目标

• 建立一个逻辑回归分类器
• 建立一个学习算法的通用结构，包括初始化参数，计算代价函数和梯度，并使用梯度下降优化算法

1. 准备工作

导入相关包，其中：

• numpy 是 python 中用于科学计算的基础包
• h5py 用于操作 H5 文件中存储的数据集
• matplotlib 是 python 用于绘图的库
• PIL 和 scipy 用于在本文档最后测试你自己的图片

import numpy as np
import matplotlib.pyplot as plt
import h5py
import scipy
from PIL import Image
from scipy import ndimage
from lr_utils import load_dataset

%matplotlib inline

2. 处理数据

我们在 data 文件夹中有两个 h5 文件，分别是 test_catvnoncat.h5 和 train_catvnoncat.h5，看名字就知道一个用作测试集，一个用作训练集。

其中，训练集包括 m_train 张训练图片，标签为 y = 1(cat) 和 y = 0(non-cat)；
测试集包括 m_test 张测试图片，标签为 cat 或 non-cat。

每张图片的形状为 (num_px, num_px, 3)，3 是彩色图像的 RGB 通道，宽、高为 num_px。

我们的目标就是建立一个简单的分类器，能够正确分类一张图片是猫（cat）还是不是猫（non-cat）

1）加载数据集

# Loading the data (cat/non-cat)
train_set_x_orig, train_set_y, test_set_x_orig, test_set_y, classes = load_dataset()

这里加 orig 是因为我们需要对图像数据进行预处理，在没有预处理前用 orig 表示原始数据，而标签数据不需要预处理。在 train_set_x_orig 和 test_set_x_orig 中每一行都表示一张图片。

load_dataset() 函数在 lr_utils.py 文件中：

def load_dataset():
    train_dataset = h5py.File('datasets/train_catvnoncat.h5', "r")
    train_set_x_orig = np.array(train_dataset["train_set_x"][:]) # your train set features
    train_set_y_orig = np.array(train_dataset["train_set_y"][:]) # your train set labels

    test_dataset = h5py.File('datasets/test_catvnoncat.h5', "r")
    test_set_x_orig = np.array(test_dataset["test_set_x"][:]) # your test set features
    test_set_y_orig = np.array(test_dataset["test_set_y"][:]) # your test set labels

    classes = np.array(test_dataset["list_classes"][:]) # the list of classes
    
    train_set_y_orig = train_set_y_orig.reshape((1, train_set_y_orig.shape[0]))
    test_set_y_orig = test_set_y_orig.reshape((1, test_set_y_orig.shape[0]))
    
    return train_set_x_orig, train_set_y_orig, test_set_x_orig, test_set_y_orig, classes

2）显示一张数据集中的图片及标签：

# Example of a picture
index = 5
plt.imshow(train_set_x_orig[index])
print ("y = " + str(train_set_y[:, index]) + ", it's a '" + classes[np.squeeze(train_set_y[:, index])].decode("utf-8") +  "' picture.")

>>>
y = [0], it's a 'non-cat' picture.

3）查看数据集格式：

注意 train_set_x_orig 是一个形状为 (m_train, num_px, num_px, 3) 的 numpy array

### START CODE HERE ### (≈ 3 line of code)
m_train = train_set_x_orig.shape[0]
m_test = test_set_x_orig.shape[0]
num_px = train_set_x_orig.shape[1]
### END CODE HERE ###

print ("Number of training examples: m_train = " + str(m_train))
print ("Number of testing examples: m_test = " + str(m_test))
print ("Height/Width of each image: num_px = " + str(num_px))
print ("Each image is of size: (" + str(num_px) + ", " + str(num_px) + ", 3)")
print ("train_set_x shape: " + str(train_set_x_orig.shape))
print ("train_set_y shape: " + str(train_set_y.shape))
print ("test_set_x shape: " + str(test_set_x_orig.shape))
print ("test_set_y shape: " + str(test_set_y.shape))

>>>
Number of training examples: m_train = 209
Number of testing examples: m_test = 50
Height/Width of each image: num_px = 64
Each image is of size: (64, 64, 3)
train_set_x shape: (209, 64, 64, 3)
train_set_y shape: (1, 209)
test_set_x shape: (50, 64, 64, 3)
test_set_y shape: (1, 50)

4）对图像数据形状进行变形

为了方便期间，我们需要将形状为 (num_px, num_px, 3) 的图像数据形状拉伸为 (num_px * num_px * 3, 1)。这样我们的训练数据集每一列代表一张图像，共有 m_train 列。

