文章目录
- 编程题 1
- 1. numpy 基本函数
- 1.1 编写 sigmoid 函数
- 1.2 编写 sigmoid 函数的导数
- 1.3 reshape操作
- 1.4 标准化
- 1.5 广播机制
- 2. 向量化
- 2.1 L1L2损失函数
- 编程题 2. 图片?识别
- 1. 导入包
- 2. 数据预览
- 3. 算法的一般结构
- 4. 建立算法
- 4.1 辅助函数
- 4.2 初始化参数
- 4.3 前向后向传播
- 4.4 更新参数，梯度下降
- 4.5 合并所有函数到Model
- 4.6 分析
- 4.7 用自己的照片测试模型
- 5. 总结
选择题测试，请参考 链接博文
编程题 1
1. numpy 基本函数
1.1 编写 sigmoid 函数
代码语言：javascript复制import math
def basic_sigmoid(x):
"""
Compute sigmoid of x.
Arguments:
x -- A scalar
Return:
s -- sigmoid(x)
"""
### START CODE HERE ### (≈ 1 line of code)
s = 1/(1 math.pow(math.e, -x))
# or s = 1/(1 math.exp(-x))
### END CODE HERE ###
return s- 不推荐使用 math 包，因为深度学习里很多都是向量，math 包不能对向量进行计算
### One reason why we use "numpy" instead of "math" in Deep Learning ###
x = [1, 2, 3]
basic_sigmoid(x) # you will see this give an error when you run it, because x is a vector.
# 会报错！import numpy as np
# example of np.exp
x = np.array([1, 2, 3])
print(np.exp(x)) # result is (exp(1), exp(2), exp(3))
# [ 2.71828183 7.3890561 20.08553692]
# numpy 可以对向量进行操作- 使用 numpy 编写的 sigmoid 函数
import numpy as np # this means you can access numpy functions by writing np.function() instead of numpy.function()
def sigmoid(x):
"""
Compute the sigmoid of x
Arguments:
x -- A scalar or numpy array of any size
Return:
s -- sigmoid(x)
"""
### START CODE HERE ### (≈ 1 line of code)
s = 1/(1 np.exp(-x))
### END CODE HERE ###
return sx = np.array([1, 2, 3])
sigmoid(x)
# array([0.73105858, 0.88079708, 0.95257413])1.2 编写 sigmoid 函数的导数
代码语言：javascript复制# GRADED FUNCTION: sigmoid_derivative
def sigmoid_derivative(x):
"""
Compute the gradient (also called the slope or derivative) of the sigmoid function with respect to its input x.
You can store the output of the sigmoid function into variables and then use it to calculate the gradient.
Arguments:
x -- A scalar or numpy array
Return:
ds -- Your computed gradient.
"""
### START CODE HERE ### (≈ 2 lines of code)
s = sigmoid(x)
ds = s*(1-s)
### END CODE HERE ###
return dsx = np.array([1, 2, 3])
sigmoid_derivative(x)
print ("sigmoid_derivative(x) = " str(sigmoid_derivative(x)))
# sigmoid_derivative(x) = [0.19661193 0.10499359 0.04517666]1.3 reshape操作
将照片的数据展平，不想计算的维，可以置为 -1，会自动计算
代码语言：javascript复制# GRADED FUNCTION: image2vector
def image2vector(image):
"""
Argument:
image -- a numpy array of shape (length, height, depth)
Returns:
v -- a vector of shape (length*height*depth, 1)
"""
### START CODE HERE ### (≈ 1 line of code)
v = image.reshape(-1,1)
### END CODE HERE ###
return v# This is a 3 by 3 by 2 array, typically images will be (num_px_x, num_px_y,3) where 3 represents the RGB values
image = np.array([[[ 0.67826139, 0.29380381],
[ 0.90714982, 0.52835647],
[ 0.4215251 , 0.45017551]],
[[ 0.92814219, 0.96677647],
[ 0.85304703, 0.52351845],
[ 0.19981397, 0.27417313]],
[[ 0.60659855, 0.00533165],
[ 0.10820313, 0.49978937],
[ 0.34144279, 0.94630077]]])
print ("image2vector(image) = " str(image2vector(image)))# 输出
image2vector(image) = [[0.67826139]
[0.29380381]
[0.90714982]
[0.52835647]
[0.4215251 ]
[0.45017551]
[0.92814219]
[0.96677647]
[0.85304703]
[0.52351845]
[0.19981397]
[0.27417313]
[0.60659855]
[0.00533165]
[0.10820313]
[0.49978937]
[0.34144279]
[0.94630077]]1.4 标准化
标准化通常使得梯度下降收敛更快。
# GRADED FUNCTION: normalizeRows
def normalizeRows(x):
"""
Implement a function that normalizes each row of the matrix x (to have unit length).
