神经网络与深度学习第二周-2-Logistic Regression with a Neural Network mindset

Logistic Regression with a Neural Network mindset

Welcome to your first (required) programming assignment! You will build a logistic regression classifier to recognize cats. This assignment will step you through how to do this with a Neural Network mindset, and so will also hone your intuitions about deep learning.

Instructions:
- Do not use loops (for/while) in your code, unless the instructions explicitly ask you to do so.(不要使用for/while循环)

You will learn to:
- Build the general architecture of a learning algorithm, including:
- Initializing parameters(初始化参数)
- Calculating the cost function and its gradient(计算代价函数和它的梯度)
- Using an optimization algorithm (gradient descent) (使用优化算法(梯度下降))
- Gather all three functions above into a main model function, in the right order.

1 - Packages

First, let’s run the cell below to import all the packages that you will need during this assignment.
- numpy is the fundamental package for scientific computing with Python.

(numpy是用Python进行科学计算的基本软件包)
- h5py is a common package to interact with a dataset that is stored on an H5 file.

(h5py是与H5文件中存储的数据集交互的常用软件包)
- matplotlib is a famous library to plot graphs in Python.

(matplot是python中一个著名的绘图包)
- PIL and scipy are used here to test your model with your own picture at the end.

(PIL和scipy在这里用来测试你自己的模型和最后的图片)

import numpy as npimport matplotlib.pyplot as pltimport h5pyimport scipyfrom PIL import Imagefrom scipy import ndimagefrom lr_utils import load_dataset%matplotlib inline

2 - Overview of the Problem set

Problem Statement: You are given a dataset (“data.h5”) containing:
- a training set of m_train images labeled as cat (y=1) or non-cat (y=0)

(*标记为猫（y = 1）或者不为猫（y = 0）的m_train（*训练*）图像的训练集合*)- a test set of m_test images labeled as cat or non-cat(*标记为猫（y = 1）或者不为猫（y = 0）的m_test（*测试*）图像的训练集合*)- each image is of shape (num_px, num_px, 3) where 3 is for the 3 channels (RGB). Thus, each image is square (height = num_px) and (width = num_px).(*每个图像的形状是（num_px，num_px，3），其中3是3个通道（RGB）。因此，每个图像是正方形（height = num_px）和（width = num_px）。*)

You will build a simple image-recognition algorithm that can correctly classify pictures as cat or non-cat.

(建立一个简单的逻辑回归算法来正确的分辨出是不是猫)

Let’s get more familiar with the dataset. Load the data by running the following code.

# 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load_dataset()源代码如下,保存在lr_utils.py中

import numpy as npimport h5pydef 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

print(train_set_x_orig.shape)print(train_set_y.shape)print(test_set_x_orig.shape)print(test_set_y.shape)

(209, 64, 64, 3)(1, 209)(50, 64, 64, 3)(1, 50)

We added “_orig” at the end of image datasets (train and test) because we are going to preprocess them. After preprocessing, we will end up with train_set_x and test_set_x (the labels train_set_y and test_set_y don’t need any preprocessing).

(我们在图像数据集（训练和测试）的最后添加了“_orig”，因为我们要对它们进行预处理。经过预处理后，我们将以train_set_x和test_set_x结束（标签train_set_y和test_set_y不需要任何预处理）。)

Each line of your train_set_x_orig and test_set_x_orig is an array representing an image. You can visualize an example by running the following code. Feel free also to change the index value and re-run to see other images.

(train_set_x_orig和test_set_x_orig的每一行都是表示图像的数组。您可以通过运行以下代码来可视化示例。随意更改索引值并重新运行以查看其他图像)

# Example of a pictureindex = 24 #可以为0~208之间的任意一个数，训练集中一共有209组数据plt.imshow(train_set_x_orig[index]) ##绘制第index个图片的图像print ("y = " + str(train_set_y[:, index]) + ", it's a '" + classes[np.squeeze(train_set_y[:, index])].decode("utf-8") +  "' picture.")

y = [1], it's a 'cat' picture.

Many software bugs in deep learning come from having matrix/vector dimensions that don’t fit. If you can keep your matrix/vector dimensions straight you will go a long way toward eliminating many bugs.

