Calculate weighted statistical moments in Python

Question

I've been looking for a function or package that would allow me to calculate the skew and kurtosis of a distribution in a weighted way, as I have histogram data.

For instance I have the data

import numpy as np

np.array([[1, 2],
          [2, 5],
          [3, 6],
          [4,12],
          [5, 1])

where the first column [1,2,3,4,5] are the values and the second column [2,5,6,12,1] are the frequencies of the values.

I have found out how to do the first two moments (mean, standard deviation) in a weighted way using the weighted_avg_and_std function specified in this thread, but I was not quite sure how I could extend this to both the skew and kurtosis, or even the nth statistical moment.

I have found the definitions themselves here and could manually write functions to implement this from scratch, but before I go and do that I was wondering if there were any existing packages or functions that might be able to do this.

Thanks

EDIT: I figured it out, the following code works (please note that this is for population moments)

skewnewss = np.average(((values-average)/np.sqrt(variance))**3, weights=weights)

and

kurtosis=np.average(((values-average)/np.sqrt(variance))**4-3, weights=weights)

score 2 · Answer 1 · answered Dec 24 '20 at 04:46

Taken from here

Here is the code

def weighted_mean(var, wts):
    """Calculates the weighted mean"""
    return np.average(var, weights=wts)


def weighted_variance(var, wts):
    """Calculates the weighted variance"""
    return np.average((var - weighted_mean(var, wts))**2, weights=wts)


def weighted_skew(var, wts):
    """Calculates the weighted skewness"""
    return (np.average((var - weighted_mean(var, wts))**3, weights=wts) /
            weighted_variance(var, wts)**(1.5))

def weighted_kurtosis(var, wts):
    """Calculates the weighted skewness"""
    return (np.average((var - weighted_mean(var, wts))**4, weights=wts) /
            weighted_variance(var, wts)**(2))

FBruzzesi · Accepted Answer · 2020-04-30T12:10:17.320

I think you have already listed all the ingredients that you need, following the formulas in the link you provided:

import numpy as np

a = np.array([[1,2],[2,5],[3,6],[4,12],[5,1]])
values, weights = a.T

def n_weighted_moment(values, weights, n):

    assert n>0 & (values.shape == weights.shape)
    w_avg = np.average(values, weights = weights)
    w_var = np.sum(weights * (values - w_avg)**2)/np.sum(weights)

    if n==1:
        return w_avg
    elif n==2:
        return w_var
    else:
        w_std = np.sqrt(w_var)
        return np.sum(weights * ((values - w_avg)/w_std)**n)/np.sum(weights)
              #Same as np.average(((values - w_avg)/w_std)**n, weights=weights)

Which results in:

for n in range(1,5):
    print(f'Moment {n} value is {n_weighted_moment(values, weights, n)}')

Moment 1 value is 3.1923076923076925
Moment 2 value is 1.0784023668639053
Moment 3 value is -0.5962505715592139
Moment 4 value is 2.384432138280637

Notice that while you are calculating the excess kurtosis, the formula implemented for a generic n-moment doesn't account for that.

Calculate weighted statistical moments in Python

2 Answers2