Gautier, Athénaïs (2023). Modelling and predicting distribution-valued fields with applications to inversion under uncertainty. (Thesis). Universität Bern, Bern
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23gautier_a.pdf - Thesis Available under License Creative Commons: Attribution (CC-BY 4.0). Download (5MB) | Preview |
Abstract
Capturing the dependence between a random response and predictors is a fundamental task in statistics and stochastic modelling. The focus of this work is on density regression, which entails estimating response distributions given predictor values. It enables the derivation of various statistical quantities, including the conditional mean, threshold exceedance probabilities, and quantiles. This thesis presents a flexible approach, based upon the class of so-called Spatial Logistic Gaussian Processes (SLGPs). The SLGP framework utilizes a well-behaved latent Gaussian Process that undergoes a non-linear transformation, resulting in a class of models suitable for capturing spatially-dependent probability measures. SLGP models overcome limitations associated with strong distributional assumptions (e.g. shapes constraints, log-concavity, Gaussianity, etc.), varying sample sizes, and changes in target density shapes and modalities. The first part of this work is dedicated to the development of SLGP models and gaining a deep understanding of the associated mathematical concepts. We introduce SLGPs from the perspective of random measures and their densities, and investigate links between properties of SLGPs and underlying processes. We show that SLGP models can be characterized by their log-increments and leverage this characterization to establish theoretical results with a main focus on spatial regularity. We then focus on applicability of our approach, and propose an implementation relying on finite rank Gaussian Processes. We demonstrate it on synthetic examples and on temperature distributions at meteorological stations. Finally, we address the potential of SLGPs for statistical inference, focusing on their potential in stochastic optimization and stochastic inverse problems. Notably, for inverse problems, an Approximate Bayesian Computation (ABC) framework is introduced, leveraging SLGP-surrogated likelihoods to accommodate situations with limited to moderate data. This methodology, inspired by GP-ABC methods, harnesses the probabilistic nature of SLGPs to guide data acquisition, thereby facilitating accelerated inference. We illustrate these approaches on synthetic examples as well as on a hydrogeological inverse problem in which a contaminant source is sought under uncertain geological scenario.
| Item Type: | Thesis |
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| Dissertation Type: | Single |
| Date of Defense: | 19 May 2023 |
| Subjects: | 500 Science > 510 Mathematics |
| Institute / Center: | 08 Faculty of Science > Department of Mathematics and Statistics > Institute of Mathematical Statistics and Actuarial Science |
| Depositing User: | Hammer Igor |
| Date Deposited: | 19 Jun 2023 14:41 |
| Last Modified: | 19 May 2024 22:25 |
| URI: | https://boristheses.unibe.ch/id/eprint/4377 |
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