Aleatoric Uncertainty Modelling in Regression Problems using Deep Learning

Abstract

Nowadays, we live in an intrinsically uncertain world from our perspective. We do not know what will happen in the future but, to infer it, we build the so-called models. These models are abstractions of the world we live which allow us to conceive how the world works and that are, essentially, validated from our previous experience and discarded if their predictions prove to be incorrect in the future. This common scientific process of inference has several non-deterministic steps. First of all, our measuring instruments could be inaccurate. That is, the information we use a priori to know what will happen may already contain some irreducible error. Besides, our past experience in building the model could be biased (and, therefore, we would incorrectly infer the future, as the models would be based on unrepresentative data). On the other hand, our model itself may be an oversimplification of the reality (which would lead us to unrealistic generalizations). Furthermore, the overall task of inferring the future may be downright non-deterministic. This often happens when the information we have a priori to infer the future is incomplete or partial for the task to be performed (i.e. it depends on factors we cannot observe at the time of prediction) and we are, consequently, obliged to consider that what we want to predict is not a deterministic value. One way to model all of these uncertainties is through a probabilistic approach that mathematically formalizes these sources of uncertainty in order to create specific methods that capture them. Accordingly, the general aim of this thesis is to define a probabilistic approach that contributes to artificial intelligence-based systems (specifically, deep learning) becoming robust and reliable systems capable of being applied to high-risk problems, where having generic good performance is not enough but also to ensure that critical errors with high costs are avoided. In particular, the thesis shows the current divergence in the literature - when it comes to dividing and naming the different types of uncertainty - by proposing a procedure to follow. In addition, based on a real problem case arising from the industrial nature of the current thesis, the importance of investigating the last type of uncertainty is emphasized, which arises from the lack of a priori information in order to infer deterministically the future, the so-called aleatoric uncertainty. The current thesis delves into different literature models in order to capture aleatoric uncertainty using deep learning and analyzes their limitations. In addition, it proposes new state-of-the-art approaches that allow to solve the limitations exposed during the thesis. As a result of applying the aleatoric uncertainty modelling in real-world problems, the uncertainty modelling of a black box systems problem arises. Generically, a Black box system is a pre-existing predictive system which originally do not model uncertainty and where no requirements or assumptions are made about its internals. Therefore, the goal is to build a new system that wrappers the black box and models the uncertainty of this original system. In this scenario, not all previously introduced aleatoric uncertainty modelling approaches can be considered and this implies that flexible methods such as Quantile Regression ones need to be modified in order to be applied in this context. Subsequently, the Quantile Regression study brings the need to solve one critical literature problem in the QR literature, the so-called crossing quantile, which motivates the proposal of new additional models to solve it. Finally, all of the above research will be summarized in visualization and evaluation methods for the predicted uncertainty to produce uncertainty-tailored methods.

Estem rodejats d’incertesa. Cada decisió que prenem té una probabilitat de sortir com un espera i, en funció d’aquesta, molts cops condicionem les nostres decisions. De la mateixa manera, els sistemes autònoms han de saber interpretar aquests escenaris incerts. Tot i això, actualment, malgrat els grans avenços en el camp de la intel·ligència artificial, ens trobem en un moment on la incapacitat d'aquests sistemes per poder identificar a priori un escenari de major risc impedeix la seva inclusió com a part de solucions que podrien revolucionar la societat tal i com la coneixem. El repte és significatiu i, per això, és essencial que aquests sistemes aprenguin a modelar i gestionar totes les fonts de la incertesa. Partint d'un enfocament probabilístic, aquesta tesi proposa formalitzar els diferents tipus d'incerteses i, en particular, centra la seva recerca en un tipus anomenada com incertesa aleatòrica, ja que va ser detectada com la principal incertesa decisiva a tractar en el problema financer original que va motivar el present doctorat industrial. A partir d'aquesta investigació, la tesi proposa nous models per millorar l'estat de l'art en la modelització de la incertesa aleatòrica, així com introdueix un nou problema, a partir d’una necessitat real industrial, que apareix quan hi ha un sistema predictiu en producció que no modela la incertesa i es vol modelar la incertesa a posteriori de forma independent. Aquest problema es denotarà com la modelització de la incertesa d'un sistema de caixa negra i motivarà la proposta de nous models especialitzats en mantenir els avantatges predictius, com ara la Regressió Quantílica (RQ), adaptant-los al problema de la caixa negra. Posteriorment, la investigació en RQ motivarà la proposta de nous models per resoldre un problema fonamental de la literatura en RQ conegut com el fenomen del creuament de quantils, que apareix quan, a l’hora de predir simultàniament diferents quantils, l’ordre entre quantils no es conserva. Finalment, tota la investigació anterior es resumirà en mètodes de visualització i avaluació de la incertesa reportada per tal de produir mètodes que mitjançant aquesta informació extra prenguin decisions més robustes.