arXiv subjects
Publications and source records attributed to Denys Pommeret.
We introduce a new copula class, called Lancaster copulas, built from orthogonal expansions of continuous Lancaster probabilities. We derive infinite-series representations for the copula and its density, study truncation effects, and show in numerical experiments that low-order truncations already provide accurate approximation.
Learning from an imbalanced distribution presents a major challenge in predictive modeling, as it generally leads to a reduction in the performance of standard algorithms. Various approaches exist to address this issue, but many of them concern classification problems, with a limited focus on regression. In this paper, we introduce a novel method aimed at enhancing learning on tabular data in the Imbalanced Regression (IR) framework, which remains a significant problem. We propose to use variational autoencoders (VAE) which are known as a powerful tool for synthetic data generation, offering an interesting approach to modeling and capturing latent representations of complex distributions. However, VAEs can be inefficient when dealing with IR. Therefore, we develop a novel approach for generating data, combining VAE with a smoothed bootstrap, specifically designed to address the challenges of IR. We numerically investigate the scope of this method by comparing it against its competitors on simulations and datasets known for IR.
The field of imbalanced self-supervised learning, especially in the context of tabular data, has not been extensively studied. Existing research has predominantly focused on image datasets. This paper aims to fill this gap by examining the specific challenges posed by data imbalance in self-supervised learning in the domain of tabular data, with a primary focus on autoencoders. Autoencoders are widely employed for learning and constructing a new representation of a dataset, particularly for dimensionality reduction. They are also often used for generative model learning, as seen in variational autoencoders. When dealing with mixed tabular data, qualitative variables are often encoded using a one-hot encoder with a standard loss function (MSE or Cross Entropy). In this paper, we analyze the drawbacks of this approach, especially when categorical variables are imbalanced. We propose a novel metric to balance learning: a Multi-Supervised Balanced MSE. This approach reduces the reconstruction error by balancing the influence of variables. Finally, we empirically demonstrate that this new metric, compared to the standard MSE: i) outperforms when the dataset is imbalanced, especially when the learning process is insufficient, and ii) provides similar results in the opposite case.
A smooth test to simultaneously compare $K$ copulas, where $K \geq 2$ is proposed. The $K$ observed populations can be paired, and the test statistic is constructed based on the differences between moment sequences, called copula coefficients. These coefficients characterize the copulas, even when the copula densities may not exist. The procedure employs a two-step data-driven procedure. In the initial step, the most significantly different coefficients are selected for all pairs of populations. The subsequent step utilizes these coefficients to identify populations that exhibit significant differences. To demonstrate the effectiveness of the method, we provide illustrations through numerical studies and application to two real datasets.
In supervised learning, it is quite frequent to be confronted with real imbalanced datasets. This situation leads to a learning difficulty for standard algorithms. Research and solutions in imbalanced learning have mainly focused on classification tasks. Despite its importance, very few solutions exist for imbalanced regression. In this paper, we propose a data augmentation procedure, the GOLIATH algorithm, based on kernel density estimates which can be used in classification and regression. This general approach encompasses two large families of synthetic oversampling: those based on perturbations, such as Gaussian Noise, and those based on interpolations, such as SMOTE. It also provides an explicit form of these machine learning algorithms and an expression of their conditional densities, in particular for SMOTE. New synthetic data generators are deduced. We apply GOLIATH in imbalanced regression combining such generator procedures with a wild-bootstrap resampling technique for the target values. We evaluate the performance of the GOLIATH algorithm in imbalanced regression situations. We empirically evaluate and compare our approach and demonstrate significant improvement over existing state-of-the-art techniques.
In this work, we consider the problem of imbalanced data in a regression framework when the imbalanced phenomenon concerns continuous or discrete covariates. Such a situation can lead to biases in the estimates. In this case, we propose a data augmentation algorithm that combines a weighted resampling (WR) and a data augmentation (DA) procedure. In a first step, the DA procedure permits exploring a wider support than the initial one. In a second step, the WR method drives the exogenous distribution to a target one. We discuss the choice of the DA procedure through a numerical study that illustrates the advantages of this approach. Finally, an actuarial application is studied.
We propose a clustering procedure to group K populations into subgroups with the same dependence structure. The method is adapted to paired population and can be used with panel data. It relies on the differences between orthogonal projection coefficients of the K density copulas estimated from the K populations. Each cluster is then constituted by populations having significantly similar dependence structures. A recent test statistic from Ngounou-Bakam and Pommeret (2022) is used to construct automatically such clusters. The procedure is data driven and depends on the asymptotic level of the test. We illustrate our clustering algorithm via numerical studies and through two real datasets: a panel of financial datasets and insurance dataset of losses and allocated loss adjustment expense.
In this paper we study nonparametric estimators of copulas and copula densities. We first focus our study on a density copula estimator based on a polynomial orthogonal projection of the joint density. A new copula estimator is then deduced. Its asymptotic properties are studied: we provide a large functional class for which this construction is optimal in the minimax and maxiset sense and we propose a method selection for the smoothing parameter. An intensive simulation study shows the very good performance of both copulas and copula densities estimators which we compare to a large panel of competitors. A real dataset in actuarial science illustrates this approach.