提示：当我们想要将一个形如 (a, b, c, d) 的矩阵拉伸为 (b * c * d, a)，可以执行如下语句：
X_flatten = X.reshape(X.shape[0], -1).T # X.T is the transpose of X

# Reshape the training and test examples
### START CODE HERE ### (≈ 2 line of code)
train_set_x_flatten = train_set_x_orig.reshape(train_set_x_orig.shape[0], -1).T
test_set_x_flatten = test_set_x_orig.reshape(test_set_x_orig.shape[0], -1).T
### END CODE HERE ###

print ("train_set_x_flatten shape: " + str(train_set_x_flatten.shape))
print ("train_set_y shape: " + str(train_set_y.shape))
print ("test_set_x_flatten shape: " + str(test_set_x_flatten.shape))
print ("test_set_y shape: " + str(test_set_y.shape))
print ("sanity check after reshaping: " + str(train_set_x_flatten[0:5,0]))

>>>
train_set_x_flatten shape: (12288, 209)
train_set_y shape: (1, 209)
test_set_x_flatten shape: (12288, 50)
test_set_y shape: (1, 50)
sanity check after reshaping: [17 31 56 22 33]

5）对数据进行预处理

在机器学习中一种最常见的预处理就是归一化数据，一般的归一化是对每个数据减去它们的均值再除以整个数组的标准差，而对于图片数据更好的归一化方法是直接除以 255（通道的最大值）。

train_set_x = train_set_x_flatten/255
test_set_x = test_set_x_flatten/255

【总结】数据预处理的步骤：

• 获取数据格式
• 对数据进行变形处理
• 数据归一化处理

3. 建立模型

建立一个逻辑回归算法，下图解释了逻辑回归算法实际上就是一个非常简单的神经网络。

算法的数学表达式：

设有一个输入图片 $x^{(i)}$x(i):
$z^{(i)} = w^T x^{(i)} + b \tag{1}$z(i)=wTx(i)+b(1)

$\hat{y}^{(i)} = a^{(i)} = sigmoid(z^{(i)})\tag{2}$y^​(i)=a(i)=sigmoid(z(i))(2)

$\mathcal{L}(a^{(i)}, y^{(i)}) = - y^{(i)} \log(a^{(i)}) - (1-y^{(i)} ) \log(1-a^{(i)})\tag{3}$L(a(i),y(i))=−y(i)log(a(i))−(1−y(i))log(1−a(i))(3)

代价函数为所有损失函数和的平均值：
$J = \frac{1}{m} \sum_{i=1}^m \mathcal{L}(a^{(i)}, y^{(i)})\tag{6}$J=m1​i=1∑m​L(a(i),y(i))(6)

建立步骤：

• 定义模型结构
• 初始化模型参数
• 循环计算 loss（前向传播），gradient（反向传播），更新参数
• 利用习得的参数做预测
• 分析结果并得出结论

通常 1 - 3 步我们分别完成后，再整合到一起形成一个完整的 model。

3.1 Helper function

我们需要 sigmoid 函数进行预测，先写出 sigmoid 函数。

# GRADED FUNCTION: sigmoid

def sigmoid(z):
    """
    Compute the sigmoid of z

    Arguments:
    z -- A scalar or numpy array of any size.

    Return:
    s -- sigmoid(z)
    """
	### START CODE HERE ### (≈ 1 line of code)
	s = 1/(1 + np.exp(-z))
	### END CODE HERE ###
    
    return s

print ("sigmoid([0, 2]) = " + str(sigmoid(np.array([0,2]))))
>>>
sigmoid([0, 2]) = [ 0.5         0.88079708]

3.2 Initializing parameters

我们需要对参数 $w$w 和 $b$b 进行初始化，尝试用 np.zeros() 对 $w$w 和 $b$b 进行初始化。

# GRADED FUNCTION: initialize_with_zeros

def initialize_with_zeros(dim):
    """
    This function creates a vector of zeros of shape (dim, 1) for w and initializes b to 0.
    
