Lab01 linear regression with one variable
import numpy as np
import matplotlib.pyplot as plt
plt.style.use('./deeplearning.mplstyle')- NumPy, a popular library for scientific computing
- Matplotlib, a popular library for plotting data
plt.style.use()是 Matplotlib 库中用于设置绘图样式的函数。通过使用
plt.style.use()函数，我们可以方便地应用各种预定义的风格或自定义的样式表。Matplotlib 提供了许多内置风格表，包括'default'(默认)、'ggplot'、'seaborn'等。此外，你还可以创建和使用自己的样式表。
# x_train is the input variable (size in 1000 square feet)
# y_train is the target (price in 1000s of dollars)
x_train = np.array([1.0, 2.0])
y_train = np.array([300.0, 500.0])
print(f"x_train = {x_train}")
print(f"y_train = {y_train}")NDArray和list的区别：
内存占用：ndarray 占用的内存相对较小，因为它是在一个连续的块中分配并保存数据的。列表则不同，它由多个对象所组成，并且保存每个对象的引用。这意味着，对于相同数量的数据，列表需要更多的内存空间。
可变性：列表是可变对象，它允许添加、删除或修改元素。相反，ndarray 是不可变对象，你只能通过创建一个新的数组来改变其内容。 这使得 ndarray 更加适用于数值计算，因为它们可以充分发挥硬件支持的优化技术（如向量化运算），从而提高代码的执行效率。
访问速度：由于 ndarray 存储数据是连续和紧凑的，因此它的访问速度要快于列表。与此相比，列表的元素存储位置可能会散布在内存中不同的位置，因此每次访问列表时都需要先遍历列表中的元素指针，然后到内存中找到相应的数据才能读取。
数据类型：ndarray 可以保存同一种数据类型的序列，这使得许多针对数值计算优化的库（如 NumPy 和 SciPy 等）可以更加高效地处理数据。而在列表中，则可以包含不同类型的对象，例如字符串、整数和浮点数等。
x_train和y_train是NDArray
np.array() 创建数组的几种常见方式：
1.从 Python 列表或元组创建：
import numpy as np # 从 Python 列表创建一维数组 a = np.array([1, 2, 3]) print(a) # 输出: [1 2 3] # 从 Python 列表创建二维数组（矩阵） b = np.array([[1, 2, 3], [4, 5, 6]]) print(b) # 输出： # [[1 2 3] # [4 5 6]]2.使用函数
np.zeros()和np.ones()创建全零或全一数组：import numpy as np # 创建一个长度为5的全零数组 a = np.zeros(5) print(a) # 输出: [0. 0. 0. 0. 0.] # 创建一个形状为 2×3 的全一数组（矩阵） b = np.ones((2,3)) print(b) # 输出： # [[1. 1. 1.] # [1. 1. 1.]]3.使用函数
np.arange()创建等差数列：import numpy as np # 创建一个值域为[0,9]的整数数组 a = np.arange(10) print(a) # 输出: [0 1 2 3 4 5 6 7 8 9] # 创建一个值域为[1,10]，公差为2的整数数组 b = np.arange(1, 11, 2) print(b) # 输出: [1 3 5 7 9]4.使用函数
np.linspace()创建等分数列：import numpy as np # 创建一个值域为[0,1]，长度为5的等分数列 a = np.linspace(0, 1, 5) print(a) # 输出: [0. 0.25 0.5 0.75 1. ] # 创建一个值域为[1,10]，长度为4的等分数列 b = np.linspace(1, 10, 4) print(b) # 输出: [ 1. 4. 7. 10.]5.NumPy 还提供了很多其他创建数组的方法，例如随机数生成、从文件读取数据等。
# m is the number of training examples
print(f"x_train.shape: {x_train.shape}")
m = x_train.shape[0]
print(f"Number of training examples is: {m}")
#另一种方法
# m is the number of training examples
m = len(x_train)
print(f"Number of training examples is: {m}")Numpy arrays have a `.shape` parameter.
`x_train.shape` returns a python tuple with an entry for each dimension.
`x_train.shape[0]` is the length of the array and number of examples as shown below.
i = 0 # Change this to 1 to see (x^1, y^1)
x_i = x_train[i]
y_i = y_train[i]
print(f"(x^({i}), y^({i})) = ({x_i}, {y_i})")(x^(0), y^(0)) = (1.0, 300.0)
x^(0)代表第0组数据的input
# Plot the data points
plt.scatter(x_train, y_train, marker='*', c='r')
# Set the title
plt.title("Housing Prices")
# Set the y-axis label
plt.ylabel('Price (in 1000s of dollars)')
# Set the x-axis label
plt.xlabel('Size (1000 sqft)')
plt.show()The function arguments `marker` and `c` show the points as red crosses (the default is blue dots).
