一. 多项式拟合

import numpy as np
import matplotlib.pyplot as plt


def cost_function(w, b, data):
    total_cost = 0
    M = len(data)
    for i in range(M):
        x = data[i, 0]
        y = data[i, 1]
        total_cost = 0.5/M * (np.square(y-w*x-b)).sum()
    return total_cost


def step_gradient(b_current, w_current, data, alpha):
    b_sum = 0.0
    w_sum = 0.0
    M = float(len(data))

    # 对每个点代入公式求和
    for i in range(0, len(data)):
        x = data[i, 0]
        y = data[i, 1]
        tmp = w_current * x + b_current - y
        b_sum += tmp
        w_sum += tmp * x

    # 用公式求当前梯度
    grad_w = 2 / M * w_sum
    grad_b = 2 / M * b_sum

    # 梯度下降，更新当前的w和b
    b_new = b_current - alpha * grad_b
    w_new = w_current - alpha * grad_w
    return [b_new, w_new]


def gradient_descent(data, b_start, w_start, alpha, num_iterations):
    b = b_start
    w = w_start
    # 保存所有的损失函数值，用来显示下降过程
    cost_list=[]
    for i in range(num_iterations):
        cost_list.append(cost_function(w, b, data))
        b_new, w_new = step_gradient(b, w, data, alpha)
        b = b_new
        w = w_new
    return [b, w, cost_list]


#  create data
b_start = 1
w_start = 1
alpha = float(input("步长："))
num_iterations = int(input("迭代次数："))
data = [
    [0.0, 0],
    [0.9, 10],
    [1.9, 30],
    [3.0, 50],
    [3.9, 80],
    [5.0, 110]
]
data = np.array(data)
x = data[:, 0]
y = data[:, 1]

# train
b_now, w_now, cost_list = gradient_descent(data, b_start, w_start, alpha, num_iterations)
print("w is :", w_now)
print("b is :", b_now)
print("运动方程S=%f*t+(%f)"%(w_now, b_now))

# 精确值
p = np.polyfit(x, y, deg=1)
print("参考系数:", p)

# plot
plt.scatter(x, y)
ori_y = x * w_start + b_start
plt.plot(x, ori_y, color="green", label="origin")
pre_y = w_now * x + b_now
plt.plot(x, pre_y, color="red", label="fit")
plt.xlabel("x")
plt.ylabel("y")
plt.legend()
plt.figure()
cost = cost_function(w_now, b_now, data)
# print("cost_list:", cost_list)
print("cost is:", cost)
plt.plot(cost_list)

plt.show()

二. 曲线拟合

import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit

# 原始散点坐标
x = [70,100,120,150,180, 200]
y = [0.95,0.85,0.5,0.2,0.05, 0]

xd = np.array([70,100,120,150,180])
yd = np.log(np.array([0.95,0.85,0.5,0.2,0.05]))

# 表达式
def func(x, d, n):
    return - np.power(x / d, n)


# 非线性最小二乘拟合
popt, pcov = curve_fit(func, xd, yd)


# 获取popt里的拟合系数
d = popt[0]
n = popt[1]

# 拟合yd的值

yvals = np.exp(func(x, d, n))
print("d:", d)
print("n:", n)


# 绘图
plt.figure()
plt.plot(x, y, 'd', label="Original Values")
xx = np.linspace(70,200,50)
y_vals = np.exp(func(xx, d, n))
plt.plot(xx, y_vals, 'r', label="$fit: y_d = e^{-(\\frac{d}{%5.5f})^{%5.5f}}$" % tuple(popt))
plt.xlabel("diameter")
plt.ylabel("mass fraction")
plt.legend(loc=1) # 图例位置
plt.show()

自定义离散点进行指定多项式函数拟合

用户自己自己输入坐标点，拟合函数可根据用户输入的多项式的最高次方进行自动拟合函数，拟合方法采用最小二乘法进行函数拟合。

(1,2),(2,5),(3,10),(4,17),(5,26),(6,37)(7,50),(8,65),(9,82)
很显然是函数为二次函数，y=x^2+1
代码中，f1 = np.polyfit(x, y, 2)，2表示使用二次函数进行拟合

import numpy as np
import matplotlib.pyplot as plt

#定义x、y散点坐标
x = [1,2,3,4,5,6,7,8,9]
x = np.array(x)
print('x is :\n',x)

num = [2,5,10,17,26,37,50,65,82]

y = np.array(num)
print('y is :\n',y)

#用多次多项式拟合
f1 = np.polyfit(x, y, 2)
print('f1 is :\n',f1)
p1 = np.poly1d(f1)
print('p1 is :\n',p1)

#也可使用yvals=np.polyval(f1, x)
yvals = p1(x) #拟合y值
print('yvals is :\n',yvals)

#绘图
plot1 = plt.plot(x, y, 's',label='original values')
plot2 = plt.plot(x, yvals, 'r',label='polyfit values')
plt.xlabel('grade')
plt.ylabel('value')
plt.legend(loc=4) #指定标识的位置(1为右上角，2为左上角，3为左下角，4为右下角)
plt.title('Color fastness grade')
plt.show()

运行效果如下：