Lasso and Ridge Regression with Caret in R

While the glmnet package is the workhorse behind these methods in R, directly using it can sometimes be a bit cumbersome for tasks like hyperparameter tuning and cross-validation. This is where the caret (Classification And REgression Training) package shines! caret provides a unified interface for training a vast array of machine learning models, including Lasso and Ridge, making the entire process much more efficient and robust.

1 Understanding Lasso and Ridge Regression

Both Lasso (Least Absolute Shrinkage and Selection Operator) and Ridge regression are extensions of ordinary least squares (OLS) regression that add a penalty term to the loss function during model fitting. This penalty shrinks the regression coefficients towards zero.

• Ridge Regression (L2 Penalty): Adds a penalty proportional to the sum of the squared magnitudes of the coefficients. The penalty term is given by: \[ \lambda \sum_{j=1}^{p} \beta_j^2 \]

  • Effect: Shrinks coefficients towards zero, but rarely to exactly zero. It’s good for handling multicollinearity by spreading the impact of correlated predictors.

  • Utility: Reduces variance, especially when predictors are highly correlated.

• Lasso Regression (L1 Penalty): Adds a penalty proportional to the sum of the absolute magnitudes of the coefficients. The penalty term is given by:
  \[ \lambda \sum_{j=1}^{p} |\beta_j| \]

  • Effect: Shrinks coefficients towards zero, and can shrink some coefficients exactly to zero, effectively performing feature selection.

  • Utility: Useful for models with many predictors where only a subset are truly relevant. It leads to sparser models.

The strength of this penalty is controlled by a tuning parameter, often denoted as \(\lambda\) (lambda). In glmnet, the alpha parameter controls the mix between Ridge and Lasso, where alpha = 0 is pure Ridge, alpha = 1 is pure Lasso, and values between 0 and 1 represent Elastic Net. Finding the optimal \(\lambda\) is crucial, and this is typically done via cross-validation.

2 Setting Up Our Environment

First, let’s load the necessary packages. If you don’t have them installed, you’ll need to run install.packages("package_name").

# Install packages if you haven't already
# install.packages("glmnet")
# install.packages("caret")
# install.packages("Matrix") # Often a dependency for glmnet
# install.packages("mlbench") # For the BostonHousing dataset

# Load necessary libraries
library(glmnet) # The core package for Lasso and Ridge
library(caret)  # For streamlined model training and tuning
library(Matrix) # For sparse matrix operations, often used by glmnet
library(dplyr)  # For data manipulation
library(ggplot2) # For plotting
library(mlbench) # To load the BostonHousing dataset

3 Loading and Preparing Data from mlbench

We will use the BostonHousing dataset from the mlbench package. This dataset contains various housing-related features and the median value of owner-occupied homes (medv) in Boston suburbs.

# Load the BostonHousing dataset
data(BostonHousing)

# Inspect the data
head(BostonHousing)

str(BostonHousing)

'data.frame':   506 obs. of  14 variables:
 $ crim   : num  0.00632 0.02731 0.02729 0.03237 0.06905 ...
 $ zn     : num  18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
 $ indus  : num  2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
 $ chas   : Factor w/ 2 levels "0","1": 1 1 1 1 1 1 1 1 1 1 ...
 $ nox    : num  0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
 $ rm     : num  6.58 6.42 7.18 7 7.15 ...
 $ age    : num  65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
 $ dis    : num  4.09 4.97 4.97 6.06 6.06 ...
 $ rad    : num  1 2 2 3 3 3 5 5 5 5 ...
 $ tax    : num  296 242 242 222 222 222 311 311 311 311 ...
 $ ptratio: num  15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
 $ b      : num  397 397 393 395 397 ...
 $ lstat  : num  4.98 9.14 4.03 2.94 5.33 ...
 $ medv   : num  24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...

# The response variable is 'medv' (median value of owner-occupied homes)
# All other columns will be used as predictors.
data_df <- BostonHousing

4 Fitting Lasso and Ridge with glmnet (Directly)

Before diving into caret, let’s briefly see how you would fit these models directly using glmnet. This will help us appreciate caret’s abstraction.

When using glmnet, you typically need to:

1. Prepare your data as a matrix for predictors (x) and a vector for the response (y).

2. Specify alpha (0 for Ridge, 1 for Lasso).

3. Use cv.glmnet to perform cross-validation to find the optimal lambda.

# Prepare data for glmnet
# Exclude the 'medv' column from predictors and use it as the response
x_matrix <- as.matrix(data_df %>% select(-medv))
y_vector <- data_df$medv

# --- Ridge Regression with glmnet ---
# alpha = 0 for Ridge
ridge_glmnet <- cv.glmnet(x_matrix, y_vector, alpha = 0, standardize = TRUE)

# Plot the cross-validation curve
plot(ridge_glmnet)
# Use base R title function for glmnet plots as they are base R plots
title("Ridge Regression CV Curve (glmnet)", line = 2.5)

# Get the optimal lambda (lambda.min)
cat("Optimal lambda for Ridge (glmnet):", ridge_glmnet$lambda.min, "\n")

Optimal lambda for Ridge (glmnet): 0.6777654 

# Get coefficients at optimal lambda
coef(ridge_glmnet, s = "lambda.min")

14 x 1 sparse Matrix of class "dgCMatrix"
                       s1
(Intercept)  28.001475824
crim         -0.087572712
zn            0.032681030
indus        -0.038003639
chas          2.899781645
nox         -11.913360479
rm            4.011308385
age          -0.003731470
dis          -1.118874607
rad           0.153730052
tax          -0.005751054
ptratio      -0.854984614
b             0.009073740
lstat        -0.472423800

# --- Lasso Regression with glmnet ---
# alpha = 1 for Lasso
lasso_glmnet <- cv.glmnet(x_matrix, y_vector, alpha = 1, standardize = TRUE)

# Plot the cross-validation curve
plot(lasso_glmnet)
# Use base R title function for glmnet plots as they are base R plots
title("Lasso Regression CV Curve (glmnet)", line = 2.5)

# Get the optimal lambda (lambda.min)
cat("Optimal lambda for Lasso (glmnet):", lasso_glmnet$lambda.min, "\n")

Optimal lambda for Lasso (glmnet): 0.02118502 

# Get coefficients at optimal lambda
coef(lasso_glmnet, s = "lambda.min")

14 x 1 sparse Matrix of class "dgCMatrix"
                       s1
(Intercept)  34.880894915
crim         -0.100714832
zn            0.042486737
indus         .          
chas          2.693903097
nox         -16.562664331
rm            3.851646315
age           .          
dis          -1.419168850
rad           0.263725830
tax          -0.010286456
ptratio      -0.933927773
b             0.009089735
lstat        -0.522521473

While glmnet is powerful, you manually manage the alpha parameter, interpret the CV plots, and extract coefficients. For more complex workflows (e.g., comparing multiple models, pre-processing, feature engineering), this can become tedious.

