Hypercomplex-valued Least Squares: Modeling and Prediction with the Lorenz System

• Grant number: 25/02901-2
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: August 01, 2025
• End date: July 31, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Applied Mathematics
• Principal Investigator: Marcos Eduardo Ribeiro Do Valle Mesquita
• Grantee: Thiago Gradvohl de Oliveira
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil

Abstract

The use of hypercomplex algebras has attracted the attention of the scientific community in recent years, as they result in models with interesting algebraic and geometric properties. In fact, complex numbers are essential for the proper treatment of the phase and the information contained therein. Quaternions are widely used in computer graphics due to their efficiency in describing geometric transformations, such as rotations and translations. Fitting models to error-prone observational data represents a central challenge in science and engineering, with wide applications in statistics, data science, signal processing, and control. In this context, the least squares method offers an optimal solution for curve fitting, minimizing the weighted sum of squares of the residuals. In this research project, we will study the formulation and methods for solving least squares problems with hypercomplex values. The proposed methodology will be validated through a problem of predicting the future position of a point in space, described by the Lorenz system.