LOESS (or LOWESS), which stands for Locally Weighted Scatterplot Smoothing, is a flexible method for fitting a smooth curve to a scatterplot. Unlike traditional regression that fits a single equation to the data, LOESS fits multiple regressions in localized subsets to produce a curve. This allows for the capturing of non-linear relationships between variables. In this comprehensive article, we will explore LOESS regression, its advantages, and how to perform it in R.
Introduction to LOESS Regression
LOESS regression is a non-parametric technique that does not assume a particular functional form for the relationship between the dependent and independent variables. Instead, it relies on localized subsets of the data to construct a curve that fits the observed data points.
How Does LOESS Work?
- Weighted Regression: For each data point, a subset of neighboring points is chosen, and a weighted least squares regression is applied. The data point at which the regression is centered receives the highest weight, with weights decreasing for points further away.
- Smoothing Parameter: This determines the fraction of data points used for each localized regression. A smaller fraction results in a more wiggly curve, while a larger fraction produces a smoother curve.
- Degree of the Polynomial: Typically, either a first-degree (linear) or second-degree (quadratic) polynomial is used for the localized regressions.
Advantages of LOESS:
- It’s flexible and can capture complex non-linear relationships.
- It doesn’t require the specification of a functional form like linear or logistic regression.
- It’s intuitive and visually interpretable.
Performing LOESS Regression in R
We’ll use the
mtcars dataset, which comes with R, to demonstrate LOESS regression.
Visualizing the Data:
Before fitting the model, let’s visualize the relationship between the weight (
wt) of cars and their miles-per-gallon (
library(ggplot2) ggplot(mtcars, aes(x = wt, y = mpg)) + geom_point() + ggtitle("Scatterplot of mpg against wt")
Fitting the LOESS Model:
loess() function, we can fit a LOESS model:
loess_fit <- loess(mpg ~ wt, data = mtcars)
Visualizing the LOESS Fit:
Overlay the LOESS curve on the scatterplot:
ggplot(mtcars, aes(x = wt, y = mpg)) + geom_point() + geom_smooth(method = "loess", se = FALSE, color = "red") + ggtitle("LOESS Regression of mpg on wt")
geom_smooth() with the method set to “loess” adds the LOESS curve to the plot.
Adjusting the Smoothing Parameter:
span argument controls the smoothing parameter. A smaller value of
span makes the curve wigglier:
ggplot(mtcars, aes(x = wt, y = mpg)) + geom_point() + geom_smooth(method = "loess", se = FALSE, color = "red", span = 0.5) + ggtitle("LOESS Regression with span = 0.5")
mpg for specific
new_data <- data.frame(wt = c(2.5, 3.5)) predictions <- predict(loess_fit, new_data) print(predictions)
It’s essential to check the assumptions and performance of the LOESS fit. One way is by examining the residuals:
residuals <- resid(loess_fit)
You can plot the residuals against the fitted values to check for any patterns or heteroscedasticity.
Caveats and Considerations:
- Overfitting: Since LOESS is highly flexible, it can sometimes overfit the data, especially with a small smoothing parameter.
- Computational Intensity: For large datasets, LOESS can be computationally demanding.
- Interpretability: While LOESS provides a good fit, it may not provide as clear an interpretation as parametric models.
LOESS regression offers a versatile method to capture non-linear relationships in data without making strong assumptions about the functional form of the relationship. R provides intuitive and efficient tools for performing and visualizing LOESS regression, making it an invaluable tool for data scientists and researchers working with non-linear data. As with all models, it’s crucial to understand its strengths and limitations to use it effectively.