5 Fitting Lasso and Ridge with caret

Now, let’s see how caret simplifies this process. caret provides a unified train() function that handles:

• Data Splitting: Creating training and testing sets.

• Pre-processing: (e.g., centering, scaling).

• Resampling: Cross-validation or bootstrapping.

• Hyperparameter Tuning: Searching for the best lambda (and alpha for Elastic Net).

• Model Training: Fitting the model.

• Performance Evaluation: Calculating metrics like RMSE, R-squared.

caret uses the glmnet package internally for Lasso and Ridge, but it wraps the functionality in a user-friendly way.

5.1 Training Control (Cross-Validation Setup)

First, we define our training control, which specifies the resampling method (e.g., k-fold cross-validation).

# Define training control for cross-validation
# We'll use 10-fold cross-validation, repeated 3 times
fitControl <- trainControl(
  method = "repeatedcv",
  number = 10,
  repeats = 3,
  verboseIter = TRUE # Show training progress
)

5.2 Fitting Ridge Regression with caret

For Ridge regression, caret’s glmnet method requires alpha to be fixed at 0. caret will then tune \(\lambda\) for us.

# --- Ridge Regression with caret ---
# The 'glmnet' method in caret can handle Ridge (alpha=0) and Lasso (alpha=1)
# For Ridge, we specify a tuneGrid where alpha is fixed at 0.
# caret will then search for the best lambda within the specified range.

# Define a grid for lambda (caret will choose from these)
# It's good practice to provide a range of lambda values, often on a log scale.
# caret will automatically select an appropriate range if tuneGrid is not provided,
# but explicit control can be useful.
ridgeGrid <- expand.grid(alpha = 0,
                         lambda = 10^seq(-3, 1, length = 100)) # A range of lambda values

cat("\n--- Training Ridge Regression with caret ---\n")

--- Training Ridge Regression with caret ---

ridge_caret <- train(
  medv ~ .,                # Formula interface: response ~ all predictors
  data = data_df,         # Our data frame
  method = "glmnet",      # Specify the glmnet model
  tuneGrid = ridgeGrid,   # Our custom tuning grid for alpha and lambda
  trControl = fitControl, # Our defined cross-validation control
  preProcess = c("center", "scale") # Center and scale predictors (important for regularization)
)

+ Fold01.Rep1: alpha=0, lambda=10 
- Fold01.Rep1: alpha=0, lambda=10 
+ Fold02.Rep1: alpha=0, lambda=10 
- Fold02.Rep1: alpha=0, lambda=10 
+ Fold03.Rep1: alpha=0, lambda=10 
- Fold03.Rep1: alpha=0, lambda=10 
+ Fold04.Rep1: alpha=0, lambda=10 
- Fold04.Rep1: alpha=0, lambda=10 
+ Fold05.Rep1: alpha=0, lambda=10 
- Fold05.Rep1: alpha=0, lambda=10 
+ Fold06.Rep1: alpha=0, lambda=10 
- Fold06.Rep1: alpha=0, lambda=10 
+ Fold07.Rep1: alpha=0, lambda=10 
- Fold07.Rep1: alpha=0, lambda=10 
+ Fold08.Rep1: alpha=0, lambda=10 
- Fold08.Rep1: alpha=0, lambda=10 
+ Fold09.Rep1: alpha=0, lambda=10 
- Fold09.Rep1: alpha=0, lambda=10 
+ Fold10.Rep1: alpha=0, lambda=10 
- Fold10.Rep1: alpha=0, lambda=10 
+ Fold01.Rep2: alpha=0, lambda=10 
- Fold01.Rep2: alpha=0, lambda=10 
+ Fold02.Rep2: alpha=0, lambda=10 
- Fold02.Rep2: alpha=0, lambda=10 
+ Fold03.Rep2: alpha=0, lambda=10 
- Fold03.Rep2: alpha=0, lambda=10 
+ Fold04.Rep2: alpha=0, lambda=10 
- Fold04.Rep2: alpha=0, lambda=10 
+ Fold05.Rep2: alpha=0, lambda=10 
- Fold05.Rep2: alpha=0, lambda=10 
+ Fold06.Rep2: alpha=0, lambda=10 
- Fold06.Rep2: alpha=0, lambda=10 
+ Fold07.Rep2: alpha=0, lambda=10 
- Fold07.Rep2: alpha=0, lambda=10 
+ Fold08.Rep2: alpha=0, lambda=10 
- Fold08.Rep2: alpha=0, lambda=10 
+ Fold09.Rep2: alpha=0, lambda=10 
- Fold09.Rep2: alpha=0, lambda=10 
+ Fold10.Rep2: alpha=0, lambda=10 
- Fold10.Rep2: alpha=0, lambda=10 
+ Fold01.Rep3: alpha=0, lambda=10 
- Fold01.Rep3: alpha=0, lambda=10 
+ Fold02.Rep3: alpha=0, lambda=10 
- Fold02.Rep3: alpha=0, lambda=10 
+ Fold03.Rep3: alpha=0, lambda=10 
- Fold03.Rep3: alpha=0, lambda=10 
+ Fold04.Rep3: alpha=0, lambda=10 
- Fold04.Rep3: alpha=0, lambda=10 
+ Fold05.Rep3: alpha=0, lambda=10 
- Fold05.Rep3: alpha=0, lambda=10 
+ Fold06.Rep3: alpha=0, lambda=10 
- Fold06.Rep3: alpha=0, lambda=10 
+ Fold07.Rep3: alpha=0, lambda=10 
- Fold07.Rep3: alpha=0, lambda=10 
+ Fold08.Rep3: alpha=0, lambda=10 
- Fold08.Rep3: alpha=0, lambda=10 
+ Fold09.Rep3: alpha=0, lambda=10 
- Fold09.Rep3: alpha=0, lambda=10 
+ Fold10.Rep3: alpha=0, lambda=10 
- Fold10.Rep3: alpha=0, lambda=10 
Aggregating results
Selecting tuning parameters
Fitting alpha = 0, lambda = 0.614 on full training set

