Principal component analysis is applied to a large data set with four variables. Loadings for the first four principal components are estimated.

Determine which of the following statements is/are true with respect the loadings.

• The loadings are unique.
• For a given principal component, the sum of the squares of the loadings across the four variables is one.
• Together, the four principal components explain 100% of the variance.

• None
• I and II only
• I and III only
• II and III only
• The correct answer is not given by (A), (B), (C), or (D).

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Key: D

I is false because the loadings are unique only up to a sign flip.

II is true. Principal components are designed to maximize variance. If there are no constraints on the magnitude of the loadings, the variance can be made arbitrarily large. The PCA algorithm’s constraint is that the sum of the squares of the loadings equals 1.

III is true because four components can capture all the variation in four variables, provided there are at least four data points (note that the problem states that the data set is large).

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.