DeepLearing学习笔记-从逻辑回归出发

背景：

从逻辑回归出发，介绍单层神经网络在模式分类中的简单应用。本文将阐述如何用逻辑回归进行猫的识别。从中，我们将创建一个常见的简单的算法模型：
1：参数初始化
2：计算代价函数及其梯度
3：采用优化算法，如梯度下降算法

准备工作：

• numpy 是科学计算的常用库。
• h5py 是python中用于处理H5文件的接口。
• matplotlibpython中常用的图像绘制库。
• PIL and scipy 在本文是用于测试自己训练的模型。
  如上上述依赖库都正常安装的话，那么下面的代码对于库的加载操作就可以正常执行。

import numpy as npimport matplotlib.pyplot as pltimport h5pyimport scipyfrom PIL import Imagefrom scipy import ndimagefrom lr_utils import load_dataset#数据的读取，自定义的文件%matplotlib inline'''首先讲讲这句话的作用，matplotlib是最著名的Python图表绘制扩展库，它支持输出多种格式的图形图像，并且可以使用多种GUI界面库交互式地显示图表。使用%matplotlib命令可以将matplotlib的图表直接嵌入到Notebook之中，或者使用指定的界面库显示图表，它有一个参数指定matplotlib图表的显示方式。inline表示将图表嵌入到Notebook中。'''

数据简介：

数据集基本信息：

本文所采用的数据是data.h是一种h5的数据存储方式，包括：

• 训练数据集及其真实值，是cat则（y=1），非cat则（y=0）。
• 测试数据集及其标注。
• 每张图片的尺寸是(num_px, num_px, 3)，其中的3表示图像是RGB图像，有三个通道。且图像是正方形的，(height = num_px) and (width = num_px)。

数据加载：

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

其中train_set_x_orig和test_set_x_orig就是原始的训练数据集和测试数据集（之后将对其进行预处理操作）。train_set_x_orig和test_set_x_orig中的每一行都是一个矩阵，这里的行根据index游标进行索引就可以遍历。

数据可视化：

# Example of a pictureindex = 25plt.imshow(train_set_x_orig[index])print("type=train_set_x_orig",type(train_set_x_orig))print("shape of image = ",train_set_x_orig[index].shape)print(train_set_y[:, index])#结果是一个矩阵print("type of ",type(train_set_y[:, index]))print("squeeze=",np.squeeze(train_set_y[:, index]))#获取对应的y值print ("y = " + str(train_set_y[:, index]) + ", it's a '" + classes[np.squeeze(train_set_y[:, index])].decode("utf-8") +  "' picture.")

运行结果：

type=train_set_x_orig <class 'numpy.ndarray'>shape of image =  (64, 64, 3)[1]<class 'numpy.ndarray'>squeeze= 1y = [1], it's a 'cat' picture.

获取图像各维度尺寸信息：

我们记：
- m_train (训练样本数量)
- m_test (测试样本数量)
- num_px (训练数据集的长和宽)
需要铭记 train_set_x_orig 是一个numpy-array ，其尺寸是(m_train, num_px, num_px, 3)。所以我们可以通过以下方式train_set_x_orig.shape[0]访问 m_train 的第一个样本。

### START CODE HERE ### (≈ 3 lines of code)m_train = len(train_set_x_orig)m_test = len(test_set_x_orig)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))

运行结果：

Number of training examples: m_train = 209Number of testing examples: m_test = 50Height/Width of each image: num_px = 64Each 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)

从中我们可以知道m_train, m_test and num_px值如下：

m_train 209 m_test 50 num_px 64

为了方便起见，我们对于 尺寸为(num_px, num_px, 3) 的图像进行reshape，使其转化为 尺寸为(num_px ∗∗ num_px ∗∗ 3, 1)的numpy-array 。再转化之后，训练数据集和测试数据集的每一列都是一个扁平化的样本数据，样本数则对应的是列数量。
我们可以通过reshape来实现样本数据的扁平化操作：

X_flatten = X.reshape(X.shape[0], -1).T      # X.T is the transpose of X

具体例子如下：

# Reshape the training and test examples### START CODE HERE ### (≈ 2 lines of code)print("train_set_x_orig shape=",train_set_x_orig.shape)train_set_x_flatten = train_set_x_orig.reshape(train_set_x_orig.shape[0],-1).Ttest_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]))

运行结果：

train_set_x_orig shape= (209, 64, 64, 3)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]

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]

另外，一般在对数据的预处理过程中，我们还常常对数据进行中心化和标准化，即每个样本的像素值减去整体的像素均值，再除以整体的标准差。对于图像数据的话，可以简单除以最大的像素值即可：

train_set_x = train_set_x_flatten/255.test_set_x = test_set_x_flatten/255.

