of the hidden layer
  11.     n_y -- the size of the output layer
  12.     """
  13.     ### START CODE HERE ### (≈ 3 lines of code)
  14.     n_x = X.shape[0]  # size of input layer
  15.     n_h = 4
  16.     n_y = Y.shape[0]  # size of output layer
  17.     ### END CODE HERE ###
  18.     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

  1. #现在输出来看看我们的网络结构
  2. X_assess, Y_assess = layer_sizes_test_case() #这里是将数据导入
  3. (n_x, n_h, n_y) = layer_sizes(X_assess, Y_assess)
  4. print("The size of the input layer is: n_x = " + str(n_x))
  5. print("The size of the hidden layer is: n_h = " + str(n_h))
  6. print("The size of the output layer is: n_y = " + str(n_y))

输出是这样(就不截图了。。。)

  1. The size of the input layer is: n_x = 5
  2. The size of the hidden layer is: n_h = 4
  3. 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）

  1. #接下来是初始化函数
  2. # GRADED FUNCTION: initialize_parameters
  3. def initialize_parameters(n_x, n_h, n_y):
  4.     """
  5.     Argument:
  6.     n_x -- size of the input layer
  7.     n_h -- size of the hidden layer
  8.     n_y -- size of the output layer
  10.     Returns:
  11.     params -- python dictionary containing your parameters:
  12.                     W1 -- weight matrix of shape (n_h, n_x)
  13.                     b1 -- bias vector of shape (n_h, 1)
  14.                     W2 -- weight matrix of shape (n_y, n_h)
  15.                     b2 -- bias vector of shape (n_y, 1)
  16.     """
  18.     np.random.seed(2)  # we set up a seed so that your output matches ours although the initialization is random.
  20.     ### START CODE HERE ### (≈ 4 lines of code)
  21.     W1 = np.random.randn(n_h , n_x) * 0.01
  22.     b1 = np.zeros((n_h , 1)) #记着这里是两个括号
  23.     W2 = np.random.randn(n_y , n_h) * 0.01
  24.     b2 = np.zeros((n_y , 1))
  25.     ### END CODE HERE ###
  27.     assert (W1.shape == (n_h, n_x))
  28.     assert (b1.shape == (n_h, 1))
  29.     assert (W2.shape == (n_y, n_h))
  30.     assert (b2.shape == (n_y, 1))
  32.     parameters = {"W1": W1,
  33.                   "b1": b1,
  34.                   "W2": W2,
  35.                   "b2": b2}
  37.     return parameters

然后现在来看看初始化的样子
代码长这样：

  1. #现在可以看看初始化结构咋样
  2. n_x, n_h, n_y = initialize_parameters_test_case()
  4. parameters = initialize_parameters(n_x, n_h, n_y)
  5. print("W1 = " + str(parameters["W1"]))
  6. print("b1 = " + str(parameters["b1"]))
  7. print("W2 = " + str(parameters["W2"]))
  8. print("b2 = " + str(parameters["b2"]))

输出是

  1. W1 = [[-0.00416758 -0.00056267]
  2.  [-0.02136196  0.01640271]
  3.  [-0.01793436 -0.00841747]
  4.  [ 0.00502881 -0.01245288]]
  5. b1 = [[ 0.]
  6.  [ 0.]
  7.  [ 0.]
  8.  [ 0.]]
  9. W2 = [[-0.01057952 -0.00909008  0.00551454  0.02292208]]
  10. b2 = [[ 0.]]
  1. 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
  1. #OK，现在开始正向传播啦
  2. # GRADED FUNCTION: forward_propagation​
  3. def forward_propagation(X, parameters):
  4.     """
  5.     Argument:
  6.     X -- input data of size (n_x, m)
  7.     parameters -- python dictionary containing your parameters (output of initialization function)
  9.     Returns:
  10.     A2 -- The sigmoid output of the second activation
  11.     cache -- a dictionary containing "Z1", "A1", "Z2" and "A2"
  12.     """
  13.     # Retrieve each parameter from the dictionary "parameters"
  14.     ### START CODE HERE ### (≈ 4 lines of code)
  15.     W1 = parameters['W1']
  16.     b1 = parameters['b1']
  17.     W2 = parameters['W2']
  18.     b2 = parameters['b2']
  19.     ### END CODE HERE ###
  21.     # Implement Forward Propagation to calculate A2 (probabilities)
  22.     ### START CODE HERE ### (≈ 4 lines of code)
  23.     Z1 = np.dot(W1 , X) + b1
  24.     A1 = np.tanh(Z1)
  25.     Z2 = np.dot(W2 , A1) + b2
  26.     A2 = sigmoid(Z2)
  27.     ### END CODE HERE ###
  29.     assert (A2.shape == (1, X.shape[1]))
  31.     cache = {"Z1": Z1,
  32.              "A1": A1,
  33.              "Z2": Z2,
  34.              "A2": A2}
  36.     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：

