Logistic Regression

Logistic Regression

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

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

展示图片

# Example of a picture
index = 100
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.")

张量train_set_x_orig的形状：

(m_train, num_px, num_px, 3)
用例数量*x方向像素数*y方向像素数*3（RGB）

数据集相关信息

### START CODE HERE ### (≈ 3 lines 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))

压平+标准化

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(m_test,-1).T

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

sigmoid函数

def sigmoid(z):
	#z为向量
    s = 1/(1+np.exp(-z))
    return s

初始化函数
初始化w,b
dim为图片向量的维数即nx
w为一个nx维的向量，b为一个实数，二者初始化均为0

def initialize_with_zeros(dim):
    w =np.zeros((dim,1))
    b = 0
    assert(w.shape == (dim, 1))
    assert(isinstance(b, float) or isinstance(b, int))
    return w, b

正向传播函数：
参数：
w,b为神经元参量
X矩阵为m组测试用例的合集
Y是答案向量（0/1），用于和Y_hat进行比对
返回：
向量w的变化率dw
b的变化率db
与真实值的差距cost

def propagate(w, b, X, Y):
    m = X.shape[1]#m组用例（nx=X.shape[0]）
    A = sigmoid(np.dot(w.T,X)+b)                                    
    cost = 1/(-m)*np.sum(Y*np.log(A)+(1-Y)*np.log(1-A))                              
    dw = 1/m*np.dot(X,(A-Y).T)
    db = 1/m*np.sum(A-Y)
    
    assert(dw.shape == w.shape)
    assert(db.dtype == float)
    cost = np.squeeze(cost)
    assert(cost.shape == ())
    grads = {"dw": dw,
             "db": db}
    
    return grads, cost

迭代更新函数

def optimize(w, b, X, Y, num_iterations, learning_rate, print_cost = False):
    """
    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)
    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.
    """
    
    costs = []
    for i in range(num_iterations):
        grads, cost = propagate(w,b,X,Y)
        dw = grads["dw"]
        db = grads["db"]
        
        w = w-learning_rate*dw
        b = b-learning_rate*db
        
        if i % 100 == 0:
            costs.append(cost)
        
        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

预测函数

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]#m用例个数
    Y_prediction = np.zeros((1,m))#生成一个m维的预测结果行向量
    w = w.reshape(X.shape[0], 1)
    
    A = sigmoid(np.dot(w.T,X)+b)

    for i in range(A.shape[1]):
    #若<=0.5则视为不是猫的图片
        if A[0,i]<=0.5:
            Y_prediction[0,i]=0
        else:
            Y_prediction[0,i]=1
    assert(Y_prediction.shape == (1, m)
    return Y_prediction#返回预测结果向量

完整模型
Builds the logistic regression model by calling the function you’ve implemented previously

Arguments:
X_train -- 训练集矩阵，training set represented by a numpy array of shape (num_px * num_px * 3, m_train)
Y_train -- 训练集答案行向量，training labels represented by a numpy array (vector) of shape (1, m_train)
X_test -- 测试集矩阵，test set represented by a numpy array of shape (num_px * num_px * 3, m_test)
Y_test -- 测试集答案行向量，test labels represented by a numpy array (vector) of shape (1, m_test)
num_iterations -- 迭代次数，hyperparameter representing the number of iterations to optimize the parameters
learning_rate -- 学习率，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.

def model(X_train, Y_train, X_test, Y_test, num_iterations = 2000, learning_rate = 0.5, print_cost = False):
    # 初始化参量
    w, b = initialize_with_zeros(X_train.shape[0])
    # Gradient descent
    parameters, grads, costs = optimize(w,b,X_train,Y_train,num_iterations,learning_rate)
    # 获得训练后的参量w,b
    w = parameters["w"]
    b = parameters["b"]
    # 在测试集、训练集上进行预测
    Y_prediction_test = predict(w,b,X_test)
    Y_prediction_train = predict(w,b,X_train)
    # 打印结果
    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

运行模型

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