Univariate Distributions

All the examples in the introduction focused exclusively on the Normal (or Gaussian) Distribution over a single random variable, but lots of other univariate distributions exist.

The Copulas library supports several of them through the Univariate subclasses defined within the copulas.univariate package:

• copulas.univariate.BetaUnivariate: Implements a Beta distribution.

• copulas.univariate.GammaUnivariate: Implements a Gamma distribution.

• copulas.univariate.GaussianKDE: Implements a Kernel-Density Estimate using a Gaussian kernel.

• copulas.univariate.GaussianUnivariate: Implements a Gaussian (or Normal) distribution.

• copulas.univariate.TruncatedGaussian: Implements a Truncated Gaussian distribution.

Let’s explore an example of how to use a Univariate subclass.

Univariate Usage Example

In this example we will be focusing on the BetaUnivariate class, which implements a Beta Distribution.

This distribution is parameterized by two positive shape parameters, \(α\) and \(β\), which in our implementation are called \(a\) and \(b\).

Additionally, our implementation also uses the same loc and scale arguments as the underlying scipy.stats.beta model that we are using.

For this example, we will be using a simulated dataset that follows the beta distribution with parameters \(a = 3\), \(b = 1\), \(loc = 4\) and \(scale = 1\).

from copulas.datasets import sample_univariate_beta

data = sample_univariate_beta()

data.head()

0    4.796025
1    4.935189
2    4.637677
3    4.945320
4    4.726815
dtype: float64

from copulas.visualization import hist_1d

hist_1d(data)

<matplotlib.axes._subplots.AxesSubplot at 0x7f2f587b86a0>

Fitting the model

The first step to use our BetaUnivariate model is to fit it to the data by passing the data to its fit method.

from copulas.univariate import BetaUnivariate

beta = BetaUnivariate()
beta.fit(data)

After the model has been fitted, we can observe the parameters that it has aproximated from the data.

beta._params

{'loc': 4.095558923695414,
 'scale': 0.9044258869808379,
 'a': 2.552323190906777,
 'b': 0.9041091899047261}

Sampling new data

Once the model is fitted, we can generate new data that follows a similar distribution.

To do this, we can call the sample method of our beta object, passing the number of samples that we want to generate.

sampled = beta.sample(1000)

We can compare the distribution of the real data to the sampled one by plotting both histograms together.

from copulas.visualization import compare_1d

compare_1d(data, sampled)

Probability Density

The probability density of a Beta distribution is defined by

:nbsphinx-math:`begin{equation} frac{x^{alpha -1}(1-x)^{beta -1}}

{B(alpha ,beta)}

end{equation}`

where \(B(\alpha,\beta)\) is the Beta function defined as

:nbsphinx-math:`begin{equation} B(alpha, beta) = frac{Gamma (alpha )Gamma (beta )}

{Gamma (alpha +beta )}

end{equation}`

and \(\Gamma\) is the Gamma function.

The probability density can be computed for an array of data points using the probability_density method.

probability_density = beta.pdf(sampled)

probability_density[0:5]

array([3.96772858, 1.67572128, 1.90369638, 3.51125542, 0.71732109])

If we plot the probability densities sorted by the sampled values we will get a better notion of their behavior.

import pandas as pd

pd.DataFrame({
    'data': sampled,
    'probability_density': probability_density
}).sort_values('data').set_index('data').plot()

<matplotlib.axes._subplots.AxesSubplot at 0x7f2f55d6fd30>

Cumulative Distribution

The cumulative distribution of Beta distribution is defined by:

\begin{equation} p = F(x, a, b) = \frac{1}{B(a, b)}\int_{0}^{x}t^{a - 1}{(1 - t)}^{b - 1}dt \end{equation}

where \(B(\alpha,\beta)\) is the Beta function defined above.

The cumulative distribution can be computed for an array of data points using the cumulative_distribution method.

cumulative_distribution = beta.cumulative_distribution(sampled)

cumulative_distribution[0:5]

array([0.98185868, 0.4148939 , 0.49927947, 0.95037356, 0.11102835])

Like with probability densities, if we plot the cumulative distribution values sorted by the input data, we will get a better notion of how it behaves

pd.DataFrame({
    'data': sampled,
    'cumulative distribution': cumulative_distribution
}).sort_values('data').set_index('data').plot()

<matplotlib.axes._subplots.AxesSubplot at 0x7f2f55c3fb70>

to_dict and from_dict

All the Univariate classes implement a to_dict method that allows obtaining all the parameters that define the distribution in a python dictionary.

parameters = beta.to_dict()

parameters

{'loc': 4.095558923695414,
 'scale': 0.9044258869808379,
 'a': 2.552323190906777,
 'b': 0.9041091899047261,
 'type': 'copulas.univariate.beta.BetaUnivariate'}

This parameters dictionary can be later on passed to the Univariate.from_dict class method, which will create an instance of our model with the same parameters as before.

from copulas.univariate import Univariate

new_beta = Univariate.from_dict(parameters)

We can sample some more data again to see how it still behaves as expected.

new_sampled = new_beta.sample(1000)

compare_1d(data, new_sampled)

Selecting the best Univariate

A part from the specific Univariate subclasses, Copulas allows you to use the Univariate parent class as a generic distribution.

Fitting a generic Univariate

When this class is used directly and fitted to the data, it will automatically search for the Univariate subclass that better fits to the data.

