import copy, math
import numpy as np
import matplotlib.pyplot as plt
plt.style.use('./deeplearning.mplstyle')
np.set_printoptions(precision=2) # reduced display precision on numpy arrays
X_train = np.array([[2104, 5, 1, 45], [1416, 3, 2, 40], [852, 2, 1, 35]])
y_train = np.array([460, 232, 178])
# data is stored in numpy array/matrix
print(f"X Shape: {X_train.shape}, X Type:{type(X_train)})")
print(X_train)
print(f"y Shape: {y_train.shape}, y Type:{type(y_train)})")
print(y_train)
'''
X Shape: (3, 4), X Type:<class 'numpy.ndarray'>)
[[2104 5 1 45]
[1416 3 2 40]
[ 852 2 1 35]]
y Shape: (3,), y Type:<class 'numpy.ndarray'>)
[460 232 178]
'''
b_init = 785.1811367994083
w_init = np.array([ 0.39133535, 18.75376741, -53.36032453, -26.42131618])
print(f"w_init shape: {w_init.shape}, b_init type: {type(b_init)}")
'''
w_init shape: (4,), b_init type: <class 'float'>
'''
def predict_single_loop(x, w, b):
"""
single predict using linear regression
Args:
x (ndarray): Shape (n,) example with multiple features
w (ndarray): Shape (n,) model parameters
b (scalar): model parameter
Returns:
p (scalar): prediction
"""
n = x.shape[0]
p = 0
for i in range(n):
p_i = x[i] * w[i]
p = p + p_i
p = p + b
return p
#每一个样本这样计算
# get a row from our training data
x_vec = X_train[0,:]
print(f"x_vec shape {x_vec.shape}, x_vec value: {x_vec}")
# make a prediction
f_wb = predict_single_loop(x_vec, w_init, b_init)
print(f"f_wb shape {f_wb.shape}, prediction: {f_wb}")
'''
x_vec shape (4,), x_vec value: [2104 5 1 45]
f_wb shape (), prediction: 459.9999976194083
'''
def predict(x, w, b):
"""
single predict using linear regression
Args:
x (ndarray): Shape (n,) example with multiple features
w (ndarray): Shape (n,) model parameters
b (scalar): model parameter
Returns:
p (scalar): prediction
"""
p = np.dot(x, w) + b
return p
# get a row from our training data
x_vec = X_train[0,:]
print(f"x_vec shape {x_vec.shape}, x_vec value: {x_vec}")
# make a prediction
f_wb = predict(x_vec,w_init, b_init)
print(f"f_wb shape {f_wb.shape}, prediction: {f_wb}")
def compute_gradient(X, y, w, b):
"""
Computes the gradient for linear regression
Args:
X (ndarray (m,n)): Data, m examples with n features
y (ndarray (m,)) : target values
w (ndarray (n,)) : model parameters
b (scalar) : model parameter
Returns:
dj_dw (ndarray (n,)): 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.
"""
m,n = X.shape #(number of examples, number of features)
#m : row n:column
dj_dw = np.zeros((n,)) #creat a n dimensions list to store the differentiations
dj_db = 0.
for i in range(m): #m templates so need m loops
err = (np.dot(X[i], w) + b) - y[i]
for j in range(n):
dj_dw[j] = dj_dw[j] + err * X[i, j] #n features so need n loops
dj_db = dj_db + err
dj_dw = dj_dw / m
dj_db = dj_db / m
return dj_db, dj_dw
#Compute and display gradient
tmp_dj_db, tmp_dj_dw = compute_gradient(X_train, y_train, w_init, b_init)
print(f'dj_db at initial w,b: {tmp_dj_db}')
print(f'dj_dw at initial w,b: \n {tmp_dj_dw}')
'''
dj_db at initial w,b: -1.673925169143331e-06
dj_dw at initial w,b:
[-2.73e-03 -6.27e-06 -2.22e-06 -6.92e-05]
'''
def gradient_descent(X, y, w_in, b_in, cost_function, gradient_function, alpha, num_iters):
"""
Performs batch gradient descent to learn w and b. Updates w and b by taking
num_iters gradient steps with learning rate alpha
Args:
X (ndarray (m,n)) : Data, m examples with n features
y (ndarray (m,)) : target values
w_in (ndarray (n,)) : initial model parameters
b_in (scalar) : initial model parameter
cost_function : function to compute cost
gradient_function : function to compute the gradient
alpha (float) : Learning rate
num_iters (int) : number of iterations to run gradient descent
Returns:
w (ndarray (n,)) : Updated values of parameters
b (scalar) : Updated value of parameter
"""
# An array to store cost J and w's at each iteration primarily for graphing later
J_history = []
w = copy.deepcopy(w_in) #avoid modifying global w within function
b = b_in
for i in range(num_iters):
# Calculate the gradient and update the parameters
dj_db,dj_dw = gradient_function(X, y, w, b) ##None
# Update Parameters using w, b, alpha and gradient
w = w - alpha * dj_dw ##None
b = b - alpha * dj_db ##None
# Save cost J at each iteration
if i<100000: # prevent resource exhaustion
J_history.append( cost_function(X, y, 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:4d}: Cost {J_history[-1]:8.2f} ")
return w, b, J_history #return final w,b and J history for graphing
# initialize parameters
initial_w = np.zeros_like(w_init)
initial_b = 0.
