高斯曲线拟合遇exp溢出警告及拟合效果差的解决求助

问题描述 #

运行高斯曲线拟合代码时出现RuntimeWarning: overflow encountered in exp，对应代码行是y = A*np.exp(-1*B*x**2)，拟合生成的曲线效果极差。数据已归一化到1，推测是数值范围问题导致报错。原代码如下：

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

def GaussFit():

    xdata_raw = [0,24,22,20,18,16,14,12,10,8,6,4,2,-24,-22,-20,-18,-16,-14,-12,-10,-8,-6,-4,-2]
    ydata_raw =[0.398,0.061,0.066,0.076,0.095,0.115,0.148,0.183,0.211,0.270,0.330,0.361,0.391,0.061,0.066,0.076,0.095,0.115,0.148,0.183,0.211,0.270,0.330,0.361,0.391]
    y_norm = []

    for i in range(len(ydata_raw)):
        temp = ydata_raw[i]/0.398
        y_norm.append(temp)
    
    plt.plot(xdata_raw, y_norm, 'o')
    xdata = np.asarray(xdata_raw)
    ydata = np.asarray(y_norm)
    
    def Gauss(x, A, B):
        y = A*np.exp(-1*B*x**2)
        return y
    parameters, covariance = curve_fit(Gauss, xdata, ydata)

    fit_A = parameters[0]
    fit_B = parameters[1]

    fit_y = Gauss(xdata, fit_A, fit_B)
    plt.plot(xdata, ydata, 'o', label='data')
    plt.plot(xdata, fit_y, '-', label='fit')
    plt.grid(color='grey', linestyle='-', linewidth=0.25, alpha=0.5)
    plt.legend()
    plt.xlabel('Winkel der Auslenkung in °')
    plt.ylabel('Intensität [I]')
    plt.title('vertikale Ausrichtung')

GaussFit()

问题原因 #

1. 初始参数不合理：curve_fit默认初始参数为[1, 1]，当B=1时，x=24的平方是576，-B*x²=-576，exp(-576)几乎为0；若拟合过程中B变为负数，-B*x²会变成大正数，直接导致exp计算溢出。
2. 无参数约束：未限制B的取值范围，拟合过程中可能出现负的B值，既引发溢出，又导致曲线形状完全偏离预期。

解决方案 #

1. 提供合理初始参数 #

根据归一化后的数据，峰值A接近1；通过x=24时的y值估算B≈0.003，设置初始参数p0=[1, 0.003]，引导拟合收敛到正确方向。

2. 约束参数范围 #

用bounds参数限制A和B为非负数，彻底避免出现导致溢出的负B值。

3. 简化数据处理 #

用numpy数组操作替代循环，提升代码效率和可读性。

修改后的完整代码 #

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

def GaussFit():
    xdata_raw = [0,24,22,20,18,16,14,12,10,8,6,4,2,-24,-22,-20,-18,-16,-14,-12,-10,-8,-6,-4,-2]
    ydata_raw = [0.398,0.061,0.066,0.076,0.095,0.115,0.148,0.183,0.211,0.270,0.330,0.361,0.391,0.061,0.066,0.076,0.095,0.115,0.148,0.183,0.211,0.270,0.330,0.361,0.391]
    
    # 用numpy简化归一化操作
    y_norm = np.array(ydata_raw) / 0.398
    xdata = np.array(xdata_raw)
    ydata = y_norm
    
    def Gauss(x, A, B):
        return A * np.exp(-B * x**2)
    
    # 设置初始参数与参数约束范围
    initial_params = [1, 0.003]
    param_bounds = ([0, 0], [np.inf, np.inf])  # 约束A、B均非负
    parameters, covariance = curve_fit(Gauss, xdata, ydata, p0=initial_params, bounds=param_bounds)
    
    fit_A, fit_B = parameters
    fit_y = Gauss(xdata, fit_A, fit_B)
    
    plt.plot(xdata, ydata, 'o', label='data')
    plt.plot(xdata, fit_y, '-', label='fit')
    plt.grid(color='grey', linestyle='-', linewidth=0.25, alpha=0.5)
    plt.legend()
    plt.xlabel('Winkel der Auslenkung in °')
    plt.ylabel('Intensität [I]')
    plt.title('vertikale Ausrichtung')
    plt.show()

GaussFit()

效果说明 #

修改后，curve_fit会从合理的初始值开始收敛，参数约束避免了溢出问题，拟合曲线会与数据点高度吻合。

内容的提问来源于stack exchange，提问作者Wolle_Wolle