chapter03.tex

\chapter{\uppercase{Data-Driven Maps and Consistent Inversion For Parameter Estimation} \label{chapter:mud}}

In this chapter, we extend the data-consistent framework to solve problems involving epistemic uncertainties in model parameters.
This requires constructing QoI maps in an {\em a posteriori} fashion using noisy data, which is a novel way to define QoI maps in the data-consistent framework.
The theory of existence, uniqueness, and convergence of maximal updated density (MUD) points are also provided for linear data-constructed QoI maps using typical assumptions found in the Bayesian literature.
Comparisons to other parameter estimates such as those obtained using a least-squares or Bayesian approach are also provided.
The applicability of the approach to nonlinear data-constructed QoI maps is also numerically demonstrated for ordinary and partial differential equation models of dynamical and stationary systems.

While there are several on-going and future research directions that are outside the scope of this thesis, we do not wish to hide certain technical challenges in implementing this approach in certain scenarios.
We therefore address several technical challenges where appropriate to both point out these challenges as well as the heuristics we currently employ to handle these challenges.


The rest of this chapter is outlined as follows.
We first compare the estimators induced by the solutions to the DIP and SIP by taking the point of maximal density of the respective distributions in Section~\ref{sec:estimation}.
There, we present closed-form derivations for the novel estimator presented in this work: the MUD point.
In Section~\ref{sec:high-dim-linear-example}, we present an example involving high-dimensional linear maps, connecting rank and dimension to the accuracy and convergence of the estimators.
Following the example, we show in \ref{sec:data-maps} how a QoI map can be constructed from an arbitrary collection of noisy data.
We leverage the theory of the MUD point from the earlier section to provide a general framework for reducing epistemic uncertainty in a point-estimate as more data are collected.
Finally, in Section~\ref{sec:Parameter-identification}, we demonstrate that the presented QoI map can be used to form accurate point estimates by solving the SIP even on nonlinear maps and initial densities which violate the assumptions of the theoretical construction.

\input{body.tex}

\FloatBarrier
\subsection{Conclusions}\label{sec:conclusions}
We demonstrate that the QoI map and $\updated$ can be modified so that the Data-Consistent SIP can be used for parameter estimation.
% Much like how Hierarchical Bayesian approaches cast a distribution-estimation problem in a parameter-identification context, we have shown that the complement can also be accomplished.
Specifically, the definition of a particular form of QoI map is utilized that can incorporate an arbitrary stream of data.
Then, we recast the problem of solving for a true parameter estimate into one involving the accurate identification of a distribution on the residuals errors predicted by a single parameter.
The role of the QoI is central in accomplishing this reformulation, as the distribution being estimated is a direct consequence of the statistical properties of the form of the QoI map.

In Chapter~\ref{chapter:geometry}, we study in abstract the role of QoI maps on the solution to the SIP in the natural context of distribution-estimation.
We introduce measures of the QoI map's geometric properties that are directly related to our ability to accurately approximate the distribution using numerical methods.
Later, in Chapter~\ref{chapter:vector-valued} we leverage this awareness of the geometric properties of the QoI maps and connect them to the goal of parameter identification in the context of the map given in \eqref{eq:qoi_WME}.
We demonstrate that the way in which we aggregate available data to construct the QoI map has a direct consequence on the accuracy of the parameter estimate as more data are incorporated, which is hinted at by the last example above.