Description: The Least Squares method is a statistical technique used to estimate the parameters of a linear model. Its main objective is to minimize the sum of the squares of the differences between observed values and the values predicted by the model. This approach is based on the premise that discrepancies between data and the model can be interpreted as random errors, and by minimizing these errors, a better representation of the relationship between variables is obtained. The method is widely used in various disciplines, including economics, engineering, and social sciences, due to its simplicity and effectiveness. Additionally, it is fundamental in the context of machine learning, where it is used to fit models to data and make predictions. The Least Squares technique is not only applied to simple linear models but also extends to more complex models, including multiple and polynomial regressions, making it a versatile tool in data analysis.
History: The Least Squares method was developed by the French mathematician Adrien-Marie Legendre in 1805, although it is also attributed to Carl Friedrich Gauss, who used it in his work on astronomy. Throughout the 19th century, the method became established as an essential tool in statistics and error theory, being adopted in various scientific and technical applications. Its popularity grew with the advancement of statistics and the development of regression methods, becoming a fundamental pillar in data analysis.
Uses: The Least Squares method is primarily used in linear regression to fit models to observed data. It is common in fields such as economics to predict market trends, in engineering to analyze relationships between physical variables, and in social sciences to study behaviors and patterns. It is also applied in machine learning to optimize predictive models and in anomaly detection, where it helps identify data that significantly deviates from an expected pattern.
Examples: A practical example of the Least Squares method is its use in predicting housing prices, where a linear model is fitted to historical data of prices and property characteristics. Another example is in the calibration of measuring instruments, where experimental data is used to adjust a model that relates the instrument reading to the actual value. In the field of anomaly detection, it can be applied to identify unusual financial transactions by comparing expected spending patterns with observed ones.
