The estimation of dependencies between multiple variables is a central problem in the analysis of financial time series. A common approach is to express these dependencies in terms of a copula function. Typically the copula function is assumed to be constant but this may be innacurate when there are covariates that could have a large influence on the dependence structure of the data. To account for this, a Bayesian framework for the estimation of conditional copulas is proposed. In this framework the parameters of a copula are non-linearly related to some arbitrary conditioning variables.

A new framework based on the theory of copulas is proposed to address semisupervised domain adaptation problems. The presented method factorizes any multivariate density into a product of marginal distributions and bivariate copula functions. Therefore, changes in each of these factors can be detected and corrected to adapt a density model accross different learning domains. Importantly, we introduce a novel vine copula model, which allows for this factorization in a non-parametric manner. Experimental results on regression problems with real-world data illustrate the efficacy of the proposed approach when compared to state-of-the-art techniques.

We present a self-supervised variational autoencoder (VAE) to jointly learn disentangled and dependent hidden factors and then enhance disentangled representation learning by a self-supervised classifier to eliminate coupled representations in a contrastive manner. To this end, a Contrastive Copula VAE (C$^2$VAE) is introduced without relying on prior knowledge about data in the probabilistic principle and involving strong modeling assumptions on the posterior in the neural architecture. C$^2$VAE simultaneously factorizes the posterior (evidence lower bound, ELBO) with total correlation (TC)-driven decomposition for learning factorized disentangled representations and extracts the dependencies between hidden features by a neural Gaussian copula for copula coupled representations. Then, a self-supervised contrastive classifier differentiates the disentangled representations from the coupled representations, where a contrastive loss regularizes this contrastive classification together with the TC loss for eliminating entangled factors and strengthening disentangled representations. C$^2$VAE demonstrates a strong effect in enhancing disentangled representation learning. C$^2$VAE further contributes to improved optimization addressing the TC-based VAE instability and the trade-off between reconstruction and representation.

The estimation of dependencies between multiple variables is a central problem in the analysis of financial time series. A common approach is to express these dependencies in terms of a copula function. Typically the copula function is assumed to be constant but this may be innacurate when there are covariates that could have a large influence on the dependence structure of the data. To account for this, a Bayesian framework for the estimation of conditional copulas is proposed. In this framework the parameters of a copula are non-linearly related to some arbitrary conditioning variables.

Change detection (CD) in heterogeneous remote sensing images is a practical and challenging issue for real-life emergencies. In the past decade, the heterogeneous CD problem has significantly benefited from the development of deep neural networks (DNN). However, the data-driven DNNs always perform like a black box where the lack of interpretability limits the trustworthiness and controllability of DNNs in most practical CD applications. As a strong knowledge-driven tool to measure correlation between random variables, Copula theory has been introduced into CD, yet it suffers from non-robust CD performance without manual prior selection for Copula functions. To address the above issues, we propose a knowledge-data-driven heterogeneous CD method (NN-Copula-CD) based on the Copula-guided interpretable neural network. In our NN-Copula-CD, the mathematical characteristics of Copula are designed as the losses to supervise a simple fully connected neural network to learn the correlation between bi-temporal image patches, and then the changed regions are identified via binary classification for the correlation coefficients of all image patch pairs of the bi-temporal images. We conduct in-depth experiments on three datasets with multimodal images (e.g., Optical, SAR, and NIR), where the quantitative results and visualized analysis demonstrate both the effectiveness and interpretability of the proposed NN-Copula-CD.