# Print the model results
print(ridge_caret)

glmnet 

506 samples
 13 predictor

Pre-processing: centered (13), scaled (13) 
Resampling: Cross-Validated (10 fold, repeated 3 times) 
Summary of sample sizes: 456, 456, 454, 455, 455, 456, ... 
Resampling results across tuning parameters:

  lambda        RMSE      Rsquared   MAE     
   0.001000000  4.806746  0.7301647  3.317381
   0.001097499  4.806746  0.7301647  3.317381
   0.001204504  4.806746  0.7301647  3.317381
   0.001321941  4.806746  0.7301647  3.317381
   0.001450829  4.806746  0.7301647  3.317381
   0.001592283  4.806746  0.7301647  3.317381
   0.001747528  4.806746  0.7301647  3.317381
   0.001917910  4.806746  0.7301647  3.317381
   0.002104904  4.806746  0.7301647  3.317381
   0.002310130  4.806746  0.7301647  3.317381
   0.002535364  4.806746  0.7301647  3.317381
   0.002782559  4.806746  0.7301647  3.317381
   0.003053856  4.806746  0.7301647  3.317381
   0.003351603  4.806746  0.7301647  3.317381
   0.003678380  4.806746  0.7301647  3.317381
   0.004037017  4.806746  0.7301647  3.317381
   0.004430621  4.806746  0.7301647  3.317381
   0.004862602  4.806746  0.7301647  3.317381
   0.005336699  4.806746  0.7301647  3.317381
   0.005857021  4.806746  0.7301647  3.317381
   0.006428073  4.806746  0.7301647  3.317381
   0.007054802  4.806746  0.7301647  3.317381
   0.007742637  4.806746  0.7301647  3.317381
   0.008497534  4.806746  0.7301647  3.317381
   0.009326033  4.806746  0.7301647  3.317381
   0.010235310  4.806746  0.7301647  3.317381
   0.011233240  4.806746  0.7301647  3.317381
   0.012328467  4.806746  0.7301647  3.317381
   0.013530478  4.806746  0.7301647  3.317381
   0.014849683  4.806746  0.7301647  3.317381
   0.016297508  4.806746  0.7301647  3.317381
   0.017886495  4.806746  0.7301647  3.317381
   0.019630407  4.806746  0.7301647  3.317381
   0.021544347  4.806746  0.7301647  3.317381
   0.023644894  4.806746  0.7301647  3.317381
   0.025950242  4.806746  0.7301647  3.317381
   0.028480359  4.806746  0.7301647  3.317381
   0.031257158  4.806746  0.7301647  3.317381
   0.034304693  4.806746  0.7301647  3.317381
   0.037649358  4.806746  0.7301647  3.317381
   0.041320124  4.806746  0.7301647  3.317381
   0.045348785  4.806746  0.7301647  3.317381
   0.049770236  4.806746  0.7301647  3.317381
   0.054622772  4.806746  0.7301647  3.317381
   0.059948425  4.806746  0.7301647  3.317381
   0.065793322  4.806746  0.7301647  3.317381
   0.072208090  4.806746  0.7301647  3.317381
   0.079248290  4.806746  0.7301647  3.317381
   0.086974900  4.806746  0.7301647  3.317381
   0.095454846  4.806746  0.7301647  3.317381
   0.104761575  4.806746  0.7301647  3.317381
   0.114975700  4.806746  0.7301647  3.317381
   0.126185688  4.806746  0.7301647  3.317381
   0.138488637  4.806746  0.7301647  3.317381
   0.151991108  4.806746  0.7301647  3.317381
   0.166810054  4.806746  0.7301647  3.317381
   0.183073828  4.806746  0.7301647  3.317381
   0.200923300  4.806746  0.7301647  3.317381
   0.220513074  4.806746  0.7301647  3.317381
   0.242012826  4.806746  0.7301647  3.317381
   0.265608778  4.806746  0.7301647  3.317381
   0.291505306  4.806746  0.7301647  3.317381
   0.319926714  4.806746  0.7301647  3.317381
   0.351119173  4.806746  0.7301647  3.317381
   0.385352859  4.806746  0.7301647  3.317381
   0.422924287  4.806746  0.7301647  3.317381
   0.464158883  4.806746  0.7301647  3.317381
   0.509413801  4.806746  0.7301647  3.317381
   0.559081018  4.806746  0.7301647  3.317381
   0.613590727  4.806746  0.7301647  3.317381
   0.673415066  4.807033  0.7301517  3.317428
   0.739072203  4.811518  0.7298486  3.317314
   0.811130831  4.816688  0.7294945  3.317410
   0.890215085  4.822560  0.7290988  3.318470
   0.977009957  4.829213  0.7286581  3.320186
   1.072267222  4.836750  0.7281649  3.323002
   1.176811952  4.845217  0.7276188  3.326951
   1.291549665  4.854696  0.7270178  3.331652
   1.417474163  4.865317  0.7263531  3.337287
   1.555676144  4.877231  0.7256163  3.343453
   1.707352647  4.890479  0.7248094  3.350169
   1.873817423  4.905252  0.7239185  3.358365
   2.056512308  4.921678  0.7229364  3.368157
   2.257019720  4.939879  0.7218607  3.379089
   2.477076356  4.960040  0.7206732  3.391748
   2.718588243  4.982301  0.7193676  3.406106
   2.983647240  5.006867  0.7179301  3.422477
   3.274549163  5.033853  0.7163524  3.440514
   3.593813664  5.063455  0.7146199  3.460589
   3.944206059  5.095833  0.7127234  3.482775
   4.328761281  5.131167  0.7106439  3.507956
   4.750810162  5.169606  0.7083577  3.535950
   5.214008288  5.211237  0.7058604  3.566416
   5.722367659  5.256181  0.7031371  3.599599
   6.280291442  5.304507  0.7001766  3.636717
   6.892612104  5.356264  0.6969686  3.675653
   7.564633276  5.411474  0.6935052  3.716819
   8.302175681  5.470122  0.6897812  3.761187
   9.111627561  5.532156  0.6857950  3.808851
  10.000000000  5.597482  0.6815522  3.859416

Tuning parameter 'alpha' was held constant at a value of 0
RMSE was used to select the optimal model using the
 smallest value.
The final values used for the model were alpha = 0 and
 lambda = 0.6135907.

