指定具有未知系数和数据的函数，并在 python 中查找系数

Question

我有一个函数：f(theta) = a+b*cos(theta - c) 以及采样数据。我想找到最小化均方误差的系数 a、b 和 c。知道 python 中是否有有效的方法吗？

编辑：

import numpy as np
from scipy.optimize import curve_fit

#definition of the function 
def myfunc(x, a, b, c):
    return a + b * np.cos(x - c)

#sample data
x_data = [0, 60, 120, 180, 240, 300]
y_data = [25, 40, 70, 30, 10, 15]

#the actual curve fitting procedure, a, b, c are stored in popt
popt, _pcov = curve_fit(myfunc, x_data, y_data)
print(popt)
print(np.degrees(popt[2]))

#the rest is just a graphic representation of the data points and the fitted curve
from matplotlib import pyplot as plt

#x_fit = np.linspace(-1, 6, 1000)
y_fit = myfunc(x_data, *popt)

plt.plot(x_data, y_data, "ro")
plt.plot(x_data, y_fit, "b")
plt.xlabel(r'$\theta$ (degrees)');
plt.ylabel(r'$f(\theta)$');

plt.legend()
plt.show()

这是一张显示曲线如何与点不完全吻合的图片。看起来振幅应该更高。局部最小值和最大值似乎在正确的位置。

Answer 1

scipy.optimize.curve_fit 使将数据点适合您的自定义函数变得非常容易：

import numpy as np
from scipy.optimize import curve_fit

#definition of the function 
def myfunc(x, a, b, c):
    return a + b * np.cos(x - c)

#sample data
x_data = np.arange(5)
y_data = 2.34 + 1.23 * np.cos(x_data + .23)

#the actual curve fitting procedure, a, b, c are stored in popt
popt, _pcov = curve_fit(myfunc, x_data, y_data)
print(popt)

#the rest is just a graphic representation of the data points and the fitted curve
from matplotlib import pyplot as plt

x_fit = np.linspace(-1, 6, 1000)
y_fit = myfunc(x_fit, *popt)

plt.plot(x_data, y_data, "ro", label = "data points")
plt.plot(x_fit, y_fit, "b", label = "fitted curve\na = {}\nb = {}\nc = {}".format(*popt))

plt.legend()
plt.show()

输出：

[ 2.34  1.23 -0.23]

编辑：

您的问题更新引入了几个问题。首先，您的 x 值是度数，而 np.cos expects values in radians. Therefore, we better convert the values with np.deg2rad. The reverse function would be np.rad2deg.
其次，适应不同的频率也是一个好主意，让我们为此引入一个额外的参数。
第三，拟合通常对初始猜测非常敏感。您可以为此在 scipy 中提供一个参数 p0。
第四，您将拟合曲线的分辨率更改为数据点的低分辨率，因此它看起来采样不足。如果我们解决所有这些问题：

import numpy as np
from scipy.optimize import curve_fit

#sample data
x_data = [0, 60, 120, 180, 240, 300]
y_data = [25, 40, 70, 30, 10, 15]

#definition of the function with additional frequency value d
def myfunc(x, a, b, c, d):
    return a + b * np.cos(d * np.deg2rad(x) - c)
#initial guess of parameters a, b, c, d
p_initial = [np.average(y_data), np.average(y_data), 0, 1]

#the actual curve fitting procedure, a, b, c, d are stored in popt
popt, _pcov = curve_fit(myfunc, x_data, y_data, p0 = p_initial)
print(popt)
#we have to convert the phase shift back into degrees
print(np.rad2deg(popt[2]))

#graphic representation of the data points and the fitted curve
from matplotlib import pyplot as plt
#define x_values for a smooth curve representation
x_fit = np.linspace(np.min(x_data), np.max(x_data), 1000)
y_fit = myfunc(x_fit, *popt)

plt.plot(x_data, y_data, "ro", label = "data")
plt.plot(x_fit, y_fit, "b", label = "fit")
plt.xlabel(r'$\theta$ (degrees)');
plt.ylabel(r'$f(\theta)$');

plt.legend()
plt.show()

我们得到这个输出：

[34.31293761 26.92479369  2.20852009  1.18144319]
126.53888003953764

指定具有未知系数和数据的函数，并在 python 中查找系数

Specify a function with unknown coefficients as well as data and find coefficients in python

python

scipy

curve-fitting

minimize

mse