Package for canonical vine copula trees with mixed continuous and discrete marginals.

## Project description

Package for canonical vine copula trees with mixed continuous and discrete marginals. If you use this software for publication, please cite [ONKEN2016].

## 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:

import numpy as np
is_continuous = np.full((3), True, dtype=bool)
is_continuous[1] = False

To fit a mixed vine to the samples:

from 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)

Note that for the canonical vine, the order of elements is important. Elements should be sorted according to the importance of their dependencies to other elements where elements with important dependencies to many other elements should come first. A heuristic way to select the order of elements is to calculate Kendall’s tau between all element pairs (see scipy.stats.kendalltau), to obtain a score for each element by summing the tau’s of the pairs the element occurs in and to sort elements in descending order according to their scores.

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 Copula, GaussianCopula, ClaytonCopula, \
FrankCopula
from mixedvines.mixedvine import MixedVine
import matplotlib.pyplot as plt
import itertools
# Manually construct mixed vine
dim = 3  # Dimension
vine_type = 'c-vine'  # Canonical vine type
vine = MixedVine(dim, vine_type)
# 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):
margin = np.sum(pdf, axis=len(comb)-i-1).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

[ONKEN2016] (1,2)

A. Onken and S. Panzeri (2016). Mixed vine copulas as joint models of spike counts and local field potentials. In D. D. Lee, M. Sugiyama, U. V. Luxburg, I. Guyon and R. Garnett, editors, Advances in Neural Information Processing Systems 29 (NIPS 2016), pages 1325-1333.

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/>.

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