  1. #现在来看看正向传播的结果
  2. X_assess, parameters = forward_propagation_test_case()
  4. A2, cache = forward_propagation(X_assess, parameters)
  6. # Note: we use the mean here just to make sure that your output matches ours.
  7. print(np.mean(cache['Z1']) ,np.mean(cache['A1']),np.mean(cache['Z2']),np.mean(cache['A2']))

然后，结果如下：

  1. -0.000499755777742 -0.000496963353232 0.000438187450959 0.500109546852
  1. 到了这里，我们可以来看看cost函数了，
    先看看公式：

下面是他的代码（重点关注28,29行）

  1. #现在来算算代价函数J
  2. #GRADED
  3. #FUNCTION: compute_cost
  5. def compute_cost(A2, Y, parameters):
  6.     """
  7.     Computes the cross-entropy cost given in equation (13)
  9.     Arguments:
  10.     A2 -- The sigmoid output of the second activation, of shape (1, number of examples)
  11.     Y -- "true" labels vector of shape (1, number of examples)
  12.     parameters -- python dictionary containing your parameters W1, b1, W2 and b2
  14.     Returns:
  15.     cost -- cross-entropy cost given equation (13)
  16.     """
  18.     m = Y.shape[1]  # number of example
  20.     # Retrieve W1 and W2 from parameters
  21.     ### START CODE HERE ### (≈ 2 lines of code)
  22.     W1 = parameters['W1']
  23.     W2 = parameters['W2']
  24.     ### END CODE HERE ###
  26.     # Compute the cross-entropy cost
  27.     ### START CODE HERE ### (≈ 2 lines of code)
  28.     logprobs = np.multiply(np.log(A2), Y) + np.multiply(np.log(1 - A2), 1 - Y)
  29.     cost = -np.sum(logprobs) / m
  30.     ### END CODE HERE ###
  32.     cost = np.squeeze(cost)  # makes sure cost is the dimension we expect.
  33.     # E.g., turns [[17]] into 17
  34.     assert (isinstance(cost, float))
  36.     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

  1. #现在来检测下代价函数的计算
  2. A2, Y_assess, parameters = compute_cost_test_case()
  3. print("cost = " + str(compute_cost(A2, Y_assess, parameters)))

输出是这样

  1. cost = 0.692919893776
  1. 难点来了，反向传播。(重点是32-37行，这是精髓)
  1. #难点来了，反向传播
  2. # GRADED FUNCTION: backward_propagation
  3. def backward_propagation(parameters, cache, X, Y):
  4.     """
  5.     Implement the backward propagation using the instructions above.
  7.     Arguments:
  8.     parameters -- python dictionary containing our parameters
  9.     cache -- a dictionary containing "Z1", "A1", "Z2" and "A2".
  10.     X -- input data of shape (2, number of examples)
  11.     Y -- "true" labels vector of shape (1, number of examples)
  13.     Returns:
  14.     grads -- python dictionary containing your gradients with respect to different parameters
  15.     """
  16.     m = X.shape[1]
  18.     # First, retrieve W1 and W2 from the dictionary "parameters".
  19.     ### START CODE HERE ### (≈ 2 lines of code)
  20.     W1 = parameters['W1']
  21.     W2 = parameters['W2']
  22.     ### END CODE HERE ###
  24.     # Retrieve also A1 and A2 from dictionary "cache".
  25.     ### START CODE HERE ### (≈ 2 lines of code)
  26.     A1 = cache['A1']
  27.     A2 = cache['A2']
  28.     ### END CODE HERE ###
  30.     # Backward propagation: calculate dW1, db1, dW2, db2.
  31.     ### START CODE HERE ### (≈ 6 lines of code, corresponding to 6 equations on slide above)
  32.     dZ2 = A2 - Y
  33.     dW2 = np.dot(dZ2 , A1.T) / m
  34.     db2 = np.sum(dZ2 , axis = 1 , keepdims = True) / m
  35.     dZ1 = np.dot(W2.T , dZ2) * (1 - A1**2)
  36.     dW1 = np.dot(dZ1 , X.T) / m
  37.     db1 = np.sum(dZ1 , axis = 1 , keepdims = True) / m
  38.     ### END CODE HERE ###
  40.     grads = {"dW1": dW1,
  41.              "db1": db1,
  42.              "dW2": dW2,
  43.              "db2": db2}
  45.     return grads

