歡迎來到你的第一個程式設計作業，在這次作業中你將會用邏輯迴歸去識別一個貓。並且在這次作業中你將會用神經網路的思維去一步一步的去解決這個問題和磨練你的深度學習的直覺。

說明：

• 在你的程式碼中不能使用for或while迴圈，除非說明明確要你這麼做。

你將會學習到：

1.建立一個學習演算法的一般結構，包括

• 初始化引數
• 計算代價函式和它的梯度
• 使用最優化演算法（梯度下降）

2.用正確的順序將上面三個函式集合到一個主函式模型裡。

1 程式包

h5py 是與儲存在一個H5檔案上的資料集互動的公共包。

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 問題集的概述

問題集合”data.h5”包括

• 被標記的訓練集m_train
• 未標記的測試集m_test
• 每個影象由(num_px, num_px, 3)構成，對應於(height ，width，channels)

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

我們在影象集前加了 “_orig” 表示預處理前。
我們也可以通過索引顯示出來。

# 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."
 
)

問題：找出下面的值

• m_train（訓練樣例的個數）
• m_test（測試樣例的個數）
• num_px (= height = width 的訓練圖片)

### 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))

輸出：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: (209L, 64L, 64L, 3L)
train_set_y shape: (1L, 209L)
test_set_x shape: (50L, 64L, 64L, 3L)
test_set_y shape: (1L, 50L)

問題： 重構訓練集和測試集使得一個矩陣影象扁平成一個向量。

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 lines 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]))

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

歸一化資料集

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

這部分內容需要你記住的是：
預處理一個數據集的一般步驟是

• 找出資料集的維度和形狀（m_train, m_test, num_px, …）
• 重構資料集為一個向量 (num_px * num_px * 3, 1)
• 歸一化資料

3 學習演算法的一般架構

以下過程解釋了為什麼邏輯迴歸是一個非常簡單的神經網路

演算法的數學表示式

通過對所有樣例之和，求代價函式

關鍵步驟：

• 初始化模型引數
• 通過最小化代價函式，學習模型引數
• 通過學習到的模型在測試集做預測
• 分析結果和總結

4 構建演算法的各個部分

建立神經網路的主要步驟：

1. 定義模型結構（例如輸入特徵的數量）
2. 初始化模型引數
3. 迴圈部分
   計算當前的損失函式（向前傳播）；
   計算損失函式的梯度（反向傳播）；
   更新引數（梯度下降）。

通常構建1—3個函式，並把它們集合到一個主函式main()中。

4.1 輔助函式

為了做預測，利用上次作業的sigmoid()，函式計算
。

# 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 = N1 / (1 + np.exp(-z))
    ### END CODE HERE ###

    return s

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

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

4.2 初始化引數

將w初始化為與x同維度的零向量

# 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

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

輸出：w = [[ 0.]
[ 0.]]
b = 0

如果輸入為圖片，w將會是(num_px ×× num_px ×× 3, 1)。

4.3 向前傳播和向後傳播

實現propagate()，計算代價函式和他的梯度。

# 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 / 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

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))

輸出：dw = [[ 0.]
[ 0.]]
db = 0.0
cost = 12.0001295464

用梯度下降的方法進行優化
通過最小化損失函式J，學習引數w,b。對於引數P,
更新法則是P=P - a dP,在這裡a是學習速率

# 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 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

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)

輸出：w = [[ 1.]
[ 2.]]
b = 2.0
dw = [[ 0.]
[ 0.]]
db = 0.0
[12.000129546384411]

通過w和b預測X的標籤

1. 計算
2. 以0.5為閾值，預測X的標籤。

# 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

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

輸出：predictions = [[ 1. 1.]]

你需要記住的是，你可以實現一系列的函式

• 初始化（w,b）
• 從學習到的（w,b）當中最優化
  計算損失函式和它的梯度
  利用梯度下降更新引數
• 根據學習到的（w,b）預測樣本集的標籤

5 將所有函式集合在模型當中

利用先前的符號，實現模型函式

• Y_prediction 用於預測測試集
• Y_prediction_train 用於預測訓練集
• w, costs, grads 是最優化函式optimize()的輸出

# 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

執行下面的程式，來訓練你的模型

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

輸出：Cost after iteration 0: 144.867761
Cost after iteration 100: 144.867761
Cost after iteration 200: 144.867761
Cost after iteration 300: 144.867761
Cost after iteration 400: 144.867761
Cost after iteration 500: 144.867761
Cost after iteration 600: 144.867761
Cost after iteration 700: 144.867761
Cost after iteration 800: 144.867761
Cost after iteration 900: 144.867761
Cost after iteration 1000: 144.867761
Cost after iteration 1100: 144.867761
Cost after iteration 1200: 144.867761
Cost after iteration 1300: 144.867761
Cost after iteration 1400: 144.867761
Cost after iteration 1500: 144.867761
Cost after iteration 1600: 144.867761
Cost after iteration 1700: 144.867761
Cost after iteration 1800: 144.867761
Cost after iteration 1900: 144.867761
train accuracy: 65.5502392344 %
test accuracy: 34.0 %

改變索引值，觀察對測試集的預測

# 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.")

輸出：y = 1, you predicted that it is a “cat” picture.

我們還可以畫出成本函式和梯度。

# 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()

解釋
可以看到損失函式在下降。它顯示了這些引數正在被學習。但是，也可以看到，在訓練集上對模型進行更多的訓練，嘗試增加單元中的迭代次數，並重新執行單元格。可能會看到訓練集的準確性提高了，但是測試集的準確性下降了。這就是所謂的過度擬合。

6 進一步的分析

學習速率對其的影響。學習速率過大可能會錯過最佳值，學習速率過慢，可能會導致迭代次數過多。

下面對比，在不同學習速率下，對結果的影響。

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()

輸出：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 %

解釋：

• 不同的學習速率將會有不同的代價函式，因此有不同的預測結果。
• 學習速率太大（0.01），代價函式可能出現最大的波動。
• 代價函式太低，要注意過多你喝的發生。
• 在深度學習中推薦
  降低代價函式的學習速率。
  如果模型出現過擬合，我們可以採用其他方法降低過擬合。

7 測試自己的模型

## START CODE HERE ## (PUT YOUR IMAGE NAME) 
my_image = "cat_in_iran.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.")

輸出：y = 1.0, your algorithm predicts a “cat” picture.

我們在這次的作業中需要意識到：

1. 對資料集的預處理很重要。
2. 獨立實現每一個函式（initialize(), propagate(), optimize(）），然後構建你的模型model()。
3. 學習速率將會對演算法產生巨大的影響。

另外的練習

• 試驗學習速率和迭代次數的關係。
• 嘗試不同的初始化方式，並比較其影響。
• 測試其他的預處理(資料中心，或者按其標準偏差劃分每一行)