Description

This package contains a complete framework based on canonical vine copulas for modelling multivariate data that are partly discrete and partly continuous. The resulting multivariate distributions are flexible with rich dependence structures and marginals.

For continuous marginals, implementations of the normal and the gamma distributions are provided. For discrete marginals, Poisson, binomial and negative binomial distributions are provided. As bivariate copula building blocks, the Gaussian, Frank and Clayton families as well as rotation transformed families are provided. Additional marginal and pair copula distributions can be added easily.

The package includes methods for sampling, likelihood calculation and inference, all of which have quadratic complexity. These procedures are combined to estimate entropy by means of Monte Carlo integration.

Please see [ONKEN2016] for a more detailed description of the framework.

Documentation

The full documentation for the mixedvines package is available at Read the Docs.

Requirements

The package is compatible with Python 2.7 and 3.x and additionaly requires NumPy and SciPy.

Installation

To install the mixedvines package, run:

pip install mixedvines

Usage

Suppose that data are given in a NumPy array samples with shape (n, d), where n is the number of samples and d is the number of elements per sample. First, specify which of the elements are continuous. If, for instance, the distribution has three elements and the first and last element are continuous whereas the second element is discrete:

is_continuous = [True, False, True]

To fit a mixed vine to the samples:

from mixedvines.mixedvine import MixedVine
vine = MixedVine.fit(samples, is_continuous)

vine is now a MixedVine object. To draw samples from the distribution, calculate their density and estimate the distribution entropy in units of bits:

samples = vine.rvs(size=100)
logpdf = vine.logpdf(samples)
(entropy, standard_error_mean) = vine.entropy(sem_tol=1e-2)

To manually construct and visualize a simple mixed vine model:

from scipy.stats import norm, gamma, poisson
import numpy as np
from mixedvines.copula import GaussianCopula, ClaytonCopula, FrankCopula
from mixedvines.mixedvine import MixedVine
import matplotlib.pyplot as plt
import itertools
# Manually construct mixed vine
dim = 3  # Dimension
vine = MixedVine(dim)
# Specify marginals
vine.set_marginal(0, norm(0, 1))
vine.set_marginal(1, poisson(5))
vine.set_marginal(2, gamma(2, 0, 4))
# Specify pair copulas
vine.set_copula(1, 0, GaussianCopula(0.5))
vine.set_copula(1, 1, FrankCopula(4))
vine.set_copula(2, 0, ClaytonCopula(5))
# Calculate probability density function on lattice
bnds = np.empty((3), dtype=object)
bnds[0] = [-3, 3]
bnds[1] = [0, 15]
bnds[2] = [0.5, 25]
(x0, x1, x2) = np.mgrid[bnds[0][0]:bnds[0][1]:0.05, bnds[1][0]:bnds[1][1],
                        bnds[2][0]:bnds[2][1]:0.1]
points = np.array([x0.ravel(), x1.ravel(), x2.ravel()]).T
pdf = vine.pdf(points)
pdf = np.reshape(pdf, x1.shape)
# Generate random variates
size = 100
samples = vine.rvs(size)
# Visualize 2d marginals and samples
comb = list(itertools.combinations(range(dim), 2))
for i, cmb in enumerate(comb):
    # Sum over all axes not in cmb
    cmb_inv = tuple(set(range(dim)) - set(cmb))
    margin = np.sum(pdf, axis=cmb_inv).T
    plt.subplot(2, len(comb), i + 1)
    plt.imshow(margin, aspect='auto', interpolation='none', cmap='hot',
               origin='lower', extent=[bnds[cmb[0]][0], bnds[cmb[0]][1],
                                       bnds[cmb[1]][0], bnds[cmb[1]][1]])
    plt.ylabel('$x_' + str(cmb[1]) + '$')
    plt.subplot(2, len(comb), len(comb) + i + 1)
    plt.scatter(samples[:, cmb[0]], samples[:, cmb[1]], s=1)
    plt.xlim(bnds[cmb[0]][0], bnds[cmb[0]][1])
    plt.ylim(bnds[cmb[1]][0], bnds[cmb[1]][1])
    plt.xlabel('$x_' + str(cmb[0]) + '$')
    plt.ylabel('$x_' + str(cmb[1]) + '$')
plt.tight_layout()
plt.show()

This code shows the 2d marginals and 100 samples of a 3d mixed vine.

Source code

The source code of the mixedvines package is hosted on GitHub.

References

License

Copyright (C) 2017, 2018 Arno Onken

This file is part of the mixedvines package.

The mixedvines package is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3 of the License, or (at your option) any later version.

The mixedvines package is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with this program; if not, see <http://www.gnu.org/licenses/>.