Society of Actuaries (SOA) PA Practice Exam

What is a key characteristic of the principal components generated by PCA?

1. They are correlated with the original variables

2. They are always easy to interpret

3. They are orthogonal to each other

4. They are derived from the target variable

The correct answer is: They are orthogonal to each other

Principal Component Analysis (PCA) aims to reduce the dimensionality of a dataset while retaining as much variance as possible. A key characteristic of the principal components generated by PCA is that they are orthogonal to each other. Orthogonality here means that the principal components are statistically independent of one another; thus, knowing the value of one component provides no information about the others. This property is beneficial because it simplifies the structure of the data, allowing for effective data analysis and interpretation. The orthogonal nature of principal components also means that they carry uncorrelated information, which enhances their utility in various statistical methods and algorithms that rely on independent predictors. In the context of multivariate data, this characteristic helps in identifying the directions of maximum variance in the data without redundancy. Other options do not present the same level of accuracy regarding PCA's properties. The original variables may still exhibit correlation with the principal components, and the interpretation of components can vary in ease depending on the specific data and transformations applied. Finally, principal components are derived from the combination of original input variables rather than directly from a target variable. Hence, the orthogonality of principal components is a fundamental aspect that establishes their significance in PCA.