When Three Points Aren't Enough: Parameter Estimation in Newton's Law of Cooling

Authors

• Alberto A. Condori, Florida Gulf Coast UniversityFollow
• Cara D. Brooks, Florida Gulf Coast University
• Madeline R. Goldberg, Brown University

Publication Date

9-26-2026

Keywords

Newton's Law of Cooling, Parameter Estimation, Nonlinear Regression, Inverse Problems, Least Squares

Disciplines

Applied Statistics | Numerical Analysis and Computation | Ordinary Differential Equations and Applied Dynamics

Abstract

We begin with a paradoxical three-point problem where the standard parameter estimation formula fails because temperature data must satisfy a concavity condition reflecting Newton's Law of Cooling's exponential structure. Although a closed-form solution for A exists for equally-spaced measurements, high sensitivity to error motivates the use of overdetermined systems with many measurements. The "profiling over A" technique transforms this three-parameter nonlinear problem into a sequence of simple linear regressions, providing computational efficiency and conceptual transparency. This approach can be generalized to many parameter estimation problems in science and engineering, making it a valuable tool for undergraduates interested in applied mathematics.   The problem exemplifies how mathematical modeling combines theoretical analysis, model reduction, numerical implementation, and statistical validation in a real experimental context. The exercises provide further opportunities for critical analysis and promote the discovery that algorithms can produce estimates with incompatible data, R^2 values may mislead, and that one's own insight about the physical world plays an essential role in validating mathematical models.

Recommended Citation

Condori, Alberto A.; Brooks, Cara D.; and Goldberg, Madeline R. (2026) "When Three Points Aren't Enough: Parameter Estimation in Newton's Law of Cooling," CODEE Journal: Vol. 20: Iss. 2, Article 11.
Available at: https://scholarship.claremont.edu/codee/vol20/iss2/11

Creative Commons License

This work is licensed under a Creative Commons Attribution-Noncommercial 4.0 License