Using lmoments3

L-moments estimation from sample data

The primary purpose of this library is to estimate L-moments from a sample dataset.

The function lmoments3.lmom_ratios(data, nmom)() takes an input list or numpy array data and the number of L-moments to estimate from the dataset.

import lmoments3 as lm

data = [2.0, 3.0, 4.0, 2.4, 5.5, 1.2, 5.4, 2.2, 7.1, 1.3, 1.5]
lm.lmom_ratios(data, nmom=5)
[Out]: [
    3.2363636363636363,
    1.1418181818181818,
    0.27388535031847133,
    0.023354564755838598,
    -0.042462845010615709,
]

This returns the first five sample L-moments, in the structured as l1, l2, t3, t4, t5. Where t3..5 = l3..5 / l2.

Fitting distribution functions to sample data

Sample data can be fitted directly to statistical distribution functions using the lmoments3 library.

For example, using the gamma distribution:

import lmoments3 as lm
from lmoments3 import distr

data = [2.0, 3.0, 4.0, 2.4, 5.5, 1.2, 5.4, 2.2, 7.1, 1.3, 1.5]
paras = distr.gam.lmom_fit(data)
paras
[Out]: OrderedDict(
    [("a", 2.295206110128833), ("loc", 0), ("scale", 1.4100535991436407)]
)

This returns the distribution’s parameters as an OrderedDict in the same order as a standard scipy list of distribution function parameters. The distribution parameters can be used, for example, like this:

fitted_gam = distr.gam(**paras)
median = fitted_gam.ppf(0.5)
median
[Out]: 2.7804212925067344

For full details of distribution function methods, see the scipy.stats documentation. Some useful methods include:

pdf: Probability density function

cdf: Cumulative distribution function

ppf: Inverse cumulative distribution function (also known as quantile function or percentage point function)

rvs: Random numbers generator

Computing L-moments from distribution parameters

The lmoments3 package provides two additional methods to compute the L-moments (λ1..n) or L-moment ratios (λ1, λ2, τ3..n) for a distribution with given parameters.

Example:

distr.gam.lmom(nmom=3, **paras)
[Out]: [3.2363636363636363, 1.1418181181569327, 0.24963415541016151]

distr.gam.lmom_ratios(nmom=4, **paras)
[Out]: [
    3.2363636363636363,
    1.1418181181569327,
    0.21862865148182167,
    0.13877337951549581,
]

Or using the frozen distribution:

moments = fitted_gam.lmom(nmom=3)
ratios = fitted_gam.lmom_ratios(nmom=4)

Modified implementation of negative log likelihood function

nnlf(data, *args, **kwds)()

Calculates the Negative Log Likelihood. Provide data to calculate the negeative log likelihood. If no distribution parameters are provided, the scipy defaults of loc=0 and scale=1 are used.

Example: Calculate the Negative Log Likelihood of a Gamma distribution fitted to data:

from lmoments3 import distr

paras = distr.gam.lmom_fit(data)
distr.gam.nnlf(data, **paras)
[Out]: 21.283995091031549

Example: Calculate the Negative Log Likelihood of a Gamma distribution with parameters 2.5 and 1.0 when fitted to data:

from lmoments3 import distr
from collections import OrderedDict

distr.gam.nnlf(data, a=2.5, scale=1)
[Out]: 22.166452544264637

Other statistical methods

The lmoments3.stats module provides some additional statistical parametes to evaluate fitting of data to distribution function.

AIC(data, distr_name, distr_paras)()

Calculate the Akaike Information Criterion (AIC) using the chosen dataset and distribution.

Example: Calculate the Akaike Information Criterion for the weibull distribution using the input dataset data:

from lmoments3 import stats, distr

paras = {"loc": 0.67, "scale": 2.71, "c": 1.18}
stats.AIC(data, "wei", paras)
[Out]: 47.500528639652515

Functions AICc() and BIC() have a similar structure and calculate the corrected Akaike Information Criterion and the Bayesian Information Criterion respectively.