#手动赋值
w = 100
b = 100
print(f"w: {w}")
print(f"b: {b}")将所有数据进行计算
def compute_model_output(x, w, b):
"""
Computes the prediction of a linear model
Args:
x (ndarray (m,)): Data, m examples
w,b (标量) : 模型参数
Returns
y (ndarray (m,)): target values
"""
m = x.shape[0]
f_wb = np.zeros(m)
for i in range(m):
f_wb[i] = w * x[i] + b
return f_wb数据可视化
tmp_f_wb = compute_model_output(x_train, w, b,)
# Plot our model prediction
plt.plot(x_train, tmp_f_wb, c='b',label='Our Prediction')
# Plot the data points
plt.scatter(x_train, y_train, marker='x', c='r',label='Actual Values')
# Set the title
plt.title("Housing Prices")
# Set the y-axis label
plt.ylabel('Price (in 1000s of dollars)')
# Set the x-axis label
plt.xlabel('Size (1000 sqft)')
plt.legend()
plt.show()
As you can see, setting w = 100 and b = 100 does not result in a line that fits our data.
Try w = 200 and b = 100
有了基本模型后，我们可以进行预测
w = 200
b = 100
x_i = 1.2
cost_1200sqft = w * x_i + b
print(f"${cost_1200sqft:.0f} thousand dollars")$340 thousand dollars
Lab02 Cost Function
import numpy as np
%matplotlib widget
import matplotlib.pyplot as plt
from lab_utils_uni import plt_intuition, plt_stationary, plt_update_onclick, soup_bowl
plt.style.use('./deeplearning.mplstyle')
x_train = np.array([1.0, 2.0]) #(size in 1000 square feet)
y_train = np.array([300.0, 500.0]) #(price in 1000s of dollars)转化成NDArray格式
#计算代价函数J
def compute_cost(x, y, w, b):
"""
Computes the cost function for linear regression.
Args:
x (ndarray (m,)): Data, m examples
y (ndarray (m,)): target values
w,b (scalar) : model parameters
Returns
total_cost (float): The cost of using w,b as the parameters for linear regression
to fit the data points in x and y
"""
# number of training examples
m = x.shape[0]
cost_sum = 0
for i in range(m):
f_wb = w * x[i] + b
cost = (f_wb - y[i]) ** 2
cost_sum = cost_sum + cost
total_cost = (1 / (2 * m)) * cost_sum
return total_cost定义代价函数，迭代求一组w,b中的cost，即
plt_intuition(x_train,y_train)
x_train = np.array([1.0, 1.7, 2.0, 2.5, 3.0, 3.2])
y_train = np.array([250, 300, 480, 430, 630, 730,])换更大的数据集进行实验
plt.close('all')
fig, ax, dyn_items = plt_stationary(x_train, y_train)
updater = plt_update_onclick(fig, ax, x_train, y_train, dyn_items) 
plt_stationary(x_train, y_train)返回三个变量：fig是一个 Figure 对象，表示 Matplotlib 中的绘图窗口；ax是一个 Axes 对象，表示图中的坐标轴；dyn_items是一个包含动态元素的 tuple （元组）。
plt_update_onclick(fig, ax, x_train, y_train, dyn_items)生成了一个更新函数updater来响应通过鼠标点击图形窗口所产生的事件，并将其链接到绘图对象fig和ax上。该函数还通过传递参数x_train、y_train和dyn_items将图形和原始数据链接起来，以便响应事件时，可以动态地更新数据并重新绘制图形。
soup_bowl()
-
w = 200, b = 0, cost = 1736, it is near theminimum (w = 209, b = 2.4)
Lib03 Gradient_Descent
数据导入与代价函数的计算在Lib02已经完成，这里不再赘述
# x_train = []
# y_train = []
# with open('input.txt','r',encoding='UTF-8') as f:
# content = f.readlines()
# for line in content:
# line = line.split(',')
# y_train.append(line[0])
# x_train.append(line[1])
# x_train = np.ndarray(x_train)
# y_train = np.ndarray(y_train)
x_train = np.array([1.0, 1.7, 2.0, 2.5, 3.0, 3.2])
y_train = np.array([250, 300, 480, 430, 630, 730,])输入六组数据测试，也可用文件输入
#计算对w,b的偏导数
def compute_gradient(x, y, w, b):
"""
Computes the gradient for linear regression
Args:
x (ndarray (m,)): Data, m examples
y (ndarray (m,)): target values
w,b (scalar) : model parameters
Returns
dj_dw (scalar): The gradient of the cost w.r.t. the parameters w
dj_db (scalar): The gradient of the cost w.r.t. the parameter b
"""
# Number of training examples
m = x.shape[0]
dj_dw = 0
dj_db = 0
for i in range(m):
f_wb = w * x[i] + b
dj_dw_i = (f_wb - y[i]) * x[i]
dj_db_i = f_wb - y[i]
dj_db += dj_db_i
dj_dw += dj_dw_i
dj_dw = dj_dw / m
dj_db = dj_db / m
return dj_dw, dj_db
计算偏导数的函数，对w和对b的偏导数学公式如上图
下面是推导过程
plt_gradients(x_train,y_train, compute_cost, compute_gradient)
plt.show()Let's use our `compute_gradient` function to find and plot some partial derivatives of our cost function relative to one of the parameters, .
