Standard errors and confidence intervals in inverse problems: sensitivity and associated pitfalls

Abstract

We review the asymptotic theory for standard errors in classical ordinary least squares (OLS) inverse or parameter estimation problems involving general nonlinear dynamical systems where sensitivity matrices can be used to compute the asymptotic covariance matrices. We discuss possible pitfalls in computing standard errors in regions of low parameter sensitivity and/or near a steady state solution of the underlying dynamical system. 1 Report Documentation PageForm Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number.

Stochastic and deterministic models for agricultural production networks

“…Moreover, when we try to estimate the parametersκ with data from the interval [80, 210] alone (see DS 8 , DS 9 and DS 10 ), we obtain very large errors which increase in magnitude as the number of sample points increases. Although puzzling at first view, this phenomenon is not surprising at all from the perspective of the theory presented above and that presented in [6]. Indeed, by blindly sampling more data points from the region where the generalized sensitivity functions exhibit the so called "forced-to-one" artifact and the traditional sensitivity curves are flat, we simply introduce redundancy in the sensitivity matrix, thus increasing the condition number of the Fisher information matrix for our problem.…”

“…Indeed, by blindly sampling more data points from the region where the generalized sensitivity functions exhibit the so called "forced-to-one" artifact and the traditional sensitivity curves are flat, we simply introduce redundancy in the sensitivity matrix, thus increasing the condition number of the Fisher information matrix for our problem. For an illustration and discussion of this phenomenon, see [6]. By the Cramér-Rao inequality, the consequence is that the variance of the unbiased estimator (and the corresponding standard errors) will be huge, making our estimates less useful.…”

“…This methodology is a fundamental tool in identifying those parameters in a model which require further, more in-depth investigation as well as the basis of a statistical analysis for quantifying uncertainty in estimators (e.g., see [5,6,9,11,43]) as well as information content based model selection techniques [8].…”

Markov models for ion channels: versatility versus identifiability and speed

“…Using all data also means that there are no additional data left for model validation. Furthermore, additional uncertainty for parameter estimates is added when redundant information is added to the data (Banks et al 2007).…”

Sensitivity equations for a size-structured population model

“…However, to the best of our knowledge, thus far there is no literature on sensitivity equations and the related analysis for size-structured population models. Sensitivity analysis of dynamical systems has drawn the attention of numerous researchers [1,6,9,10,11,13,14,15,16,17,20,24,25,27,28,35,38,40] for many years because the resulting sensitivity functions can be used in many areas such as optimization and design [16,26,27,34,38], computation of standard errors [9,10,19,21,36], and information theory [12] related quantities (e.g., the Fisher information matrix) as well as control theory, parameter estimation and inverse problems [5,8,9,10,11,40,41]. One of our motivations for investigating sensitivity for size-structured population models derives from our efforts reported in [7], where a shrimp biomass production system and a…”