# Plot the tuning results (RMSE vs lambda) using ggplot2 directly
ridge_plot_data <- ridge_caret$results
print(ggplot(ridge_plot_data, aes(x = lambda, y = RMSE)) +
        geom_line() +
        geom_point() +
        # Add error bars if RMSESD is available in the results
        {if("RMSESD" %in% names(ridge_plot_data)) geom_errorbar(aes(ymin = RMSE - RMSESD, ymax = RMSE + RMSESD), width = 0.01)} +
        ggplot2::labs(title = "Ridge Regression Tuning (caret)",
                      x = "Lambda",
                      y = "RMSE") +
        theme_minimal())

# Get the best lambda found by caret
cat("Optimal lambda for Ridge (caret):", ridge_caret$bestTune$lambda, "\n")

Optimal lambda for Ridge (caret): 0.6135907 

# Get the coefficients from the best model
coef(ridge_caret$finalModel, s = ridge_caret$bestTune$lambda)

14 x 1 sparse Matrix of class "dgCMatrix"
                    s1
(Intercept) 22.5328063
crim        -0.7532606
zn           0.7622018
indus       -0.2607184
chas1        0.7365273
nox         -1.3804925
rm           2.8184140
age         -0.1050366
dis         -2.3560256
rad          1.3385674
tax         -0.9692660
ptratio     -1.8509951
b            0.8283859
lstat       -3.3736074

5.3 Fitting Lasso Regression with caret

Similarly, for Lasso regression, we fix \(\alpha\) at 1.

# --- Lasso Regression with caret ---
# For Lasso, we specify a tuneGrid where alpha is fixed at 1.
# caret will then search for the best lambda.

lassoGrid <- expand.grid(alpha = 1,
                         lambda = 10^seq(-3, 1, length = 100)) # A range of lambda values

cat("\n--- Training Lasso Regression with caret ---\n")

--- Training Lasso Regression with caret ---

lasso_caret <- train(
  medv ~ .,
  data = data_df,
  method = "glmnet",
  tuneGrid = lassoGrid,
  trControl = fitControl,
  preProcess = c("center", "scale")
)

+ Fold01.Rep1: alpha=1, lambda=10 
- Fold01.Rep1: alpha=1, lambda=10 
+ Fold02.Rep1: alpha=1, lambda=10 
- Fold02.Rep1: alpha=1, lambda=10 
+ Fold03.Rep1: alpha=1, lambda=10 
- Fold03.Rep1: alpha=1, lambda=10 
+ Fold04.Rep1: alpha=1, lambda=10 
- Fold04.Rep1: alpha=1, lambda=10 
+ Fold05.Rep1: alpha=1, lambda=10 
- Fold05.Rep1: alpha=1, lambda=10 
+ Fold06.Rep1: alpha=1, lambda=10 
- Fold06.Rep1: alpha=1, lambda=10 
+ Fold07.Rep1: alpha=1, lambda=10 
- Fold07.Rep1: alpha=1, lambda=10 
+ Fold08.Rep1: alpha=1, lambda=10 
- Fold08.Rep1: alpha=1, lambda=10 
+ Fold09.Rep1: alpha=1, lambda=10 
- Fold09.Rep1: alpha=1, lambda=10 
+ Fold10.Rep1: alpha=1, lambda=10 
- Fold10.Rep1: alpha=1, lambda=10 
+ Fold01.Rep2: alpha=1, lambda=10 
- Fold01.Rep2: alpha=1, lambda=10 
+ Fold02.Rep2: alpha=1, lambda=10 
- Fold02.Rep2: alpha=1, lambda=10 
+ Fold03.Rep2: alpha=1, lambda=10 
- Fold03.Rep2: alpha=1, lambda=10 
+ Fold04.Rep2: alpha=1, lambda=10 
- Fold04.Rep2: alpha=1, lambda=10 
+ Fold05.Rep2: alpha=1, lambda=10 
- Fold05.Rep2: alpha=1, lambda=10 
+ Fold06.Rep2: alpha=1, lambda=10 
- Fold06.Rep2: alpha=1, lambda=10 
+ Fold07.Rep2: alpha=1, lambda=10 
- Fold07.Rep2: alpha=1, lambda=10 
+ Fold08.Rep2: alpha=1, lambda=10 
- Fold08.Rep2: alpha=1, lambda=10 
+ Fold09.Rep2: alpha=1, lambda=10 
- Fold09.Rep2: alpha=1, lambda=10 
+ Fold10.Rep2: alpha=1, lambda=10 
- Fold10.Rep2: alpha=1, lambda=10 
+ Fold01.Rep3: alpha=1, lambda=10 
- Fold01.Rep3: alpha=1, lambda=10 
+ Fold02.Rep3: alpha=1, lambda=10 
- Fold02.Rep3: alpha=1, lambda=10 
+ Fold03.Rep3: alpha=1, lambda=10 
- Fold03.Rep3: alpha=1, lambda=10 
+ Fold04.Rep3: alpha=1, lambda=10 
- Fold04.Rep3: alpha=1, lambda=10 
+ Fold05.Rep3: alpha=1, lambda=10 
- Fold05.Rep3: alpha=1, lambda=10 
+ Fold06.Rep3: alpha=1, lambda=10 
- Fold06.Rep3: alpha=1, lambda=10 
+ Fold07.Rep3: alpha=1, lambda=10 
- Fold07.Rep3: alpha=1, lambda=10 
+ Fold08.Rep3: alpha=1, lambda=10 
- Fold08.Rep3: alpha=1, lambda=10 
+ Fold09.Rep3: alpha=1, lambda=10 
- Fold09.Rep3: alpha=1, lambda=10 
+ Fold10.Rep3: alpha=1, lambda=10 
- Fold10.Rep3: alpha=1, lambda=10 

Warning: There were missing values in resampled performance measures.

