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Using the DIRECT optimization algorithm and piecewise-linear functions for building nonlinear regression model

Nonlinear regression models are an important tool in agricultural research, as many biological processes are theoretically and experimentally described by nonlinear functions. In addition to accurately describing experimental data, nonlinear models have the property of physical interpretability of parameters and are more robust outside the domain of the studied sample. Currently, existing methods for calculating model coefficients – such as Ordinary Least Squares, Weighted Least Squares, and Generalized Least Squares – have several drawbacks. The most advanced Generalized Least Squares method relies on a large number of axioms, which are often not adhered to in real examples, and the theoretical proof is not apodictic. This article introduces a flexible, robust, and accurate method for calculating coefficients for arbitrary single-factor regression models based on the maximum likelihood estimation method. The method is theoretically justified with a minimal number of axioms, and examples of results from the software implementation are provided for the logistic function and the Michaelis function using synthetic test data and experimental samples of dry grass mass production depending on the volume of nitrogen fertilizers. The main advantage of the method lies in the simplicity of theoretical proof and the small number of theoretical constraints on the input parameters of the problem. Unlike Generalized Least Squares, the proposed method deterministically converges to the absolute minimum, thanks to the use of the DIRECT algorithm. It can account for heteroscedasticity and does not require manual tuning of optimization parameters to ensure convergence. Considerations for possible extensions of the method to multifactorial regression analysis and potential improvements for heteroscedasticity estimation are also presented. Read more...