使用Python的Scipy Curve Fit如何拟合Beta分布

Question

我对使用scipy的curve_fit函数还比较新。我有很多分布看起来像y，但实际上并不像y。大多数看起来像y的分布是贝塔分布。我的方法是，如果我可以在所有具有不同分布的唯一ID上拟合beta函数，我就可以从beta函数中找到系数，然后查看系数的大小接近的情况，那么我就可以有效地过滤掉所有看起来像y的分布。

然而，我开始有些困难。

y = array([[ 0.50423378,  0.50423378,  0.50423378,  0.50254455,  0.50423378, 0.50254455,  0.50423378,  0.50507627,  0.50507627,  0.50423378,0.50507627,  0.50507627,  0.50423378,  0.50423378,  0.50423378, 0.50423378,  0.50423378,  0.50423378,  0.50254455,  0.50254455, 0.50254455,  0.50423378,  0.50423378,  0.50507627,  0.50507627,0.50507627,  0.50507627,  0.50507627,  0.50423378,  0.50423378, 0.50423378,  0.50507627,  0.50507627,  0.50423378,  0.50507627, 0.50507627,  0.50507627,  0.50423378,  0.50423378,  0.50423378,0.50423378,  0.50423378,  0.50254455,  0.50254455,  0.5, 0.50254455,  0.50254455,  0.50254455,  0.50423378,  0.50423378,0.50423378,  0.50423378,  0.50423378,  0.50254455,  0.50423378, 0.50254455,  0.50254455,  0.50423378,  0.50423378,  0.50254455,0.5       ,  0.5       ,  0.50254455,  0.50254455,  0.5       ,0.49658699,  0.49228746,  0.49228746,  0.48707792,  0.48092881,0.48707792,  0.48092881,  0.48092881,  0.48092881,  0.48092881,0.48092881,  0.48092881,  0.47380354,  0.47380354,  0.48092881,0.48707792,  0.48707792,  0.48092881,  0.48092881,  0.48092881,0.48092881,  0.48092881,  0.48092881,  0.47380354,  0.48092881,0.48092881,  0.48092881,  0.48707792,  0.48707792,  0.48707792,0.49228746,  0.49228746,  0.49228746,  0.49228746,  0.48707792,0.48707792,  0.48707792,  0.49228746,  0.48707792,  0.48707792,0.48707792,  0.48707792,  0.48707792,  0.49228746,  0.49228746,0.48707792,  0.48707792,  0.49228746,  0.49658699,  0.49658699,0.49658699,  0.49228746,  0.49228746,  0.49658699,  0.49228746,0.49658699,  0.5       ,  0.50254455,  0.50423378,  0.50423378,0.50254455,  0.50423378,  0.50423378,  0.50254455,  0.5       ,0.5       ,  0.5       ,  0.5       ,  0.5       ,  0.50254455,0.50254455,  0.5       ,  0.50254455,  0.5       ,  0.5       ,0.5       ,  0.5       ,  0.5       ,  0.5       ,  0.49658699,0.49228746,  0.48707792,  0.48707792,  0.48707792,  0.49228746,0.49228746,  0.48707792,  0.48707792,  0.49228746,  0.48707792,0.48707792,  0.48707792,  0.48092881,  0.48092881,  0.48707792,0.48707792,  0.48092881,  0.47380354,  0.48092881,  0.48092881,0.48707792,  0.49228746,  0.48707792,  0.49228746,  0.48707792,0.48092881,  0.47380354,  0.46565731,  0.46565731,  0.46565731,0.45643546,  0.45643546,  0.45643546,  0.45643546,  0.45643546,0.45643546,  0.45643546,  0.46565731,  0.45643546,  0.45643546,0.45643546,  0.44607129,  0.45643546,  0.45643546,  0.45643546,0.44607129,  0.44607129,  0.43448304,  0.43448304,  0.43448304,0.44607129,  0.45643546,  0.45643546,  0.45643546,  0.46565731,0.47380354,  0.48092881,  0.48092881, 29.38186886, 29.38186886,29.38186886, 29.37898909, 29.45299206, 29.52449116, 29.74083063,29.73771398, 29.73771398, 29.74083063, 29.74083063, 29.74083063,29.74083063, 29.73771398, 29.74083063, 29.73771398, 29.73771398,29.73771398, 29.73771398, 29.74083063, 29.74083063, 29.74083063,30.12527698, 30.48367189, 30.8169243 , 30.8169243 , 30.8169243 ,30.8169243 , 30.82153203, 30.8169243 , 30.81230208, 30.81230208,30.80766536, 30.81230208, 30.81230208, 30.80766536, 30.80301414,30.80301414, 30.80301414, 30.80301414, 30.80301414, 30.80766536,30.81230208, 30.81230208, 30.81230208, 30.81230208, 30.8169243 ,30.82153203, 30.82612528, 10.51949923, 10.51949923, 10.51436497,10.51436497, 10.22456193,  9.91464422,  9.36922158,  9.37416663,9.36922158,  9.36922158,  9.36922158,  9.37416663,  9.37906375,9.383913  ,  9.383913  ,  9.38871446,  9.383913  ,  9.37906375,9.37416663,  9.36922158,  9.36422851,  9.35918734,  7.72711675,5.53121937,  0.5       ,  0.50254455,  0.50254455,  0.50254455,0.50254455,  0.50254455,  0.5       ,  0.5       ,  0.49658699,0.5       ,  0.5       ,  0.5       ,  0.49658699,  0.49658699,0.5       ,  0.50254455,  0.50423378,  0.50423378,  0.50423378,0.50507627,  0.50507627,  0.50423378,  0.50423378,  0.50423378,0.50423378,  0.50423378,  0.50254455,  0.50254455,  0.5       ,0.5       ,  0.5       ,  0.49658699,  0.5       ,  0.49658699,0.49658699,  0.49658699,  0.49658699,  0.49658699,  0.49658699,0.49658699,  0.49658699,  0.49228746,  0.48707792,  0.48707792,0.48092881,  0.47380354,  0.47380354,  0.46565731,  0.46565731,0.47380354,  0.46565731,  0.47380354,  0.47380354,  0.47380354, 0.47380354,  0.48092881]])