Let’s see what happens when we fit it to the data that we were using above.

from copulas.univariate import Univariate

univariate = Univariate()
univariate.fit(data)

In order to see which subclass has been chosen and with which parameters, we can call the to_dict method.

parameters = univariate.to_dict()

parameters

{'loc': 4.095558923695414,
 'scale': 0.9044258869808379,
 'a': 2.552323190906777,
 'b': 0.9041091899047261,
 'type': 'copulas.univariate.beta.BetaUnivariate'}

We can observe here how the Univariate class automatically selected the BetaUnivariate as the best distribution for our data, and how it learned the same parameters as before.

Let’s go a bit further and see what happens when we fit Univariate instances on data with different distributions:

from copulas.datasets import sample_univariates

data = sample_univariates()

data.head()

	bernoulli	bimodal	uniform	normal	degenerate	exponential	beta
0	0.0	11.399355	0.498160	1.496714	0.37454	3.469268	4.796025
1	0.0	10.924634	2.802857	0.861736	0.37454	6.010121	4.935189
2	0.0	10.059630	1.927976	1.647689	0.37454	4.316746	4.637677
3	0.0	9.353063	1.394634	2.523030	0.37454	3.912943	4.945320
4	1.0	-0.234153	-0.375925	0.765847	0.37454	3.169625	4.726815

synth_data = pd.DataFrame()
distributions = []

for column in data.columns:
    real_data = data[column]
    univariate = Univariate()
    univariate.fit(real_data)
    synth_data[column] = univariate.sample(len(real_data))
    distributions.append(univariate.to_dict()['type'])

synth_data.head()

	bernoulli	bimodal	uniform	normal	degenerate	exponential	beta
0	0.126054	1.950079	0.833157	1.154327	0.37454	3.014643	4.990397
1	-0.216558	13.461737	2.131687	-0.031510	0.37454	3.205208	4.781987
2	0.158551	1.011978	0.274088	1.563969	0.37454	4.097697	4.982117
3	0.151797	9.476947	0.527291	0.698796	0.37454	3.031567	4.809642
4	1.023261	0.982005	1.266706	0.890644	0.37454	5.713887	4.724303

distributions

['copulas.univariate.gaussian_kde.GaussianKDE',
 'copulas.univariate.gaussian_kde.GaussianKDE',
 'copulas.univariate.beta.BetaUnivariate',
 'copulas.univariate.gaussian_kde.GaussianKDE',
 'copulas.univariate.beta.BetaUnivariate',
 'copulas.univariate.gamma.GammaUnivariate',
 'copulas.univariate.beta.BetaUnivariate']

compare_1d(data, synth_data)

Recreating the Univariate

As we learned before, the Univariate.from_dict class method can be used to recreate our model.

univariate = Univariate()
univariate.fit(data['beta'])
new_model = Univariate.from_dict(parameters)

Notice, though, how in this case the model will not be an instance of the class Univariate anymore, but rather an instance of the corresponding subclass, BetaUnivariate.

new_model.__class__

copulas.univariate.beta.BetaUnivariate

Univariate Families

In some cases, we might want to add some constraints in order to restrict the search to only some Univariate distributions.

This can be achieved by creating and passing a list of candidate subclasses to our Univariate class.

For example, let’s restrict the search over the bimodal column of our dataset restricting the search to only the BetaUnivariate, the GaussianUnivariate and the GammaUnivariate.

from copulas.univariate import GaussianUnivariate, GammaUnivariate

candidates = [BetaUnivariate, GaussianUnivariate, GammaUnivariate]

univariate = Univariate(candidates=candidates)
univariate.fit(data['bimodal'])

univariate.to_dict()

{'loc': 6.830066099361434,
 'scale': 4.811082463652564,
 'type': 'copulas.univariate.gaussian.GaussianUnivariate'}

However, the Copulas Univariate subclasses are organized in families to make the application of these constraints to the search easier.

The Univariate families are organized in ways:

• PARAMETRIC: Distributions can be either non parametric or parametric.

• BOUNDED: Distributions can be either unbounded, semi-bounded or bounded.

We can see these properties by examining the corresponding attributes of each Univariate subclass.

from copulas.univariate import GaussianKDE, TruncatedGaussian

univariates = [
    BetaUnivariate,
    GammaUnivariate,
    GaussianKDE,
    GaussianUnivariate,
    TruncatedGaussian
]

pd.DataFrame([
    [uni.__name__, uni.PARAMETRIC, uni.BOUNDED]
    for uni in univariates
], columns=['Distribution', 'Parametric', 'Bounded'])

	Distribution	Parametric	Bounded
0	BetaUnivariate	ParametricType.PARAMETRIC	BoundedType.BOUNDED
1	GammaUnivariate	ParametricType.PARAMETRIC	BoundedType.SEMI_BOUNDED
2	GaussianKDE	ParametricType.NON_PARAMETRIC	BoundedType.UNBOUNDED
3	GaussianUnivariate	ParametricType.PARAMETRIC	BoundedType.UNBOUNDED
4	TruncatedGaussian	ParametricType.PARAMETRIC	BoundedType.BOUNDED

When searching for the best distribution, instead of building and passing the list of candidates by hand, we can simply pass the parametric or bounded value that we want use.

For example, let’s search again for the best distribution over the bimodal column of our dataset, but restricting the search to only PARAMETRIC and BOUNDED distributions.

from copulas.univariate import ParametricType, BoundedType

univariate = Univariate(
    parametric=ParametricType.PARAMETRIC,
    bounded=BoundedType.BOUNDED
)
univariate.fit(data['bimodal'])

univariate.to_dict()

{'a': -1.5932065697344446,
 'b': 1.2398855571316814e-14,
 'loc': 13.193107687054022,
 'scale': 10.315282059796969,
 'type': 'copulas.univariate.truncated_gaussian.TruncatedGaussian'}