The speaker will give an introduction to dependence modelling based on copulas. Basic concepts will first be presented in a data analytic perspective. Broad strategies for copula model building will then be described, and some of the most common constructions will be critically reviewed. In the second part of his talk, the speaker will show how graphical tools and nonparametric inference techniques can be used for copula model fitting and validation. Procedures will be illustrated with concrete data; special attention will be paid to phenomena exhibiting joint extreme behavior.
Slides (pdf)
Introductory
paper (pdf)
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Graphical models and copulas are two sets of tools for
multivariate analysis. Both are in some sense pathways to the
construction of multivariate distributions using modular
representations. The former focuses on languages to express
conditional independence constraints, factorizations and efficient
inference algorithms. The latter allows for the encoding of some
marginal features of the joint distribution (univariate marginals,
in particular) directly, without resorting to an inference
algorithm. In this talk we exploit copula parameterizations in two
graphical modeling tasks: parameterizing decomposable models and
building proposal distributions for inference with Markov chain
Monte Carlo; parameterizing directed mixed graph models and
providing simple estimation algorithms based on composite likelihood
methods.
Slides (pdf)
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In this talk, we introduce the main concepts of vine
pair-copula constructions. This framework uses bivariate copulas as
building blocks to obtain higher-dimensional distributions. As these
bivariate copulas can be selected from a wide range of families, the
vine approach leads to a more flexible model compared to traditional
approaches. For the estimation of vine pair-copulas, we introduce a
mathematically elegant framework that joins (a) graph theory, to
determine the dependency structure of the data, and (b)
maximum-likelihood estimation, to fit bivariate copulas.
Web
site
Slides (pdf)
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With the ``discovery'' of copulas by machine learning
researchers, several works have emerged that focus on the
high-dimensional scenario. This talk will provide a brief overview
of these works and cover tree-averaged distributions (Kirshner), the
nonparanormal (Liu, Lafferty and Wasserman), copula processes (Wilson
and Ghahramani), kernel-based copula processes (Jaimungal and Ng),
and copula networks (Elidan). Special emphasis will be given to the
high level similarities and differences between these works.
Slides (pdf)
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We introduce a Dirichlet prior mixture of
meta-Gaussian distributions to perform dependency-seeking clustering
when co-occurring samples from different data sources are
available. The model extends Bayesian mixtures of Canonical
Correlation Analysis clustering methods to multivariate data
distributed with arbitrary continuous margins. Using meta-Gaussian
distributions gives the freedom to specify each margin separately
and thereby also enables clustering in the joint space when the data
are differently distributed in the different views. The Bayesian
mixture formulation retains the advantages of using a Dirichlet
prior. We do not need to specify the number of clusters and the
model is less prone to overfitting than non-Bayesian
alternatives. Inference is carried out using a Markov chain sampling
method for Dirichlet process mixture models with non-conjugate prior
adapted to the copula mixture model. Results on different simulated
data sets show significant improvement compared to a Dirichlet prior
Gaussian mixture and a mixture of CCA model.
Slides (pdf) Poster (pdf)
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We present a semi-parametric method for the estimation
of the copula of two random variables X and Y when conditioning to
an additional covariate Z. The conditional bivariate copula is
described using a parametric model fully specified in terms of
Kendall's tau. The dependence of the conditional copula on Z is
captured by expressing tau as a function of Z. In particular, tau is
obtained by filtering a non-linear latent function, which is
evaluated on Z, through a sigmoid-like function. A Gaussian process
prior is assumed for the latent function and approximate Bayesian
inference is performed using expectation propagation. A series of
experiments with simulated and real-world data illustrate the
advantages of the proposed approach.
Slides (pdf)
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A fundamental problem in statistics is the estimation of dependence
between random variables. While information theory provides standard
measures of dependence (e.g. Shannon-, Renyi-, Tsallis-mutual
information), it is still unknown how to estimate these quantities
from i.i.d. samples in the most efficient way. In this presentation we
review some of our recent results on copula based nonparametric
dependence estimators and demonstrate their robustness to outliers
both theoretically in terms of finite-sample breakdown points and by
numerical experiments in independent subspace analysis and image
registration.
Slides (pdf)
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- New Classes of Copula Gaussian Graphical Models
Justin Dauwels, Hang Yu, Xueou Wang, Xu Zhang and Shiyan Xu
Spotlight (PDF)
- Applications of Copulas in Neuroscience
Arno Onken, Steffen Grunewalder, Matthias Munk and Klaus Obermayer
Spotlight (PDF)
- Modeling Cell Populations in High Content Screening using Copulas
Edouard Pauwels
Spotlight
(PDF)
- Transfer Learning with Copulas
David Lopez Paz and Jose Miguel Hernandez-Lobato
Spotlight (PDF)
- Copula functions for learning multimodal densities with non-linear dependencies
Ashutosh Tewari, Madhusudana Shashanka, Michael Giering
Spotlight (PDF) Poster (PDF)
- C-Vines
Nicole Kraemer and Ulf Schepsmeier
Spotlight (PDF)
- Copulas for Context Sensitive Classification
Ashish Kapoor
Spotlight (PDF)
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