可以利用 plt_gradients函数寻找并绘制成本函数相对于其中一个参数w0的一些偏导数
plt_gradients函数原型：
def gradient_descent(x, y, w_in, b_in, alpha, num_iters, cost_function, gradient_function):
"""
Performs gradient descent to fit w,b. Updates w,b by taking
num_iters gradient steps with learning rate alpha
Args:
x (ndarray (m,)) : Data, m examples
y (ndarray (m,)) : target values
w_in,b_in (scalar): initial values of model parameters
alpha (float): Learning rate
num_iters (int): number of iterations to run gradient descent
cost_function: function to call to produce cost
gradient_function: function to call to produce gradient
Returns:
w (scalar): Updated value of parameter after running gradient descent
b (scalar): Updated value of parameter after running gradient descent
J_history (List): History of cost values
p_history (list): History of parameters [w,b]
"""
w = copy.deepcopy(w_in) # avoid modifying global w_in
# An array to store cost J and w's at each iteration primarily for graphing later
J_history = []
p_history = []
b = b_in
w = w_in
for i in range(num_iters):
# Calculate the gradient and update the parameters using gradient_function
dj_dw, dj_db = gradient_function(x, y, w , b)
# Update Parameters using equation (3) above
b = b - alpha * dj_db
w = w - alpha * dj_dw
# Save cost J at each iteration
if i<100000: # prevent resource exhaustion
J_history.append( cost_function(x, y, w , b))
p_history.append([w,b])
# Print cost every at intervals 10 times or as many iterations if < 10
if i% math.ceil(num_iters/10) == 0:
print(f"Iteration {i:4}: Cost {J_history[-1]:0.2e} ",
f"dj_dw: {dj_dw: 0.3e}, dj_db: {dj_db: 0.3e} ",
f"w: {w: 0.3e}, b:{b: 0.5e}")
return w, b, J_history, p_history #return w and J,w history for graphing官方梯度下降函数，J_history记录每次迭代的cost值，p_history记录每次迭代选取的w和b值，迭代num_iters次，每次修改w,b的值
# initialize parameters
w_init = 0
b_init = 0
# some gradient descent settings
iterations = 10000
tmp_alpha = 1.0e-2
# run gradient descent
w_final, b_final, J_hist, p_hist = gradient_descent(x_train ,y_train, w_init, b_init, tmp_alpha,
iterations, compute_cost, compute_gradient)
print(f"(w,b) found by gradient descent: ({w_final:8.4f},{b_final:8.4f})")w,b初始化为0，迭代10000次，学习速率α设为0.01
# plot cost versus iteration
fig, (ax1, ax2) = plt.subplots(1, 2, constrained_layout=True, figsize=(12,4))
ax1.plot(J_hist[:100])
ax2.plot(1000 + np.arange(len(J_hist[1000:])), J_hist[1000:])
ax1.set_title("Cost vs. iteration(start)"); ax2.set_title("Cost vs. iteration (end)")
ax1.set_ylabel('Cost') ; ax2.set_ylabel('Cost')
ax1.set_xlabel('iteration step') ; ax2.set_xlabel('iteration step')
plt.show()画图观察迭代前后代价函数 的图像
print(f"1000 sqft house prediction {w_final*1.0 + b_final:0.1f} Thousand dollars")
print(f"1200 sqft house prediction {w_final*1.2 + b_final:0.1f} Thousand dollars")
print(f"2000 sqft house prediction {w_final*2.0 + b_final:0.1f} Thousand dollars") 
利用梯度下降算法得到的w,b进行预测
以下是一些数据的可视化，红色箭头的梯度下降的过程文章来源：https://www.toymoban.com/news/detail-464338.html
文章来源地址https://www.toymoban.com/news/detail-464338.html
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