    Argument:
    dim -- size of the w vector we want (or number of parameters in this case)
    
    Returns:
    w -- initialized vector of shape (dim, 1)
    b -- initialized scalar (corresponds to the bias)
    """
    ### START CODE HERE ### (≈ 2 line of code)
    w = np.zeros((dims, 1))
    b = 0
    ### END CODE HERE ###

    assert(w.shape == (dim, 1))
    assert(isinstance(b, float) or isinstance(b, int))
    
    return w, b

dim = 2
w, b = initialize_with_zeros(dim)
print ("w = " + str(w))
print ("b = " + str(b))
>>>
w = [[ 0.]
 [ 0.]]
b = 0

3.3 Forward and Backward propagation

前向传播：

• 输入 X
• $A = \sigma(w^T X + b) = (a^{(0)}, a^{(1)}, ..., a^{(m-1)}, a^{(m)})$A=σ(wTX+b)=(a(0),a(1),...,a(m−1),a(m))
• $J = -\frac{1}{m}\sum_{i=1}^{m}y^{(i)}\log(a^{(i)})+(1-y^{(i)})\log(1-a^{(i)})$J=−m1​∑i=1m​y(i)log(a(i))+(1−y(i))log(1−a(i))

反向传播时需要用到下面两个公式：
$\frac{\partial J}{\partial w} = \frac{1}{m}X(A-Y)^T\tag{7}$∂w∂J​=m1​X(A−Y)T(7)

$\frac{\partial J}{\partial b} = \frac{1}{m} \sum_{i=1}^m (a^{(i)}-y^{(i)})\tag{8}$∂b∂J​=m1​i=1∑m​(a(i)−y(i))(8)

# GRADED FUNCTION: propagate

def propagate(w, b, X, Y):
    """
    Implement the cost function and its gradient for the propagation explained above

    Arguments:
    w -- weights, a numpy array of size (num_px * num_px * 3, 1)
    b -- bias, a scalar
    X -- data of size (num_px * num_px * 3, number of examples)
    Y -- true "label" vector (containing 0 if non-cat, 1 if cat) of size (1, number of examples)

    Return:
    cost -- negative log-likelihood cost for logistic regression
    dw -- gradient of the loss with respect to w, thus same shape as w
    db -- gradient of the loss with respect to b, thus same shape as b
    
    Tips:
    - Write your code step by step for the propagation. np.log(), np.dot()
    """
    
    m = X.shape[1]
    
    # FORWARD PROPAGATION (FROM X TO COST)
    ### START CODE HERE ### (≈ 2 line of code)
    A = sigmoid(np.dot(w.T, X) + b)
    cost = -1 / m * np.sum(Y * np.log(A) + (1 - Y) * np.log(1 - A))
    ### END CODE HERE ###
    
    # BACKWARD PROPAGATION (TO FIND GRAD)
    ### START CODE HERE ### (≈ 2 line of code)
    dw = 1/m * np.dot(X, (A - Y).T)
    db = 1/m * np.sum(A - Y)
    ### END CODE HERE ###
    assert(dw.shape == w.shape)
    assert(db.dtype == float)
    cost = np.squeeze(cost)
    assert(cost.shape == ())
    
    grads = {"dw": dw,
             "db": db}
    
    return grads, cost

w, b, X, Y = np.array([[1],[2]]), 2, np.array([[1,2],[3,4]]), np.array([[1,0]])
grads, cost = propagate(w, b, X, Y)
print ("dw = " + str(grads["dw"]))
print ("db = " + str(grads["db"]))
print ("cost = " + str(cost))
>>>
dw = [[ 0.99993216]
 [ 1.99980262]]
db = 0.499935230625
cost = 6.00006477319

3.4 Optimization

优化参数 $\theta$θ 的计算公式：$\theta = \theta - \alpha d\theta$θ=θ−αdθ，其中 $\alpha$α 为学习率。

# GRADED FUNCTION: optimize

def optimize(w, b, X, Y, num_iterations, learning_rate, print_cost = False):
    """
    This function optimizes w and b by running a gradient descent algorithm
    
    Arguments:
    w -- weights, a numpy array of size (num_px * num_px * 3, 1)
    b -- bias, a scalar
    X -- data of shape (num_px * num_px * 3, number of examples)
    Y -- true "label" vector (containing 0 if non-cat, 1 if cat), of shape (1, number of examples)
    num_iterations -- number of iterations of the optimization loop
    learning_rate -- learning rate of the gradient descent update rule
    print_cost -- True to print the loss every 100 steps
    
    Returns:
    params -- dictionary containing the weights w and bias b
    grads -- dictionary containing the gradients of the weights and bias with respect to the cost function
    costs -- list of all the costs computed during the optimization, this will be used to plot the learning curve.
    
