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Lasso regression is a method we can use to fit a regression model when multicollinearity is present in the data.

In a nutshell, least squares regression tries to find coefficient estimates that minimize the sum of squared residuals (RSS):

**RSS = Î£(y _{i}Â â€“ Å·_{i})2**

where:

**Î£**: A greek symbol that meansÂ*sum***y**: The actual response value for the i_{i}^{th}Â observation**Å·**: The predicted response value based on the multiple linear regression model_{i}

Conversely, lasso regression seeks to minimize the following:

**RSS + Î»Î£|Î² _{j}|**

whereÂ *j*Â ranges from 1 toÂ *p* predictor variables and Î» â‰¥ 0.

This second term in the equation is known as aÂ *shrinkage penalty*. In lasso regression, we select a value for Î» that produces the lowest possible test MSE (mean squared error).

This tutorial provides a step-by-step example of how to perform lasso regression in R.

**Step 1: Load the Data**

For this example, weâ€™ll use the R built-in dataset calledÂ **mtcars**. Weâ€™ll useÂ **hp** as the response variable and the following variables as the predictors:

- mpg
- wt
- drat
- qsec

To perform lasso regression, weâ€™ll use functions from the **glmnet** package. This package requires the response variable to be a vector and the set of predictor variables to be of the class **data.matrix**.

The following code shows how to define our data:

**#define response variable
y #define matrix of predictor variables
**x

**Step 2: Fit the Lasso Regression Model**

Next, weâ€™ll use theÂ **glmnet()** function to fit the lasso regression model and specify **alpha=1**.

Note that setting alpha equal to 0 is equivalent to using ridge regression and setting alpha to some value between 0 and 1 is equivalent to using an elastic net.**Â **

To determine what value to use for lambda, weâ€™ll perform k-fold cross-validation and identify the lambda value that produces the lowest test mean squared error (MSE).

Note that the function **cv.glmnet()** automatically performs k-fold cross validation using k = 10 folds.

library(glmnet) #perform k-fold cross-validation to find optimal lambda value cv_model glmnet(x, y, alpha = 1) #find optimal lambda value that minimizes test MSE best_lambda lambda.min best_lambda [1] 5.616345 #produce plot of test MSE by lambda value plot(cv_model)

The lambda value that minimizes the test MSE turns out to beÂ **5.616345**.

**Step 3: Analyze Final Model**

Lastly, we can analyze the final model produced by the optimal lambda value.

We can use the following code to obtain the coefficient estimates for this model:

**#find coefficients of best model
best_model 1**, lambda = best_lambda)
coef(best_model)
5 x 1 sparse Matrix of class "dgCMatrix"
s0
(Intercept) 484.20742
mpg -2.95796
wt 21.37988
drat .
qsec -19.43425

No coefficient is shown for the predictorÂ **drat** because the lasso regression shrunk the coefficient all the way to zero. This means it was completely dropped from the model because it wasnâ€™t influential enough.

Note that this is a key difference between ridge regression and lasso regression. Ridge regression shrinks all coefficientsÂ *towards* zero, but lasso regression has the potential to remove predictors from the model by shrinking the coefficientsÂ *completely* to zero.

We can also use the final lasso regression model to make predictions on new observations. For example, suppose we have a new car with the following attributes:

- mpg: 24
- wt: 2.5
- drat: 3.5
- qsec: 18.5

The following code shows how to use the fitted lasso regression model to predict the value for *hp*Â of this new observation:

**#define new observation
new = matrix(c(24, 2.5, 3.5, 18.5), nrow=1, ncol=4)
#use lasso regression model to predict response value
predict(best_model, s = best_lambda, newx = new)
[1,] 109.0842**

Based on the input values, the model predicts this car to have anÂ *hp*Â value ofÂ **109.0842**.

Lastly, we can calculate the R-squared of the model on the training data:

#use fitted best model to make predictions y_predicted predict(best_model, s = best_lambda, newx = x) #find SST and SSE sst sum((y - mean(y))^2) sse sum((y_predicted - y)^2) #find R-Squared rsq

The R-squared turns out to beÂ **0.8047064**. That is, the best model was able to explain **80.47%** of the variation in the response values of the training data.

You can find the complete R code used in this example here.