PolyMoment: High-order Moments of Multivariate Polynomials

PolyMoment is now an online tool at http://polymoment.com/.

This is the implementation of the thesis: Moment-Based Uncertainty Propagation Using Multivariate Polynomials: Advances in Probabilistic Engineering Design.

Usage

The example below demonstrates how PolyMoment can be used to compute the high-order moment, mean, variance, standard deviation, skewness, or kurtosis of polynomial x0^2+x1^2 with random variables x0 and x1 following normal distribution with means m0 and m1, respectively, and standard deviations s0 and s1, respectively.

The supported distributions and their respective probability density functions can be found in this file.

from polymoment import PolyMoment

pm = PolyMoment(
    poly='x0**2+x1**2',
    dist={
        'x0': {
            'distribution': 'normal',	# str
            'type': 'symmetrical',      # str
            'translation': 'm0',        # str | float
            'scale': 's0',              # str | float
            'beta1': None,              # str | float
            'beta2': None,              # str | float
        },
        'x1': {
            'distribution': 'normal',   # str
            'type': 'symmetrical',      # str
            'translation': 'm1',        # str | float
            'scale': 's1',              # str | float
            'beta1': None,              # str | float
            'beta2': None,              # str | float
        }
    }
)

# mean
print(pm.mean())

# variance
print(pm.var())

# standard deviation
print(pm.std())

# skewness
print(pm.skew())

# kurtosis (excess kurtosis + 3)
print(pm.kurt())

# n-th order moment of poly
print(pm.moment(order=1))

Supported Distribution Types

'symmetrical',
'one_sided_right',
'one_sided_left'

Supported Distributions

Further information on the probability density functions listed below can be found in this file.

'uniform',
'trapezoidal',
'triangular',
'beta',
'normal',
'student',
'laplace',
'gamma',
'weibull',
'maxwell'