我有一些数据，想拟合给定的心理测量函数 p。
我对拟合参数和错误也很感兴趣。通过使用 scipy 包中的 curve_fit 函数的“经典”方法，很容易获得 p 的参数和错误。但是我想使用最大似然估计（MLE）来做同样的事情。从输出和图中您可以看到两种方法提供的参数略有不同。实现 MLE 不是问题，但我不知道如何使用此方法获取错误。有没有简单的方法来获得它们？我的似然函数 L 是：
我无法调整此处描述的代码http://rlhick.people.wm.edu/posts/estimating-custom-mle.html但这可能是一个解决方案。我该如何实施？或者这个还有其他方法吗？
这里使用 scipy stats 模型拟合了一个类似的函数：https ://stats.stackexchange.com/questions/66199/maximum-likelihood-curve-model-fitting-in-python 。然而，参数的误差也没有被计算。
负对数似然函数是正确的，因为它提供了正确的参数，但我想知道这个函数是否依赖于 y 数据？负对数似然函数 l 显然是 l = -ln(L)。这是我的代码：
#!/usr/bin/env python
# -*- coding: utf-8 -*-
## libary
import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit
from scipy.optimize import minimize
def p(x,x50,s50):
"""return y value of psychometric function p"""
return 1./(1+np.exp(4.*s50*(x50-x)))
def initialparams(x,y):
"""return initial fit parameters for function p with given dataset"""
midpoint = np.mean(x)
slope = (np.max(y)-np.min(y))/(np.max(x)-np.min(x))
return [midpoint, slope]
def cfit_error(pcov):
"""return errors of fir from covariance matrix"""
return np.sqrt(np.diag(pcov))
def neg_loglike(params):
"""analytical negative log likelihood function. This function is dependend on the dataset (x and y) and the two parameters x50 and s50."""
x50 = params[0]
s50 = params[1]
i = len(xdata)
prod = 1.
for i in range(i):
#print prod
prod *= p(xdata[i],x50,s50)**(ydata[i]*5) * (1-p(xdata[i],x50,s50))**((1.-ydata[i])*5)
return -np.log(prod)
xdata = [0.,-7.5,-9.,-13.500001,-12.436171,-16.208617,-13.533123,-12.998025,-13.377527,-12.570075,-13.320075,-13.070075,-11.820075,-12.070075,-12.820075,-13.070075,-12.320075,-12.570075,-11.320075,-12.070075]
ydata = [1.,0.6,0.8,0.4,1.,0.,0.4,0.6,0.2,0.8,0.4,0.,0.6,0.8,0.6,0.2,0.6,0.,0.8,0.6]
intparams = initialparams(xdata, ydata)## guess some initial parameters
## normal curve fit using least squares algorithm
popt, pcov = curve_fit(p, xdata, ydata, p0=intparams)
print('scipy.optimize.curve_fit:')
print('x50 = {:f} +- {:f}'.format(popt[0], cfit_error(pcov)[0]))
print('s50 = {:f} +- {:f}\n'.format(popt[1], cfit_error(pcov)[1]))
## fitting using maximum likelihood estimation
results = minimize(neg_loglike, initialparams(xdata,ydata), method='Nelder-Mead')
print('MLE with self defined likelihood-function:')
print('x50 = {:f}'.format(results.x[0]))
print('s50 = {:f}'.format(results.x[1]))
#print results
## ploting the data and results
xfit = np.arange(-20,1,0.1)
fig = plt.figure()
ax = fig.add_subplot(1,1,1)
ax.plot(xdata, ydata, 'xb', label='measured data')
ax.plot(xfit, p(xfit, *popt), '-r', label='curve fit')
ax.plot(xfit, p(xfit, *results.x), '-g', label='MLE')
plt.legend()
plt.show()
输出是：
scipy.optimize.curve_fit:
x50 = -12.681586 +- 0.252561
s50 = 0.264371 +- 0.117911
MLE with self defined likelihood-function:
x50 = -12.406544
s50 = 0.107389
拟合和测量数据都可以在这里看到：
我的 Python 版本是 Debian Stretch 上的 2.7。谢谢您的帮助。