Clustering mixed data presents numerous challenges inherent to the very heterogeneous nature of the variables. A clustering algorithm should be able, despite of this heterogeneity, to extract discriminant pieces of information from the variables in order to design groups. In this work we introduce a multilayer architecture model-based clustering method called Mixed Deep Gaussian Mixture Model (MDGMM) that can be viewed as an automatic way to merge the clustering performed separately on continuous and non-continuous data. This architecture is flexible and can be adapted to mixed as well as to continuous or non-continuous data. In this sense we generalize Generalized Linear Latent Variable Models and Deep Gaussian Mixture Models. We also design a new initialisation strategy and a data driven method that selects the best specification of the model and the optimal number of clusters for a given dataset "on the fly". Besides, our model provides continuous low-dimensional representations of the data which can be a useful tool to visualize mixed datasets. Finally, we validate the performance of our approach comparing its results with state-of-the-art mixed data clustering models over several commonly used datasets.
Approximate Bayesian Computational (ABC) methods (or likelihood-free methods) have appeared in the past fifteen years as useful methods to perform Bayesian analyses when the likelihood is analytically or computationally intractable. Several ABC methods have been proposed: Monte Carlo Markov Chains (MCMC) methods have been developped by Marjoramet al. (2003) and by Bortotet al. (2007) for instance, and sequential methods have been proposed among others by Sissonet al. (2007), Beaumont et al. (2009) and Del Moral et al. (2009). Until now, while ABC-MCMC methods remain the reference, sequential ABC methods have appeared to outperforms them (see for example McKinley et al. (2009) or Sisson et al. (2007)). In this paper a new algorithm combining population-based MCMC methods with ABC requirements is proposed, using an analogy with the Parallel Tempering algorithm (Geyer, 1991). Performances are compared with existing ABC algorithms on simulations and on a real example.
The Equi-Energy Sampler (EES) introduced by Kou et al [2006] is based on a population of chains which are updated by local moves and global moves, also called equi-energy jumps. The state space is partitioned into energy rings, and the current state of a chain can jump to a past state of an adjacent chain that has energy level close to its level. This algorithm has been developed to facilitate global moves between different chains, resulting in a good exploration of the state space by the target chain. This method seems to be more efficient than the classical Parallel Tempering (PT) algorithm. However it is difficult to use in combination with a Gibbs sampler and it necessitates increased storage. In this paper we propose an adaptation of this EES that combines PT with the principle of swapping between chains with same levels of energy. This adaptation, that we shall call Parallel Tempering with Equi-Energy Moves (PTEEM), keeps the original idea of the EES method while ensuring good theoretical properties, and practical implementation even if combined with a Gibbs sampler. Performances of the PTEEM algorithm are compared with those of the EES and of the standard PT algorithms in the context of mixture models, and in a problem of identification of gene regulatory binding motifs.
In the Bayesian stochastic search variable selection framework, a common prior distribution for the regression coefficients is the g-prior of Zellner (1986). However, there are two standard cases in which the associated covariance matrix does not exist, and the conventional prior of Zellner can not be used: if the number of observations is lower than the number of variables (large p and small n paradigm), or if some variables are linear combinations of others. In such situations a prior distribution derived from the prior of Zellner can be used, by introducing a ridge parameter. This prior introduced by Gupta and Ibrahim (2007) is a flexible and simple adaptation of the g-prior. In this paper we study the influence of the ridge parameter on the selection of variables. A simple way to choose the associated hyper-parameters is proposed. The method is valid for any generalized linear mixed model and we focus on the case of probit mixed models when some variables are linear combinations of others. The method is applied to both simulated and real datasets obtained from Affymetrix microarray experiments. Results are compared to those obtained with the Bayesian Lasso.
Consider two random variables contaminated by two unknown transformations. The aim of this paper is to test the equality of those transformations. Two cases are distinguished: first, the two random variables have known distributions. Second, they are unknown but observed before contaminations. We propose a nonparametric test statistic based on empirical cumulative distribution functions. Monte Carlo studies are performed to analyze the level and the power of the test. An illustration is presented through a real data set.
We consider the problem of testing whether two samples of contaminated data, possibly paired, are from the same distribution. Is is assumed that the contaminations are additive noises with known moments of all orders. The test statistic is based on the polynomials moments of the difference between observations and noises. . A data driven selection is proposed to choose automatically the number of involved polynomials. We present a simulation study in order to investigate the power of the proposed test within discrete and continuous cases. A real-data example is presented to demonstrate the method.
This paper considers the problem of comparing two processes with panel data. A nonparametric test is proposed for detecting a monotone change in the link between the two process distributions. The test statistic is of CUSUM type, based on the empirical distribution functions. The asymptotic distribution of the proposed statistic is derived and its finite sample property is examined by bootstrap procedures through Monte Carlo simulations.
In this paper we consider a random variable $Y$ contamined by an independent additive noise $Z$. We assume that $Z$ has known distribution. Our purpose is to test the distribution of the unobserved random variable $Y$. We propose a data driven statistic based on a development of the density of $Y+Z$, which is valid in the discrete case and in the continuous case. The test is illustrated in both cases.