使用高斯核密度（Python）计算值与平均值的差异

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100% cdfv kdev 128 26 0.129688 0.499376 512 55 0.275874 0.499671 1024 91 0.454159 0.499936 2048 12 0.062298 0.499150 3072 0 0.001556 0.498364 2800 1 0.004954 0.498573

1条回答

网友

1楼 · 发布于 2024-04-26 11:42:46

一些一般性的评论，而不涉及statsmodels的细节。在

statsmodels也有cdf内核，但我不记得它们工作得有多好，我也不认为它有自动选择带宽的功能。在

关于格伦·布的回答，阿里·穆在评论中提到：

随着样本的增长，cdf估计收敛到真实分布比密度估计快得多。为了平衡偏差-方差的权衡，我们应该为cdf核使用一个较小的带宽，这是相对于密度估计而言欠平滑的。估计值应比相应的密度估计值更准确。在

尾部观察数：

如果您在样本中的最大观察值是4，而您想知道cdf在5，那么您的数据中没有关于它的信息。对于只有很少观察值的尾部，非参数估计量（如核分布估计量）的相对方差将较大（是1e-5还是1e-20？）。在

作为核密度或核分布估计的替代方法，我们可以估计尾部的Pareto分布。例如，取最大的10%或20%的观测值并拟合帕累托分布，然后用这个来外推尾密度。有几个用于powerlaw估计的Python包，可以用于此。在

更新

下面展示了如何使用参数正态分布假设和固定带宽的高斯核密度估计来计算“外向度”。在

只有当样本来自连续分布或可以用连续分布近似时，这才是真正正确的。在这里，我们假设只有3个不同值的样本来自正态分布。本质上，计算出的cdf值类似于距离度量，而不是离散随机变量的概率。在

这使用kdescipy.stats公司使用固定带宽而不是statsmodels版本。在

我不确定带宽是如何设置在scipy的高斯分布中的，所以，我选择的固定带宽等于scale可能是错误的。如果只有三个不同的值，数据中没有足够的信息，我不知道如何选择带宽。默认带宽用于近似正态分布，或者至少是单峰分布。在

import numpy as np
from scipy import stats

# data
ram = np.array([2, <truncated from data in description>, 3])

loc = ram.mean()
scale = ram.std()

# new values
x = np.asarray([-1, 0, 2, 3, 4, 5, 100])

# assume normal distribution
cdf_val = stats.norm.cdf(x, loc=loc, scale=scale)
cdfv = np.minimum(cdf_val, 1 - cdf_val)

# use gaussian kde but fix bandwidth
kde = stats.gaussian_kde(ram, bw_method=scale)
kde_val = np.asarray([kde.integrate_box_1d(-np.inf, xx) for xx in  x])
kdev = np.minimum(kde_val, 1 - kde_val)


#print(np.column_stack((x, cdfv, kdev)))
# use pandas for prettier table
import pandas as pd
print(pd.DataFrame({'cdfv': cdfv, 'kdev': kdev}, index=x))

'''
          cdfv      kdev
-1    0.000096  0.000417
 0    0.006171  0.021262
 2    0.479955  0.482227
 3    0.119854  0.199565
 5    0.000143  0.000472
 100  0.000000  0.000000
 '''

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