Exercise: Find the values for:
- m_train (number of training examples)(训练的数量)
- m_test (number of test examples)(测试的数量)
- num_px (= height = width of a training image)(训练图片的宽和高)
Remember that train_set_x_orig is a numpy-array of shape (m_train, num_px, num_px, 3). For instance, you can access m_train by writing train_set_x_orig.shape[0].(即数组的第一个维度， 索引是从0开始 ).

### 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))

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)

Expected Output for m_train, m_test and num_px:

**m_train** 209 **m_test** 50 **num_px** 64

For convenience, you should now reshape images of shape (num_px, num_px, 3) in a numpy-array of shape (num_px ∗ num_px ∗ 3, 1). After this, our training (and test) dataset is a numpy-array where each column represents a flattened image. There should be m_train (respectively m_test) columns.

((为了方便起见，你现在应该在shape（num_px × num_px × 3，1）的numpy数组中重塑shape（num_px，num_px，3）的图像。在此之后，我们的训练（和测试）数据集是一个numpy阵列，每列代表一个扁平的图像。应该有m_train（分别是m_test）列。))

Exercise: Reshape the training and test data sets so that images of size (num_px, num_px, 3) are flattened into single vectors of shape (num_px ∗ num_px ∗ 3, 1).

(重塑训练和测试数据集，以便将大小（num_px，num_px，3）的图像展平为形状的单个向量（num_px × num_px × 3，1）。)

A trick when you want to flatten a matrix X of shape (a,b,c,d) to a matrix X_flatten of shape (b∗c∗d, a) is to use: (将每一张图片转化为一个宽为1的矩阵)

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]))

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]

Expected Output:

**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]

To represent color images, the red, green and blue channels (RGB) must be specified for each pixel, and so the pixel value is actually a vector of three numbers ranging from 0 to 255.

(为了表示彩色图像，必须为每个像素指定红色，绿色和蓝色通道（RGB），因此像素值实际上是从0到255的三个数字的向量)

One common preprocessing step in machine learning is to center and standardize your dataset, meaning that you substract the mean of the whole numpy array from each example, and then divide each example by the standard deviation of the whole numpy array. But for picture datasets, it is simpler and more convenient and works almost as well to just divide every row of the dataset by 255 (the maximum value of a pixel channel).

(机器学习中一个常见的预处理步骤是对数据集进行中心化和标准化，这意味着你从每个示例中减去整个numpy数组的平均值，然后将每个示例除以整个numpy数组的标准偏差。但是对于图片数据集来说，它更简单，更方便，几乎可以将数据集的每一行除以255（像素通道的最大值）)

Let’s standardize our dataset.

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

What you need to remember:

Common steps for pre-processing a new dataset are:
- Figure out the dimensions and shapes of the problem (m_train, m_test, num_px, …)

(找出问题的尺寸和形状（m_train，m_test，num_px，...）)

- Reshape the datasets such that each example is now a vector of size (num_px * num_px * 3, 1)

(重塑数据集，使每个示例现在是大小的向量（num_px num_px 3，1）)

- “Standardize” the data

标准化化)

3 - General Architecture of the learning algorithm

It’s time to design a simple algorithm to distinguish cat images from non-cat images.

You will build a Logistic Regression, using a Neural Network mindset. The following Figure explains why Logistic Regression is actually a very simple Neural Network!

(建立一个逻辑回归，利用神经网络的思想。下图解释了为什么Logistic回归实际上是一个非常简单的神经网络 )

Mathematical expression of the algorithm:

For one example x(i):

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

ŷ (i)=a(i)=sigmoid(z(i))(2)

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

The cost is then computed by summing over all training examples:

J=1m∑i=1m(a(i),y(i))(6)

Key steps:
In this exercise, you will carry out the following steps:
- Initialize the parameters of the model (初始化模型参数)
- Learn the parameters for the model by minimizing the cost (通过最小化代价函数来学习参数)
- Use the learned parameters to make predictions (on the test set) (使用学习的参数来进行预测)
- Analyse the results and conclude (分析结果得出结论)

4 - Building the parts of our algorithm ## (构建我们的算法部分)

The main steps for building a Neural Network are:
1. Define the model structure (such as number of input features) (定义模型结构（如输入特征的个数）)
2. Initialize the model’s parameters (初始化模型的参数)
3. Loop:
- Calculate current loss (forward propagation)(计算当前的损失（正向传播）)
- Calculate current gradient (backward propagation)(计算代价函数（反向传播）)
- Update parameters (gradient descent)(更新参数（梯度下降）)

You often build 1-3 separately and integrate them into one function we call model().

