Regularization in R Programming

Regularization is a form of regression technique that shrinks or regularizes or constraints the coefficient estimates towards 0 (or zero). In this technique, a penalty is added to the various parameters of the model in order to reduce the freedom of the given model. The concept of Regularization can be broadly classified into:

• Ridge Regression
• Lasso Regression
• Elastic Net Regression

Implementation in R

In the R language, to perform Regularization we need a handful of packages to be installed before we start working on them. The required packages are 

• glmnet package for ridge regression and lasso regression
• dplyr package for data cleaning
• psych package in order to perform or compute the trace function of a matrix
• caret package

To install these packages we have to use the install.packages() in the R Console. After installing the packages successfully, we include these packages in our R Script using the library() command. To implement the Regularization regression technique we need to follow either of the three types of regularization techniques.

Ridge Regression

The Ridge Regression is a modified version of linear regression and is also known as L2 Regularization. Unlike linear regression, the loss function is modified in order to minimize the model’s complexity and this is done by adding some penalty parameter which is equivalent to the square of the value or magnitude of the coefficient. Basically, to implement Ridge Regression in R we are going to use the “glmnet” package. The cv.glmnet() function will be used to determine the ridge regression. 

Example:

In this example, we will implement the ridge regression technique on the mtcars dataset for a better illustration. Our task is to predict the miles per gallon on the basis of other characteristics of the cars. We are going to use the set.seed() function to set seed for reproducibility. We are going to set the value of lambda in three ways:

• by performing 10 fold cross-validation
• based on the information derived
• optimal lambda based on both the criteria

R

# Regularization
# Ridge Regression in R
# Load libraries, get data & set
# seed for reproducibility
set.seed(123)
library(glmnet)
library(dplyr)
library(psych)

data("mtcars")
# Center y, X will be standardized
# in the modelling function
y <- mtcars %>% select(mpg) %>%
            scale(center = TRUE, scale = FALSE) %>%
            as.matrix()
X <- mtcars %>% select(-mpg) %>% as.matrix()

# Perform 10-fold cross-validation to select lambda
lambdas_to_try <- 10^seq(-3, 5, length.out = 100)

# Setting alpha = 0 implements ridge regression
ridge_cv <- cv.glmnet(X, y, alpha = 0,
                      lambda = lambdas_to_try,
                      standardize = TRUE, nfolds = 10)

# Plot cross-validation results
plot(ridge_cv)

# Best cross-validated lambda
lambda_cv <- ridge_cv$lambda.min

# Fit final model, get its sum of squared
# residuals and multiple R-squared
model_cv <- glmnet(X, y, alpha = 0, lambda = lambda_cv,
                   standardize = TRUE)
y_hat_cv <- predict(model_cv, X)
ssr_cv <- t(y - y_hat_cv) %*% (y - y_hat_cv)
rsq_ridge_cv <- cor(y, y_hat_cv)^2

# selecting lambda based on the information
X_scaled <- scale(X)
aic <- c()
bic <- c()
for (lambda in seq(lambdas_to_try))
{
  # Run model
  model <- glmnet(X, y, alpha = 0,
                  lambda = lambdas_to_try[lambda],
                  standardize = TRUE)

  # Extract coefficients and residuals (remove first
  # row for the intercept)
  betas <- as.vector((as.matrix(coef(model))[-1, ]))
  resid <- y - (X_scaled %*% betas)

  # Compute hat-matrix and degrees of freedom
  ld <- lambdas_to_try[lambda] * diag(ncol(X_scaled))
  H <- X_scaled %*% solve(t(X_scaled) %*% X_scaled + ld)
                                           %*% t(X_scaled)
  df <- tr(H)

  # Compute information criteria
  aic[lambda] <- nrow(X_scaled) * log(t(resid) %*% resid)
                                                   + 2 * df
  bic[lambda] <- nrow(X_scaled) * log(t(resid) %*% resid)
                           + 2 * df * log(nrow(X_scaled))
}

# Plot information criteria against tried values of lambdas
plot(log(lambdas_to_try), aic, col = "orange", type = "l",
     ylim = c(190, 260), ylab = "Information Criterion")
lines(log(lambdas_to_try), bic, col = "skyblue3")
legend("bottomright", lwd = 1, col = c("orange", "skyblue3"),
       legend = c("AIC", "BIC"))

# Optimal lambdas according to both criteria
lambda_aic <- lambdas_to_try[which.min(aic)]
lambda_bic <- lambdas_to_try[which.min(bic)]

# Fit final models, get their sum of
# squared residuals and multiple R-squared
model_aic <- glmnet(X, y, alpha = 0, lambda = lambda_aic,
                    standardize = TRUE)
y_hat_aic <- predict(model_aic, X)
ssr_aic <- t(y - y_hat_aic) %*% (y - y_hat_aic)
rsq_ridge_aic <- cor(y, y_hat_aic)^2

model_bic <- glmnet(X, y, alpha = 0, lambda = lambda_bic,
                    standardize = TRUE)
y_hat_bic <- predict(model_bic, X)
ssr_bic <- t(y - y_hat_bic) %*% (y - y_hat_bic)
rsq_ridge_bic <- cor(y, y_hat_bic)^2

# The higher the lambda, the more the
# coefficients are shrinked towards zero.
res <- glmnet(X, y, alpha = 0, lambda = lambdas_to_try,
              standardize = FALSE)
plot(res, xvar = "lambda")
legend("bottomright", lwd = 1, col = 1:6,
       legend = colnames(X), cex = .7)

Output:

Lasso Regression

Moving forward to Lasso Regression. It is also known as L1 Regression, Selection Operator, and Least Absolute Shrinkage. It is also a modified version of Linear Regression where again the loss function is modified in order to minimize the model’s complexity. This is done by limiting the summation of the absolute values of the coefficients of the model. In R, we can implement the lasso regression using the same “glmnet” package like ridge regression.

