In this section, we will learn how to execute Ridge Regression in R. We use ridge regression to tackle the multicollinearity problem. Due to multicollinearity, the model estimates (least square) see a large variance. Ridge regression is a method by which we add a degree of bias to the regression estimates.
Ridge regression is a parsimonious model that performs L2 regularization. The L2 regularization adds a penalty equivalent to the square of the magnitude of regression coefficients and tries to minimize them. The equation of ridge regression looks like as given below.
LS Obj + λ (sum of the square of coefficients)
Here the objective is as follows:
- If λ = 0, the output is similar to simple linear regression.
- If λ = very large, the coefficients will become zero.
The following diagram is the visual interpretation comparing OLS and ridge regression.
Training Ridge Regression in R
To build the ridge regression in r, we use
glmnetfunction from glmnet package in R. Let’s use ridge regression to predict the mileage of the car using mtcars dataset.
# Loaging the library library(glmnet) # Getting the independent variable x_var <- data.matrix(mtcars[, c("hp", "wt", "drat")]) # Getting the dependent variable y_var <- mtcars[, "mpg"] # Setting the range of lambda values lambda_seq <- 10^seq(2, -2, by = -.1) # Using glmnet function to build the ridge regression in r fit <- glmnet(x_var, y_var, alpha = 0, lambda = lambda_seq) # Checking the model summary(fit)
# Output Length Class Mode a0 41 -none- numeric beta 123 dgCMatrix S4 df 41 -none- numeric dim 2 -none- numeric lambda 41 -none- numeric dev.ratio 41 -none- numeric nulldev 1 -none- numeric npasses 1 -none- numeric jerr 1 -none- numeric offset 1 -none- logical call 5 -none- call nobs 1 -none- numeric
Choosing Optimal Lambda Value
glmnet function trains the model multiple times for all the different values of lambda, which we pass as a sequence of vector to the
lambda = argument in the
glmnet function. The next task is to identify the optimal value of lambda that will result in a minimum error. This can be achieved automatically by using
# Using cross validation glmnet ridge_cv <- cv.glmnet(x_var, y_var, alpha = 0, lambda = lambdas) # Best lambda value best_lambda <- ridge_cv$lambda.min best_lambda
# Output  79.43000
Extracting the best model using K-cross validation
The best model can be extracted by calling the
glmnet.fit from the cross-validation object. Once you have that, we can rebuild the model by passing lambda as 79.43000.
best_fit <- ridge_cv$glmnet.fit head(best_fit)
# Output Df %Dev Lambda [1,] 3 0.1798 100.00000 [2,] 3 0.2167 79.43000 [3,] 3 0.2589 63.10000 [4,] 3 0.3060 50.12000 [5,] 3 0.3574 39.81000 [6,] 3 0.4120 31.62000
Building the final model
# Rebuilding the model with optimal lambda value best_ridge <- glmnet(x_var, y_var, alpha = 0, lambda = 79.43000)
Checking the coefficients
# Output 4 x 1 sparse Matrix of class "dgCMatrix" s0 (Intercept) 20.099502946 hp -0.004398609 wt -0.344175261 drat 0.484807607
The next task is to use the predict function and compute the R2 value for both the train and test dataset. In this, the example we did not create the train and test split. So, I am only providing a sample code. However, you can read the linear regression chapter to understand this step in detail.
# here x is the test dataset pred <- predict(best_ridge, s = best_lambda, newx = x) # R squared formula actual <- test$Price preds <- test$PreditedPrice rss <- sum((preds - actual) ^ 2) tss <- sum((actual - mean(actual)) ^ 2) rsq <- 1 - rss/tss rsq
Bias and variance trade-off
Bias and variance trade-off is generally complicated when it comes to building ridge regression models on an actual dataset. However, following the general trend which I would like to highlight here:
- The bias increases as λ increases.
- The variance decreases as λ increases.
Assumptions of Ridge Regressions
The assumptions of ridge regression are the same as that of linear regression: linearity, constant variance, and independence. However, as ridge regression does not provide confidence limits, the distribution of errors to be normal need not be assumed.
In this chapter, we learned about ridge regression in R using functions from glmnet package. We also saw how to use cross-validation to get the best model. In the next chapter, we will learn how to use lasso regression for identifying important variables in r.