Weighted standard deviation in NumPy

Question

numpy.average() has a weights option, but numpy.std() does not. Does anyone have suggestions for a workaround?

Answer 1

How about the following short "manual calculation"?

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

There is a class in statsmodels that makes it easy to calculate weighted statistics: statsmodels.stats.weightstats.DescrStatsW.

Assuming this dataset and weights:

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

You initialize the class (note that you have to pass in the correction factor, the delta degrees of freedom at this point):

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

Then you can calculate:

• .mean the weighted mean:

  >>> weighted_stats.mean      
  1.97196261682243
  

• .std the weighted standard deviation:

  >>> weighted_stats.std       
  0.21434289609681711
  

• .var the weighted variance:

  >>> weighted_stats.var       
  0.045942877107170932
  

• .std_mean the standard error of weighted mean:

  >>> weighted_stats.std_mean  
  0.020818822467555047
  

  Just in case you're interested in the relation between the standard error and the standard deviation: The standard error is (for ddof == 0) calculated as the weighted standard deviation divided by the square root of the sum of the weights minus 1 (corresponding source for statsmodels version 0.9 on GitHub):

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

Answer 3

There doesn't appear to be such a function in numpy/scipy yet, but there is a ticket proposing this added functionality. Included there you will find Statistics.py which implements weighted standard deviations.

Answer 4

Here's one more option:

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

Answer 5

There is a very good example proposed by gaborous:

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)