对包含非矢量化函数（如定积分）的模型执行拟合（使用 LMFIT）的正确方法是什么？

Question

我想用一个包含定积分的函数对一些数据进行拟合，就拟合而言，积分极限之一也是自变量。我想具体知道如何使用 'lmfit' 来实现。考虑以下示例：

import numpy as np 
import scipy.optimize
from scipy.optimize import curve_fit 
import matplotlib.pyplot as plt 
import scipy.integrate as integrate 
from lmfit import minimize, Parameters, report_fit

def integrand(x,amp,xc,w):
    return amp*np.exp((-(x-xc)**2)/w)

def curve(x0,amp,xc,w):
    res = integrate.quad(integrand, 0, x0, args=(amp,xc,w)) #x0 is the independent variable here
    return res[0]

vcurve = np.vectorize(curve, excluded=set([1]))
# vectorizing the output of the function which includes the integration step

# Generating the data with noise
xdata = np.linspace(0,10,20)
ydata = vcurve(xdata,1,5,1) + 0.1 * np.random.randn(len(xdata))

def residual(params, x, data):
    amp = params['amp']
    xc = params['xc']
    w = params['w']
    model = vcurve(xdata,amp,xc,w)
    return data-model


# defining the parameters and providing the initial values
params = Parameters()
params.add('amp', value=1,vary=True) 
params.add('xc', value=5,vary=True)
params.add('w', value=1,vary=True)

out = minimize(residual, params, args=(xdata, ydata))

然而这会导致错误：

---> out = minimize(residual, params, args=(xdata, ydata))
... 
...
...               
TypeError: __array__() takes 1 positional argument but 2 were given

似乎参数的初始值没有被正确读取

使用 scipy 曲线拟合，我可以使其按如下方式工作：

popt, pcov = curve_fit(vcurve, xdata, ydata, p0=[2,2,2])

fig, ax = plt.subplots(1,1)
ax.plot(xdata,ydata,label='Observed',ls='',marker='o')    

#Plotting the best fit
xx = np.linspace(0,10,50)
ax.plot(xx,vcurve(xx,popt[0],popt[1],popt[2]),label='Best Fit') 
ax.legend()

print(popt)

这为我提供了最佳拟合参数的合理值。任何关于如何使用 lmfit 进行这项工作的建议将不胜感激

Answer 1

您需要在 residual 函数中解压 params 并使用正确的参数调用 vcurve -- 参数值而不是参数对象：

def residual(params, x, data):
    amp = params['amp'].value
    xc = params['xc'].value
    w = params['w'].value
    model = vcurve(xdata,amp,xc,w)
    return data-model

对包含非矢量化函数（如定积分）的模型执行拟合（使用 LMFIT）的正确方法是什么？

What is the correct way to perform fitting (using LMFIT) for a model which contains a non vectorized function such as a definite integral?

python

curve

model-fitting

lmfit

scipy-optimize