找到自定义方程的最佳参数

Question

我试图使用优化函数找到自定义方程的最佳参数，但结果不太好，我想我做错了什么，最好使用最小化方法。另外，我听说 Nelder-Mead 算法应该很适合这项任务，但我没有足够的经验在实践中实现它。

我当前的代码：

from scipy.optimize import curve_fit
        
xdata = [0.221,0.23,0.24,0.242,0.233,0.21,0.171,0.221,0.231,0.237,0.227,0.213,0.209,0.209,0.196,0.207,0.213,0.218,0.187,0.196,0.203,0.205,0.219,0.224,0.216,0.205,0.2,0.184,0.169]
zdata = [317,316.6,316.2,315.8,315.4,315,314.6,312.6,312.1,311.7,311.3,310.9,310.5,310.1,301.2,300.8,300.4,300,296.7,296.3,295.9,291,290.6,290.2,289.8,289.4,289,288.6,270.8]
ydata = [0.211,0.199,0.192,0.197,0.212,0.246,0.329,0.252,0.238,0.231,0.251,0.282,0.296,0.305,0.231,0.211,0.203,0.195,0.248,0.234,0.224,0.183,0.163,0.156,0.161,0.167,0.162,0.168,0.253]
  
def func(X,a,b,c,d):
   x,z = X
   return( (a + b * x) + (1 - (a + b * x)) * (c / z) ** (1.0 / d) )

popt, pcov = curve_fit(func, (xdata,zdata), ydata, p0=[-0.1,-0.5,0.01,5], maxfev=1000000)

x、z - 输入变量。两者始终都是积极的价值观。 y - 输出，期望，变量（总是 >= 0 且 <= 1), which I want to predict based on x and z.

a、b、c、d - 使用优化或最小化函数来拟合参数。

Answer 1

你最初的猜测不太好，除非你把它画出来，否则你无法判断。因此，提供一个更好的初始猜测，您将从图中看到它需要一个负数

c

- 如果您将其符号从表达式中取出，它仅在求幂时表现良好。

import matplotlib.pyplot as plt
import numpy as np
from matplotlib.colorbar import Colorbar
from matplotlib.tri import TriContourSet
from scipy.optimize import curve_fit


Params = tuple[float, float, float, float]


def func(xz: np.ndarray, a: float, b: float, c: float, d: float) -> np.ndarray:
    x, z = xz
    abx = a + b*x
    return abx + (1 - abx) * np.sign(c) * (np.abs(c)/z) ** (1/d)


def fit(x: np.ndarray, y: np.ndarray, z: np.ndarray) -> tuple[Params, Params]:
    p0 = (1, -5e-3, -2.5e+5, 1)
    popt, _ = curve_fit(f=func, p0=p0, xdata=(x, z), ydata=y)
    return p0, popt


def plot(xdata: np.ndarray, ydata: np.ndarray, zdata: np.ndarray, p0: Params, popt: Params) -> plt.Figure:
    fig: plt.Figure = plt.figure()
    ax_fit:   plt.Axes = fig.add_subplot(2, 2, 3)
    ax_exp:   plt.Axes = fig.add_subplot(2, 2, 1, sharex=ax_fit)
    ax_guess: plt.Axes = fig.add_subplot(2, 2, 2, sharey=ax_exp)

    vmin = ydata.min()
    vmax = ydata.max()
    contour_set: TriContourSet = ax_exp.tricontourf(xdata, zdata, ydata, vmin=vmin, vmax=vmax)
    ax_guess.tricontourf(xdata, zdata, func((xdata, zdata), *p0), vmin=vmin, vmax=vmax)
    ax_fit.tricontourf(xdata, zdata, func((xdata, zdata), *popt), vmin=vmin, vmax=vmax)

    bar: Colorbar = fig.colorbar(contour_set, ax=[ax_exp, ax_guess, ax_fit])
    bar.set_label('y')
    ax_exp.set_title('Experiment')
    ax_guess.set_title('Initial guess')
    ax_fit.set_title('Fit')
    ax_exp.set_ylabel('z')
    ax_fit.set_xlabel('x')
    ax_fit.set_ylabel('z')

    return fig


def main() -> None:
    xdata = np.array((
        0.221, 0.23, 0.24, 0.242, 0.233, 0.21, 0.171, 0.221, 0.231, 0.237, 0.227, 0.213, 0.209, 0.209, 0.196, 0.207,
        0.213, 0.218, 0.187, 0.196, 0.203, 0.205, 0.219, 0.224, 0.216, 0.205, 0.2, 0.184, 0.169))
    ydata = np.array((
        0.211, 0.199, 0.192, 0.197, 0.212, 0.246, 0.329, 0.252, 0.238, 0.231, 0.251, 0.282, 0.296, 0.305, 0.231, 0.211,
        0.203, 0.195, 0.248, 0.234, 0.224, 0.183, 0.163, 0.156, 0.161, 0.167, 0.162, 0.168, 0.253))
    zdata = np.array((
        317, 316.6, 316.2, 315.8, 315.4, 315, 314.6, 312.6, 312.1, 311.7, 311.3, 310.9, 310.5, 310.1, 301.2, 300.8,
        300.4, 300, 296.7, 296.3, 295.9, 291, 290.6, 290.2, 289.8, 289.4, 289, 288.6, 270.8))

    p0, popt = fit(xdata, ydata, zdata)
    print(popt)
    plot(xdata, ydata, zdata, p0, popt)
    plt.show()


if __name__ == '__main__':
    main()

[ 9.99844947e-01 -8.76454497e-04 -1.05139069e+05  7.56371094e-01]