Before the transition of AVs to urban roads and subsequently unprecedented changes in traffic conditions, evaluation of transportation policies and futuristic road design related to pedestrian crossing behavior is of vital importance. Recent studies analyzed the non-causal impact of various variables on pedestrian waiting time in the presence of AVs. However, we mainly investigate the causal effect of traffic density on pedestrian waiting time. We develop a Double/Debiased Machine Learning (DML) model in which the impact of confounders variable influencing both a policy and an outcome of interest is addressed, resulting in unbiased policy evaluation. Furthermore, we try to analyze the effect of traffic density by developing a copula-based joint model of two main components of pedestrian crossing behavior, pedestrian stress level and waiting time. The copula approach has been widely used in the literature, for addressing self-selection problems, which can be classified as a causality analysis in travel behavior modeling. The results obtained from copula approach and DML are compared based on the effect of traffic density. In DML model structure, the standard error term of density parameter is lower than copula approach and the confidence interval is considerably more reliable. In addition, despite the similar sign of effect, the copula approach estimates the effect of traffic density lower than DML, due to the spurious effect of confounders. In short, the DML model structure can flexibly adjust the impact of confounders by using machine learning algorithms and is more reliable for planning future policies.

The Copula is widely used to describe the relationship between the marginal distribution and joint distribution of random variables. The estimation of high-dimensional Copula is difficult, and most existing solutions rely either on simplified assumptions or on complicating recursive decompositions. Therefore, people still hope to obtain a generic Copula estimation method with both universality and simplicity. To reach this goal, a novel neural network-based method (named Neural Copula) is proposed in this paper. In this method, a hierarchical unsupervised neural network is constructed to estimate the marginal distribution function and the Copula function by solving differential equations. In the training program, various constraints are imposed on both the neural network and its derivatives. The Copula estimated by the proposed method is smooth and has an analytic expression. The effectiveness of the proposed method is evaluated on both real-world datasets and complex numerical simulations. Experimental results show that Neural Copula's fitting quality for complex distributions is much better than classical methods. The relevant code for the experiments is available on GitHub. (We encourage the reader to run the program for a better understanding of the proposed method).

The estimation of dependencies between multiple variables is a central problem in the analysis of financial time series. A common approach is to express these dependencies in terms of a copula function. Typically the copula function is assumed to be constant but this may be innacurate when there are covariates that could have a large influence on the dependence structure of the data. To account for this, a Bayesian framework for the estimation of conditional copulas is proposed. In this framework the parameters of a copula are non-linearly related to some arbitrary conditioning variables.

Due to the intractable partition function, the exact likelihood function for a Markov random field (MRF), in many situations, can only be approximated. Major approximation approaches include pseudolikelihood and Laplace approximation. In this paper, we propose a novel way of approximating the likelihood function through first approximating the marginal likelihood functions of individual parameters and then reconstructing the joint likelihood function from these marginal likelihood functions. For approximating the marginal likelihood functions, we derive a particular likelihood function from a modified scenario of coin tossing which is useful for capturing how one parameter interacts with the remaining parameters in the likelihood function. For reconstructing the joint likelihood function, we use an appropriate copula to link up these marginal likelihood functions. Numerical investigation suggests the superior performance of our approach. Especially as the size of the MRF increases, both the numerical performance and the computational cost of our approach remain consistently satisfactory, whereas Laplace approximation deteriorates and pseudolikelihood becomes computationally unbearable.

One approach for constructing copula functions is by multiplication. Given that products of cumulative distribution functions (CDFs) are also CDFs, an adjustment to this multiplication will result in a copula model, as discussed by Liebscher (J Mult Analysis, 2008). Parameterizing models via products of CDFs has some advantages, both from the copula perspective (e.g., it is well-defined for any dimensionality) and from general multivariate analysis (e.g., it provides models where small dimensional marginal distributions can be easily read-off from the parameters). Independently, Huang and Frey (J Mach Learn Res, 2011) showed the connection between certain sparse graphical models and products of CDFs, as well as message-passing (dynamic programming) schemes for computing the likelihood function of such models. Such schemes allows models to be estimated with likelihood-based methods. We discuss and demonstrate MCMC approaches for estimating such models in a Bayesian context, their application in copula modeling, and how message-passing can be strongly simplified. Importantly, our view of message-passing opens up possibilities to scaling up such methods, given that even dynamic programming is not a scalable solution for calculating likelihood functions in many models.