使用scipy中的这个例子，我如何获取x数组并将其插入以获取系数，然后在我的分布上绘制curve_fit？

import numpy as np
from scipy.optimize import curve_fit
from scipy.special import gamma as gamma

def betafunc(x,a,b,cst):
    return cst*gamma(a+b) * (x**(a-1)) * ((1-x)**(b-1))  / ( gamma(a)*gamma(b) )

x = np.array( [0.1, 0.3, 0.5, 0.7, 0.9, 1.1])
y = np.array( [0.45112234, 0.56934313, 0.3996803 , 0.28982859, 0.19682153, 0.] )

popt2,pcov2 = curve_fit(betafunc,x[:-1],y[:-1],p0=(0.5,1.5,0.5))

print(popt2)
print(pcov2)

- Starbucks

Answer 1

对于你问题的第一部分： 如果你有一组观测数据，你可以使用numpy.histogram来获取直方图。要获取每个区间的中心，请按照我的下面的代码处理。这些值可以用于拟合过程。根据你提供的数据，不能拟合beta函数，因为它根本不适合。

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


def betafunc(x,a,b,cst):
    return cst*gamma(a+b) * (x**(a-1)) * ((1-x)**(b-1))  / ( gamma(a)*gamma(b) )

y_data=np.array([[ 0.50423378,  0.50423378,  0.50423378,  0.50254455,  0.50423378, 0.50254455,  0.50423378,  0.50507627,  0.50507627,  0.50423378,0.50507627,  0.50507627,  0.50423378,  0.50423378,  0.50423378, 0.50423378,  0.50423378,  0.50423378,  0.50254455,  0.50254455, 0.50254455,  0.50423378,  0.50423378,  0.50507627,  0.50507627,0.50507627,  0.50507627,  0.50507627,  0.50423378,  0.50423378, 0.50423378,  0.50507627,  0.50507627,  0.50423378,  0.50507627, 0.50507627,  0.50507627,  0.50423378,  0.50423378,  0.50423378,0.50423378,  0.50423378,  0.50254455,  0.50254455,  0.5, 0.50254455,  0.50254455,  0.50254455,  0.50423378,  0.50423378,0.50423378,  0.50423378,  0.50423378,  0.50254455,  0.50423378, 0.50254455,  0.50254455,  0.50423378,  0.50423378,  0.50254455,0.5       ,  0.5       ,  0.50254455,  0.50254455,  0.5       ,0.49658699,  0.49228746,  0.49228746,  0.48707792,  0.48092881,0.48707792,  0.48092881,  0.48092881,  0.48092881,  0.48092881,0.48092881,  0.48092881,  0.47380354,  0.47380354,  0.48092881,0.48707792,  0.48707792,  0.48092881,  0.48092881,  0.48092881,0.48092881,  0.48092881,  0.48092881,  0.47380354,  0.48092881,0.48092881,  0.48092881,  0.48707792,  0.48707792,  0.48707792,0.49228746,  0.49228746,  0.49228746,  0.49228746,  0.48707792,0.48707792,  0.48707792,  0.49228746,  0.48707792,  0.48707792,0.48707792,  0.48707792,  0.48707792,  0.49228746,  0.49228746,0.48707792,  0.48707792,  0.49228746,  0.49658699,  0.49658699,0.49658699,  0.49228746,  0.49228746,  0.49658699,  0.49228746,0.49658699,  0.5       ,  0.50254455,  0.50423378,  0.50423378,0.50254455,  0.50423378,  0.50423378,  0.50254455,  0.5       ,0.5       ,  0.5       ,  0.5       ,  0.5       ,  