First derivatives at the optimum analysis (\textit{fdao}): An approach to estimate the uncertainty in nonlinear regression involving stochastically independent variables

TL;DR

This paper introduces a method using derivatives at the optimum to estimate uncertainty in nonlinear regression models with stochastically independent parameters, extending analytical solutions beyond linear cases.

Contribution

It proposes a novel approach to quantify parameter uncertainty in nonlinear regression by analyzing derivatives at the optimum for stochastically independent parameters.

Findings

Applicable to nonlinear models with independent parameters

Provides a way to estimate uncertainties analytically

Extends beyond linear regression solutions

Abstract

An important problem of optimization analysis surges when parameters such as $ \{\theta_j\}_{j=1,\, \dots \,,k }$, determining a function $ y=f(x\given\{\theta_j\}) $, must be estimated from a set of observables $ \{ x_i,y_i\}_{i=1,\, \dots \,,m} $. Where $ \{x_i\} $ are independent variables assumed to be uncertainty-free. It is known that analytical solutions are possible if $ y=f(x\given\theta_j) $ is a linear combination of $ \{\theta_{j=1,\, \dots \,,k} \}.$ Here it is proposed that determining the uncertainty of parameters that are not \textit{linearly independent} may be achieved from derivatives $ \tfrac{\partial f(x \given \{\theta_j\})}{\partial \theta_j} $ at an optimum, if the parameters are \textit{stochastically independent}.