Robust regression

|OLSandMM|thumb]] |ResidualPlots|thumb|left]] Robust regression is a form of regression analysis designed to overcome some limitations of traditional parametric and non-parametric methods. Regression analysis, in general, aims to model the relationship between a dependent variable and one or more independent variables. The traditional methods, such as ordinary least squares (OLS), are known to be highly sensitive to outliers in the data, which can significantly affect the model's estimates and predictions. Robust regression methods are developed to be less sensitive to outliers, providing more reliable estimates in the presence of anomalous data.

Overview

Robust regression methods were introduced as an alternative to least squares estimation for dealing with outlier data in regression models. Outliers can be a result of measurement errors, incorrect data entry, or they can be correct but extreme measurements which are not representative of the population. In any case, these outliers can leverage the regression model disproportionately and skew the results, making the model less accurate or even misleading.

Methods

Several robust regression techniques have been developed, each with its own advantages and applications. Some of the most commonly used methods include:

• Least Absolute Shrinkage and Selection Operator (LASSO): This method adds a penalty equal to the absolute value of the magnitude of coefficients to the loss function. This can both shrink some coefficients exactly to zero (providing a form of variable selection) and make the estimation process more robust to outliers.
• Ridge Regression: Similar to LASSO, ridge regression adds a penalty to the loss function, but the penalty is proportional to the square of the magnitude of the coefficients. This method is less influenced by outliers than OLS, though it does not inherently select variables.
• Huber Regression: This method is particularly popular in robust regression. It is less sensitive to outliers in data than the OLS method because it applies a linear loss to outliers, reducing their influence on the model.
• Quantile Regression: Unlike OLS, which estimates the mean of the dependent variable given the independent variables, quantile regression estimates the median or other quantiles, making it inherently more robust to outliers.

Applications

Robust regression is widely used in fields where data may be contaminated with outliers or where the distribution of the data is not normal, including:

• Econometrics
• Biostatistics
• Finance
• Environmental Science

Advantages and Disadvantages

The main advantage of robust regression is its resilience to outliers, making it possible to obtain more reliable estimates even in the presence of anomalous data. However, these methods can be more complex to understand and apply correctly compared to OLS. Additionally, the choice of the right robust method and tuning parameters (like the penalty term in LASSO and ridge regression) requires expertise and can significantly affect the results.

Conclusion

Robust regression provides an essential toolkit for analysts and researchers dealing with real-world data that is often imperfect. By minimizing the influence of outliers, robust regression methods ensure that the derived models are more representative of the underlying data, leading to more accurate predictions and insights.