0.50254455,0.50254455,  0.5       ,  0.50254455,  0.5       ,  0.5       ,0.5       ,  0.5       ,  0.5       ,  0.5       ,  0.49658699,0.49228746,  0.48707792,  0.48707792,  0.48707792,  0.49228746,0.49228746,  0.48707792,  0.48707792,  0.49228746,  0.48707792,0.48707792,  0.48707792,  0.48092881,  0.48092881,  0.48707792,0.48707792,  0.48092881,  0.47380354,  0.48092881,  0.48092881,0.48707792,  0.49228746,  0.48707792,  0.49228746,  0.48707792,0.48092881,  0.47380354,  0.46565731,  0.46565731,  0.46565731,0.45643546,  0.45643546,  0.45643546,  0.45643546,  0.45643546,0.45643546,  0.45643546,  0.46565731,  0.45643546,  0.45643546,0.45643546,  0.44607129,  0.45643546,  0.45643546,  0.45643546,0.44607129,  0.44607129,  0.43448304,  0.43448304,  0.43448304,0.44607129,  0.45643546,  0.45643546,  0.45643546,  0.46565731,0.47380354,  0.48092881,  0.48092881, 29.38186886, 29.38186886,29.38186886, 29.37898909, 29.45299206, 29.52449116, 29.74083063,29.73771398, 29.73771398, 29.74083063, 29.74083063, 29.74083063,29.74083063, 29.73771398, 29.74083063, 29.73771398, 29.73771398,29.73771398, 29.73771398, 29.74083063, 29.74083063, 29.74083063,30.12527698, 30.48367189, 30.8169243 , 30.8169243 , 30.8169243 ,30.8169243 , 30.82153203, 30.8169243 , 30.81230208, 30.81230208,30.80766536, 30.81230208, 30.81230208, 30.80766536, 30.80301414,30.80301414, 30.80301414, 30.80301414, 30.80301414, 30.80766536,30.81230208, 30.81230208, 30.81230208, 30.81230208, 30.8169243 ,30.82153203, 30.82612528, 10.51949923, 10.51949923, 10.51436497,10.51436497, 10.22456193,  9.91464422,  9.36922158,  9.37416663,9.36922158,  9.36922158,  9.36922158,  9.37416663,  9.37906375,9.383913  ,  9.383913  ,  9.38871446,  9.383913  ,  9.37906375,9.37416663,  9.36922158,  9.36422851,  9.35918734,  7.72711675,5.53121937,  0.5       ,  0.50254455,  0.50254455,  0.50254455,0.50254455,  0.50254455,  0.5       ,  0.5       ,  0.49658699,0.5       ,  0.5       ,  0.5       ,  0.49658699,  0.49658699,0.5       ,  0.50254455,  0.50423378,  0.50423378,  0.50423378,0.50507627,  0.50507627,  0.50423378,  0.50423378,  0.50423378,0.50423378,  0.50423378,  0.50254455,  0.50254455,  0.5       ,0.5       ,  0.5       ,  0.49658699,  0.5       ,  0.49658699,0.49658699,  0.49658699,  0.49658699,  0.49658699,  0.49658699,0.49658699,  0.49658699,  0.49228746,  0.48707792,  0.48707792,0.48092881,  0.47380354,  0.47380354,  0.46565731,  0.46565731,0.47380354,  0.46565731,  0.47380354,  0.47380354,  0.47380354, 0.47380354,  0.48092881]])


