如何配置来自`scipy.stats`的函数以使用`scipy.optimize.curve_fit`？

Question

我正在创建一个统一的概率向量，为区域添加权重，再次转换为概率。我想通过将beta 分布拟合到曲线来平滑曲线。我不能使用stats.beta.fit，因为我没有从这些概率中得出任何结论，只有曲线的支架。

如何配置来自scipy.stats 的函数以与scipy.optimize.curve_fit 一起使用？如果可能的话，我不想将其限制为仅beta 分布。有没有一种通用的方法可以将这些转换为可以最小化以优化参数集的形式？

本质上，我正在寻找最适合特定分布的曲线的参数集（在本例中为beta）。

有什么想法吗？

# Uniform probabilities
n = 10
x = np.linspace(0,1,n)
probs_num = np.ones(n)/n # y-values

# Update the mass around an area
idx_update = 2
probs_num[idx_update] += 0.382

# Convert fo probs
probs_num = probs_num/probs_num.sum()

# Plot
with plt.style.context("ggplot"):
    fig, ax = plt.subplots()
    ax.plot(x,probs_num)
    ax.set_ylabel("Density")
    ax.set_xlabel("$x$")


probs_num

from scipy.optimize import curve_fit
from scipy import stats
popt, pcov = curve_fit(stats.beta, x, probs_num) # NOTE: I know this is the wrong way to use it but I'm leaving it as a placeholder

回应下面的lm回答：

import lmfit
def beta_fcn(x, alpha, beta, loc):
    return stats.beta.pdf(x, alpha, beta, loc)
bmodel = lmfit.Model(beta_fcn)
params = bmodel.make_params(alpha=1, beta=1., loc=0.5)
result = bmodel.fit(probs_num, params, x=x)

# ---------------------------------------------------------------------------
# ValueError                                Traceback (most recent call last)
# <ipython-input-34-61ec32095934> in <module>()
#       4 bmodel = lmfit.Model(beta_fcn)
#       5 params = bmodel.make_params(alpha=1, beta=1., loc=0.5)
# ----> 6 result = bmodel.fit(probs_num, params, x=x)

# ~/anaconda/envs/python3/lib/python3.6/site-packages/lmfit/model.py in fit(self, data, params, weights, method, iter_cb, scale_covar, verbose, fit_kws, nan_policy, **kwargs)
#     871                              scale_covar=scale_covar, fcn_kws=kwargs,
#     872                              nan_policy=self.nan_policy, **fit_kws)
# --> 873         output.fit(data=data, weights=weights)
#     874         output.components = self.components
#     875         return output

# ~/anaconda/envs/python3/lib/python3.6/site-packages/lmfit/model.py in fit(self, data, params, weights, method, nan_policy, **kwargs)
#    1215         self.userkws.update(kwargs)
#    1216         self.init_fit = self.model.eval(params=self.params, **self.userkws)
# -> 1217         _ret = self.minimize(method=self.method)
#    1218 
#    1219         for attr in dir(_ret):

# ~/anaconda/envs/python3/lib/python3.6/site-packages/lmfit/minimizer.py in minimize(self, method, params, **kws)
#    1636                         val.lower().startswith(user_method)):
#    1637                     kwargs['method'] = val
# -> 1638         return function(**kwargs)
#    1639 
#    1640 

# ~/anaconda/envs/python3/lib/python3.6/site-packages/lmfit/minimizer.py in leastsq(self, params, **kws)
#    1288         np.seterr(all='ignore')
#    1289 
# -> 1290         lsout = scipy_leastsq(self.__residual, variables, **lskws)
#    1291         _best, _cov, infodict, errmsg, ier = lsout
#    1292         result.aborted = self._abort

# ~/anaconda/envs/python3/lib/python3.6/site-packages/scipy/optimize/minpack.py in leastsq(func, x0, args, Dfun, full_output, col_deriv, ftol, xtol, gtol, maxfev, epsfcn, factor, diag)
#     385             maxfev = 200*(n + 1)
#     386         retval = _minpack._lmdif(func, x0, args, full_output, ftol, xtol,
# --> 387                                  gtol, maxfev, epsfcn, factor, diag)
#     388     else:
#     389         if col_deriv:

# ~/anaconda/envs/python3/lib/python3.6/site-packages/lmfit/minimizer.py in __residual(self, fvars, apply_bounds_transformation)
#     489         if not self._abort:
#     490             return _nan_policy(np.asarray(out).ravel(),
# --> 491                                nan_policy=self.nan_policy)
#     492 
#     493     def __jacobian(self, fvars):

# ~/anaconda/envs/python3/lib/python3.6/site-packages/lmfit/minimizer.py in _nan_policy(arr, nan_policy, handle_inf)
#    1846 
#    1847         if contains_nan:
# -> 1848             raise ValueError("The input contains nan values")
#    1849     return arr
#    1850 

# ValueError: The input contains nan values

Answer 1

您可能想使用beta.pdf，如下所示：

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

n = 21
x = np.linspace(0, 1, n)
y = stats.beta.pdf(x, 2.5, 15, 0.3) + np.random.normal(scale=0.1, size=n)

p0 = [2.0, 10.0, 0.5]
popt, pcov = curve_fit(stats.beta.pdf, x, y, p0)
print(popt)

您可能还会发现 lmfit (https://lmfit.github.io/lmfit-py/) 很有用。为此，您必须包装 stats.beta.pdf，但这为您设置参数界限或修复参数提供了更大的灵活性：

from lmfit import Model
def beta_fcn(x, alpha, beta, loc):
    return stats.beta.pdf(x, alpha, beta, loc)

bmodel = Model(beta_fcn)

params = bmodel.make_params(alpha=2, beta=10., loc=0.5)
params['alpha'].min = 1.0

result = bmodel.fit(y, params, x=x)

print(result.fit_report())

with plt.style.context("ggplot"):
    fig, ax = plt.subplots()
    ax.plot(x, y, 'o')
    ax.plot(x, result.best_fit)
    ax.set_ylabel("Density")
    ax.set_xlabel("$x$")
plt.show()

这将打印出来

[[Model]]
    Model(beta_fcn)
[[Fit Statistics]]
    # fitting method   = leastsq
    # function evals   = 75
    # data points      = 21
    # variables        = 3
    chi-square         = 0.06244742
    reduced chi-square = 0.00346930
    Akaike info crit   = -116.177008
    Bayesian info crit = -113.043441
[[Variables]]
    alpha:  2.50445306 +/- 0.06843391 (2.73%) (init = 2)
    beta:   14.9105872 +/- 0.31198655 (2.09%) (init = 10)
    loc:    0.29906057 +/- 0.00190835 (0.64%) (init = 0.5)
[[Correlations]] (unreported correlations are < 0.100)
    C(alpha, beta) =  0.894
    C(alpha, loc)  = -0.834
    C(beta, loc)   = -0.564

（并且，需要明确的是，最佳拟合值和不确定性将与上面的 curve_fit 示例相同）和