Society of Actuaries (SOA) PA Practice Exam

Which factor contributes to the effectiveness of PCA in dimensionality reduction?

1. The preservation of target variables

2. The orthogonality of principal components

3. The use of hyperparameters for adjustment

4. The extensive correlation of components

The correct answer is: The orthogonality of principal components

The orthogonality of principal components plays a significant role in the effectiveness of Principal Component Analysis (PCA) for dimensionality reduction. In PCA, the principal components are derived in such a way that they are mutually orthogonal, meaning they are uncorrelated with each other. This orthogonality ensures that each principal component captures a different aspect of the data's variance. By maximizing variance and minimizing redundancy, PCA reduces the dimensionality of the dataset effectively while retaining as much of the original variance as possible. Maintaining orthogonality also simplifies computations and allows for a clearer interpretation of the components, as they can be seen as distinct directions in the multidimensional space that each represent different patterns in the data. In contrast, while the preservation of target variables, the use of hyperparameters, and the correlation of components might have their roles in other contexts or methods, they do not specifically contribute to the foundational effectiveness of PCA in capturing the variance structure in the data. Hence, the orthogonality of the principal components is vital for achieving effective dimensionality reduction in PCA.