Ajuste de dados usando Mínimos Quadrados Lineares

Abstract

The least squares problem is essential in computing, arising from the need to fit linear mathematical models to observations in order to reduce errors. It is an optimization technique that seeks to find the best fit for a dataset by minimizing the sum of squared differences between the observed and estimated values, known as residuals. In terms of Linear Algebra, the least squares problem involves solving an overdetermined system of equations, meaning a system with more equations than unknowns. This work aims to present the mathematical foundations of this method and introduce data fitting in a comprehensible manner using the least squares method. To illustrate the practical application of this method, examples of data fitting using linear least squares were performed, with an emphasis on real-life situations. These applications demonstrate how the least squares method can be efficiently used to find the best fit for a dataset, even in complex situations with a large amount of information. Furthermore, this work seeks to expand understanding of the importance and applications of the least squares method, showcasing how it can be a powerful tool in the field of mathematical modeling and in obtaining more precise results in various areas such as statistics, engineering, data science, and economics. Thus, this study contributes to the advancement of knowledge in the field of optimization and data fitting, providing a comprehensive and in-depth insight into the least squares method and its practical applications.