chapter04.tex

\chapter{\uppercase{Geometry of Output Quantities and Consistent Solutions} \label{chapter:geometry}}

Here, we describe how the relationship of a geometric quantity called \emph{skewness} in QoI maps impacts the accuracy of consistent solutions to SIPs approximated with random sampling.
While prior work has addressed skewness in the context of solving the SIP, its impact on parameter estimation problems formulated within the Data-Consistent framework has not been previously studied.
We demonstrate that the skewness of a map impacts the \emph{precision} of a parameter estimate.
This is subsequently utilized in Chapter~\ref{chapter:vector-valued} to aggregate data into different components of a QoI map to more precisely estimate parameter values.

In this chapter, we begin with a definition of skewness and overview of a set-based approach for constructing consistent solutions to SIPs in \ref{sec:skewness}.
That review is followed by a series of numerical examples which establish fundamental connections between the skewness of QoI maps and the difficulty of accurately approximating solutions with finite sampling.
Namely, we establish that in addition to the implied invariability of skewness to translations, rotations of maps have no impact on solution accuracy.
Furthermore, we show that the number of samples required to achieve a predefined level of error is directly proportional to the skewness of the QoI map used to solve the inverse problem.

\section{A Brief Literature Review of Skewness}\label{sec:skewness}
\input{set-based/skewness.tex}

We demonstrate that the number of samples required to approximate densities using uniform i.i.d.~sampling is proportional to the skewness of the map used for inversion, though the convergence rate of the algorithm used to solve the SIP is unaffected.
We focus on the accuracy of the consistent solutions to the SIP.
It is illustrative to begin with the original set-based approximations to solutions developed in \cite{BBE11}, \cite{BES12}, and \cite{BET+14} as the dependence of solutions on skewness is more explicit than in the density-based approach.
While the content of this chapter is concerned with estimating distributions and not individual parameter estimates, we remind the reader that the solutions to the SIP are inherently densities.
In Chapter~\ref{chapter:mud}, we used these densities to produce a parameter estimate, and so we are motivated to study the accuracy of these solutions to the SIP.


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\section{Set-Based Inversion for Consistent Measures}\label{sec:set-based}
% Intro
\input{set-based/set_derivation.tex}
%% \input{ch02/set_derivation_bayes.tex}

% Numerical Approximation
\input{set-based/set_algorithm.tex}

% Descriptions of Error
\input{set-based/set_error.tex}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% discussion of convergence

\subsection{Convergence}
To study the accuracy of the solutions to SIPs using QoI maps with different skewness values (or number of available model evaluations), we need to select a metric on the space of probability measures.
Rates of convergence depend on this choice.
Since the spaces $\pspace$ we are considering are generally bounded and finite, the Total Variation metric metrizes weak convergence (see Thm. 6 in \cite{GS02}).
The latter property of the metric is of notable importance because the QoI maps we study are indeed (component-wise) functionals on the space of model inputs $\pspace$.
Thus, convergence of a sequence of probability measures under the Total Variation metric implies that the QoIs will also converge component-wise in $\RR$.
In other words, convergence in the Total Variation metric implies the convergence of the sampled QoI map to the exact QoI map since the map is a linear functional of the probability measure.
More formally, if $\PP_{\pspace,\ndiscs,\nsamps,h}$ converges to either $\PP_{\pspace,\ndiscs,\nsamps}$, $\PP_{\pspace,\ndiscs}$, or $P_\pspace$ using the Total Variation metric, this implies that the error converged to zero in the numerically computed $\qoi(\param^{(j)})$.
Thus, convergence in the Total Variation metric implies convergence of the numerical method used to construct the QoI map.
Furthermore, recall that weak convergence $\PP_n \to \PP$ is defined to mean
\[
\int f \PP_n \to \int f \PP \text{ as } n \to \infty
\]
for bounded Lipschitz functions $f:\pspace\to\RR$.
Taking $f = \Chi_A$, this leads to the following implication:
\[
\PP_{\pspace, \ndiscs, \nsamps} \to P_\pspace \implies \PP_{\pspace, \ndiscs, \nsamps} (A) \to P_\pspace (A) \quad \forall{A\in\pborel}.
\]
%provided we rigorously define $\P\PP_{\pspace, \ndiscs, \nsamps}$ to measure sets in $\BB_\pspace$, which we proceed to do in the following section\footnote{\bf{I know this is clumsy, but I'm not exactly sure how to phrase this correctly, because it almost seems like this would be true for all $A\in\pspace$ instead. I suppose the omission of the differential operator above is intentionally vague to gloss over this detail. Can we sharpen this up?}}
We choose Total Variation as the metric used in the numerical results of Section~\ref{sec:ch03-set} because of its common use in the literature as the ``statistical distance'' between densities \cite{GS02, Silverman}, and implications for convergence.


\input{set-based/set_accuracy}
% \input{ch03/sample_accuracy}

% \input{set-based/heatrod.tex}
% \FloatBarrier
%
% \input{ch03/decay.tex}

\section{Conclusions}
In this chapter we reviewed the skewness property of QoI maps and demonstrated that for solutions to the SIP, maps which exhibit lower values of skewness exhibit lower overall finite-sampling approximation error against a reference solution.
We saw that the solutions are invariant to transformations such as rotation (which also holds for translations).
The latter observation may aid in the selection of optimal maps (i.e., one can disregard the direction of the maps' contours and focus solely on the angles among the generalized contours).
In the following chapter, we show how vector-valued QoI maps such as those in Chapter~\ref{chapter:mud} constructed from data relate to the geometry of the resulting data-spaces and subsequently impacts the solutions to the SIP.
We demonstrate that even a basic awareness and consideration of skewness can aid in the construction of maps which are more informative and thus increase the precision of parameter estimates.

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