NumPy 中的加权标准差

Question

numpy.average() 有一个权重选项，但numpy.std() 没有。有人有解决方法的建议吗？

Answer 1

以下简短的“手动计算”怎么样？

def weighted_avg_and_std(values, weights):
    """
    Return the weighted average and standard deviation.

    values, weights -- Numpy ndarrays with the same shape.
    """
    average = numpy.average(values, weights=weights)
    # Fast and numerically precise:
    variance = numpy.average((values-average)**2, weights=weights)
    return (average, math.sqrt(variance))

Answer 2

statsmodels 中有一个类可以很容易地计算加权统计：statsmodels.stats.weightstats.DescrStatsW。

假设这个数据集和权重：

import numpy as np
from statsmodels.stats.weightstats import DescrStatsW

array = np.array([1,2,1,2,1,2,1,3])
weights = np.ones_like(array)
weights[3] = 100

你初始化类（注意你必须传入校正因子，此时的delta degrees of freedom）：

weighted_stats = DescrStatsW(array, weights=weights, ddof=0)

那么你可以计算：

• .mean加权平均值：

  >>> weighted_stats.mean      
  1.97196261682243
  

• .std加权标准差：

  >>> weighted_stats.std       
  0.21434289609681711
  

• .var加权方差：

  >>> weighted_stats.var       
  0.045942877107170932
  

• .std_meanstandard error加权平均：

  >>> weighted_stats.std_mean  
  0.020818822467555047
  

  以防万一您对标准误差和标准偏差之间的关系感兴趣：标准误差（对于ddof == 0）计算为加权标准偏差除以权重总和的平方根减去1 (corresponding source for statsmodels version 0.9 on GitHub):

  standard_error = standard_deviation / sqrt(sum(weights) - 1)
  

Answer 3

这里还有一个选项：

np.sqrt(np.cov(values, aweights=weights))

Answer 4

在 numpy/scipy 中似乎还没有这样的功能，但是有一个 ticket 提出了这个附加功能。包括在那里，您会发现 Statistics.py 实现加权标准偏差。

Answer 5

gaborous提出了一个很好的例子：

import pandas as pd
import numpy as np
# X is the dataset, as a Pandas' DataFrame
mean = mean = np.ma.average(X, axis=0, weights=weights) # Computing the 
weighted sample mean (fast, efficient and precise)

# Convert to a Pandas' Series (it's just aesthetic and more 
# ergonomic; no difference in computed values)
mean = pd.Series(mean, index=list(X.keys())) 
xm = X-mean # xm = X diff to mean
xm = xm.fillna(0) # fill NaN with 0 (because anyway a variance of 0 is 
just void, but at least it keeps the other covariance's values computed 
correctly))
sigma2 = 1./(w.sum()-1) * xm.mul(w, axis=0).T.dot(xm); # Compute the 
unbiased weighted sample covariance

Correct equation for weighted unbiased sample covariance, URL (version: 2016-06-28)

Answer 6

“frequency weights”意义上的“样本”或“无偏”标准偏差的后续行动，因为“加权样本标准偏差 python”Google 搜索导致了这篇文章：

def frequency_sample_std_dev(X, n):
    """
    Sample standard deviation for X and n,
    where X[i] is the quantity each person in group i has,
    and n[i] is the number of people in group i.
    See Equation 6.4 of:
    Montgomery, Douglas, C. and George C. Runger. Applied Statistics 
     and Probability for Engineers, Enhanced eText. Available from: 
      WileyPLUS, (7th Edition). Wiley Global Education US, 2018.
    """
    n_groups = len(n)
    n_people = sum(n)
    lhs_numerator = sum([ni*Xi**2 for Xi, ni in zip(X, n)])
    rhs_numerator = sum([Xi*ni for Xi, ni in zip(X,n)])**2/n_people
    denominator = n_people-1
    var = (lhs_numerator - rhs_numerator) / denominator
    std = sqrt(var)
    return std

或者修改@Eric的答案如下：

def weighted_sample_avg_std(values, weights):
    """
    Return the weighted average and weighted sample standard deviation.

    values, weights -- Numpy ndarrays with the same shape.
    
    Assumes that weights contains only integers (e.g. how many samples in each group).
    
    See also https://en.wikipedia.org/wiki/Weighted_arithmetic_mean#Frequency_weights
    """
    average = np.average(values, weights=weights)
    variance = np.average((values-average)**2, weights=weights)
    variance = variance*sum(weights)/(sum(weights)-1)
    return (average, sqrt(variance))

print(weighted_sample_avg_std(X, n))