Argument:
x -- A numpy matrix of shape (n, m)
Returns:
x -- The normalized (by row) numpy matrix. You are allowed to modify x.
"""
### START CODE HERE ### (≈ 2 lines of code)
# Compute x_norm as the norm 2 of x. Use np.linalg.norm(..., ord = 2, axis = ..., keepdims = True)
x_norm = np.linalg.norm(x, axis=1, keepdims=True)
# Divide x by its norm.
x = x/x_norm
### END CODE HERE ###
return xx = np.array([
[0, 3, 4],
[1, 6, 4]])
print("normalizeRows(x) = " str(normalizeRows(x)))
# normalizeRows(x) = [[0. 0.6 0.8 ]
# [0.13736056 0.82416338 0.54944226]]1.5 广播机制
官方文档
# GRADED FUNCTION: softmax
def softmax(x):
"""Calculates the softmax for each row of the input x.
Your code should work for a row vector and also for matrices of shape (n, m).
Argument:
x -- A numpy matrix of shape (n,m)
Returns:
s -- A numpy matrix equal to the softmax of x, of shape (n,m)
"""
### START CODE HERE ### (≈ 3 lines of code)
# Apply exp() element-wise to x. Use np.exp(...).
x_exp = np.exp(x)
# Create a vector x_sum that sums each row of x_exp. Use np.sum(..., axis = 1, keepdims = True).
x_sum = np.sum(x_exp, axis=1, keepdims=True)
# Compute softmax(x) by dividing x_exp by x_sum. It should automatically use numpy broadcasting.
s = x_exp/x_sum
### END CODE HERE ###
return sx = np.array([
[9, 2, 5, 0, 0],
[7, 5, 0, 0 ,0]])
print("softmax(x) = " str(softmax(x)))softmax(x) = [[9.80897665e-01 8.94462891e-04 1.79657674e-02 1.21052389e-04
1.21052389e-04]
[8.78679856e-01 1.18916387e-01 8.01252314e-04 8.01252314e-04
8.01252314e-04]]2. 向量化
向量化计算更简洁，更高效
2.1 L1L2损失函数
def L1(yhat, y):
"""
Arguments:
yhat -- vector of size m (predicted labels)
y -- vector of size m (true labels)
Returns:
loss -- the value of the L1 loss function defined above
"""
### START CODE HERE ### (≈ 1 line of code)
loss = np.sum(abs(yhat-y))
### END CODE HERE ###
return lossyhat = np.array([.9, 0.2, 0.1, .4, .9])
y = np.array([1, 0, 0, 1, 1])
print("L1 = " str(L1(yhat,y)))
# L1 = 1.1
代码语言：javascript复制import numpy as np a = np.array([1, 2, 3]) np.dot(a, a) 14
# GRADED FUNCTION: L2
def L2(yhat, y):
"""
Arguments:
yhat -- vector of size m (predicted labels)
y -- vector of size m (true labels)
Returns:
loss -- the value of the L2 loss function defined above
"""
### START CODE HERE ### (≈ 1 line of code)
loss = np.dot(yhat-y, yhat-y)
### END CODE HERE ###
return lossyhat = np.array([.9, 0.2, 0.1, .4, .9])
y = np.array([1, 0, 0, 1, 1])
print("L2 = " str(L2(yhat,y)))
# L2 = 0.43编程题 2. 图片?识别
使用神经网络识别猫
1. 导入包
代码语言：javascript复制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 inline2. 数据预览
弄清楚数据的维度 reshape 数据 标准化数据
有训练集，标签为 y = 1 是猫，y = 0 不是猫 有测试集，带标签的 每个图片是 3 通道的
- 读取数据
# Loading the data (cat/non-cat)
train_set_x_orig, train_set_y, test_set_x_orig, test_set_y, classes = load_dataset()- 预览图片
# Example of a picture
index = 24
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.")y = [1], it's a 'cat' picture.