对于此，我们完成了反向传播，
现在来看看输出代码

  1. #反向传播完毕，我们来看看
  2. parameters, cache, X_assess, Y_assess = backward_propagation_test_case()
  4. grads = backward_propagation(parameters, cache, X_assess, Y_assess)
  5. print ("dW1 = "+ str(grads["dW1"]))
  6. print ("db1 = "+ str(grads["db1"]))
  7. print ("dW2 = "+ str(grads["dW2"]))
  8. print ("db2 = "+ str(grads["db2"]))

输出是这样的

  1. dW1 = [[ 0.01018708 -0.00708701]
  2.  [ 0.00873447 -0.0060768 ]
  3.  [-0.00530847  0.00369379]
  4.  [-0.02206365  0.01535126]]
  5. db1 = [[-0.00069728]
  6.  [-0.00060606]
  7.  [ 0.000364  ]
  8.  [ 0.00151207]]
  9. dW2 = [[ 0.00363613  0.03153604  0.01162914 -0.01318316]]
  10. db2 = [[ 0.06589489]]
  1. 好了，反向传播页做完了，现在开始更新参数矩阵吧！！！
  1. #反向传播完毕后，开始更新参数
  2. # GRADED FUNCTION: update_parameters​
  3. def update_parameters(parameters, grads, learning_rate=1.2):
  4.     """
  5.     Updates parameters using the gradient descent update rule given above
  7.     Arguments:
  8.     parameters -- python dictionary containing your parameters
  9.     grads -- python dictionary containing your gradients
  11.     Returns:
  12.     parameters -- python dictionary containing your updated parameters
  13.     """
  14.     # Retrieve each parameter from the dictionary "parameters"
  15.     ### START CODE HERE ### (≈ 4 lines of code)
  16.     W1 = parameters['W1']
  17.     b1 = parameters['b1']
  18.     W2 = parameters['W2']
  19.     b2 = parameters['b2']
  20.     ### END CODE HERE ###
  22.     # Retrieve each gradient from the dictionary "grads"
  23.     ### START CODE HERE ### (≈ 4 lines of code)
  24.     dW1 = grads['dW1']
  25.     db1 = grads['db1']
  26.     dW2 = grads['dW2']
  27.     db2 = grads['db2']
  28.     ## END CODE HERE ###
  30.     # Update rule for each parameter
  31.     ### START CODE HERE ### (≈ 4 lines of code)
  32.     W1 = W1 - learning_rate * dW1
  33.     b1 = b1 - learning_rate * db1
  34.     W2 = W2 - learning_rate * dW2
  35.     b2 = b2 - learning_rate * db2
  36.     ### END CODE HERE ###
  38.     parameters = {"W1": W1,
  39.                   "b1": b1,
  40.                   "W2": W2,
  41.                   "b2": b2}
  43.     return parameters

老规矩，跑一跑(代码如下)

  1. #现在验证一下反向传播算法的参数更新
  2. parameters, grads = update_parameters_test_case()
  3. parameters = update_parameters(parameters, grads)
  5. print("W1 = " + str(parameters["W1"]))
  6. print("b1 = " + str(parameters["b1"]))
  7. print("W2 = " + str(parameters["W2"]))
  8. print("b2 = " + str(parameters["b2"]))

结果长这样：

  1. W1 = [[-0.00643025  0.01936718]
  2.  [-0.02410458  0.03978052]
  3.  [-0.01653973 -0.02096177]
  4.  [ 0.01046864 -0.05990141]]
  5. b1 = [[ -1.02420756e-06]
  6.  [  1.27373948e-05]
  7.  [  8.32996807e-07]
  8.  [ -3.20136836e-06]]
  9. W2 = [[-0.01041081 -0.04463285  0.01758031  0.04747113]]
  10. 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

