用Numpy实现线性回归
现在二维平面上有一系列点point，我们要找到一个一次函数$y=wx+b$，使得所有点到这条直线的距离平方和$\sum(wx+b-y)^2$最小
因此我们可以定义损失函数$\mathcal{L} = (wx+b-y)^2$，计算损失的代码如下：
# compute loss
def compute_error_for_line_given_points(b, w, points):
totalError = 0
for i in range(len(points)):
x = points[i, 0]
y = points[i, 1]
totalError += (y - (w * x + b)) ** 2
return totalError / float(len(points)) # average然后用梯度下降法更新$w$和$b$
$$ \begin{aligned} w' &= w - lr*\frac{\partial \mathcal{L}}{\partial w}\\ b' &= b - lr*\frac{\partial \mathcal{L}}{\partial b} \end{aligned} $$
其中$\frac{\partial \mathcal{L}}{\partial w} = 2 * x * (wx + b - y),\frac{\partial \mathcal{L}}{\partial b} = 2 * (wx + b - y)$
# compute gradient
def step_gradient(b_current, w_current, points, learningRate):
b_gradient = 0
w_gradient = 0
N = float(len(points))
for i in range(len(points)):
x = points[i, 0]
y = points[i, 1]
b_gradient += 2 * ((w_current * x) + b_current - y)
w_gradient += 2 * x * ((w_current * x) + b_current - y)
b_gradient = b_gradient / N
w_gradient = w_gradient / N
new_b = b_current - (learningRate * b_gradient)
new_w = w_current - (learningRate * w_gradient)
return [new_b, new_w]最后只要设定迭代次数，不断的重复更新$w$和$b$就行了
def gradient_descent_runner(points, starting_b, starting_w, learning_rate, num_iterations): # num_iteration 迭代次数
b = starting_b
w = starting_w
for i in range(num_iterations):
b, w = step_gradient(b, w, np.array(points), learning_rate)
return [b, w]主函数
def run():
points = np.genfromtxt("data.txt", delimiter=",")
learning_rate = 0.0001
initial_b = random()
initial_w = random()
num_iterations = 1000
print("Starting gradient descent at b = {0}, w = {1}, error = {2}"
.format(initial_b, initial_w,
compute_error_for_line_given_points(initial_b, initial_w, points)))
print("Running...")
[b, w] = gradient_descent_runner(points, initial_b, initial_w, learning_rate, num_iterations)
print("After {0} iterations at b = {1}, w = {2}, error = {3}"
.format(num_iterations, b, w,
compute_error_for_line_given_points(b, w, points)))
run()
