python - 在 python 中拟合泊松直方图的问题

Question

我试图用泊松分布拟合一些数据，但它不起作用。

x = [46,71,106,126,40,27,19,103,46,89,31,70,35,43,82,128,185,47,18,36,96,30,135,36,40,72,32,86,76,116,51,23,40,121,22,107,65,93,25,74,73,73,111,56,34,28,87,14,70,54,63,50,89,62,35,59,71,39,23,46,32,56,15,68,30,69,37,41,43,106,20,35,63,44,40,32,102,28,54,32,42,19,69,31,36,86,41,57,39,53,48,121,35,51,10,68,14,140,57,50,178,37,121,35,206,26,54,5,53,17,139,49,122,110,62,81,43,83,47,62,2,50,36,190,32,124,89,60,39,156,89,26,57,34,58,29,22,96,132,59,34,43,50,58,48,56,43,54,22,26,60,43,69,58,100,122,48,55,29,55,57,36,42,51,24,81,66,73,112,34,54,45,29,53,43,60,72,13,72,85,49,80,47,40,28,43,37,48,31,60,33,75,53,71,49,142,47,28,51,80,50,33,67,28,101,80,60,80,98,39,69,27,32,11,32,62,32,77,110,45,61,22,23,73,25,27,41,42,65,23,127,128,42,44,10,50,56,73,42,63,70,148,18,109,111,54,34,18,32,50,100,41,39,58,93,42,86,70,41,27,24,57,77,81,101,48,52,146,59,87,86,120,28,23,76,52,59,31,60,32,65,49,27,106,136,23,15,77,44,96,62,66,26,41,70,13,64,124,49,44,55,68,54,58,72,41,21,80,3,49,54,35,48,38,83,59,36,80,47,32,38,16,43,196,19,80,28,56,23,81,103,45,25,42,44,34,106,23,47,53,119,56,54,108,35,20,34,39,70,61,40,35,51,104,63,55,93,22,32,48,20,121,55,76,36,32,121,58,42,101,32,49,77,23,95,32,75,53,106,194,54,31,104,69,58,66,29,66,37,28,59,60,70,95,63,103,173,47,59,27] #geiger count
bins = np.histogram_bin_edges(x)
n, bins_edges, patches = plt.hist(x,bins, density=1, facecolor='darkblue',ec='white', log=0)
print(n)
bin_middles = 0.5*(bins_edges[1:] + bins_edges[:-1])
def fit_function(k, lamb):
    return poisson.pmf(k, lamb)
parameters, cov_matrix = curve_fit(fit_function, bin_middles,n)
x_plot = np.arange(0,max(x))
plt.plot(x_plot,fit_function(x_plot, *parameters),label='Poisson')
plt.show()

我得到了这个结果，但我们可以看到它是不对的

Answer 1

您正在使用诸如 np.histogram_bin_edges 等用于连续分布的函数，而泊松分布是离散的。

根据 wikipedia , lambda 可以通过取样本的平均值来估计:

from scipy.stats import poisson
import numpy as np
from matplotlib import pyplot as plt

x = [46,71,106,126,40,27,19,103,46,89,31,70,35,43,82,128,185,47,18,36,96,30,135,36,40,72,32,86,76,116,51,23,40,121,22,107,65,93,25,74,73,73,111,56,34,28,87,14,70,54,63,50,89,62,35,59,71,39,23,46,32,56,15,68,30,69,37,41,43,106,20,35,63,44,40,32,102,28,54,32,42,19,69,31,36,86,41,57,39,53,48,121,35,51,10,68,14,140,57,50,178,37,121,35,206,26,54,5,53,17,139,49,122,110,62,81,43,83,47,62,2,50,36,190,32,124,89,60,39,156,89,26,57,34,58,29,22,96,132,59,34,43,50,58,48,56,43,54,22,26,60,43,69,58,100,122,48,55,29,55,57,36,42,51,24,81,66,73,112,34,54,45,29,53,43,60,72,13,72,85,49,80,47,40,28,43,37,48,31,60,33,75,53,71,49,142,47,28,51,80,50,33,67,28,101,80,60,80,98,39,69,27,32,11,32,62,32,77,110,45,61,22,23,73,25,27,41,42,65,23,127,128,42,44,10,50,56,73,42,63,70,148,18,109,111,54,34,18,32,50,100,41,39,58,93,42,86,70,41,27,24,57,77,81,101,48,52,146,59,87,86,120,28,23,76,52,59,31,60,32,65,49,27,106,136,23,15,77,44,96,62,66,26,41,70,13,64,124,49,44,55,68,54,58,72,41,21,80,3,49,54,35,48,38,83,59,36,80,47,32,38,16,43,196,19,80,28,56,23,81,103,45,25,42,44,34,106,23,47,53,119,56,54,108,35,20,34,39,70,61,40,35,51,104,63,55,93,22,32,48,20,121,55,76,36,32,121,58,42,101,32,49,77,23,95,32,75,53,106,194,54,31,104,69,58,66,29,66,37,28,59,60,70,95,63,103,173,47,59,27] bins = np.histogram_bin_edges(x)
n, bins_edges, patches = plt.hist(x, bins, density=1, facecolor='darkblue', ec='white', log=0)
lamd = np.mean(x)
x_plot = np.arange(0, max(x) + 1)
plt.plot(x_plot, poisson.pmf(x_plot, lamd), label='Poisson')
plt.show()

计算出的 lambda 约为 60。该图似乎表明泊松分布不是非常适合给定的样本。