[Paper Review] Parameter Selection Methods in Inverse Problem Formulation

H. T. Banks, Ariel Cintrón-Arias|arXiv (Cornell University)|Apr 15, 2020

HIV Research and Treatment60 references22 citations

TL;DR

This paper proposes a sensitivity-based parameter selection method using the Fisher Information Matrix and asymptotic standard errors to identify which parameters in high-dimensional inverse problems can be reliably estimated. By evaluating parameter sensitivity and uncertainty quantification, the method reduces estimation error and improves reliability, demonstrated in an HIV dynamics model where coefficient of variation dropped by up to four orders of magnitude when selecting only five key parameters from an initial set of 18.

ABSTRACT

We discuss methods for {\em a priori} selection of parameters to be estimated in inverse problem formulations (such as Maximum Likelihood, Ordinary and Generalized Least Squares) for dynamical systems with numerous state variables and an even larger number of parameters. We illustrate the ideas with an in-host model for HIV dynamics which has been successfully validated with clinical data and used for prediction.

Motivation & Objective

• To address the challenge of selecting a subset of parameters that can be reliably estimated in inverse problems involving large dynamical systems with many parameters and limited data.
• To improve uncertainty quantification in parameter estimation by focusing on parameters with low sensitivity and high identifiability.
• To develop a practical, computationally feasible framework for inverse problem formulation in complex biological models, such as HIV dynamics.
• To demonstrate that reducing the number of estimated parameters based on sensitivity analysis leads to dramatically improved estimation accuracy and reduced standard errors.
• To provide a quantitative, local method for parameter selection that leverages prior knowledge of parameter ranges and observation error to guide inverse problem formulation.

Proposed method

• Utilizes local sensitivity analysis and the Fisher Information Matrix (FIM) to assess the influence of each parameter on model output.
• Computes asymptotic standard errors from the inverse of the FIM to quantify uncertainty in parameter estimates.
• Applies a selection score based on the condition number of the FIM and the norm of the coefficient of variation (CV) vector to rank parameter sets.
• Employs a greedy algorithm to iteratively select the smallest subset of parameters that minimizes uncertainty and condition number.
• Uses the HIV in-host model as a testbed, applying Ordinary Least Squares (OLS) and Generalized Least Squares (GLS) for parameter estimation.
• Validates results through comparison of coefficient of variation (CV) across different parameter subsets, showing significant reductions in estimation uncertainty.

Experimental results

Research questions

• RQ1 Which parameters in a high-dimensional inverse problem can be reliably estimated given limited longitudinal data?
• RQ2 How does parameter sensitivity, as quantified by the Fisher Information Matrix, correlate with estimation uncertainty and identifiability?
• RQ3 To what extent can reducing the number of estimated parameters improve uncertainty quantification in complex biological models?
• RQ4 What is the impact of parameter selection on the condition number of the Fisher Information Matrix and the resulting standard errors?
• RQ5 Can a sensitivity-based selection framework consistently identify a minimal set of parameters that yield reliable and low-uncertainty estimates?

Key findings

• The coefficient of variation (CV) for the parameter $\lambda_1$ decreased from 84.3% to 11.5% when reducing the number of estimated parameters from 18 to 5, indicating a significant improvement in estimation reliability.
• For $N_T$, the CV dropped from 404% to 9.99%, reducing the standard error from 40,000% to 10% of the estimate, demonstrating a four-order-of-magnitude improvement in uncertainty quantification.
• The standard error for $x_1^0$ decreased from 5,000% to 4% of the estimate when reducing parameters from 18 to 5, showing a dramatic reduction in estimation uncertainty.
• The condition number of the Fisher Information Matrix dropped from $7.518 \times 10^8$ to $8.383 \times 10^1$ when reducing the parameter set from 18 to 5, indicating a major improvement in numerical conditioning.
• The selection score decreased from $1.025 \times 10^5$ to $3.990 \times 10^{-1}$, confirming that the selected five-parameter subset is significantly more identifiable and less sensitive to noise.
• Even with a reduced parameter set, some parameters like $b_E$, $K_b$, $d_E$, and $K_d$ retained high uncertainty (CVs of 11,200–40,400%), indicating that not all parameters benefit equally from reduction, and careful selection remains critical.