Parametric

The distfit library uses the goodness of fit test to determine the best probability distribution to the non-censored data. It works by comparing the observed frequency (f) to the expected frequency from the model (f-hat), and computing the residual sum of squares (RSS). Note that non-censored data is the full dataset, and not having any part deleted or suppressed as that can lead to biases. With distfit we can test up to 89 univariate distributions, derived from the scipy library, for which the best fitted distribution is returned with the loc, scale, arg parameters.

Distributions

The distributions to be tested can be specified at initialization using the distr parameter. The distributions options are 1. Manually specifying one or multiple distribution 2. popular set of distributions 3. full set of distributions

Manually specifying: Manually specifying can be for one or multiple distributions. See example below how its done.

# Load library
from distfit import distfit
# Initialize model and test only for normal distribution
dfit = distfit(distr='norm')
# Set multiple distributions to test for
dfit = distfit(distr=['norm','t'])

The popular set of PDFs contains the following set of distributions and can be used as depicted below:

norm

genextreme

expon

gamma

pareto

lognorm

dweibull

beta

t

uniform

# Initialize model and select popular distributions
dfit = distfit(distr='popular')

The full set contains the following set of distributions:

alpha

betaprime

chi2

expon

fatiguelife

anglit

bradford

cosine

exponnorm

fisk

arcsine

burr

dgamma

exponweib

foldcauchy

arcsine

cauchy

dweibull

exponpow

foldnorm

beta

chi

erlang

f

frechet_r[x]

gilbrat

gompertz

gumbel_r

gumbel_l

halfcauchy

halfgennorm

hypsecant

invgamma

invgauss

invweibull

laplace

levy

levy_l [X]

levy_stable[X]

logistic

lognorm

lomax

maxwell

mielke

nakagami

pearson3

powerlaw

powerlognorm

powernorm

rdist

rice

recipinvgauss

semicircular

t

triang

tukeylambda

uniform

vonmises

vonmises_line

wald

wrapcauchy

gengamma

genlogistic

frechet_l[x]

halfnorm

genexpon

genextreme

gennorm

gausshyper

genpareto

gamma

genhalflogistic

halflogistic

johnsonsb

johnsonsu

loggamma

loglaplace

norm

pareto

rayleigh

reciprocal

truncexpon

truncnorm

weibull_min

weibull_max

Note that levy_l and levy_stable are removed from the full list because it is too slow. The distributions frechet_r and frechet_l are also not supported anymore.

# Initialize model and select all distributions
dfit = distfit(distr='full')

Residual Sum of Squares (RSS)

The RSS describes the deviation predicted from actual empirical values of data. Or in other words, the differences in the estimates. It is a measure of the discrepancy between the data and an estimation model. A small RSS indicates a tight fit of the model to the data. RSS is computed by:

Where yi is the ith value of the variable to be predicted, xi is the i-th value of the explanatory variable, and f(xi) is the predicted value of yi (also termed y-hat).

Goodness-of-fit

Besides RSS, there are various other approaches to determine the goodness-of-fit, such as the maximum likelihood estimation (mle), moment matching estimation (mme), quantile matching estimation (qme) or maximizing goodness-of-fit estimation (mge). distfit may be extended with more approaches in future versions.

Probabilities and multiple test correction

The predict function: distfit.distfit.distfit.predict() will compute the probability of samples in the fitted PDF. Each probability will by default be corrected for multiple testing. Multiple testing correction refers to re-calculating probabilities obtained from a statistical test which was repeated multiple times. In order to retain a prescribed family-wise error rate alpha in an analysis involving more than one comparison, the error rate for each comparison must be more stringent than alpha. Note that, due to multiple testing approaches, it can occur that samples can be located outside the confidence interval but not marked as significant. See section Algorithm -> Multiple testing for more information.

The following output variables are available. More information can be found under return in the docstring.

dfit.predict

dfit.results[‘y_proba’]
dfit.results[‘y_pred’]
dfit.results[‘df’]
dfit.summary

The output variable y_proba is by default corrected for multiple testing using the false discovery rate (fdr). FDR-controlling procedures are designed to control the expected proportion of “discoveries” that are false. If desired, other multiple test methods can be choosen, each with its own properties.

# Initialize
dfit = distfit(multtest='holm', alpha=0.01)

None	No multiple testing
bonferroni	one-step correction
sidak	one-step correction
holm-sidak	step down method using Sidak adjustments
holm	step-down method using Bonferroni adjustments
simes-hochberg	step-up method (independent)
hommel	closed method based on Simes tests (non-negative)
fdr_bh	Benjamini/Hochberg (non-negative)
fdr_by	Benjamini/Yekutieli (negative)
fdr_tsbh	two stage fdr correction (non-negative)
fdr_tsbky	two stage fdr correction (non-negative)

Input parameters

Various input parameters can be specified at the initialization of distfit.

Variable name	type	Default	Description
method	str	‘parametric’	Specify the method type: ‘parametric’, ‘empirical’
alpha	float	0.05	Significance alpha.
multtest	str	‘fdr_bh’	Multiple test correction method
bins	int	50	To determine the empirical historgram
bound	int	‘both’	Directionality to test for significance Upper and lowerbounds: ‘both’ Upperbounds: ‘up’, ‘high’, ‘right’ Lowerbounds: ‘down’, ‘low’, ‘left’
distr	str	‘popular’	The (set) of distribution to test. ‘popular’, ‘full’ ‘t’ : user specified ‘norm’ : user specified etc
n_perm	int	10000	Number of permutations to model null-distribution in case of method is ‘empirical’

Output variables

There are many output parameters provided by distfit. It all starts with the initialization:

# Initialize model and select popular distributions
dfit = distfit(alpha=0.01)

The object now returns variables that are set by default, except for the alpha parameter (nothing else is provided). For more details, see the returns in the docstrings at distfit.distfit.distfit(). In the next step, input-data X can be provided:

# Initialize model and select popular distributions
dfit.fit_transform(X)

The object can now be feeded with data X, using fit and transform function, that will add more output variables to the object. Instead of using the two functions seperately, it can also be performed with fit_transform: distfit.distfit.distfit.fit_transform().

The fit_transform outputs the variables summary, distributions and model

dfit.summary: The summary of the fits across the distributions.

print(dfit.summary)
#   name         RSS  ...      scale                                      arg
# 0       gamma  0.00185211  ...  0.0370159                     (3004.147964288284,)
# 1           t  0.00186936  ...    2.02883                     (2517332.591227023,)
# 2        norm  0.00186945  ...    2.02882                                       ()
# 3        beta  0.00186949  ...    37.7852  (39.068072383763294, 46.06165256503778)
# 4     lognorm  0.00197359  ...    57.4149                   (0.03537982752374607,)
# 5  genextreme  0.00297519  ...     2.0106                    (0.2437702978900108,)
# 6    dweibull  0.00695379  ...    1.73297                    (1.2545534252305621,)
# 7     uniform    0.241881  ...    14.1011                                       ()
# 8       expon    0.353202  ...    6.99491                                       ()
# 9      pareto    0.634924  ...    1.42095                    (0.5384782616155881,)

dfit.distributions is a list containing the extracted pdfs from scipy

The collected distributions.

dfit.model contains information regarding the best scoring pdf:

dfit.model[‘RSS’]
dfit.model[‘name’]
dfit.model[‘model’]
dfit.model[‘params’]
dfit.model[‘loc’]
dfit.model[‘scale’]
dfit.model[‘arg’]