4.1 - Helper functions

Exercise: Using your code from “Python Basics”, implement sigmoid(). As you’ve seen in the figure above, you need to compute sigmoid(wTx+b)=11+e−(wTx+b) to make predictions. Use np.exp().

# 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 = None    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]

Expected Output:

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

4.2 - Initializing parameters

Exercise: Implement parameter initialization in the cell below. You have to initialize w as a vector of zeros. If you don’t know what numpy function to use, look up np.zeros() in the Numpy library’s documentation.

(在下面的单元格中实现参数初始化。你必须将w初始化为零向量。如果您不知道要使用哪个numpy函数，请在Numpy库文档中查找np.zeros（）。)

# 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 = None    b = None    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 = 2w, b = initialize_with_zeros(dim)print ("w = " + str(w))print ("b = " + str(b))

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

Expected Output:

** w ** [[ 0.] [ 0.]] ** b ** 0

For image inputs, w will be of shape (num_px × num_px × 3, 1).

4.3 - Forward and Backward propagation(正向传播和反向传播)

Now that your parameters are initialized, you can do the “forward” and “backward” propagation steps for learning the parameters.

(现在您的参数已初始化，您可以执行“前进”和“后退”传播步骤来学习参数)

Exercise: Implement a function propagate() that computes the cost function and its gradient.

(*实现一个函数`propagate（）`来计算代价函数及其渐变*)

Hints:

Forward Propagation:
- You get X
- You compute A=σ(wTX+b)=(a(0),a(1),...,a(m−1),a(m))
- You calculate the cost function: J=−1m∑mi=1y(i)log(a(i))+(1−y(i))log(1−a(i))

Here are the two formulas you will be using:

∂J∂w=1mX(A−Y)T(7)

∂J∂b=1m∑i=1m(a(i)−y(i))(8)

# GRADED FUNCTION: propagatedef propagate(w, b, X, Y):    """    Implement the cost function and its gradient for the propagation explained above    (*实现上述传播的代价函数及其梯度*)    Arguments:    w -- weights, a numpy array of size (num_px * num_px * 3, 1)    b -- bias, a scalar(*偏执单元，一个标量*)    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(Y * np.log(A) + (1 - Y)*np.log(1 - A))/(-m)                                  # compute cost    ### 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)/m    ### 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 = 0.499935230625cost = 6.00006477319

print("w = " + str(w))print("b = " + str(b))print("X = " + str(X))print("Y = " + str(Y))

w = [[1] [2]]b = 2X = [[1 2] [3 4]]Y = [[1 0]]

Expected Output:

** dw ** [[ 0.99993216] [ 1.99980262]] ** db ** 0.499935230625 ** cost ** 6.000064773192205

d) Optimization(优化)

• You have initialized your parameters.(你已经初始化了你的参数)
• You are also able to compute a cost function and its gradient.(你也能计代价函数和它的梯度)
• Now, you want to update the parameters using gradient descent.(现在你想用梯度下降更新你的参数)

Exercise: Write down the optimization function. The goal is to learn w and b by minimizing the cost function J. For a parameter θ, the update rule is θ=θ−α dθ, where α is the learning rate.
(写下优化函数，目的是通过最小化代价函数来学习参数w和b。)

# 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

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"]))

w = [[ 0.1124579 ] [ 0.23106775]]b = 1.55930492484dw = [[ 0.90158428] [ 1.76250842]]db = 0.430462071679

Expected Output:

**w** [[ 0.1124579 ] [ 0.23106775]] **b** 1.55930492484 **dw** [[ 0.90158428] [ 1.76250842]] **db** 0.430462071679

Exercise: The previous function will output the learned w and b. We are able to use w and b to predict the labels for a dataset X. Implement the predict() function. There is two steps to computing predictions:
(前面的函数将输出学习到的w和b。我们可以使用w和b来预测数据集X的标签。实现predict（）函数)