Aggregating results
Selecting tuning parameters
Fitting alpha = 1, lambda = 0.0313 on full training set

# Print the model results
print(lasso_caret)

glmnet 

506 samples
 13 predictor

Pre-processing: centered (13), scaled (13) 
Resampling: Cross-Validated (10 fold, repeated 3 times) 
Summary of sample sizes: 456, 455, 456, 455, 455, 455, ... 
Resampling results across tuning parameters:

  lambda        RMSE      Rsquared   MAE     
   0.001000000  4.827425  0.7335316  3.393066
   0.001097499  4.827425  0.7335316  3.393066
   0.001204504  4.827425  0.7335316  3.393066
   0.001321941  4.827425  0.7335316  3.393066
   0.001450829  4.827425  0.7335316  3.393066
   0.001592283  4.827425  0.7335316  3.393066
   0.001747528  4.827425  0.7335316  3.393066
   0.001917910  4.827425  0.7335316  3.393066
   0.002104904  4.827425  0.7335316  3.393066
   0.002310130  4.827425  0.7335316  3.393066
   0.002535364  4.827425  0.7335316  3.393066
   0.002782559  4.827425  0.7335316  3.393066
   0.003053856  4.827425  0.7335316  3.393066
   0.003351603  4.827425  0.7335316  3.393066
   0.003678380  4.827425  0.7335316  3.393066
   0.004037017  4.827425  0.7335316  3.393066
   0.004430621  4.827425  0.7335316  3.393066
   0.004862602  4.827425  0.7335316  3.393066
   0.005336699  4.827425  0.7335316  3.393066
   0.005857021  4.827406  0.7335340  3.393038
   0.006428073  4.827260  0.7335475  3.392831
   0.007054802  4.827098  0.7335620  3.392556
   0.007742637  4.826881  0.7335818  3.392098
   0.008497534  4.826607  0.7336092  3.391507
   0.009326033  4.826304  0.7336394  3.390856
   0.010235310  4.826006  0.7336691  3.390167
   0.011233240  4.825662  0.7337037  3.389439
   0.012328467  4.825308  0.7337402  3.388642
   0.013530478  4.824912  0.7337794  3.387781
   0.014849683  4.824455  0.7338255  3.386805
   0.016297508  4.824026  0.7338671  3.385767
   0.017886495  4.823597  0.7339097  3.384665
   0.019630407  4.823147  0.7339534  3.383476
   0.021544347  4.822767  0.7339890  3.382235
   0.023644894  4.822450  0.7340148  3.380885
   0.025950242  4.822233  0.7340258  3.379502
   0.028480359  4.822119  0.7340269  3.378029
   0.031257158  4.822114  0.7340157  3.376497
   0.034304693  4.822354  0.7339813  3.374991
   0.037649358  4.822854  0.7339164  3.373470
   0.041320124  4.823551  0.7338280  3.371835
   0.045348785  4.824612  0.7337017  3.370324
   0.049770236  4.826082  0.7335323  3.368953
   0.054622772  4.828061  0.7333088  3.367784
   0.059948425  4.830641  0.7330209  3.366864
   0.065793322  4.833864  0.7326627  3.366030
   0.072208090  4.837730  0.7322322  3.365426
   0.079248290  4.842752  0.7316706  3.365236
   0.086974900  4.849078  0.7309704  3.365654
   0.095454846  4.856825  0.7301116  3.366740
   0.104761575  4.866181  0.7290690  3.368768
   0.114975700  4.877414  0.7278118  3.372438
   0.126185688  4.890990  0.7262832  3.377892
   0.138488637  4.906864  0.7244850  3.385528
   0.151991108  4.923308  0.7226186  3.394208
   0.166810054  4.938754  0.7209047  3.403435
   0.183073828  4.951737  0.7195315  3.412342
   0.200923300  4.962284  0.7184644  3.420201
   0.220513074  4.973371  0.7173569  3.428915
   0.242012826  4.986318  0.7160580  3.438947
   0.265608778  5.001966  0.7144936  3.451218
   0.291505306  5.020676  0.7125915  3.466796
   0.319926714  5.040717  0.7105349  3.484102
   0.351119173  5.062932  0.7082395  3.502757
   0.385352859  5.088999  0.7055459  3.523538
   0.422924287  5.119854  0.7023088  3.549734
   0.464158883  5.150742  0.6990987  3.577273
   0.509413801  5.177733  0.6963939  3.600444
   0.559081018  5.199051  0.6945347  3.618083
   0.613590727  5.217934  0.6932181  3.633864
   0.673415066  5.236341  0.6921782  3.649264
   0.739072203  5.257714  0.6910335  3.667368
   0.811130831  5.282367  0.6897834  3.688690
   0.890215085  5.310458  0.6884365  3.713712
   0.977009957  5.342468  0.6869322  3.743012
   1.072267222  5.377386  0.6855324  3.774360
   1.176811952  5.415443  0.6844058  3.805175
   1.291549665  5.458257  0.6834713  3.837310
   1.417474163  5.507601  0.6825933  3.873493
   1.555676144  5.566556  0.6814855  3.918551
   1.707352647  5.636855  0.6799977  3.973028
   1.873817423  5.720373  0.6780110  4.036070
   2.056512308  5.819382  0.6753206  4.109432
   2.257019720  5.936463  0.6716188  4.194332
   2.477076356  6.074523  0.6664300  4.297626
   2.718588243  6.235495  0.6592842  4.422737
   2.983647240  6.415520  0.6509253  4.559803
   3.274549163  6.598890  0.6464106  4.692162
   3.593813664  6.799916  0.6446798  4.830492
   3.944206059  7.035107  0.6420236  4.992616
   4.328761281  7.308930  0.6374380  5.186985
   4.750810162  7.626249  0.6285255  5.418035
   5.214008288  7.990396  0.6091422  5.693873
   5.722367659  8.387284  0.5752620  6.009139
   6.280291442  8.789860  0.5588197  6.345285
   6.892612104  9.150170  0.5006470  6.649038
   7.564633276  9.152032        NaN  6.650629
   8.302175681  9.152032        NaN  6.650629
   9.111627561  9.152032        NaN  6.650629
  10.000000000  9.152032        NaN  6.650629

Tuning parameter 'alpha' was held constant at a value of 1
RMSE was used to select the optimal model using the
 smallest value.
The final values used for the model were alpha = 1 and
 lambda = 0.03125716.