算法流程设计：

逻辑回归如下：

逻辑回归是一种简单的神经网络思路实现。

逻辑回归的数学表达式:

对于其中的一个样本 x(i):

z(i)=wTx(i)+b(1)

y^(i)=a(i)=sigmoid(z(i))(2)

L(a(i),y(i))=−y(i)log(a(i))−(1−y(i))log(1−a(i))(3)

整个训练数据集的代价函数:

J=1m∑i=1mL(a(i),y(i))(6)

关键步骤:

• 模型参数初始化
• 通过最小代价函数学习获取模型的参数
• 对于学习到的模型参数，用来对测试数据集进行预测，实现模型参数的校验
• 分析结果

算法实现：

创建一个神经网络的主要步骤：

1. 定义模型的结构 (如输入特征的数量)
2. 初始化模型参数
3. 做以下循环:
   
   • 计算当前的损失，即前向传播 (forward propagation)
   • 计算当前的梯度，即后向传播 (backward propagation)
   • 更新参数，即采用梯度下降法对参数进行优化更新 (gradient descent)

一般我们可以分开独立地实现1-3步骤，再将其集成到一个模型函数中如model()。

激活函数

由于我们之前分析的的激活函数是 sigmoid()，所以可以利用sigmoid(wTx+b)=11+e−(wTx+b) 进行对输入进行预测。

# GRADED FUNCTION: sigmoiddef 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

测试上述函数：

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

输出：

sigmoid([0, 2]) = [ 0.5         0.88079708]

第一：参数初始化

在这里由于是逻辑回归，我们可以将w初始为全0向量（采用np.zeros()）

代码实现：

# GRADED FUNCTION: initialize_with_zerosdef 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 ### (≈ 1 line of code)    w = np.zeros((dim,1))    b = 0    ### END CODE HERE ###    assert(w.shape == (dim, 1))    assert(isinstance(b, float) or isinstance(b, int))    return w, b

测试代码：

dim = 2w, b = initialize_with_zeros(dim)print ("w = " + str(w))print ("b = " + str(b))

执行结果：

w = [[ 0.] [ 0.]]b = 0

注意，在使用np.zero（）的时候我们是可以通过dtype指定数据类型的，如np.int
对于本文的图像数据，w的尺寸是(num_px × num_px × 3, 1)，wT的尺寸是(1，num_px × num_px × 3)

第二：前向和后向传播

在参数初始化之后，我们采用前向和后向传播来学习模型的参数。

前向传播(Forward Propagation):

• 确定输入X
• 计算 A=σ(wTX+b)=(a(0),a(1),...,a(m−1),a(m))
  -计算代价函数cost function: J=−1m∑mi=1y(i)log(a(i))+(1−y(i))log(1−a(i))

以下的两个公式直接给出，在此处忽略证明过程，后续再补充：

∂J∂w=1mX(A−Y)T(7)

∂J∂b=1m∑i=1m(a(i)−y(i))(8)

代码实现：

# GRADED FUNCTION: propagatedef 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 lines of code)    A = sigmoid(np.dot(w.T,X)+b)            # compute activation    cost = -np.sum(np.dot(Y,np.log(A).T) + np.dot(1-Y,np.log(1-A).T))/m         # compute cost 这里需要转置操作，可以从维度来考虑。Y的维度是(1，m),A的维度也是（1，m）输出结果是（1,1），所以A的log操作是需要转置的    ### END CODE HERE ###    # BACKWARD PROPAGATION (TO FIND GRAD)    ### START CODE HERE ### (≈ 2 lines of code)    dw = np.dot(X,(A-Y).T)/m    db = np.sum(A-Y,axis = 1, keepdims=True)/m#db的维度是（1,1），所以求和，是行方向的求和    ### 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

测试代码：

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))

运行结果：

dw = [[ 0.99993216] [ 1.99980262]]db = [[ 0.49993523]]cost = 6.00006477319

第三：优化

在上述我们：初始化了参数，计算了代价函数和梯度。现在我们需要采用梯度下降来进行参数的更新。
我们的目标是通过最小化cost function J来学习获得 w and b 。对于参数θ, 更新规则： θ=θ−α dθ, 此处的α 是学习率。
每次的迭代都需要计算正向传播和反向传播，从而获取梯度和代价，之后在对模型参数w和b进行更新。

代码实现：

# GRADED FUNCTION: optimizedef 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 lines 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 lines 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

测试代码：

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"]))

运行结果：

w = [[ 0.1124579 ] [ 0.23106775]]b = [[ 1.55930492]]dw = [[ 0.90158428] [ 1.76250842]]db = [[ 0.43046207]]

第四：预测

通过第三步骤的optimize 函数可以获得模型参数 w 和b，我们可以用来对数据集进行预测。本文采用 predict() 函数来进行预测，主要有以下两步：

1. 计算Y^=A=σ(wTX+b)

2. 对于激活输出结果，根据其函数曲线，我们可以将输出值<= 0.5记为0，激活函数输出值 > 0.5时，记为1，新的结果存储于 Y_prediction矩阵中。