    Tips:
    You basically need to write down two steps and iterate through them:
        1) Calculate the cost and the gradient for the current parameters. Use propagate().
        2) Update the parameters using gradient descent rule for w and b.
    """
    
    costs = []
    for i in range(num_iterations):
    	# Cost and gradient calculation (≈ 1-4 line of code)
        ### START CODE HERE ### 
        grads, cost = propagate(w, b, X, Y)
        ### END CODE HERE ###
        
        # Retrieve derivatives from grads
        dw = grads["dw"]
        db = grads["db"]
        
        # update rule (≈ 2 line of code)
        ### START CODE HERE ###
        w = w - learning_rate * dw
        b = b - learning_rate * db
        ### END CODE HERE ###
        
        # Record the costs
        if i % 100 == 0:
            costs.append(cost)
        
        # Print the cost every 100 training examples
        if print_cost and i % 100 == 0:
            print ("Cost after iteration %i: %f" %(i, cost))
    
    params = {"w": w,
              "b": b}
    
    grads = {"dw": dw,
             "db": db}
    
    return params, grads, costs

params, grads, costs = optimize(w, b, X, Y, num_iterations= 100, learning_rate = 0.009, print_cost = False)

print ("w = " + str(params["w"]))
print ("b = " + str(params["b"]))
print ("dw = " + str(grads["dw"]))
print ("db = " + str(grads["db"]))
print(costs)
>>>
w = [[ 0.1124579 ]
 [ 0.23106775]]
b = 1.55930492484
dw = [[ 0.90158428]
 [ 1.76250842]]
db = 0.430462071679
[6.0000647731922054]

3.5 Prediction

1. 计算 $\hat{Y} = A = \sigma(w^T X + b)$Y^=A=σ(wTX+b)
2. 将预测输出转换为 0 或 1，假设 activation <= 0.5 预测为 0，activation >= 0.5 预测为 1

# GRADED FUNCTION: predict

def predict(w, b, X):
    '''
    Predict whether the label is 0 or 1 using learned logistic regression parameters (w, b)
    
    Arguments:
    w -- weights, a numpy array of size (num_px * num_px * 3, 1)
    b -- bias, a scalar
    X -- data of size (num_px * num_px * 3, number of examples)
    
    Returns:
    Y_prediction -- a numpy array (vector) containing all predictions (0/1) for the examples in X
    '''
    
    m = X.shape[1]
    Y_prediction = np.zeros((1,m))
    w = w.reshape(X.shape[0], 1)
    
    # Compute vector "A" predicting the probabilities of a cat being present in the picture
    ### START CODE HERE ### (≈ 1 line of code)
    A = sigmoid(np.dot(w.T, X) + b)
    ### END CODE HERE ###

    for i in range(A.shape[1]):
        # Convert probabilities A[0,i] to actual predictions p[0,i]
        ### START CODE HERE ### (≈ 4 line of code)
        if A[0, i] <= 0.5:
            Y_prediction[0, i] = 0
        else:
            Y_prediction[0, i] = 1
        ### END CODE HERE ###
    
    assert(Y_prediction.shape == (1, m))
    
    return Y_prediction     

print ("predictions = " + str(predict(w, b, X)))
>>>
predictions = [[ 1.  1.]]

4. 合并功能函数

# GRADED FUNCTION: model

def model(X_train, Y_train, X_test, Y_test, num_iterations = 2000, learning_rate = 0.5, print_cost = False):
    """
    Builds the logistic regression model by calling the function you've implemented previously
    
    Arguments:
    X_train -- training set represented by a numpy array of shape (num_px * num_px * 3, m_train)
    Y_train -- training labels represented by a numpy array (vector) of shape (1, m_train)
    X_test -- test set represented by a numpy array of shape (num_px * num_px * 3, m_test)
    Y_test -- test labels represented by a numpy array (vector) of shape (1, m_test)
    num_iterations -- hyperparameter representing the number of iterations to optimize the parameters
    learning_rate -- hyperparameter representing the learning rate used in the update rule of optimize()
    print_cost -- Set to true to print the cost every 100 iterations
    
    Returns:
    d -- dictionary containing information about the model.
    """
    
    ### START CODE HERE ###
    # initialize parameters with zeros (≈ 1 line of code)
    w, b = initialize_with_zeros(X_train.shape[0])
    # Gradient descent (≈ 1 line of code)
    parameters, grads, costs = optimize(w, b, X_train, Y_train, num_iterations, learning_rate, print_cost)
    
    # Retrieve parameters w and b from dictionary "parameters"
    w = parameters["w"]
    b = parameters["b"]

	# Predict test/train set examples (≈ 2 line of code)
	Y_prediction_test = predict(w, b, X_test)
    Y_prediction_train = predict(w, b, X_train)