1. Calculate Ŷ =A=σ(wTX+b)

2. Convert the entries of a into 0 (if activation <= 0.5) or 1 (if activation > 0.5), stores the predictions in a vector Y_prediction. If you wish, you can use an if/else statement in a for loop (though there is also a way to vectorize this).
   (将a的条目转换为0（如果激活<= 0.5）或1（如果激活> 0.5），则将预测存储在向量“Y_prediction”中。如果你愿意的话，你可以在for循环中使用if /else语句（虽然也有一种方法可以将其向量化）)

# 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    '''    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] = 1        else:            Y_prediction[0][i] = 0        ### END CODE HERE ###    assert(Y_prediction.shape == (1, m))    return Y_prediction

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

predictions = [[ 1.  1.]]

Expected Output:

**predictions** [[ 1. 1.]]

What to remember:
You’ve implemented several functions that:
- Initialize (w,b)
- Optimize the loss iteratively to learn parameters (w,b):
- computing the cost and its gradient
- updating the parameters using gradient descent
- Use the learned (w,b) to predict the labels for a given set of examples

5 - Merge all functions into a model ##(将所有功能合并到模型中)

You will now see how the overall model is structured by putting together all the building blocks (functions implemented in the previous parts) together, in the right order.
(现在，你将看到整个模型是如何构建的，将所有构建模块（前面部分中实现的功能）以正确的顺序放在一起。)

Exercise: Implement the model function. Use the following notation:
- Y_prediction for your predictions on the test set
- Y_prediction_train for your predictions on the train set
- w, costs, grads for the outputs of optimize()

# 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 ###    # 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

Run the following cell to train your model.

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: 0.693147Cost after iteration 100: 0.584508Cost after iteration 200: 0.466949Cost after iteration 300: 0.376007Cost after iteration 400: 0.331463Cost after iteration 500: 0.303273Cost after iteration 600: 0.279880Cost after iteration 700: 0.260042Cost after iteration 800: 0.242941Cost after iteration 900: 0.228004Cost after iteration 1000: 0.214820Cost after iteration 1100: 0.203078Cost after iteration 1200: 0.192544Cost after iteration 1300: 0.183033Cost after iteration 1400: 0.174399Cost after iteration 1500: 0.166521Cost after iteration 1600: 0.159305Cost after iteration 1700: 0.152667Cost after iteration 1800: 0.146542Cost after iteration 1900: 0.140872train accuracy: 99.04306220095694 %test accuracy: 70.0 %

Expected Output:

**Train Accuracy** 99.04306220095694 % **Test Accuracy** 70.0 %

Comment: Training accuracy is close to 100%. This is a good sanity check: your model is working and has high enough capacity to fit the training data. Test error is 68%. It is actually not bad for this simple model, given the small dataset we used and that logistic regression is a linear classifier. But no worries, you’ll build an even better classifier next week!

Also, you see that the model is clearly overfitting the training data. Later in this specialization you will learn how to reduce overfitting, for example by using regularization. Using the code below (and changing the index variable) you can look at predictions on pictures of the test set.

# Example of a picture that was wrongly classified.index = 1plt.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(np.squeeze(d["Y_prediction_test"][0,index]))].decode("utf-8") +  "\" picture.")

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

Let’s also plot the cost function and the gradients.

# 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()

Interpretation:
You can see the cost decreasing. It shows that the parameters are being learned. However, you see that you could train the model even more on the training set. Try to increase the number of iterations in the cell above and rerun the cells. You might see that the training set accuracy goes up, but the test set accuracy goes down. This is called overfitting.
(你可以看到成本下降。它显示参数正在被学习。但是，你看到你可以在训练集上进一步训练模型。尝试增加上面单元格中的迭代次数，然后重新运行单元格。您可能会看到训练集的准确度上升，但测试集准确度下降。这被称为过度拟合。)

6 - Further analysis (optional/ungraded exercise)

Congratulations on building your first image classification model. Let’s analyze it further, and examine possible choices for the learning rate α.

