In this article, we will discuss what local regression is and how we implement it in the R Programming Language.

What is Local Regression in R?

Local regression is also known as LOESS (locally estimated scatterplot smoothing) regression. It is a flexible non-parametric method for fitting regression models to data. Local regression models adapt to the local structure of the data and make them particularly useful for analyzing complex relationships and nonlinear patterns. Now we implement Local Regression in R step by step.

Step 1. Installing and Loading Required Packages

Before diving in let, us talk about the necessary packages to be installed in your R environment. The ‘stats’ package, which comes with base R provides functions for performing local regression. Additional packages like ‘ggplot2’ and ‘dplyr’ are commonly used for data visualization and manipulation.

# Install and load required packages
install.packages("ggplot2")
install.packages("dplyr")

library(ggplot2)
library(dplyr)

Step 2. Preparing Data for local Regression

Data preparation is essential before conducting local regression because dataset should be clean and well-structured, with the relevant variables rightly formatted. So, handle missing values and outliers appropriately to prevent bias in the analysis. Get the Dataset here.

data <- read.csv("local_regression_data.csv")

# Handle missing values
data <- na.omit(data)

# Compute outlier threshold for response_variable
outlier_threshold <- 3 * sd(data$response_variable)  # Adjust multiplier as needed
# Handle outliers
data <- filter(data, response_variable < outlier_threshold)
data

Output:

  predictor_variable response_variable
1                1.2               3.4
2                2.3               4.5
3                3.4               5.6
4                4.5               6.7
5                5.6               7.8

Step 3. Performing Local Regression

Performing local regression in R is very straightforward using the ‘loess()’ function. And simply specifying predictor and response variables. The bandwidth parameter in local regression controls the degree of smoothing applied to the data. It determines the size of the local neighborhood used to estimate the regression function. Choosing an appropriate bandwidth is crucial for achieving a balance between bias and variance in the model.

# Perform local regression
loess_model <- loess(response_variable ~ predictor_variable, data = data)

#bandwidth parameter
bandwidth <- 0.1  # Adjust as needed

summary(loess_model)

Output:

Call:
loess(formula = response_variable ~ predictor_variable, data = data)

Number of Observations: 5 
Equivalent Number of Parameters: 5 
Residual Standard Error: Inf 
Trace of smoother matrix: 5  (exact)

Control settings:
  span     :  0.75 
  degree   :  2 
  family   :  gaussian
  surface  :  interpolate      cell = 0.2
  normalize:  TRUE
 parametric:  FALSE
drop.square:  FALSE 

The loess function in R fits a smooth curve to data points using local regression. In this case, there are only 5 data points, and the model uses a quadratic fit to these points. The output shows that the model has as many parameters as data points, indicating a high flexibility. However, the fit is poor, as indicated by the infinite residual standard error, meaning the model does not capture the data’s pattern well. The settings like span, degree, and family control how the curve is fitted locally. More data would likely improve the fit and provide a more reliable model.

Step 4. Visualizing Local Regression Results

Visualizing local regression helps in interpreting the fitted model and identifying any nonlinear relationships in the data. Using scatter plots overlaid with smoothed curves to visualize the relationship between the predictor and response variables.

# Visualize local regression
ggplot(data, aes(x = predictor_variable, y = response_variable)) +
  geom_point() +
  geom_smooth(method = "loess")

Output:

Local Regression in R

The plot shows the result of fitting a loess model to a small dataset with 5 observations. Here’s a simple explanation of what it depicts:

1. Data Points: The black dots represent the original data points for the predictor variable on the x-axis and the response variable on the y-axis.
2. Fitted Curve: The blue line is the smooth curve fitted by the loess model. This curve captures the trend of the data points, showing how the response variable changes with the predictor variable.
3. Smooth Fit: Despite the small number of data points, the loess model has managed to fit a smooth, non-linear curve that follows the pattern of the data.
4. Quadratic Fit: The curve uses a quadratic polynomial for the local fits, meaning it can capture bends and curves in the data, as seen in the smooth, wavy nature of the blue line.
5. Interpolation: Since there are very few points, the curve closely follows the data points, suggesting an interpolation approach where the fit goes through or very near each point.

Overall, the plot demonstrates how loess can fit a flexible, smooth curve to a small set of data points, effectively capturing the underlying trend.

Conclusion

Local regression offers a flexible and powerful approach for analyzing complex relationships and nonlinear patterns in data. By adapting to the local structure of the data local regression models provide valuable insights into the underlying relationships between variables. Whether it’s exploring recent trends or predicting financial markets local regression analysis in R equips analysts with the tools needed to uncover insights and make informed decisions.