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