# Plot the tuning results (RMSE vs lambda) using ggplot2 directly
lasso_plot_data <- lasso_caret$results
print(ggplot(lasso_plot_data, aes(x = lambda, y = RMSE)) +
        geom_line() +
        geom_point() +
        # Add error bars if RMSESD is available in the results
        {if("RMSESD" %in% names(lasso_plot_data)) geom_errorbar(aes(ymin = RMSE - RMSESD, ymax = RMSE + RMSESD), width = 0.01)} +
        ggplot2::labs(title = "Lasso Regression Tuning (caret)",
                      x = "Lambda",
                      y = "RMSE") +
        theme_minimal())

# Get the best lambda found by caret
cat("Optimal lambda for Lasso (caret):", lasso_caret$bestTune$lambda, "\n")

Optimal lambda for Lasso (caret): 0.03125716 

# Get the coefficients from the best model
coef(lasso_caret$finalModel, s = lasso_caret$bestTune$lambda)

14 x 1 sparse Matrix of class "dgCMatrix"
                    s1
(Intercept) 22.5328063
crim        -0.8364201
zn           0.9551219
indus        .        
chas1        0.6809208
nox         -1.8755555
rm           2.7219580
age          .        
dis         -2.9166593
rad          2.1553510
tax         -1.6200257
ptratio     -2.0093438
b            0.8212497
lstat       -3.7311343

6 Contrasting caret with glmnet

Here’s a summary of why caret is often preferred for these tasks:

Feature	glmnet (Direct)	caret (with glmnet method)
Interface	Requires x (matrix) and y (vector) as inputs.	Uses formula interface (y ~ .) or x and y directly.
Pre-processing	Manual (e.g., scale()).	Automated (preProcess argument, e.g., c("center", "scale")).
Cross-Validation	cv.glmnet() function.	Integrated into train() via trControl.
Hyperparameter Tuning	Manual interpretation of plot(cv.glmnet) and selection of lambda.min/lambda.1se.	Automated search for optimal \(\lambda\) (and alpha for Elastic Net) based on tuneGrid and trControl.
Model Comparison	Requires separate calls and manual comparison.	Unified train() output makes it easy to compare different models and tuning results.
Output	glmnet object, requires specific functions (coef, predict).	train object, provides consistent methods (predict, plot, print).
Workflow	More granular control, but more manual steps.	Streamlined, automated, and reproducible workflow.
Flexibility	Highly flexible for glmnet specific tasks.	Highly flexible for any model, providing a consistent API.

In essence, caret acts as a powerful wrapper around glmnet (and many other packages), abstracting away much of the boilerplate code involved in model training, tuning, and evaluation. This allows you to focus more on the modeling problem and less on the implementation details of individual algorithms.

7 Making Predictions

When evaluating a model, it’s crucial to estimate its performance on unseen data. This is often referred to as out-of-sample error.

While you could manually split your data into training and test sets and then make predictions on the test set, caret’s train() function, when used with resampling methods like cross-validation, automatically provides robust estimates of out-of-sample performance.

The print() function on a caret train object (e.g., print(ridge_caret)) will display the cross-validated performance metrics (like RMSE, R-squared) for each lambda value tried, as well as for the optimal lambda. These metrics are calculated from the held-out folds during cross-validation and are excellent indicators of how the model generalizes to new data.

Let’s re-examine the output of print(ridge_caret) and print(lasso_caret):

# Re-printing the caret model results to highlight cross-validated metrics
cat("\n--- Ridge Regression Caret Model Summary (Cross-Validated Results) ---\n")

--- Ridge Regression Caret Model Summary (Cross-Validated Results) ---

print(ridge_caret)

glmnet 

506 samples
 13 predictor

Pre-processing: centered (13), scaled (13) 
Resampling: Cross-Validated (10 fold, repeated 3 times) 
Summary of sample sizes: 456, 456, 454, 455, 455, 456, ... 
Resampling results across tuning parameters:

  lambda        RMSE      Rsquared   MAE     
   0.001000000  4.806746  0.7301647  3.317381
   0.001097499  4.806746  0.7301647  3.317381
   0.001204504  4.806746  0.7301647  3.317381
   0.001321941  4.806746  0.7301647  3.317381
   0.001450829  4.806746  0.7301647  3.317381
   0.001592283  4.806746  0.7301647  3.317381
   0.001747528  4.806746  0.7301647  3.317381
   0.001917910  4.806746  0.7301647  3.317381
   0.002104904  4.806746  0.7301647  3.317381
   0.002310130  4.806746  0.7301647  3.317381
   0.002535364  4.806746  0.7301647  3.317381
   0.002782559  4.806746  0.7301647  3.317381
   0.003053856  4.806746  0.7301647  3.317381
   0.003351603  4.806746  0.7301647  3.317381
   0.003678380  4.806746  0.7301647  3.317381
   0.004037017  4.806746  0.7301647  3.317381
   0.004430621  4.806746  0.7301647  3.317381
   0.004862602  4.806746  0.7301647  3.317381
   0.005336699  4.806746  0.7301647  3.317381
   0.005857021  4.806746  0.7301647  3.317381
   0.006428073  4.806746  0.7301647  3.317381
   0.007054802  4.806746  0.7301647  3.317381
   0.007742637  4.806746  0.7301647  3.317381
   0.008497534  4.806746  0.7301647  3.317381
   0.009326033  4.806746  0.7301647  3.317381
   0.010235310  4.806746  0.7301647  3.317381
   0.011233240  4.806746  0.7301647  3.317381
   0.012328467  4.806746  0.7301647  3.317381
   0.013530478  4.806746  0.7301647  3.317381
   0.014849683  4.806746  0.7301647  3.317381
   0.016297508  4.806746  0.7301647  3.317381
   0.017886495  4.806746  0.7301647  3.317381
   0.019630407  4.806746  0.7301647  3.317381
   0.021544347  4.806746  0.7301647  3.317381
   0.023644894  4.806746  0.7301647  3.317381
   0.025950242  4.806746  0.7301647  3.317381
   0.028480359  4.806746  0.7301647  3.317381
   0.031257158  4.806746  0.7301647  3.317381
   0.034304693  4.806746  0.7301647  3.317381
   0.037649358  4.806746  0.7301647  3.317381
   0.041320124  4.806746  0.7301647  3.317381
   0.045348785  4.806746  0.7301647  3.317381
   0.049770236  4.806746  0.7301647  3.317381
   0.054622772  4.806746  0.7301647  3.317381
   0.059948425  4.806746  0.7301647  3.317381
   0.065793322  4.806746  0.7301647  3.317381
   0.072208090  4.806746  0.7301647  3.317381
   0.079248290  4.806746  0.7301647  3.317381
   0.086974900  4.806746  0.7301647  3.317381
   0.095454846  4.806746  0.7301647  3.317381
   0.104761575  4.806746  0.7301647  3.317381
   0.114975700  4.806746  0.7301647  3.317381
   0.126185688  4.806746  0.7301647  3.317381
   0.138488637  4.806746  0.7301647  3.317381
   0.151991108  4.806746  0.7301647  3.317381
   0.166810054  4.806746  0.7301647  3.317381
   0.183073828  4.806746  0.7301647  3.317381
   0.200923300  4.806746  0.7301647  3.317381
   0.220513074  4.806746  0.7301647  3.317381
   0.242012826  4.806746  0.7301647  3.317381
   0.265608778  4.806746  0.7301647  3.317381
   0.291505306  4.806746  0.7301647  3.317381
   0.319926714  4.806746  0.7301647  3.317381
   0.351119173  4.806746  0.7301647  3.317381
   0.385352859  4.806746  0.7301647  3.317381
   0.422924287  4.806746  0.7301647  3.317381
   0.464158883  4.806746  0.7301647  3.317381
   0.509413801  4.806746  0.7301647  3.317381
   0.559081018  4.806746  0.7301647  3.317381
   0.613590727  4.806746  0.7301647  3.317381
   0.673415066  4.807033  0.7301517  3.317428
   0.739072203  4.811518  0.7298486  3.317314
   0.811130831  4.816688  0.7294945  3.317410
   0.890215085  4.822560  0.7290988  3.318470
   0.977009957  4.829213  0.7286581  3.320186
   1.072267222  4.836750  0.7281649  3.323002
   1.176811952  4.845217  0.7276188  3.326951
   1.291549665  4.854696  0.7270178  3.331652
   1.417474163  4.865317  0.7263531  3.337287
   1.555676144  4.877231  0.7256163  3.343453
   1.707352647  4.890479  0.7248094  3.350169
   1.873817423  4.905252  0.7239185  3.358365
   2.056512308  4.921678  0.7229364  3.368157
   2.257019720  4.939879  0.7218607  3.379089
   2.477076356  4.960040  0.7206732  3.391748
   2.718588243  4.982301  0.7193676  3.406106
   2.983647240  5.006867  0.7179301  3.422477
   3.274549163  5.033853  0.7163524  3.440514
   3.593813664  5.063455  0.7146199  3.460589
   3.944206059  5.095833  0.7127234  3.482775
   4.328761281  5.131167  0.7106439  3.507956
   4.750810162  5.169606  0.7083577  3.535950
   5.214008288  5.211237  0.7058604  3.566416
   5.722367659  5.256181  0.7031371  3.599599
   6.280291442  5.304507  0.7001766  3.636717
   6.892612104  5.356264  0.6969686  3.675653
   7.564633276  5.411474  0.6935052  3.716819
   8.302175681  5.470122  0.6897812  3.761187
   9.111627561  5.532156  0.6857950  3.808851
  10.000000000  5.597482  0.6815522  3.859416

Tuning parameter 'alpha' was held constant at a value of 0
RMSE was used to select the optimal model using the
 smallest value.
The final values used for the model were alpha = 0 and
 lambda = 0.6135907.

cat("\n--- Lasso Regression Caret Model Summary (Cross-Validated Results) ---\n")

--- Lasso Regression Caret Model Summary (Cross-Validated Results) ---

print(lasso_caret)

glmnet 

506 samples
 13 predictor

Pre-processing: centered (13), scaled (13) 
Resampling: Cross-Validated (10 fold, repeated 3 times) 
Summary of sample sizes: 456, 455, 456, 455, 455, 455, ... 
Resampling results across tuning parameters:

