Lasso Regression Analysis with K-Fold Cross Validation

Linear regression models are widely used for predicting a response variable based on one or more predictor variables. However, when dealing with a large number of predictors, the model can become complex and prone to overfitting. Lasso regression, a shrinkage and variable selection method, provides a solution to this issue. In this blog post, we will explore the principles of lasso regression and apply it to identify a subset of predictors from a larger pool of variables.

Understanding Lasso Regression

Lasso regression aims to minimize prediction error for a quantitative response variable while simultaneously performing variable selection. It achieves this by imposing a constraint on the model parameters, causing the regression coefficients of some variables to shrink toward zero. Variables with a regression coefficient equal to zero after the shrinkage process are excluded from the model. The remaining variables, with non-zero coefficients, are the ones most strongly associated with the response variable.

The flexibility of lasso regression makes it suitable for a variety of predictor types, including quantitative, categorical, or a mix of both.

Exploratory Data Analysis (EDA)

To understand the relationships between variables, we’ll start with some exploratory data analysis (EDA) visualizations.

# Scatter plots for selected variables
ggplot(random_data, aes(x = predictor_matrix[, 1], y = response_variable)) +
  geom_point() +
  labs(title = "Scatter Plot of X1 vs Response Variable",
       x = "X1",
       y = "Response Variable")

Explore additional visualizations for a comprehensive EDA.

Lasso Regression Analysis

Now, let’s perform lasso regression with k-fold cross-validation to identify the optimal subset of predictors.

# Set up k-fold cross-validation
cv <- cv.glmnet(x = as.matrix(random_data[, -c(1)]),
                y = random_data$response_variable,
                alpha = 1,  # 1 for lasso regression
                nfolds = 10)

# Visualize cross-validation results
plot(cv)

The plot above displays the cross-validated error as a function of the number of predictors. We’ll use this information to determine the optimal number of predictors for our lasso regression model.

Final Model and Interpretation

Once we identify the optimal subset of predictors, we can build our final lasso regression model and interpret the results.

# Extract the optimal lambda value
optimal_lambda <- cv$lambda.min

# Build the final lasso regression model
lasso_model <- glmnet(x = as.matrix(random_data[, -c(1)]),
                      y = random_data$response_variable,
                      alpha = 1,  # 1 for lasso regression,
                      lambda = optimal_lambda)

# Display coefficients
print(coef(lasso_model))

## 11 x 1 sparse Matrix of class "dgCMatrix"
##                     s0
## (Intercept)  2.7390677
## V2           1.8831599
## V3          -1.3263326
## V4           .        
## V5           .        
## V6           0.0353103
## V7           .        
## V8           .        
## V9           .        
## V10          .        
## V11          .

The coefficients indicate the strength and direction of the relationship between each predictor and the response variable.