Sensitivity Functions and Their Uses in Inverse Problems

Abstract

In this note we present a critical review of the some of the positive features as well as some of the shortcomings of the generalized sensitivity functions (GSF) of Thomaseth-Cobelli in comparison to traditional sensitivity functions (TSF). We do this from a computational perspective of ordinary least squares estimation or inverse problems using two illustrative examples: the Verhulst-Pearl logistic growth model and a recently developed agricultural production network model. Because GSF provide information on the relevance of data measurements for the identification of certain parameters in a typical parameter estimation problems, we argue that they provide the basis for new tools for investigators in design of inverse problem studies.Keywords: Inverse problems, sensitivity and generalized sensitivity functions, Fisher information matrix, logistic growth model, agricultural production networks. In this note we present a critical review of the some of the positive features as well as some of the shortcomings of the generalized sensitivity functions (GSF) of Thomaseth-Cobelli in comparison to traditional sensitivity functions (TSF). We do this from a computational perspective of ordinary least squares estimation or inverse problems using two illustrative examples: the Verhulst-Pearl logistic growth model and a recently developed agricultural production network model. Because GSF provide information on the relevance of data measurements for the identification of certain parameters in a typical parameter estimation problems, we argue that they provide the basis for new tools for investigators in design of inverse problem studies.

An inverse problem for a class of conditional probability measure-dependent evolution equations

“…For the last few values of t f , the error does appear to exhibit an increasing trend, possibly due a lack of sufficient resolution in time. This result is consistent with the literature [11, 52, 75] in that the size and resolution of the observation interval can have a substantial effect on identifiability of parameters. For instance, Thomaseth and Cobelli [75] developed generalized sensitivity functions that can be used for the qualitative analysis of the impact of the observation intervals on identifiability of parameters in dynamical systems.…”

“…These sensitivity functions help to identify the most relevant data and time subdomains for identification of certain parameters. Later, Banks et al [12, 13] offered a quantitative means to choose the duration t f required for an optimal experiment design. Moreover, Keck and Bortz [52], provided an extension of this sensitivity functions to the size-structured population models.…”

Sensitivity Analysis for the EEG Forward Problem

“…In this frame the Fisher information matrix becomes $F_{ij}(T,\theta )={\int }_0^T\frac{1}{v^{2}false(tfalse)}\frac{\partial hfalse(t,\theta false)}{\partial {\theta }_i}\frac{\partial hfalse(t,\theta false)}{\partial {\theta }_j}dP(t)$ and its inverse is, asymptotically and approximately, the covariant matrix of the estimator $\hat{\theta }$ of θ. The generalized sensitivity at t  ∈ [0, T ] is the diagonal of $F{(P,{\theta }_0)}^{-1}F(P_{[0,t]},{\theta }_0)$ (see Banks et al, 2008). …”

“…They can help to determine the time instants where the output of a dynamical system has more information about the value of its parameters in order to carry on an estimation process. Both functions were considered by some authors who compared their results for different dynamical systems (see Banks and Bihari, 2001; Kappel and Batzel, 2006; Banks et al, 2008). In this work we apply the TSF and the GSF to analyze the sensitivity of the 3D Poisson-type equation with interfaces of the forward problem of electroencephalography.…”

Standard error computations for uncertainty quantification in inverse problems: Asymptotic theory vs. bootstrapping

“…In addition to computing estimates $\hat{\theta }$ for the unknown parameters using observations { y j }, it is widely accepted that quantifying the uncertainty in these parameter estimates is equally important. A standard method [3, 4, 6, 8, 14, 16, 25] to do this involves computation of standard errors (SE) to be used in confidence intervals (CI) for the parameter estimates. Discussions of the fundamental ideas and methods that are accessible to non-statisticians can be found in [3, 8].…”