To evaluate how well a model is able to fit a dataset, we must analyze how the model performs on observations it has never seen before.
One of the most common ways to do this is by using k-fold cross-validation, which uses the following approach:
1. Randomly divide a dataset into k groups, or “folds”, of roughly equal size.
2. Choose one of the folds to be the holdout set. Fit the model on the remaining k-1 folds. Calculate the test MSE on the observations in the fold that was held out.
3. Repeat this process k times, using a different set each time as the holdout set.
4. Calculate the overall test MSE to be the average of the k test MSE’s.
The easiest way to perform k-fold cross-validation in R is by using the trainControl() and train() functions from the caret library in R.
The trainControl() function is used to specify the parameters for training (e.g. the type of cross-validation to use, the number of folds to use, etc.) and the train() function is used to actually fit the model to the data.
The following example shows how to use the trainControl() and train() functions in practice.
Example: How to Use trainControl() in R
Suppose we have the following dataset in R:
#create data frame df <- data.frame(y=c(6, 8, 12, 14, 14, 15, 17, 22, 24, 23), x1=c(2, 5, 4, 3, 4, 6, 7, 5, 8, 9), x2=c(14, 12, 12, 13, 7, 8, 7, 4, 6, 5)) #view data frame df y x1 x2 6 2 14 8 5 12 12 4 12 14 3 13 14 4 7 15 6 8 17 7 7 22 5 4 24 8 6 23 9 5
Now suppose we use the lm() function to fit a multiple linear regression model to this dataset, using x1 and x2 as the predictor variables and y as the response variable:
#fit multiple linear regression model to data fit <- lm(y ~ x1 + x2, data=df) #view model summary summary(fit) Call: lm(formula = y ~ x1 + x2, data = df) Residuals: Min 1Q Median 3Q Max -3.6650 -1.9228 -0.3684 1.2783 5.0208 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 21.2672 6.9927 3.041 0.0188 * x1 0.7803 0.6942 1.124 0.2981 x2 -1.1253 0.4251 -2.647 0.0331 * --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Residual standard error: 3.093 on 7 degrees of freedom Multiple R-squared: 0.801, Adjusted R-squared: 0.7441 F-statistic: 14.09 on 2 and 7 DF, p-value: 0.003516
Using the coefficients in the model output, we can write the fitted regression model:
y = 21.2672 + 0.7803*(x1) – 1.1253(x2)
To get an idea of how well this model would perform on unseen observations, we can use k-fold cross validation.
The following code shows how to use the trainControl() function from the caret package to specify a k-fold cross validation (method = “cv”) that uses 5 folds (number = 5).
We then pass this trainControl() function to the train() function to actually perform the k-fold cross validation:
library(caret) #specify the cross-validation method ctrl <- trainControl(method = "cv", number = 5) #fit a regression model and use k-fold CV to evaluate performance model <- train(y ~ x1 + x2, data = df, method = "lm", trControl = ctrl) #view summary of k-fold CV print(model) Linear Regression 10 samples 2 predictor No pre-processing Resampling: Cross-Validated (5 fold) Summary of sample sizes: 8, 8, 8, 8, 8 Resampling results: RMSE Rsquared MAE 3.612302 1 3.232153 Tuning parameter 'intercept' was held constant at a value of TRUE
From the output we can see that the model was fit 5 times using a sample size of 8 observations each time.
Each time the model was then used to predict the values of the 2 observations that were held out and the following metrics were calculated each time:
- RMSE: The root mean squared error. This measures the average difference between the predictions made by the model and the actual observations. The lower the RMSE, the more closely a model can predict the actual observations.
- MAE: The mean absolute error. This is the average absolute difference between the predictions made by the model and the actual observations. The lower the MAE, the more closely a model can predict the actual observations.
The average of the RMSE and MAE values for the five folds is shown in the output:
- RMSE: 3.612302
- MAE: 3.232153
These metrics give us an idea of how well the model performs on previously unseen data.
In practice, we typically fit several different models and compare these metrics to determine which model performs best on unseen data.
For example, we might proceed to fit a polynomial regression model and perform K-fold cross validation on it to see how the RMSE and MAE metrics compare to the multiple linear regression model.
Note #1: In this example we chose to use k=5 folds, but you can choose however many folds you’d like. In practice, we typically choose between 5 and 10 folds because this turns out to be the optimal number of folds that produce reliable test error rates.
Note #2: The trainControl() function accepts many potential arguments. You can find the complete documentation for this function here.
The following tutorials provide additional information about model training:
Introduction to K-Fold Cross-Validation
Introduction to Leave-One-Out Cross-Validation
What is Overfitting in Machine Learning?