  1. #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
  2. # GRADED FUNCTION: nn_model​
  3. def nn_model(X, Y, n_h, num_iterations=10000, print_cost=False):
  4.     """
  5.     Arguments:
  6.     X -- dataset of shape (2, number of examples)
  7.     Y -- labels of shape (1, number of examples)
  8.     n_h -- size of the hidden layer
  9.     num_iterations -- Number of iterations in gradient descent loop
  10.     print_cost -- if True, print the cost every 1000 iterations
  12.     Returns:
  13.     parameters -- parameters learnt by the model. They can then be used to predict.
  14.     """
  16.     np.random.seed(3)
  17.     n_x = layer_sizes(X, Y)[0]
  18.     n_y = layer_sizes(X, Y)[2]
  20.     # Initialize parameters, then retrieve W1, b1, W2, b2. Inputs: "n_x, n_h, n_y". Outputs = "W1, b1, W2, b2, parameters".
  21.     ### START CODE HERE ### (≈ 5 lines of code)
  22.     parameters = initialize_parameters(n_x , n_h , n_y)
  23.     W1 = parameters['W1']
  24.     b1 = parameters['b1']
  25.     W2 = parameters['W2']
  26.     b2 = parameters['b2']
  27.     ### END CODE HERE ###
  29.     # Loop (gradient descent)
  30.     import pdb
  32.     for i in range(0, num_iterations):
  34.         ### START CODE HERE ### (≈ 4 lines of code)
  35.         # Forward propagation. Inputs: "X, parameters". Outputs: "A2, cache".
  36.         A2, cache = forward_propagation(X , parameters)
  38.         # Cost function. Inputs: "A2, Y, parameters". Outputs: "cost".
  39.         cost = compute_cost(A2 , Y , parameters)
  41.         # Backpropagation. Inputs: "parameters, cache, X, Y". Outputs: "grads".
  42.         grads = backward_propagation(parameters, cache, X, Y)
  44.         # Gradient descent parameter update. Inputs: "parameters, grads". Outputs: "parameters".
  45.         parameters = update_parameters(parameters, grads)
  47.         ### END CODE HERE ###
  49.         # Print the cost every 1000 iterations
  50.         if print_cost and i % 1000 == 0:
  51.             print ("Cost after iteration %i: %f" % (i, cost))
  53.     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)

  1. #现在我们来跑一跑这个模型
  2. X_assess, Y_assess = nn_model_test_case()
  4. parameters = nn_model(X_assess, Y_assess, 4, num_iterations=10000, print_cost=False)
  5. print("W1 = " + str(parameters['W1']))
  6. print("b1 = " + str(parameters['b1']))
  7. print("W2 = " + str(parameters['W2']))
  8. 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：

  1. W1 = [[-4.18493855  5.33220875]
  2.  [-7.52989335  1.24306212]
  3.  [-4.19297397  5.32630669]
  4.  [ 7.52983568 -1.24309519]]
  5. b1 = [[ 2.32926551]
  6.  [ 3.79459096]
  7.  [ 2.33002105]
  8.  [-3.79469147]]
  9. W2 = [[-6033.83673001 -6008.12981298 -6033.1009627   6008.06639333]]
  10. b2 = [[-52.66607002]]

最后，跑一下预测函数

  1. #现在来写一写预测函数
  2. # GRADED FUNCTION: predict
  3. def predict(parameters, X):
  4.     """
  5.     Using the learned parameters, predicts a class for each example in X
  7.     Arguments:
  8.     parameters -- python dictionary containing your parameters
  9.     X -- input data of size (n_x, m)
  11.     Returns
  12.     predictions -- vector of predictions of our model (red: 0 / blue: 1)
  13.     """
  15.     # Computes probabilities using forward propagation, and classifies to 0/1 using 0.5 as the threshold.
  16.     ### START CODE HERE ### (≈ 2 lines of code)
  17.     A2, cache = forward_propagation(X , parameters)
  18.     predictions = np.array([0 if i <= 0.5 else 1 for i in np.squeeze(A2)])
  19.     ### END CODE HERE ###
  21.     return predictions

看看预测值吧

  1. #来看看预测值
  2. parameters, X_assess = predict_test_case()
  3. predictions = predict(parameters, X_assess)
  4. print("predictions mean = " + str(np.mean(predictions)))

结果长这样：

  1. 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

  1. #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
  2. parameters = nn_model(X, Y, n_h = 4, num_iterations = 10000, print_cost=True)
  3. # Plot the decision boundary
  4. plot_decision_boundary(lambda x: predict(parameters, x.T), X, Y)
  5. plt.title("Decision Boundary for hidden layer size " + str(4))
  6. pylab.show()
  7. predictions=predict(parameters,X)
  8. print ('Accuracy: %d' % float((np.dot(Y,predictions.T) + np.dot(1-Y,1-predictions.T))/float(Y.size)*100) + '%')
  9. #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

  1. Cost after iteration 0: 0.693048
  2. Cost after iteration 1000: 0.288083
  3. Cost after iteration 2000: 0.254385
  4. Cost after iteration 3000: 0.233864
  5. Cost after iteration 4000: 0.226792
  6. Cost after iteration 5000: 0.222644
  7. Cost after iteration 6000: 0.219731
  8. Cost after iteration 7000: 0.217504
  9. Cost after iteration 8000: 0.219501
  10. Cost after iteration 9000: 0.218620
  11. 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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