  lambda        RMSE      Rsquared   MAE     
   0.001000000  4.827425  0.7335316  3.393066
   0.001097499  4.827425  0.7335316  3.393066
   0.001204504  4.827425  0.7335316  3.393066
   0.001321941  4.827425  0.7335316  3.393066
   0.001450829  4.827425  0.7335316  3.393066
   0.001592283  4.827425  0.7335316  3.393066
   0.001747528  4.827425  0.7335316  3.393066
   0.001917910  4.827425  0.7335316  3.393066
   0.002104904  4.827425  0.7335316  3.393066
   0.002310130  4.827425  0.7335316  3.393066
   0.002535364  4.827425  0.7335316  3.393066
   0.002782559  4.827425  0.7335316  3.393066
   0.003053856  4.827425  0.7335316  3.393066
   0.003351603  4.827425  0.7335316  3.393066
   0.003678380  4.827425  0.7335316  3.393066
   0.004037017  4.827425  0.7335316  3.393066
   0.004430621  4.827425  0.7335316  3.393066
   0.004862602  4.827425  0.7335316  3.393066
   0.005336699  4.827425  0.7335316  3.393066
   0.005857021  4.827406  0.7335340  3.393038
   0.006428073  4.827260  0.7335475  3.392831
   0.007054802  4.827098  0.7335620  3.392556
   0.007742637  4.826881  0.7335818  3.392098
   0.008497534  4.826607  0.7336092  3.391507
   0.009326033  4.826304  0.7336394  3.390856
   0.010235310  4.826006  0.7336691  3.390167
   0.011233240  4.825662  0.7337037  3.389439
   0.012328467  4.825308  0.7337402  3.388642
   0.013530478  4.824912  0.7337794  3.387781
   0.014849683  4.824455  0.7338255  3.386805
   0.016297508  4.824026  0.7338671  3.385767
   0.017886495  4.823597  0.7339097  3.384665
   0.019630407  4.823147  0.7339534  3.383476
   0.021544347  4.822767  0.7339890  3.382235
   0.023644894  4.822450  0.7340148  3.380885
   0.025950242  4.822233  0.7340258  3.379502
   0.028480359  4.822119  0.7340269  3.378029
   0.031257158  4.822114  0.7340157  3.376497
   0.034304693  4.822354  0.7339813  3.374991
   0.037649358  4.822854  0.7339164  3.373470
   0.041320124  4.823551  0.7338280  3.371835
   0.045348785  4.824612  0.7337017  3.370324
   0.049770236  4.826082  0.7335323  3.368953
   0.054622772  4.828061  0.7333088  3.367784
   0.059948425  4.830641  0.7330209  3.366864
   0.065793322  4.833864  0.7326627  3.366030
   0.072208090  4.837730  0.7322322  3.365426
   0.079248290  4.842752  0.7316706  3.365236
   0.086974900  4.849078  0.7309704  3.365654
   0.095454846  4.856825  0.7301116  3.366740
   0.104761575  4.866181  0.7290690  3.368768
   0.114975700  4.877414  0.7278118  3.372438
   0.126185688  4.890990  0.7262832  3.377892
   0.138488637  4.906864  0.7244850  3.385528
   0.151991108  4.923308  0.7226186  3.394208
   0.166810054  4.938754  0.7209047  3.403435
   0.183073828  4.951737  0.7195315  3.412342
   0.200923300  4.962284  0.7184644  3.420201
   0.220513074  4.973371  0.7173569  3.428915
   0.242012826  4.986318  0.7160580  3.438947
   0.265608778  5.001966  0.7144936  3.451218
   0.291505306  5.020676  0.7125915  3.466796
   0.319926714  5.040717  0.7105349  3.484102
   0.351119173  5.062932  0.7082395  3.502757
   0.385352859  5.088999  0.7055459  3.523538
   0.422924287  5.119854  0.7023088  3.549734
   0.464158883  5.150742  0.6990987  3.577273
   0.509413801  5.177733  0.6963939  3.600444
   0.559081018  5.199051  0.6945347  3.618083
   0.613590727  5.217934  0.6932181  3.633864
   0.673415066  5.236341  0.6921782  3.649264
   0.739072203  5.257714  0.6910335  3.667368
   0.811130831  5.282367  0.6897834  3.688690
   0.890215085  5.310458  0.6884365  3.713712
   0.977009957  5.342468  0.6869322  3.743012
   1.072267222  5.377386  0.6855324  3.774360
   1.176811952  5.415443  0.6844058  3.805175
   1.291549665  5.458257  0.6834713  3.837310
   1.417474163  5.507601  0.6825933  3.873493
   1.555676144  5.566556  0.6814855  3.918551
   1.707352647  5.636855  0.6799977  3.973028
   1.873817423  5.720373  0.6780110  4.036070
   2.056512308  5.819382  0.6753206  4.109432
   2.257019720  5.936463  0.6716188  4.194332
   2.477076356  6.074523  0.6664300  4.297626
   2.718588243  6.235495  0.6592842  4.422737
   2.983647240  6.415520  0.6509253  4.559803
   3.274549163  6.598890  0.6464106  4.692162
   3.593813664  6.799916  0.6446798  4.830492
   3.944206059  7.035107  0.6420236  4.992616
   4.328761281  7.308930  0.6374380  5.186985
   4.750810162  7.626249  0.6285255  5.418035
   5.214008288  7.990396  0.6091422  5.693873
   5.722367659  8.387284  0.5752620  6.009139
   6.280291442  8.789860  0.5588197  6.345285
   6.892612104  9.150170  0.5006470  6.649038
   7.564633276  9.152032        NaN  6.650629
   8.302175681  9.152032        NaN  6.650629
   9.111627561  9.152032        NaN  6.650629
  10.000000000  9.152032        NaN  6.650629

Tuning parameter 'alpha' was held constant at a value of 1
RMSE was used to select the optimal model using the
 smallest value.
The final values used for the model were alpha = 1 and
 lambda = 0.03125716.

In the output above, The “RMSE” and “Rsquared” columns represent the average performance across the cross-validation folds for different lambda values. The row corresponding to the bestTune (optimal lambda) provides the best estimate of your model’s out-of-sample performance.

While caret handles the out-of-sample error estimation through cross-validation, you can still use the predict() function to get predictions on new data. For demonstration purposes, we’ll predict on the original data_df, but in a real-world scenario, this would be a completely separate test set.

# Make predictions on the original data (for demonstration)
# In a real scenario, you would predict on new, unseen test data.

# Predictions from Ridge model
ridge_predictions <- predict(ridge_caret, newdata = data_df)
cat("\nFirst 5 Ridge predictions:\n")

First 5 Ridge predictions:

print(head(ridge_predictions))

       1        2        3        4        5        6 
30.26572 25.02806 30.53628 28.96496 28.41959 25.56159 

# Predictions from Lasso model
lasso_predictions <- predict(lasso_caret, newdata = data_df)
cat("\nFirst 5 Lasso predictions:\n")

First 5 Lasso predictions:

print(head(lasso_predictions))

       1        2        3        4        5        6 
30.21874 25.05759 30.65066 28.77710 28.12243 25.38625 

# Evaluate performance (e.g., RMSE) on the training data
# (Again, in practice, do this on a separate test set)
ridge_rmse <- sqrt(mean((data_df$medv - ridge_predictions)^2))
lasso_rmse <- sqrt(mean((data_df$medv - lasso_predictions)^2))

cat("\nRMSE for Ridge (on training data):", ridge_rmse, "\n")

RMSE for Ridge (on training data): 4.734318 

cat("RMSE for Lasso (on training data):", lasso_rmse, "\n")

RMSE for Lasso (on training data): 4.684567 

# Compare coefficients (Lasso's sparsity vs Ridge's shrinkage)
cat("\n--- Coefficients from Lasso (best lambda) ---\n")

--- Coefficients from Lasso (best lambda) ---

print(coef(lasso_caret$finalModel, s = lasso_caret$bestTune$lambda))

14 x 1 sparse Matrix of class "dgCMatrix"
                    s1
(Intercept) 22.5328063
crim        -0.8364201
zn           0.9551219
indus        .        
chas1        0.6809208
nox         -1.8755555
rm           2.7219580
age          .        
dis         -2.9166593
rad          2.1553510
tax         -1.6200257
ptratio     -2.0093438
b            0.8212497
lstat       -3.7311343

cat("\n--- Coefficients from Ridge (best lambda) ---\n")

--- Coefficients from Ridge (best lambda) ---

print(coef(ridge_caret$finalModel, s = ridge_caret$bestTune$lambda))

14 x 1 sparse Matrix of class "dgCMatrix"
                    s1
(Intercept) 22.5328063
crim        -0.7532606
zn           0.7622018
indus       -0.2607184
chas1        0.7365273
nox         -1.3804925
rm           2.8184140
age         -0.1050366
dis         -2.3560256
rad          1.3385674
tax         -0.9692660
ptratio     -1.8509951
b            0.8283859
lstat       -3.3736074

Notice how Lasso has driven some coefficients exactly to zero, performing feature selection, while Ridge has shrunk them but kept them non-zero.