Python中的曲线拟合指数增长函数

Question

我有以下数据点想要曲线拟合：

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

t = np.array([15474.6, 15475.6, 15476.6, 15477.6, 15478.6, 15479.6, 15480.6,
              15481.6, 15482.6, 15483.6, 15484.6, 15485.6, 15486.6, 15487.6,
              15488.6, 15489.6, 15490.6, 15491.6, 15492.6, 15493.6, 15494.6,
              15495.6, 15496.6, 15497.6, 15498.6, 15499.6, 15500.6, 15501.6,
              15502.6, 15503.6, 15504.6, 15505.6, 15506.6, 15507.6, 15508.6,
              15509.6, 15510.6, 15511.6, 15512.6, 15513.6])

v = np.array([4.082, 4.133, 4.136, 4.138, 4.139, 4.14, 4.141, 4.142, 4.143,
              4.144, 4.144, 4.145, 4.145, 4.147, 4.146, 4.147, 4.148, 4.148,
              4.149, 4.149, 4.149, 4.15, 4.15, 4.15, 4.151, 4.151, 4.152,
              4.152, 4.152, 4.153, 4.153, 4.153, 4.153, 4.154, 4.154, 4.154,
              4.154, 4.154, 4.155, 4.155])

我想拟合数据的指数函数是：

表示上述公式的 Python 函数以及与数据拟合的关联曲线详述如下：

def func(t, a, b, alpha):
    return a - b * np.exp(-alpha * t)


# scale vector to start at zero otherwise exponent is too large
t_scale = t - t[0]

# initial guess for curve fit coefficients
a0 = v[-1]
b0 = v[0]
alpha0 = 1/t_scale[-1]

# coefficients and curve fit for curve
popt4, pcov4 = curve_fit(func, t_scale, v, p0=(a0, b0, alpha0))

a, b, alpha = popt4
v_fit = func(t_scale, a, b, alpha)

ss_res = np.sum((v - v_fit) ** 2)       # residual sum of squares
ss_tot = np.sum((v - np.mean(v)) ** 2)  # total sum of squares
r2 = 1 - (ss_res / ss_tot)              # R squared fit, R^2

与曲线拟合相比的数据绘制如下。还提供了参数和 R 平方值。

a0 = 4.1550   b0 = 4.0820   alpha0 = 0.0256
a = 4.1490    b = 0.0645    alpha = 0.9246
R² = 0.8473

是否可以使用上述方法更好地拟合数据，或者我是否需要使用不同形式的指数方程？

我也不确定初始值 ( a0, b0, alpha0)使用什么。在示例中，我从数据中选择了点，但这可能不是最好的方法。关于曲线拟合系数的初始猜测使用什么的任何建议？

Answer 1

对我来说，这看起来更适合多个组件，而不是单个指数。

def func(t, a, b, c, d, e):
    return a*np.exp(-t/b) + c*np.exp(-t/d) + e


# scale vector to start at zero otherwise exponent is too large
t_scale = t - t[0]

# initial guess for curve fit coefficients
guess = [1, 1, 1, 1, 0]

# coefficients and curve fit for curve
popt, pcov = curve_fit(func, t_scale, v, p0=guess)

v_fit = func(t_scale, *popt)