Choice of learning rate

Reminder:
In order for Gradient Descent to work you must choose the learning rate wisely. The learning rate α determines how rapidly we update the parameters. If the learning rate is too large we may “overshoot” the optimal value. Similarly, if it is too small we will need too many iterations to converge to the best values. That’s why it is crucial to use a well-tuned learning rate.
(为了渐变下降，你必须明智地选择学习速度。学习率 alpha决定了我们更新参数的速度。如果学习率太高，我们可能会“超过”最优值。同样，如果它太小，我们将需要太多迭代来收敛到最佳值。这就是为什么使用良好的学习速度至关​​重要。)

Let’s compare the learning curve of our model with several choices of learning rates. Run the cell below. This should take about 1 minute. Feel free also to try different values than the three we have initialized the learning_rates variable to contain, and see what happens.
(我们来比较一下我们模型的学习曲线和几个学习速率的选择。运行下面的单元格。这应该需要大约1分钟。随意尝试不同的值，我们已经初始化的learning_rates变量包含，看看会发生什么。)

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: 0.01train accuracy: 99.52153110047847 %test accuracy: 68.0 %-------------------------------------------------------learning rate is: 0.001train accuracy: 88.99521531100478 %test accuracy: 64.0 %-------------------------------------------------------learning rate is: 0.0001train accuracy: 68.42105263157895 %test accuracy: 36.0 %-------------------------------------------------------

Interpretation:
- Different learning rates give different costs and thus different predictions results.(不同的学习率给不同的成本，从而不同的预测结果。)
- If the learning rate is too large (0.01), the cost may oscillate up and down. It may even diverge (though in this example, using 0.01 still eventually ends up at a good value for the cost). (如果学习率太高（0.01），成本可能会上下波动。它甚至可能会发生分歧（尽管在这个例子中，使用0.01仍然最终会以成本为代价）)
- A lower cost doesn’t mean a better model. You have to check if there is possibly overfitting. It happens when the training accuracy is a lot higher than the test accuracy.(较低的成本并不意味着更好的模式。你必须检查是否有可能过度配合。这种情况发生在训练的准确性远远高于测试精度的情况下。)
- In deep learning, we usually recommend that you:
- Choose the learning rate that better minimizes the cost function.(选择更好地最小化成本函数的学习率)
- If your model overfits, use other techniques to reduce overfitting. (We’ll talk about this in later videos.)(如果你的模型适合，使用其他技术来减少过度拟合。)

7 - Test with your own image (optional/ungraded exercise)

Congratulations on finishing this assignment. You can use your own image and see the output of your model. To do that:
1. Click on “File” in the upper bar of this notebook, then click “Open” to go on your Coursera Hub.
2. Add your image to this Jupyter Notebook’s directory, in the “images” folder
3. Change your image’s name in the following code
4. Run the code and check if the algorithm is right (1 = cat, 0 = non-cat)!

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

/usr/local/lib/python3.5/dist-packages/ipykernel_launcher.py:7: DeprecationWarning: `imread` is deprecated!`imread` is deprecated in SciPy 1.0.0.Use ``matplotlib.pyplot.imread`` instead.  import sys/usr/local/lib/python3.5/dist-packages/ipykernel_launcher.py:8: DeprecationWarning: `imresize` is deprecated!`imresize` is deprecated in SciPy 1.0.0, and will be removed in 1.2.0.Use ``skimage.transform.resize`` instead.y = 0.0, your algorithm predicts a "non-cat" picture.

What to remember from this assignment:
1. Preprocessing the dataset is important.
2. You implemented each function separately: initialize(), propagate(), optimize(). Then you built a model().
3. Tuning the learning rate (which is an example of a “hyperparameter”) can make a big difference to the algorithm. You will see more examples of this later in this course!

Finally, if you’d like, we invite you to try different things on this Notebook. Make sure you submit before trying anything. Once you submit, things you can play with include:
- Play with the learning rate and the number of iterations
- Try different initialization methods and compare the results
- Test other preprocessings (center the data, or divide each row by its standard deviation)

Bibliography:
- http://www.wildml.com/2015/09/implementing-a-neural-network-from-scratch/
- https://stats.stackexchange.com/questions/211436/why-do-we-normalize-images-by-subtracting-the-datasets-image-mean-and-not-the-c