Example:

Again in this example, we are using the mtcars dataset. Here also we are going to set the lambda value like the previous example.

R

# Regularization
# Lasso Regression
# Load libraries, get data & set
# seed for reproducibility
set.seed(123)
library(glmnet)
library(dplyr)
library(psych)

data("mtcars")
# Center y, X will be standardized in the modelling function
y <- mtcars %>% select(mpg) %>% scale(center = TRUE,
                                      scale = FALSE) %>%
                                      as.matrix()
X <- mtcars %>% select(-mpg) %>% as.matrix()


# Perform 10-fold cross-validation to select lambda
lambdas_to_try <- 10^seq(-3, 5, length.out = 100)

# Setting alpha = 1 implements lasso regression
lasso_cv <- cv.glmnet(X, y, alpha = 1,
                      lambda = lambdas_to_try,
                      standardize = TRUE, nfolds = 10)

# Plot cross-validation results
plot(lasso_cv)

# Best cross-validated lambda
lambda_cv <- lasso_cv$lambda.min

# Fit final model, get its sum of squared
# residuals and multiple R-squared
model_cv <- glmnet(X, y, alpha = 1, lambda = lambda_cv,
                   standardize = TRUE)
y_hat_cv <- predict(model_cv, X)
ssr_cv <- t(y - y_hat_cv) %*% (y - y_hat_cv)
rsq_lasso_cv <- cor(y, y_hat_cv)^2

# The higher the lambda, the more the
# coefficients are shrinked towards zero.
res <- glmnet(X, y, alpha = 1, lambda = lambdas_to_try,
              standardize = FALSE)
plot(res, xvar = "lambda")
legend("bottomright", lwd = 1, col = 1:6,
       legend = colnames(X), cex = .7)

Output:

If we compare Lasso and Ridge Regression techniques we will notice that both the techniques are more or less the same. But there are few characteristics where they differ from each other.

• Unlike Ridge, Lasso can set some of its parameters to zero.
• In ridge the coefficient of the predictor that is correlated is similar. While in lasso only one of the coefficient of predictor is larger and the rest tends to zero.
• Ridge works well if there exist many huge or large parameters that are of the same value. While lasso works well if there exist only a small number of definite or significant parameters and rest tending to zero.

Elastic Net Regression

We shall now move on to Elastic Net Regression. Elastic Net Regression can be stated as the convex combination of the lasso and ridge regression. We can work with the glmnet package here even. But now we shall see how the package caret can be used to implement the Elastic Net Regression.

Example:

R

# Regularization
# Elastic Net Regression
library(caret)

# Set training control
train_control <- trainControl(method = "repeatedcv",
                              number = 5,
                              repeats = 5,
                              search = "random",
                              verboseIter = TRUE)

# Train the model
elastic_net_model <- train(mpg ~ .,
                           data = cbind(y, X),
                           method = "glmnet",
                           preProcess = c("center", "scale"),
                           tuneLength = 25,
                           trControl = train_control)

# Check multiple R-squared
y_hat_enet <- predict(elastic_net_model, X)
rsq_enet <- cor(y, y_hat_enet)^2

print(y_hat_enet)
print(rsq_enet)

Output:

> print(y_hat_enet)
          Mazda RX4       Mazda RX4 Wag          Datsun 710      Hornet 4 Drive   Hornet Sportabout             Valiant 
         2.13185747          1.76214273          6.07598463          0.50410531         -3.15668592          0.08734383 
         Duster 360           Merc 240D            Merc 230            Merc 280           Merc 280C          Merc 450SE 
        -5.23690809          2.82725225          2.85570982         -0.19421572         -0.16329225         -4.37306992 
         Merc 450SL         Merc 450SLC  Cadillac Fleetwood Lincoln Continental   Chrysler Imperial            Fiat 128 
        -3.83132657         -3.88886320         -8.00151118         -8.29125966         -8.08243188          6.98344302 
        Honda Civic      Toyota Corolla       Toyota Corona    Dodge Challenger         AMC Javelin          Camaro Z28 
         8.30013895          7.74742320          3.93737683         -3.13404917         -2.56900144         -5.17326892 
   Pontiac Firebird           Fiat X1-9       Porsche 914-2        Lotus Europa      Ford Pantera L        Ferrari Dino 
        -4.02993835          7.36692700          5.87750517          6.69642869         -2.02711333          0.06597788 
      Maserati Bora          Volvo 142E 
        -5.90030273          4.83362156 
> print(rsq_enet)
         [,1]
mpg 0.8485501