hist=np.histogram(y_data[0],bins=20)
x=(hist[1][1:]+hist[1][:-1])/2
y=hist[0]

print(x,y)

plt.step(x,y,label='Manual calculation of the center of the bins')
plt.hist(y_data[0],bins=20,histtype='bar',label='Automatic plot with plt.hist')
plt.legend()
plt.show()

popt2,pcov2 = curve_fit(betafunc,x[:-1],y[:-1],p0=(0.5,1.5,0.5))

关于你问题的第二个部分： 要绘制具有最佳拟合参数的函数，您只需要添加我在最后添加的四行代码。

import numpy as np
from scipy.optimize import curve_fit
from scipy.special import gamma as gamma


def betafunc(x,a,b,cst):
    return cst*gamma(a+b) * (x**(a-1)) * ((1-x)**(b-1))  / ( gamma(a)*gamma(b) )



x = np.array( [0.1, 0.3, 0.5, 0.7, 0.9, 1.1])
y = np.array( [0.45112234, 0.56934313, 0.3996803 , 0.28982859, 0.19682153, 0.] )

popt2,pcov2 = curve_fit(betafunc,x[:-1],y[:-1],p0=(0.5,1.5,0.5))

print(popt2)
print(pcov2)

from matplotlib import pyplot as plt
plt.plot(x,betafunc(x,*popt2))
plt.plot(x,y)
plt.show()

- flabons

---

我不确定将数据分组是否正确：任何聚合都包含对数据的任意修改。使用两个不同数量的箱子，您可能会得到两个不同的拟合结果。这是一种为解释变量拟合模型的方法，但是_distribution_拟合的方法是不同的：我喜欢维基百科中的概述。 - Vojta F