# some gradient descent settings
iterations = 1000
alpha = 5.0e-7
# run gradient descent
w_final, b_final, J_hist = gradient_descent(X_train, y_train, initial_w, initial_b,
compute_cost, compute_gradient,
alpha, iterations)
print(f"b,w found by gradient descent: {b_final:0.2f},{w_final} ")
m,_ = X_train.shape
for i in range(m):
print(f"prediction: {np.dot(X_train[i], w_final) + b_final:0.2f}, target value: {y_train[i]}")
# initialize parameters
initial_w = np.zeros_like(w_init)
initial_b = 0.
# some gradient descent settings
iterations = 1000
alpha = 5.0e-7
# run gradient descent
w_final, b_final, J_hist = gradient_descent(X_train, y_train, initial_w, initial_b,
compute_cost, compute_gradient,
alpha, iterations)
print(f"b,w found by gradient descent: {b_final:0.2f},{w_final} ")
m,_ = X_train.shape
for i in range(m):
print(f"prediction: {np.dot(X_train[i], w_final) + b_final:0.2f}, target value: {y_train[i]}")
'''
Iteration 0: Cost 2529.46
Iteration 100: Cost 695.99
Iteration 200: Cost 694.92
Iteration 300: Cost 693.86
Iteration 400: Cost 692.81
Iteration 500: Cost 691.77
Iteration 600: Cost 690.73
Iteration 700: Cost 689.71
Iteration 800: Cost 688.70
Iteration 900: Cost 687.69
b,w found by gradient descent: -0.00,[ 0.2 0. -0.01 -0.07]
prediction: 426.19, target value: 460
prediction: 286.17, target value: 232
prediction: 171.47, target value: 178
'''
# plot cost versus iteration
fig, (ax1, ax2) = plt.subplots(1, 2, constrained_layout=True, figsize=(12, 4))
ax1.plot(J_hist) #坐标轴1
ax2.plot(100 + np.arange(len(J_hist[100:])), J_hist[100:])
ax1.set_title("Cost vs. iteration"); ax2.set_title("Cost vs. iteration (tail)")
ax1.set_ylabel('Cost') ; ax2.set_ylabel('Cost')
ax1.set_xlabel('iteration step') ; ax2.set_xlabel('iteration step')
plt.show()
copy，deepcopy()用法
'''
copy.deepcopy()的用法是将某一个变量的值赋值给另一个变量(此时两个变量地址不同)，因为地址不同，所以可以防止变量间相互干扰。
'''
a = [1, 2, 3]
d = a # a和d的地址相同， 看第5行的输出
a[0] = 2
print(d)
print(id(a), id(b)) # id() 输出a和d变量的地址
'''
[2, 2, 3]
2063948923080 2063948923080
'''
import copy
a = [1, 2, 3]
d = copy.deepcopy(a) # a和d的地址不相同
a[0] = 2
print(d)
print(id(a), id(d))
'''
[1, 2, 3]
2793378037960 2793379617288
'''
zeros_like()用法
eg.W_update=np.zeros_like (W); 函数主要是想实现构造一个矩阵W_update，其维度与矩阵W一致，并为其初始化为全0；这个函数方便的构造了新矩阵，无需参数指定shape大小；
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