概述

本文主要內(nèi)容：如何利用Python的來實(shí)現(xiàn)Logistic函數(shù)。包括：初始化、計(jì)算代價(jià)函數(shù)和梯度、使用梯度下降算法進(jìn)行優(yōu)化等并把他們整合成為一個函數(shù)。本文將通過訓(xùn)練來判斷一副圖像是否為貓。

準(zhǔn)備

在這個過程中，我們將會用到如下庫：
numpy：Python科學(xué)計(jì)算中最重要的庫
h5py：Python與H5文件交互的庫
mathplotlib：Python畫圖的庫
PIL：Python圖像相關(guān)的庫
scipy：Python科學(xué)計(jì)算相關(guān)的庫

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

數(shù)據(jù)集

在訓(xùn)練之前，首先需要讀取數(shù)據(jù)，讀取數(shù)據(jù)的代碼如下：

def load_dataset():
    """
    # 加載數(shù)據(jù)集
    """
    train_dataset = h5py.File('E:/python/week2/train_catvnoncat.h5', "r")  #讀取H5文件
    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('E:/python/week2/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]))  #對訓(xùn)練集和測試集標(biāo)簽進(jìn)行reshape
    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
    
train_set_x_orig, train_set_y, test_set_x_orig, test_set_y, classes = load_dataset()

數(shù)據(jù)說明：

對于訓(xùn)練集的標(biāo)簽而言，對于貓，標(biāo)記為1，否則標(biāo)記為0。每一個圖像的維度都是(num_px, num_px, 3)，其中，長寬相同，3表示是RGB圖像。train_set_x_orig和test_set_x_orig中，包含_orig是由于我們稍候需要對圖像進(jìn)行預(yù)處理，預(yù)處理后的變量將會命名為train_set_x和train_set_y。train_set_x_orig中的每一個元素對于這一副圖像，我們可以用如下代碼將圖像顯示出來：

# Example of a picture
index = 25
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.")

接下來，根據(jù)圖像集來計(jì)算出訓(xùn)練集的大小、測試集的大小以及圖片的大?。?/p>

### 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))

Ps：其中X_flatten = X.reshape(X.shape[0], -1).T可以將一個維度為(a,b,c,d)的矩陣轉(zhuǎn)換為一個維度為(b??c??d, a)的矩陣。

接下來，對圖像值進(jìn)行歸一化。

由于圖像的原始值在0到255之間，最簡單的方式是直接除以255即可。

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

logistics介紹

對于每個訓(xùn)練樣本x，其誤差函數(shù)的計(jì)算方式如下：

而整體的代價(jià)函數(shù)計(jì)算如下：

實(shí)現(xiàn)

接下來，我們將按照如下步驟來實(shí)現(xiàn)Logistic：

1. 定義模型結(jié)構(gòu)
2. 初始化模型參數(shù)
3. 循環(huán)
3.1 前向傳播
3.2 反向傳遞
3.3 更新參數(shù)
4. 整合成為一個完整的模型

Step1：實(shí)現(xiàn)sigmod函數(shù)

# 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

Step2：初始化參數(shù)

# 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 ### (≈ 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

Step3：前向傳播與反向傳播

計(jì)算公式如下：

# 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 lines of code)
    A = sigmoid(np.dot(w.T,X)+b)                                    # compute activation
    cost = -1.0/m*(np.sum(Y*np.log(A)+(1-Y)*np.log(1-A)))                                 # compute cost
    ### END CODE HERE ###
    
    # BACKWARD PROPAGATION (TO FIND GRAD)
    ### START CODE HERE ### (≈ 2 lines 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

Step4：更新參數(shù)

更新參數(shù)的公式如下：

完整代碼如下：

# 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 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

Step5：利用訓(xùn)練好的模型對測試集進(jìn)行預(yù)測：

計(jì)算公式如下：

當(dāng)輸入大于0.5時，我們認(rèn)為其預(yù)測認(rèn)為結(jié)果是貓，否則不是貓。

# 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 lines 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

Step5：將以上功能整合到一個模型中：

# 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 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

測試一下該模型吧：

# 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.")

此時觀察打印結(jié)果，測試準(zhǔn)確率已經(jīng)可以達(dá)到70.0%。

而對于訓(xùn)練集，其準(zhǔn)確性達(dá)到了99%。這表明了模型有著一定的過擬合。