# Principal Components Regression in R (Step-by-Step)

Given a set of p predictor variables and a response variable, multiple linear regression uses a method known as least squares to minimize the sum of squared residuals (RSS):

where:

• Σ: A greek symbol that means sum
• yi: The actual response value for the ith observation
• ŷi: The predicted response value based on the multiple linear regression model

However, when the predictor variables are highly correlated then multicollinearity can become a problem. This can cause the coefficient estimates of the model to be unreliable and have high variance.

One way to avoid this problem is to instead use principal components regression, which finds M linear combinations (known as “principal components”) of the original p predictors and then uses least squares to fit a linear regression model using the principal components as predictors.

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

### Step 1: Load Necessary Packages

The easiest way to perform principal components regression in R is by using functions from the pls package.

```#install pls package (if not already installed)
install.packages("pls")

library(pls)
```

### Step 2: Fit PCR Model

For this example, we’ll use the built-in R dataset called mtcars which contains data about various types of cars:

```#view first six rows of mtcars dataset

mpg cyl disp  hp drat    wt  qsec vs am gear carb
Mazda RX4         21.0   6  160 110 3.90 2.620 16.46  0  1    4    4
Mazda RX4 Wag     21.0   6  160 110 3.90 2.875 17.02  0  1    4    4
Datsun 710        22.8   4  108  93 3.85 2.320 18.61  1  1    4    1
Hornet 4 Drive    21.4   6  258 110 3.08 3.215 19.44  1  0    3    1
Hornet Sportabout 18.7   8  360 175 3.15 3.440 17.02  0  0    3    2
Valiant           18.1   6  225 105 2.76 3.460 20.22  1  0    3    1
```

For this example we’ll fit a principal components regression (PCR) model using hp as the response variable and the following variables as the predictor variables:

• mpg
• disp
• drat
• wt
• qsec

The following code shows how to fit the PCR model to this data. Note the following arguments:

• scale=TRUE: This tells R that each of the predictor variables should be scaled to have a mean of 0 and a standard deviation of 1. This ensures that no predictor variable is overly influential in the model if it happens to be measured in different units.
• validation=”CV”: This tells R to use k-fold cross-validation to evaluate the performance of the model. Note that this uses k=10 folds by default. Also note that you can specify “LOOCV” instead to perform leave-one-out cross-validation.
```#make this example reproducible
set.seed(1)

#fit PCR model
model <- pcr(hp~mpg+disp+drat+wt+qsec, data=mtcars, scale=TRUE, validation="CV")```

### Step 3: Choose the Number of Principal Components

Once we’ve fit the model, we need to determine the number of principal components worth keeping.

The way to do so is by looking at the test root mean squared error (test RMSE) calculated by the k-fold cross-validation:

```#view summary of model fitting
summary(model)

Data: 	X dimension: 32 5
Y dimension: 32 1
Fit method: svdpc
Number of components considered: 5

VALIDATION: RMSEP
Cross-validated using 10 random segments.
(Intercept)  1 comps  2 comps  3 comps  4 comps  5 comps
CV           69.66    44.56    35.64    35.83    36.23    36.67
adjCV        69.66    44.44    35.27    35.43    35.80    36.20

TRAINING: % variance explained
1 comps  2 comps  3 comps  4 comps  5 comps
X     69.83    89.35    95.88    98.96   100.00
hp    62.38    81.31    81.96    81.98    82.03
```

There are two tables of interest in the output:

1. VALIDATION: RMSEP

This table tells us the test RMSE calculated by the k-fold cross validation. We can see the following:

• If we only use the intercept term in the model, the test RMSE is 69.66.
• If we add in the first principal component, the test RMSE drops to 44.56.
• If we add in the second principal component, the test RMSE drops to 35.64.

We can see that adding additional principal components actually leads to an increase in test RMSE. Thus, it appears that it would be optimal to only use two principal components in the final model.

2. TRAINING: % variance explained

This table tells us the percentage of the variance in the response variable explained by the principal components. We can see the following:

• By using just the first principal component, we can explain 69.83% of the variation in the response variable.
• By adding in the second principal component, we can explain 89.35% of the variation in the response variable.

Note that we’ll always be able to explain more variance by using more principal components, but we can see that adding in more than two principal components doesn’t actually increase the percentage of explained variance by much.

We can also visualize the test RMSE (along with the test MSE and R-squared) based on the number of principal components by using the validationplot() function.

```#visualize cross-validation plots
validationplot(model)
validationplot(model, val.type="MSEP")
validationplot(model, val.type="R2")```   In each plot we can see that the model fit improves by adding in two principal components, yet it tends to get worse when we add more principal components.

Thus, the optimal model includes just the first two principal components.

### Step 4: Use the Final Model to Make Predictions

We can use the final PCR model with two principal components to make predictions on new observations.

The following code shows how to split the original dataset into a training and testing set and use the PCR model with two principal components to make predictions on the testing set.

```#define training and testing sets
train <- mtcars[1:25, c("hp", "mpg", "disp", "drat", "wt", "qsec")]
y_test <- mtcars[26:nrow(mtcars), c("hp")]
test <- mtcars[26:nrow(mtcars), c("mpg", "disp", "drat", "wt", "qsec")]

#use model to make predictions on a test set
model <- pcr(hp~mpg+disp+drat+wt+qsec, data=train, scale=TRUE, validation="CV")
pcr_pred <- predict(model, test, ncomp=2)

#calculate RMSE
sqrt(mean((pcr_pred - y_test)^2))

 56.86549
```

We can see that the test RMSE turns out to be 56.86549. This is the average deviation between the predicted value for hp and the observed value for hp for the observations in the testing set.

The complete R code use in this example can be found here.