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Unbiased estimators for multivariate statistical moments

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PyMoments is a toolkit for unbiased estimation of multivariate statistical moments. In the current version (1.0.0), only multivariate k-statistics are implemented, allowing for the unbiased estimation of cumulants. An implementation of h-statistics (for unbiased estimation of central moments) is planned for a future release.


PyMoments can be installed either from the GitHub source, or via PyPI. To install the package from GitHub, first clone the repository:

$ git clone

Navigate to the root directory, which contains the file. You can then use setuptools to install the package and run the unit tests:

$ python install
$ python test

Alternatively, PyMoments can be installed using PyPI:

$ pip install PyMoments

Note that PyMoments requires NumPy. I have only tested the package with NumPy version 1.16.5, but it should be compatible with older versions.

Basic Usage

The simplest use case is to compute a multivariate k-statistic from a 2D array of data. For example, generate a random sample from a multivariate normal distribution:

import numpy as np
mu = np.zeros(3,)
sigma = np.array([
    [2, 0, 1],
    [0, 2, -1],
    [1, -1, 2]
data = np.random.multivariate_normal(mu, sigma, size=(1000,))

The data variable is a 1000 x 3 array, where each column corresponds to one of the three random variables in the joint distribution, and each row is an observation.

The first k-statistic is identical to the sample mean. Thus, we can compare the first-order k-statistics for each column of the array, to the simple average:

from PyMoments import kstat

first_order_kstats = [
    kstat(data, (0,)),
    kstat(data, (1,)),
    kstat(data, (2,))

sample_means = np.mean(data, axis=0)

print('First-order k-statistics:', first_order_kstats)
print('Sample means:', sample_means) 

The method kstat(data, (0,)) computes the first-order k-statistic, from the first column of the data. As the console output of this example reveals, the first-order k-statistics are indeed equivalent to the sample means, up to floating point error.

Second-order k-statistics are identical to covariances. Therefore, we can compare the values of the kstat() function to the covariance matrix of the data:

second_order_kstats = np.zeros((3, 3))
for i in range(3):
    for j in range(3):
        second_order_kstats[i, j] = kstat(data, (i, j))

print('Second-order k-statistics:')
print('Covariance matrix:')

Higher-order k-statistics become difficult to express in terms of familiar statistics, but they still provide insight into the underlying distribution. Third-order k-statistics, for example, are related to the skewness of a distribution. Normal distributions have zero skew, and therefore, all third-order k-statistics should be fairly close to zero:

print('Sample third-order k-statistics:') 
print(kstat(data, (0, 1, 2)))
print(kstat(data, (0, 0, 1)))
print(kstat(data, (2, 2, 2)))

The second argument to the kstat() method specifies a multiset of column indices (as a sequence of integers), which encodes the particular k-statistic to be computed. The order of the k-statistic is equal to the length of this multiset. Note that kstat() is symmetric with respect to this argument, i.e., permuting the indices will have no affect on the output. Repeated indices are allowed; in fact, repeated indices are required to compute classical univariate k-statistics:

new_data = np.random.randn(1000, 1)

print('First univariate k-stat:')
print(kstat(new_data, (0,)))

print('Second univariate k-stat:')
print(kstat(new_data, (0, 0)))

print('Third univariate k-stat:')
print(kstat(new_data, (0, 0, 0)))

print('Fourth univariate k-stat:')
print(kstat(new_data, (0, 0, 0, 0)))

It should also be pointed out that k-statistics become noisy and difficult to compute with increasing order. For example, try evaluating a 9th-order univariate cumulant on a new sample several times:

kstat(np.random.randn(1000, 1), (0,) * 9)

Not only does the function take several seconds to return, but repeats of this experiment can lead to widely different values of the k-statistic, despite the fact that normal distributions have ninth-order cumulants of zero. The runtime of kstat() on nth-order indices scales with Bell's number B(n), so... I do not recommend trying k-statistics of order 10 or higher.

License, Citation, and Acknowledgements

PyMoments by Kevin D. Smith is licensed under a non-commercial Creative Commons license (CC BY-NC 4.0). When possible, please cite this package:

  author = {Kevin D. Smith},
  title = {PyMoments: A Python toolkit for unbiased estimation of multivariate statistical moments},
  howpublished = {\texttt{}},
  year = 2020 

This work was supported in part by the U.S. Defense Threat Reduction Agency, under grant HDTRA1-19-1-0017.

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