# Cross Validation

Directions: Follow along with the slides, completing the questions in blue on your computer, and answering the questions in red in your journal.

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# What is cross-validation?

• In the previous two labs, we learned how to:
• Create a linear model predicting height from the arm_span data (4A).
• See how well our model predicts height on the arm_span data by computing mean squared error (MSE) (4B).
• In this lab, we will see how well our model predicts the heights of people we haven’t yet measured.
• To do this, we will use a method called cross-validation.
• Cross-validation consists of three steps:
• Step 1: Split the data into training and test sets.
• Step 2: Create a model using the training set.
• Step 3: Use this model to make predictions on the test set.

# Step 1: training-test split

• Waiting for new observations can take a long time. The U.S. takes a census of its population once every 10 years, for example.
• Instead of waiting for new observations, data scientists will take their current data and divide it into two distinct sets.
• Split the arm_span data into training and test sets using the following two steps.
• First, fill in the blanks below to randomly select which rows of arm_span will go into the training set.
set.seed(123)
training_rows <- sample(1:____, size = 85)
• Second, use the slice function to create two dataframes: one called training consisting of the training_rows, and another called test consisting of the remaining rows of arm_span.
training <- slice(arm_span, ____)
test <- slice(____, - ____)
• Explain these lines of code and describe the training and test datasets.

# Aside: set.seed()

• When we split data, we’re randomly separating our observations into training and test sets.
• It’s important to notice that no single observation will be placed in both sets.
• Because we’re splitting the datasets randomly, our models can also vary slightly, person-to-person.
• This is why it’s important to use set.seed.
• By using set.seed, we’re able to reproduce the random splitting so that each person’s model outputs the same results.

Whenever you split data into training and test, always use set.seed first.

# Aside: training-test ratio

• When splitting data into training and test sets, we need to have enough observations in our data so that we can build a good model.
• This is why we kept 85 observations in our training data.
• As datasets grow larger, we can use a larger proportion of the data to test with.

# Step 2: training the model

• Step 2 is to create a linear model relating height and armspan using the training data.
• Fit a line of best fit model to our training data and assign it the name best_training.
• Recall that the slope and intercept of our linear model are chosen to minimize MSE.
• Since the MSE being minimized is from the training data, we can call it training MSE.

# Step 3: test the model

• Step 3 is to use the model we built on the training data to make predictions on the test data.
• Note that we are NOT recomputing the slope and intercept to fit the test data best. We use the same slope and intercept that were computed in step 2.
• Because we’re using the line of best fit, we can use the predict() function we introduced in the last lab to make predictions.
• Fill in the blanks below to add predicted heights to our test data:
test <- mutate(test, ____ = predict(best_training, newdata = ____))
• Hint: the predict function without the argument newdata will output predictions on the training data. To output predictions on the test data, supply the test data to the newdata argument.
• Calculate the test MSE in the same way as you did in the previous lab (test MSE is simply MSE of the predictions on the test data).

# Recap

• Another way to describe the three steps is
• Step 1: Split the data into training and test sets.
• Step 2: Choose a slope and intercept that minimize training MSE.
• Step 3: Using the same slope and intercept from step 2, make predictions on the test set, and use these predictions to compute test MSE.
• This begs the question, why do we care about test MSE?

# Why cross-validate?

• Why go to all this trouble to compute test MSE when we could just compute MSE on the original dataset?
• When we compute MSE on the original dataset, we are measuring the ability of a model to make predictions on the current batch of data.
• Relying on a single dataset can lead to models that are so specific to the current batch of data that they’re unable to make good predictions for future observations.
• This phenomenon is known as overfitting.
• By splitting the data into a training and test set, we are hiding a proportion of the data from the model. This emulates future observations, which are unseen.
• Test MSE estimates the ability of a model to make predictions on future observations.

# Example of overfitting

• The following example motivates cross-validation by illustrating the dangers of overfitting.
• We randomly select 7 points from the arm_span dataset and fit two models: a linear model, and a polynomial model.
• You will learn how to fit a polynomial model in the next lab.
• Below is a plot of these 7 training points, and two curves representing the value of height each model would predict given a value of armspan.

• Which model does a better job of predicting the 7 training points?
• Which model do you think will do a better job of predicting the rest of the data?

# Example of overfitting, continued

• Below is a plot of the rest of the arm_span dataset, along with the predictions each model would make.

• Which model does a better job of generalizing to the rest of the arm_span dataset?