This two-part tutorial will introduce the framework of conformal prediction, and will provide an overview of both theoretical foundations and practical methodologies in this field. In the first part of the tutorial, we will cover methods including holdout set methods, full conformal prediction, cross-validation based methods, calibration procedures, and more, with emphasis on how these methods can be adapted to a range of settings to achieve robust uncertainty quantification without compromising on accuracy. In the second part, we will cover some recent extensions that allow the methodology to be applied in broader settings, such as weighted conformal prediction, localized methods, online conformal prediction, and outlier detection.[-]

In the fist part of the talk, we will look to some statistical inverse problems for which the natural framework is no more an Euclidian one.
In the second part we will try to give the initial construction of (not orthogonal) wavelets -of the 80 - by Frazier, Jawerth,Weiss, before the Yves Meyer ORTHOGONAL wavelets theory.
In the third part we will propose a construction of a geometric wavelet theory. In the Euclidian case, Fourier transform plays a fundamental role. In the geometric situation this role is given to some "Laplacian operator" with some properties.
In the last part we will show that the previous theory could help to revisit the topic of regularity of Gaussian processes, and to give a criterium only based on the regularity of the covariance operator.[-]

There is an emerging consensus in the transdiciplinary literature that the ultimate goal of regression analysis is to model the conditional distribution of an outcome, given a set of explanatory variables or covariates. This new approach is called "distributional regression", and marks a clear break from the classical view of regression, which has focused on estimating a conditional mean or quantile only. Isotonic Distributional Regression (IDR) learns conditional distributions that are simultaneously optimal relative to comprehensive classes of relevant loss functions, subject to monotonicity constraints in terms of a partial order on the covariate space. This IDR solution is exactly computable and does not require approximations nor implementation choices, except for the selection of the partial order. Despite being an entirely generic technique, IDR is strongly competitive with state-of-the-art methods in a case study on probabilistic precipitation forecasts from a leading numerical weather prediction model.

Joint work with Alexander Henzi and Johanna F. Ziegel.[-]

Many modern applications seek to understand the relationship between an outcome variable of interest and a high-dimensional set of covariates. Often the first question asked is which covariates are important in this relationship, but the immediate next question, which in fact subsumes the first, is \emph{how} important each covariate is in this relationship. In parametric regression this question is answered through confidence intervals on the parameters. But without making substantial assumptions about the relationship between the outcome and the covariates, it is unclear even how to \emph{measure} variable importance, and for most sensible choices even less clear how to provide inference for it under reasonable conditions. In this paper we propose \emph{floodgate}, a novel method to provide asymptotic inference for a scalar measure of variable importance which we argue has universal appeal, while assuming nothing but moment bounds about the relationship between the outcome and the covariates. We take a model-X approach and thus assume the covariate distribution is known, but extend floodgate to the setting that only a \emph{model} for the covariate distribution is known and also quantify its robustness to violations of the modeling assumptions. We demonstrate floodgate's performance through extensive simulations and apply it to data from the UK Biobank to quantify the effects of genetic mutations on traits of interest.[-]

In this paper we study asymptotic properties of random forests within the framework of nonlinear time series modeling. While random forests have been successfully applied in various fields, the theoretical justification has not been considered for their use in a time series setting. Under mild conditions, we prove a uniform concentration inequality for regression trees built on nonlinear autoregressive processes and, subsequently, use this result to prove consistency for a large class of random forests. The results are supported by various simulations. (This is joint work with Mikkel Slot Nielsen.)[-]

Estimation of conditional quantiles is requiered for many purposes, in particular when the conditional mean is not suffisiant to describe the impact of covariates on the dependent variable. For example, one may estimate the quantile of one financial index (e.g. WisdomTree Japan Hedged Equity Fund) knowing financial indeces from other countries. It is also requiered to estimated conditional quantiles in Quantile Oriented Sensitivity Analysis (QOSA). QOSA indices are relevant in order to quantify uncertainty on quantiles, for example in insurance operational risk contexts. We shall present several view points on conditional quantile estimation: quantile regression and improvements, Kernel based estimation, random forest estimation. We shall focus on applications to QOSA.[-]

Estimation of conditional quantiles is requiered for many purposes, in particular when the conditional mean is not suffisiant to describe the impact of covariates on the dependent variable. For example, one may estimate the quantile of one financial index (e.g. WisdomTree Japan Hedged Equity Fund) knowing financial indeces from other countries. It is also requiered to estimated conditional quantiles in Quantile Oriented Sensitivity Analysis (QOSA). QOSA indices are relevant in order to quantify uncertainty on quantiles, for example in insurance operational risk contexts. We shall present several view points on conditional quantile estimation: quantile regression and improvements, Kernel based estimation, random forest estimation. We shall focus on applications to QOSA.[-]

Estimation of conditional quantiles is requiered for many purposes, in particular when the conditional mean is not suffisiant to describe the impact of covariates on the dependent variable. For example, one may estimate the quantile of one financial index (e.g. WisdomTree Japan Hedged Equity Fund) knowing financial indeces from other countries. It is also requiered to estimated conditional quantiles in Quantile Oriented Sensitivity Analysis (QOSA). QOSA indices are relevant in order to quantify uncertainty on quantiles, for example in insurance operational risk contexts. We shall present several view points on conditional quantile estimation: quantile regression and improvements, Kernel based estimation, random forest estimation. We shall focus on applications to QOSA.[-]

In extreme value statistics, the tail index is used to measure the occurrence and the intensity of extreme events. In many applied fields, the tail behavior of such events depends on explanatory variables. This article proposes an ensemble learning method for tail index regression which is called Hill random forests and combines Hill's approach on tail index estimation (Hill (1975)) with the aggregation of randomized decision trees based on the gamma deviance. We prove a consistency result when the tail index function is a multiplicative function.[-]