Description: The term ‘residual’ refers to the difference between the observed value and the predicted value in a regression model. In the context of statistics and machine learning, residuals are fundamental for evaluating the quality of a model. A positive residual indicates that the model has underestimated the actual value, while a negative residual suggests an overestimation. Residual analysis allows researchers to identify patterns not captured by the model, which may indicate the need for adjustments or the inclusion of additional variables. Furthermore, residuals are used to verify model assumptions, such as homoscedasticity (constancy of residual variance) and normality. In summary, residuals are a crucial tool for validating and improving predictive models, providing valuable insights into their performance and the relationships between the analyzed variables.
History: The concept of residual dates back to the early days of statistics, but its formalization is associated with the development of linear regression in the 19th century. Francis Galton and Karl Pearson were pioneers in using regression to analyze relationships between variables. As statistics evolved, residual analysis became an essential tool for assessing model quality. In the 20th century, with the rise of computing, residual analysis became more accessible and was integrated into statistical software, facilitating its use across various disciplines.
Uses: Residuals are primarily used in the validation of statistical and machine learning models. They allow analysts to identify whether a model fits the data adequately and whether there are uncaptured patterns. They are also useful for detecting outliers that may influence model performance. In practice, residual analysis is applied in fields such as economics, biology, and engineering, where a precise understanding of relationships between variables is required.
Examples: A practical example of using residuals can be found in predicting housing prices. When building a regression model to predict price based on features such as size and location, the residuals will help identify whether the model is underestimating or overestimating prices in certain areas. Another case is in sales forecasting, where residual analysis can reveal seasonal patterns not accounted for in the initial model.
