【深度学习】吴恩达网易公开课练习(class1 week3)

知识点梳理

python工具使用：

sklearn: 数据挖掘，数据分析工具，内置logistic回归
matplotlib：做图工具，可绘制等高线等
绘制散点图： plt.scatter(X[0, :], X[1, :], c=np.squeeze(Y), s=40, cmap=plt.cm.Spectral); s：绘制点大小 cmap：颜色集
绘制等高线：先做网格，计算结果，绘图

     x_min, x_max = X[0, :].min() - 1, X[0, :].max() + 1

     y_min, y_max = X[1, :].min() - 1, X[1, :].max() + 1

     h = 0.01

     xx, yy = np.meshgrid(np.arange(x_min, x_max, h), np.arange(y_min, y_max, h))

     Z = model(np.c_[xx.ravel(), yy.ravel()])

     Z = Z.reshape(xx.shape)

     #xx是x轴值， yy是y轴值， Z是预测结果值， cmap表示采用什么颜色

     plt.contourf(xx, yy, Z, cmap=plt.cm.Spectral)

关键变量：

m: 训练样本数量
n_x：一个训练样本的输入数量，输入层大小
n_h：隐藏层大小
方括号上标[l]: 第l层
圆括号上标(i): 第i个样本

$$
X =
\left[
\begin{matrix}
\vdots & \vdots & \vdots & \vdots \\
x^{(1)} & x^{(2)} & \vdots & x^{(m)} \\
\vdots & \vdots & \vdots & \vdots \\
\end{matrix}
\right]_{(n\_x, m)}
$$

$$
W^{[1]} =
\left[
\begin{matrix}
\cdots & w^{[1] T}_1 & \cdots \\
\cdots & w^{[1] T}_2 & \cdots \\
\cdots & \cdots & \cdots \\
\cdots & w^{[1] T}_{n\_h} & \cdots \\
\end{matrix}
\right]_{(n\_h, n\_x)}
$$

$$
b^{[1]} =
\left[
\begin{matrix}
b^{[1]}_1 \\
b^{[1]}_2 \\
\vdots \\
b^{[1]}_{n\_h} \\
\end{matrix}
\right]_{(n\_h, 1)}
$$

$$
A^{[1]}=
\left[
\begin{matrix}
\vdots & \vdots & \vdots & \vdots \\
a^{[1](1)} & a^{[1](2)} & \vdots & a^{[1](m)} \\
\vdots & \vdots & \vdots & \vdots \\
\end{matrix}
\right]_{(n\_h, m)}
$$

$$
Z^{[1]}=
\left[
\begin{matrix}
\vdots & \vdots & \vdots & \vdots \\
z^{[1](1)} & z^{[1](2)} & \vdots & z^{[1](m)} \\
\vdots & \vdots & \vdots & \vdots \\
\end{matrix}
\right]_{(n\_h, m)}
$$

***

单隐层神经网络关键公式：

前向传播：

$$Z^{[1]}=W^{[1]}X+b^{[1]}$$
$$A^{[1]}=g^{[1]}(Z^{[1]})$$
$$Z^{[2]}=W^{[2]}A^{[1]}+b^{[2]}$$
$$A^{[2]}=g^{[2]}(Z^{[2]})$$

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

    A1 = np.tanh(Z1)

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

    A2 = sigmoid(Z2)

反向传播

    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)

cost计算

\[J = - \frac{1}{m} \sum\limits_{i = 0}^{m} \large{(} \small y^{(i)}\log\left(a^{[2] (i)}\right) + (1-y^{(i)})\log\left(1- a^{[2] (i)}\right) \large{)} \small
\]

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

    cost = - 1 / m * np.sum(logprobs)

单隐层神经网络代码：

# Package imports

import numpy as np

import matplotlib.pyplot as plt

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

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.

    W1 = np.random.randn(n_h, n_x) * 0.01

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

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

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

    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

def forward_propagation(X, parameters):

    # 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

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

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

    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

def update_parameters(parameters, grads, learning_rate = 0.8):

    # 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 = 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

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

    np.random.seed(3)

    n_x = X.shape[0]

    n_y = Y.shape[0]

    # 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)

        # Print the cost every 1000 iterations

        if print_cost and i % 1000 == 0:

            print ("Cost after iteration %i: %f" %(i, cost))

    return parameters

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

X, Y = load_planar_dataset()

# 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, np.squeeze(Y))

plt.title("Decision Boundary for hidden layer size " + str(4))

# planar_utils.py

import matplotlib.pyplot as plt

import numpy as np

import sklearn

import sklearn.datasets

import sklearn.linear_model

def plot_decision_boundary(model, X, y):

    # Set min and max values and give it some padding

    x_min, x_max = X[0, :].min() - 1, X[0, :].max() + 1

    y_min, y_max = X[1, :].min() - 1, X[1, :].max() + 1

    h = 0.01

    # Generate a grid of points with distance h between them

    # 创造网格，以0.01为间隔划分整个区间

    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 grid

    # 计算每个网格点上的预测结果

    Z = model(np.c_[xx.ravel(), yy.ravel()])

    # 将预测结果变形为与网格形式一致

    Z = Z.reshape(xx.shape)

    # Plot the contour and training examples

    # xx是x轴值， yy是y轴值， Z是预测结果值， cmap表示采用什么颜色

    plt.contourf(xx, yy, Z, cmap=plt.cm.Spectral) #等位线

    plt.ylabel('x2')

    plt.xlabel('x1')

    plt.scatter(X[0, :], X[1, :], c=y, cmap=plt.cm.Spectral)

def sigmoid(x):

    """

    Compute the sigmoid of x

    Arguments:

    x -- A scalar or numpy array of any size.

    Return:

    s -- sigmoid(x)

    """

    s = 1/(1+np.exp(-x))

    return s

def load_planar_dataset():

    np.random.seed(1)

    m = 400 # number of examples

    N = int(m/2) # number of points per class

    D = 2 # dimensionality

    X = np.zeros((m,D)) # data matrix where each row is a single example

    Y = np.zeros((m,1), dtype='uint8') # labels vector (0 for red, 1 for blue)

    a = 4 # maximum ray of the flower

    for j in range(2):

        ix = range(N*j,N*(j+1))

        t = np.linspace(j*3.12,(j+1)*3.12,N) + np.random.randn(N)*0.2 # theta

        r = a*np.sin(4*t) + np.random.randn(N)*0.2 # radius

        X[ix] = np.c_[r*np.sin(t), r*np.cos(t)]

        Y[ix] = j

    X = X.T

    Y = Y.T

    return X, Y

def load_extra_datasets():

    N = 200

    noisy_circles = sklearn.datasets.make_circles(n_samples=N, factor=.5, noise=.3)

    noisy_moons = sklearn.datasets.make_moons(n_samples=N, noise=.2)

    blobs = sklearn.datasets.make_blobs(n_samples=N, random_state=5, n_features=2, centers=6)

    gaussian_quantiles = sklearn.datasets.make_gaussian_quantiles(mean=None, cov=0.5, n_samples=N, n_features=2, n_classes=2, shuffle=True, random_state=None)

    no_structure = np.random.rand(N, 2), np.random.rand(N, 2)

    return noisy_circles, noisy_moons, blobs, gaussian_quantiles, no_structure