- 数据大小
### 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 = 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: (209, 64, 64, 3)
train_set_y shape: (1, 209)
test_set_x shape: (50, 64, 64, 3)
test_set_y shape: (1, 50)- 将样本图片矩阵展平
# Reshape the training and test examples
### START CODE HERE ### (≈ 2 lines of code)
train_set_x_flatten = train_set_x_orig.reshape(m_train, -1).T
test_set_x_flatten = test_set_x_orig.reshape(m_test, -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]- 图片的矩阵数值为 0 - 255，标准化数据
train_set_x = train_set_x_flatten/255.
test_set_x = test_set_x_flatten/255.3. 算法的一般结构
用神经网络的思路，建立一个 Logistic 回归
4. 建立算法
定义模型结构（如，输入的特征个数） 初始化模型参数 循环迭代：
- 计算当前损失（前向传播）
- 计算当前梯度（后向传播）
- 更新参数（梯度下降）
4.1 辅助函数
sigmoid函数
# 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 s4.2 初始化参数
逻辑回归的参数可以都设置为 0（神经网络不可以）
代码语言：javascript复制# 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, b4.3 前向后向传播
前向传播：
# 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
# w 是列向量， A 行向量，dot 矩阵乘法
cost = np.sum(Y*np.log(A) (1-Y)*np.log(1-A))/(-m) # compute cost
# Y 行向量，* 对应位置相乘
### 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)/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, costw, 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 = [0.49993523]
cost = 6.0000647731922054.4 更新参数，梯度下降
代码语言：javascript复制# 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, costsparams, 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 = [1.55930492]
dw = [[0.90158428]
[1.76250842]]
db = [0.43046207]- 可以利用学习到的参数来进行预测
计算预测值 Y^=A=σ(wTX b) 根据预测值进行分类，<= 0.5 标记为0，否则为1
代码语言：javascript复制# 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)
Y_prediction[0][i] = 0 if A[0][i] <= 0.5 else 1
### END CODE HERE ###
assert(Y_prediction.shape == (1, m))
return Y_predictionprint ("predictions = " str(predict(w, b, X)))predictions = [[1. 1.]]4.5 合并所有函数到Model
代码语言：javascript复制# 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 = 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 dd = 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: 0.693147
Cost after iteration 100: 0.584508
Cost after iteration 200: 0.466949
Cost after iteration 300: 0.376007
Cost after iteration 400: 0.331463
Cost after iteration 500: 0.303273
Cost after iteration 600: 0.279880
Cost after iteration 700: 0.260042
Cost after iteration 800: 0.242941
Cost after iteration 900: 0.228004
Cost after iteration 1000: 0.214820
Cost after iteration 1100: 0.203078
Cost after iteration 1200: 0.192544
Cost after iteration 1300: 0.183033
Cost after iteration 1400: 0.174399
Cost after iteration 1500: 0.166521
Cost after iteration 1600: 0.159305
Cost after iteration 1700: 0.152667
Cost after iteration 1800: 0.146542
Cost after iteration 1900: 0.140872
train accuracy: 99.04306220095694 %
test accuracy: 70.0 %- 模型在训练集上表现的很好，在测试集上一般，存在过拟合现象
# Example of a picture that was wrongly classified.
index = 24
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.")y = 1, you predicted that it is a "cat" picture.
更改 index 可以查看 测试集的 预测值和真实值
- 绘制代价函数、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()
- 增加训练迭代次数为 3000（上面是2000）
train accuracy: 99.52153110047847 %
test accuracy: 68.0 %
训练集上的准确率上升，但是测试集上准确率下降，这就是过拟合了
4.6 分析
- 不同学习率下的对比
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: 0.01
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 %
-------------------------------------------------------
- 学习率太大的话，容易引起震荡，导致不收敛（本例子0.01，不算太坏，最后收敛了）
- 低的
cost不意味着好的模型，要检查是否过拟合（训练集很好，测试集很差）
4.7 用自己的照片测试模型
代码语言：javascript复制## START CODE HERE ## (PUT YOUR IMAGE NAME)
my_image = "cat1.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 = Image.open(fname)
my_image = np.array(image.resize((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.")
5. 总结
- 处理数据很重要，数据维度，数据标准化
- 各个独立的函数，初始化，前后向传播，梯度下降更新参数
- 组成模型
- 调节学习率等
超参数
