Estimating the coefficients of variation of Freundlich parameters with weighted least squares analysis

TL;DR

This paper develops an analytical method for estimating the coefficients of variation of Freundlich isotherm parameters using weighted least squares, improving accuracy over unweighted methods especially when CVs of initial and equilibrium concentrations are unknown.

Contribution

It introduces an analytical approximation for weights in weighted least squares to better estimate Freundlich parameters' CVs, validated through Monte-Carlo simulations.

Findings

Analytical weights decrease with concentration, Freundlich exponent, and CVs.

Weighted least squares outperforms unweighted least squares in CV estimation.

Accuracy diminishes when CVs are estimated from data but remains superior to unweighted methods.

Abstract

The Freundlich isotherm has been used widely to describe sorption of solutes to soils for many decades. The Freundlich parameters are often estimated using unweighted least squares (ULS) analysis after log-log transformation. Estimating the accuracy of these parameters (characterized by their coefficient of variation, CV) is an essential element of the use of these parameters. Accurate CVs can be derived with weighted least squares (WLS), but only if proper weights are assigned to the residuals in the fitting procedure. This work presents the derivation of an analytical approximation of these weights which were found to decrease with increasing concentration, increasing Freundlich exponent, and increasing CVs of the initial and equilibrium concentrations. Monte-Carlo simulations for a wide range of Freundlich systems based on known values of the CVs of the initial and equilibrium concentrations confirmed the accuracy of this analytical approximation. Unfortunately, in practice, the CVs of the initial and equilibrium concentrations are unknown a priori. Simulations showed that the accuracy of the estimated CVs of the Freundlich parameters was distinctly lower if the CVs of the initial and equilibrium concentrations were estimated from the isotherm data. However, this accuracy was still considerably better than when using ULS. It is recommended to use this analytical approximation whenever these CVs are relevant for the further use and interpretation of the estimated Freundlich parameters.