    ### END CODE HERE ###

    # Print train/test Errors
    print("train accuracy: {} %".format(100 - np.mean(np.abs(Y_prediction_train - Y_train)) * 100))
    print("test accuracy: {} %".format(100 - np.mean(np.abs(Y_prediction_test - Y_test)) * 100))

    
    d = {"costs": costs,
         "Y_prediction_test": Y_prediction_test, 
         "Y_prediction_train" : Y_prediction_train, 
         "w" : w, 
         "b" : b,
         "learning_rate" : learning_rate,
         "num_iterations": num_iterations}
    
    return d
    
d = model(train_set_x, train_set_y, test_set_x, test_set_y, num_iterations = 2000, learning_rate = 0.005, print_cost = True)
>>>
Cost after iteration 0: 0.693147
Cost after iteration 100: 0.584508
Cost after iteration 200: 0.466949
Cost after iteration 300: 0.376007
Cost after iteration 400: 0.331463
Cost after iteration 500: 0.303273
Cost after iteration 600: 0.279880
Cost after iteration 700: 0.260042
Cost after iteration 800: 0.242941
Cost after iteration 900: 0.228004
Cost after iteration 1000: 0.214820
Cost after iteration 1100: 0.203078
Cost after iteration 1200: 0.192544
Cost after iteration 1300: 0.183033
Cost after iteration 1400: 0.174399
Cost after iteration 1500: 0.166521
Cost after iteration 1600: 0.159305
Cost after iteration 1700: 0.152667
Cost after iteration 1800: 0.146542
Cost after iteration 1900: 0.140872
train accuracy: 99.04306220095694 %
test accuracy: 70.0 %

5. 测试图像：

# Example of a picture that was wrongly classified.
index = 1
plt.imshow(test_set_x[:,index].reshape((num_px, num_px, 3)))
print ("y = " + str(test_set_y[0,index]) + ", you predicted that it is a \"" + classes[int(d["Y_prediction_test"][0,index])].decode("utf-8") +  "\" picture.")

>>>
y = 1, you predicted that it is a "cat" picture.

6. 绘制代价函数曲线

# Plot learning curve (with costs)
costs = np.squeeze(d['costs'])
plt.plot(costs)
plt.ylabel('cost')
plt.xlabel('iterations (per hundreds)')
plt.title("Learning rate =" + str(d["learning_rate"]))
plt.show()

如果增大迭代次数，我们可能看到训练集上准确率上升，但测试集上的准确率下降，这时就发生了过拟合。

进一步分析，我们可以尝试不同的学习率对学习曲线的影响。学习率决定了我们更新的步伐有多大。如果学习率过大，我们可能会越过最优值而无法收敛；如果学习率过小，我们需要迭代很多次才能达到一个最优值，所以学习率的选择很重要。

下面比较了不同的学习率所呈现的学习曲线：

learning_rates = [0.01, 0.001, 0.0001]
models = {}
for i in learning_rates:
    print ("learning rate is: " + str(i))
    models[str(i)] = model(train_set_x, train_set_y, test_set_x, test_set_y, num_iterations = 1500, learning_rate = i, print_cost = False)
    print ('\n' + "-------------------------------------------------------" + '\n')

for i in learning_rates:
    plt.plot(np.squeeze(models[str(i)]["costs"]), label= str(models[str(i)]["learning_rate"]))

plt.ylabel('cost')
plt.xlabel('iterations')

legend = plt.legend(loc='upper center', shadow=True)
frame = legend.get_frame()
frame.set_facecolor('0.90')
plt.show()

>>>
learning rate is: 0.01
train accuracy: 99.52153110047847 %
test accuracy: 68.0 %

-------------------------------------------------------

learning rate is: 0.001
train accuracy: 88.99521531100478 %
test accuracy: 64.0 %

-------------------------------------------------------

learning rate is: 0.0001
train accuracy: 68.42105263157895 %
test accuracy: 36.0 %

-------------------------------------------------------

7. 测试自己的数据

把自己的图片放在 /image 文件夹下，然后修改代码中 my_image 指向的图片，就可以对自己的图片进行测试了。

## START CODE HERE ## (PUT YOUR IMAGE NAME) 
my_image = "cat.jpg"   # change this to the name of your image file 
## END CODE HERE ##

# We preprocess the image to fit your algorithm.
fname = "images/" + my_image
image = np.array(ndimage.imread(fname, flatten=False))
my_image = scipy.misc.imresize(image, size=(num_px,num_px)).reshape((1, num_px*num_px*3)).T
my_predicted_image = predict(d["w"], d["b"], my_image)

plt.imshow(image)
print("y = " + str(np.squeeze(my_predicted_image)) + ", your algorithm predicts a \"" + classes[int(np.squeeze(my_predicted_image)),].decode("utf-8") +  "\" picture.")