Towards a decision-centric framework for uncertainty propagation and data assimilation
Terejanu, Gabriel A.
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Decision makers increasingly rely on mathematical models in choosing the right set of actions in critical situations. Such situations are often encountered in deployment of emergency responders in response to extreme events such as accidental or covert release of hazardous material, storm surge due to a hurricane, wild fire etc. Based on the outcome of the prediction system, decisions can be made on deploying emergency responders, evacuating cities, sheltering or medical gear caching. The models used are usually driven by random noise or aleatory uncertainty and the lack of knowledge about the model parameters is characterized as epistemic uncertainty. Hence, to make the right decision it is necessary to solve the nontrivial task of having an accurate aleatory and epistemic uncertainty propagation and sensor data assimilation. In this work, uncertainty propagation is viewed from both the producer’s perspective and the user’s perspective. Usually uncertainty propagation problem is seen strictly from the producer’s perspective with algorithms used to solve the problem derived based on statistical measures independent of user’s decision needs. However, the uncertainty evolution given by a particular method may be irrelevant to the user or the decision maker, who takes decisions based on an implicit or explicit utility function. While in a static environment, one may be able to select an appropriate method for uncertainty propagation, in a dynamic environment with an ever-changing utility function this becomes a challenging task. The goal of the present work is to reconcile the two views into a decision-centric framework which provides a better global approximation to the relevant probability density functions while simultaneously offering a more accurate expected utility value for the decision maker. A novel method to propagate aleatory uncertainty through a nonlinear dynamical system is derived from the producer’s non-interactive perspective and extended to combine model predictions with sensor measurements in a data assimilation context. Because solving for the exact conditional probability density function requires in general an infinite number of parameters, researchers are using different propagation algorithms based on finite approximations. Here, we propose an Adaptive Gaussian Mixture model approach for accurate uncertainty propagation through a general nonlinear system and its application to nonlinear filtering problem is also presented. When propagated through the nonlinear dynamical system, the weights of the Gaussian components are modified based on the error in the Fokker-Planck-Kolmogorov equation for continuous-time dynamical systems and on the error in the Chapman-Kolmogorov equation for discrete-time dynamical systems. We show that even though we are able to improve the propagation method, chances are that we may still underestimate the tails of the distribution due to our finite parameterization. This becomes misleading for the decision maker when expected utility theory is used, particularly when there are extreme utility events embedded in the tails of these distributions. Hence, building on the previous method, we develop an interaction level between the user’s decision making level and the producer’s uncertainty propagation level. The interaction level incorporates the contextual information held by the decision maker into the prediction process using a novel Progressive Selection of Gaussian Components algorithm. The progressive selection method is designed to add new Gaussian components to the initial mixture, such that probabilistic support covers the region of interest at the decision time. The new probability density function obtained will address the decision makers’ region of interest and will provide overall a better approximation to the true conditional probability density function within it. When ignorance, modeled within the paradigm of epistemic uncertainty, is present in the model parameters or in different probability characterizations such as prior, process noise or measurement noise, a new framework is necessary to propagate both aleatory and epistemic uncertainty. The new framework proposed here, combines both the epistemic and aleatory uncertainty into two level hierarchical model. On the first level we model the aleatory uncertainty by choosing a representative approximation to the actual probability density function with a finite number of parameters. The second level is the epistemic uncertainty about the parameters of probability density functions. This way we are able to model the ignorance when the knowledge about the system is incomplete. The output of the system is a Dempster-Shafer structure on probability sets which can be transformed into a pair of a singleton cumulative distribution function using Smets’ pignistic transformation and an ignorance function. The previous two functions and the utility function are used to take actions using a modified expected utility decision framework which is ignorance sensitive.