Answer 2

如果您不受限于使用curve_fit，我建议您查看scipy.stats.beta。一种可能的解决方案是：

from scipy.stats import beta

y = array([[ 0.50423378,  0.50423378,  0.50423378,  0.50254455,  0.50423378, 0.50254455,  0.50423378,  0.50507627,  0.50507627,  0.50423378,0.50507627,  0.50507627,  0.50423378,  0.50423378,  0.50423378, 0.50423378,  0.50423378,  0.50423378,  0.50254455,  0.50254455, 0.50254455,  0.50423378,  0.50423378,  0.50507627,  0.50507627,0.50507627,  0.50507627,  0.50507627,  0.50423378,  0.50423378, 0.50423378,  0.50507627,  0.50507627,  0.50423378,  0.50507627, 0.50507627,  0.50507627,  0.50423378,  0.50423378,  0.50423378,0.50423378,  0.50423378,  0.50254455,  0.50254455,  0.5, 0.50254455,  0.50254455,  0.50254455,  0.50423378,  0.50423378,0.50423378,  0.50423378,  0.50423378,  0.50254455,  0.50423378, 0.50254455,  0.50254455,  0.50423378,  0.50423378,  0.50254455,0.5       ,  0.5       ,  0.50254455,  0.50254455,  0.5       ,0.49658699,  0.49228746,  0.49228746,  0.48707792,  0.48092881,0.48707792,  0.48092881,  0.48092881,  0.48092881,  0.48092881,0.48092881,  0.48092881,  0.47380354,  0.47380354,  0.48092881,0.48707792,  0.48707792,  0.48092881,  0.48092881,  0.48092881,0.48092881,  0.48092881,  0.48092881,  0.47380354,  0.48092881,0.48092881,  0.48092881,  0.48707792,  0.48707792,  0.48707792,0.49228746,  0.49228746,  0.49228746,  0.49228746,  0.48707792,0.48707792,  0.48707792,  0.49228746,  0.48707792,  0.48707792,0.48707792,  0.48707792,  0.48707792,  0.49228746,  0.49228746,0.48707792,  0.48707792,  0.49228746,  0.49658699,  0.49658699,0.49658699,  0.49228746,  0.49228746,  0.49658699,  0.49228746,0.49658699,  0.5       ,  0.50254455,  0.50423378,  0.50423378,0.50254455,  0.50423378,  0.50423378,  0.50254455,  0.5       ,0.5       ,  0.5       ,  0.5       ,  0.5       ,  0.50254455,0.50254455,  0.5       ,  0.50254455,  0.5       ,  0.5       ,0.5       ,  0.5       ,  0.5       ,  0.5       ,  0.49658699,0.49228746,  0.48707792,  0.48707792,  0.48707792,  0.49228746,0.49228746,  0.48707792,  0.48707792,  0.49228746,  0.48707792,0.48707792,  0.48707792,  0.48092881,  0.48092881,  0.48707792,0.48707792,  0.48092881,  0.47380354,  0.48092881,  0.48092881,0.48707792,  0.49228746,  0.48707792,  0.49228746,  0.48707792,0.48092881,  0.47380354,  0.46565731,  0.46565731,  0.46565731,0.45643546,  0.45643546,  0.45643546,  0.45643546,  0.45643546,0.45643546,  0.45643546,  0.46565731,  0.45643546,  0.45643546,0.45643546,  0.44607129,  0.45643546,  0.45643546,  0.45643546,0.44607129,  0.44607129,  0.43448304,  0.43448304,  0.43448304,0.44607129,  0.45643546,  0.45643546,  0.45643546,  0.46565731,0.47380354,  0.48092881,  0.48092881, 29.38186886, 29.38186886,29.38186886, 29.37898909, 29.45299206, 29.52449116, 29.74083063,29.73771398, 29.73771398, 29.74083063, 29.74083063, 29.74083063,29.74083063, 29.73771398, 29.74083063, 29.73771398, 29.73771398,29.73771398, 29.73771398, 29.74083063, 29.74083063, 29.74083063,30.12527698, 30.48367189, 30.8169243 , 30.8169243 , 30.8169243 ,30.8169243 , 30.82153203, 30.8169243 , 30.81230208, 30.81230208,30.80766536, 30.81230208, 30.81230208, 30.80766536, 30.80301414,30.80301414, 30.80301414, 30.80301414, 30.80301414, 30.80766536,30.81230208, 30.81230208, 30.81230208, 30.81230208, 30.8169243 ,30.82153203, 30.82612528, 10.51949923, 10.51949923, 10.51436497,10.51436497, 10.22456193,  9.91464422,  9.36922158,  9.37416663,9.36922158,  9.36922158,  9.36922158,  9.37416663,  9.37906375,9.383913  ,  9.383913  ,  9.38871446,  9.383913  ,  9.37906375,9.37416663,  9.36922158,  9.36422851,  9.35918734,  7.72711675,5.53121937,  0.5       ,  0.50254455,  0.50254455,  0.50254455,0.50254455,  0.50254455,  0.5       ,  0.5       ,  0.49658699,0.5       ,  0.5       ,  0.5       ,  0.49658699,  0.49658699,0.5       ,  0.50254455,  0.50423378,  0.50423378,  0.50423378,0.50507627,  0.50507627,  0.50423378,  0.50423378,  0.50423378,0.50423378,  0.50423378,  0.50254455,  0.50254455,  0.5       ,0.5       ,  0.5       ,  0.49658699,  0.5       ,  0.49658699,0.49658699,  0.49658699,  0.49658699,  0.49658699,  0.49658699,0.49658699,  0.49658699,  0.49228746,  0.48707792,  0.48707792,0.48092881,  0.47380354,  0.47380354,  0.46565731,  0.46565731,0.47380354,  0.46565731,  0.47380354,  0.47380354,  0.47380354, 0.47380354,  0.48092881]]) 

params = beta.fit(y)

y2 = np.loadtxt("other_data_file.dat")   # other distribution file
params2 = beta.fit(y2)

然后可以通过比较params和params2来逐个比较参数。请注意，scipy.stats.beta在定义概率密度函数时使用标准化形式。

- Arkadipta Sarkar