# From Outliers to Inliers: Robust Non-Parametric Regression with Median-of-Means

Regression analysis is a widely used statistical tool for predicting a continuous dependent variable based on one or more independent variables. However, traditional regression methods, such as linear and polynomial regression, can be sensitive to outliers and make incorrect predictions if the assumptions of normality and homoscedasticity are violated. To address these limitations, researchers have developed robust regression techniques that are more resistant to outliers and do not make any assumptions about the distribution of the data. One such technique is median-of-means (MoM) regression, which provides a non-parametric solution to robust regression analysis.

In this article, we will explore the concept of median-of-means (MoM) regression and its application to robust non-parametric regression analysis. We will begin by defining the problem of outliers in regression analysis and explaining why traditional regression methods are sensitive to them. Then, we will introduce the concept of non-parametric regression and explain how median-of-means (MoM) regression can be used to provide a robust solution. Finally, we will provide a step-by-step guide to performing MoM regression in R and present an example of a real-world application of MoM regression in finance.

##### Problem of Outliers in Regression Analysis

Outliers are extreme values that deviate significantly from the other observations in the data. In regression analysis, outliers can have a significant impact on the estimates of the regression coefficients and the predictions of the dependent variable. For example, a single outlier observation can drastically change the slope of the regression line and cause the predictions to be far off from the actual values.

Traditional regression methods, such as linear and polynomial regression, are sensitive to outliers because they are based on the least squares method, which minimizes the sum of squared residuals. This method is sensitive to outliers because the squared residuals of the outliers are much larger than those of the other observations, which results in the outliers having a disproportionate influence on the regression coefficients.

##### Introduction to Non-Parametric Regression

Non-parametric regression is a type of regression analysis that does not make any assumptions about the distribution of the data. Unlike traditional regression methods, non-parametric regression methods are not based on the least squares method, which makes them less sensitive to outliers. Non-parametric regression methods include techniques such as spline regression, kernel regression, and local regression.

##### Median-of-Means (MoM) Regression

Median-of-means (MoM) regression is a non-parametric regression technique that provides a robust solution to regression analysis. MoM regression is based on the concept of dividing the data into subgroups, computing the median of each subgroup, and then computing the mean of these medians. This process is repeated multiple times to obtain a final estimate of the regression coefficients.

One advantage of MoM regression is that it is less sensitive to outliers than traditional regression methods. This is because the median of a subgroup is less sensitive to outliers than the mean of the entire group. By computing the median of multiple subgroups, MoM regression provides a more robust estimate of the regression coefficients that is less influenced by outliers.

##### Step-by-Step Guide to Performing MoM Regression in R

To perform MoM regression in R, we will use the “robustbase” package, which provides functions for performing robust regression analysis. In this example, we will use the “lmrob” function to perform MoM regression. The “lmrob” function has the same syntax as the “lm” function in R, which is used for traditional linear regression.

Step 1: Load the “robustbase” package in R

To load the “robustbase” package in R, you can use the following command:

`library(robustbase)`

In this example, we will use the “mtcars” dataset that is built into R. The “mtcars” dataset contains information on 32 different car models and their specifications, including the weight of the car, the number of cylinders, and the miles per gallon (mpg) rating. We will use the weight of the car (wt) as the independent variable and the mpg rating as the dependent variable. To load the “mtcars” dataset, you can use the following command:

`data(mtcars)`

Step 3: Perform MoM regression

To perform MoM regression, you can use the “lmrob” function from the “robustbase” package. The syntax for the “lmrob” function is the same as the “lm” function in R. In this example, we will perform MoM regression to predict the mpg rating based on the weight of the car (wt). To perform MoM regression, you can use the following command:

`mo_reg <- lmrob(mpg ~ wt, data = mtcars)`

Step 4: Summarize the results

To summarize the results of the MoM regression, you can use the “summary” function in R. The “summary” function provides information on the estimated regression coefficients, the residuals, and the goodness-of-fit statistics. To summarize the results of the MoM regression, you can use the following command:

`summary(mo_reg)`
##### Example of MoM Regression in Finance

MoM regression can also be applied in the field of finance. For example, MoM regression can be used to predict stock prices based on the historical data. In finance, it is common to have outliers in the data due to sudden changes in the market, such as a stock market crash. Traditional regression methods can be sensitive to these outliers, which can result in incorrect predictions of stock prices.

To address this issue, MoM regression can be used to provide a robust solution to predicting stock prices. The process is the same as described above, but the independent and dependent variables would be the historical stock prices and the predicted stock prices, respectively.

##### Conclusion

Median-of-means (MoM) regression provides a robust solution to regression analysis that is less sensitive to outliers than traditional regression methods. By dividing the data into subgroups, computing the median of each subgroup, and then computing the mean of these medians, MoM regression provides a more robust estimate of the regression coefficients that is less influenced by outliers. In this article, we provided a step-by-step guide to performing MoM regression in R and presented an example of a real-world application of MoM regression in finance.