Robust Regression

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21 Aug, 2024

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Robust Regression is a statistical approach that reduces the impact of violations of assumptions and outliers on the regression analysis. It addresses and reduces the impact of outliers by assigning weight to each data point, reducing the influence of outliers and influential observations.

It detects impactful observation and accounts for the assumption’s ordinary least squares (OLS) regression violations. This regression technique helps predict disease risk, fraud, customer churn, and the gravity of natural calamities, as it can make reliable and accurate inferences. It is available in multiple methods like least trimmed squares (LTS), median regression, m-estimators, and quantile regression.

What is robust regression?

Robust regression effectively handles and diminishes the influence of outliers by assigning weights to individual data points.
It is a method employed for datasets that demonstrate non-linear patterns and where the assumptions of the dataset are anticipated to change in the future.
Its applications include outlier detection, heteroscedasticity, missing data handling, non-linear relationships, model selection, forecasting, machine learning applications, robust regression in r, robust regression in Python, and robust regression Stata.
It is vigorous to outliers but can become less efficient than OLS in their absence.

Robust Regression Explained

Robust regression is a technique utilized for such datasets that exhibit non-linear trends while the underlying assumptions of the dataset might change in the future. Since conventional least square data has remained sensitive toward noise, even a single outlier can distort the results. Hence, the robust regression model is usually aligned to weighted data, offering metrics less affected by extreme observations. As a result, it estimates regression coefficients, making correct inferences despite noisy data or the assumptions of OLS.

It works by assigning different algorithms to estimate the regression coefficients like M-estimation. As such, the least squares are iteratively reweighted. Therefore, these techniques adjust the data points' weights based on their residuals, with outliers down-weighted and less influential observations up-weighted. Such an insistent procedure continues until it achieves convergence, leading to robust estimates.

Consequently, it poses many implications, like providing highly accurate estimates and predictions even when outliers and violations of assumptions exist. It decreases the bias due to influential observations and outliers, assuring less influence on results by extreme values. This regression is also more resistant to heteroscedasticity, where the variance of the residuals is not the same throughout different layers of the independent variables.

This concept finds usability in economics, social sciences, engineering, and finance. It comes to the fore in handling changing assumptions, missing data, and non-linear relationships. In finance, it is common to find outliers and violations of assumptions like heteroscedasticity and non-linear patterns. So, by using robust regression, banks assess the borrowing risk, investment firms develop trading strategies, regulators monitor financial markets and determine potential risks. Lastly, it also plays an important role in asset pricing models, portfolio optimization, and risk management.

Examples

Let us use a few examples to understand the topic.

Example #1

The publication Robust Discovery of Regression Models of April 2023 presents a technique for locating robust regression models for equations. It deals with missing variables, non-linearity, outliers, incorrectly described dynamics, and unsuitable conditioning assumptions that arise when modeling observational data. The regression methods can create a preliminary generic model with candidate variables and non-linear representations. The authors then manage outliers and non-constant parameters, choose pertinent effects, and remove redundant variables using a machine-learning multi-block search technique called Autometrics.

The study emphasizes the significance of maintaining subject-matter theory and collaborative decision-making in model selection. Two real-world examples show how well the suggested method works to enhance model selection and inference. The study gives a workable method for identifying regression models in practical applications and sheds light on the difficulties associated with modeling observational data.

Example #2

Suppose Dr. Alice, the data scientist in the hypothetical Acme Corporation, is working on a project to forecast customer attrition. She possesses an extensive collection of consumer data, encompassing demographics, past purchases, and feedback from customer surveys. However, she recognizes that the data is likely to contain outliers, such as consumers who have only made one transaction or customers who have given a high satisfaction rating but later churned. Dr. Alice chooses to employ robust regression to construct a churn prediction model. A statistical technique that is resilient to outliers is robust regression. This implies that a few extreme numbers won't have an undue impact on the model.

Dr. Alice builds a churn prediction model using many robust regression approaches. She discovers that the most effective model combines random forests with lasso regression. One regularization method that lessens overfitting is lasso regression. An ensemble learning technique, random forests, combines several decision trees to get more accurate predictions. Even in the face of outliers, Alice's churn prediction algorithm can reliably identify which customers are most likely to leave. Acme Corporation may utilize this data to create focused marketing campaigns and customer retention plans.

Applications

It finds application in different fields using different methods. Some of its important applications are as follows:

Heteroscedasticity: It considers weighted residuals to reduce the impact of heteroscedasticity on regression estimates.
Non-linear Relationships: It can model non-linear relationships between dependent and independent variables, providing more accurate predictions than linear regression.
Model Selection: It can help identify the most relevant variables and functional forms for a regression model by selecting significant predictors.
Forecasting and Time Series Analysis: It can improve forecasting accuracy by accounting for outliers, non-linear trends, and changing assumptions over time.
Machine Learning Applications: It can boost machine learning model performance and robustness by mitigating outliers and handling non-linear relationships.

However, it is applied only by using a two-step process. First, fit a preliminary model and identify outliers. Second, remove those outliers and fit a robust regression model. Hence, as a result, some of its applications are:

Robust regression analysis is a statistical method that is resistant to outliers and violations of assumptions.
Robust regression in R can be performed using a variety of packages, such as robustbase and rlm.
Robust regression in Python can be performed using the statsmodels library.
Robust regression in Stata can be performed using the rreg command.

Pros And Cons

Although robust regression is a powerful tool with limitations, it requires careful selection of a method for the best results. Here are the major benefits and challenges of robust regression:

Pros	Cons
Vigorous to outliers	In the absence of outliers, it can become less efficient than OLS.
Handles non-linear relationships adequately.	Is expensive than OLS regression computation-wise
Influential detection is done by it.	Remains sensitive towards the methods of regression chosen.
If heteroscedasticity is present, then it gives a more accurate estimate.	Becomes hard to interpret.
Can easily handle missing data and violation of assumptions.	Requires large sample data to get more accurate
Adaptable to changing assumptions	Coefficient interpretation could be challenging.

Frequently Asked Questions (FAQs)

1. When to use robust regression?

It can be employed in situations where the data contains outliers or broken assumptions. Because the impact of outliers is lessened, the regression coefficient estimations are more trustworthy. In circumstances when ordinary least squares (OLS) regression might yield skewed findings, this regression is especially helpful.

2. How to run robust regression in Stata?

Using the robust option in conjunction with the regress command in Stata allows to do robust regression. A robust regression analysis, with y as the dependent variable and x as the independent variable, may be done, for instance, by typing regress y x, robust. The coefficients will then be estimated by Stata using strong standard errors.

3. Is robust regression in r?

Indeed, R has robust regression available. Robust regression analysis functions are available in a number of R packages, including lmrob from the robustbase package and "rlm" from the "MASS" package. These functions may provide the right settings and inputs to execute robust regression.

4. What is robust regression in SAS?

SAS offers robust regression techniques like PROC ROBUSTREG and PROC QUANTREG for estimating linear regression model parameters, handling various data formats and specifying robust estimators.