代码实现：

# GRADED FUNCTION: predictdef 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    '''    print("size of X=",X.shape)    m = X.shape[1]    Y_prediction = np.zeros((1,m))    print("shape size of w=",w.shape)    w = w.reshape(X.shape[0], 1)    print("reshape size of w=",w.shape)    # 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 ###    print("size of A=",A.shape)    for i in range(A.shape[1]):        # Convert probabilities A[0,i] to actual predictions p[0,i]        ### START CODE HERE ### (≈ 4 lines of code)        Y_prediction[0, i] = 1 if A[0,i] >= 0.5 else 0        ### END CODE HERE ###    assert(Y_prediction.shape == (1, m))    return Y_prediction

测试代码：

print ("predictions = " + str(predict(w, b, X)))

运行结果：

size of X= (2, 2)shape size of w= (2, 1)reshape size of w= (2, 1)size of A= (1, 2)predictions = [[ 1.  1.]]

小结：

至此，我们梳理下此前的步骤：

• 初始化参数（w，b）
• 迭代方式优化代价函数，以学习获取最优的参数(w，b)
  
  • 计算代价函数及其梯度
  • 利用梯度下降法对梯度进行更新
• 采用学习到的最优参数（w，b）对测试数据集进行预测

第五：创建完整的逻辑回归模型

之前的1到4步骤，我们实现了各个独立的函数，所以现在，我们需要将各个模块融合到一个模块中，以创建一个完整的模型。
在模型函数中，我们做以下的符号约定:

• Y_prediction是测试数据集的预测结果
• Y_prediction_train是训练数据集的预测结果
• w, costs, grads是优化函数optimize()的输出结果

代码实现：

# GRADED FUNCTION: modeldef 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 ###    #1： initialize parameters with zeros (≈ 1 line of code)    w, b = initialize_with_zeros(X_train.shape[0])    #2： 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 lines 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

测试代码：

d = model(train_set_x, train_set_y, test_set_x, test_set_y, num_iterations = 2000, learning_rate = 0.005, print_cost = False)

输出结果：

size of X= (12288, 50)shape size of w= (12288, 1)reshape size of w= (12288, 1)size of A= (1, 50)size of X= (12288, 209)shape size of w= (12288, 1)reshape size of w= (12288, 1)size of A= (1, 209)train accuracy: 99.04306220095694 %test accuracy: 70.0 %

从输出结果我们可以看出，模型对训练集拟合得很好，准确率接近100%，而测试数据集的准确率在70%。鉴于数据集较小，这个结果对于线性分类器的逻辑回归这种简单模型来说，不算太坏。此外，我们需要注意的是，训练数据集接近100%的准确率，说明存在过拟合，我们可以通过归一化来处理这个问题。本文暂不展开说明，后续补充。

测试数据集结果查看：

我们可以查看测试数据集的预测结果：

# Example of a picture that was wrongly classified.index = 1plt.imshow(test_set_x[:,index].reshape((num_px, num_px, 3)))print(d["Y_prediction_test"][0,index])#是个floatprint(type(d["Y_prediction_test"][0,index]))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.")

运行结果：

1.0<class 'numpy.float64'>y = 1, you predicted that it is a "cat" picture.

绘制学习曲线：

代码实现：

# 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()

从上图我们可以看出代价函数是在下降的。如果我们增加迭代过程，那么训练数据的准确率会进一步提高，但是测试数据集的准确率可能会明显下降，这就是由于过拟合造成的。

第六： 进一步的分析

学习率的选择：

如果学习率过大，则可能会直接跨越最优值，从而来回震荡；而如果过小的话，收敛速度过慢，迭代次数过多。

我们先对比下不同学习率对应下的学习效果。
代码：

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()

输出结果：

size of X= (12288, 50)shape size of w= (12288, 1)reshape size of w= (12288, 1)size of A= (1, 50)size of X= (12288, 209)shape size of w= (12288, 1)reshape size of w= (12288, 1)size of A= (1, 209)train accuracy: 68.42105263157895 %test accuracy: 36.0 %

从上图可以看出：

• 当学习率过大 (0.01)，代价函数出现上下震荡，甚至可能出现偏离。本文选取的0.01最终幸运地收敛到了一个比较好的结果值，纯属运气。
• 过小的学习率可能会产生过拟合。特别是当训练数据集的准确率远大于测试数据集的时候。

第七：用自己的数据进行验证

至此，我们已经训练出了一个模型，那么我们可以用自己的图片输入到该模型，让模型做判断。

## START CODE HERE ## (PUT YOUR IMAGE NAME) my_image = "my_image2.jpg"   # change this to the name of your image file ## END CODE HERE ### We preprocess the image to fit your algorithm.fname = "images/" + my_imageimage = 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)).Tmy_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.")

输出结果：

size of X= (12288, 1)shape size of w= (12288, 1)reshape size of w= (12288, 1)size of A= (1, 1)y = 1.0, your algorithm predicts a "cat" picture.