Society of Actuaries (SOA) PA Practice Exam

How do the principal components from PCA typically function in terms of variance representation?

1. They capture low proportions of original variable variance

2. They capture a high proportion of the variance of original variables

3. They consistently preserve the variance of the original variables

4. They are solely based on the first principal component

The correct answer is: They capture a high proportion of the variance of original variables

The principal components resulting from Principal Component Analysis (PCA) are designed to capture a high proportion of the variance present in the original variables. This method transforms the original correlated variables into a new set of uncorrelated variables called principal components. The first principal component accounts for the largest possible variance from the original dataset. Subsequent components capture the remaining variance but with each subsequent component accounting for progressively less. This characteristic is fundamental to PCA, as it seeks to reduce dimensionality while retaining those characteristics of the dataset that contribute most to its variance. By going this route, PCA helps reveal the underlying structure of the data, making it easier to analyze complex datasets without losing significant information. In contrast, capturing low proportions of the original variable variance or solely relying on the first principal component would undermine the purpose of PCA, which is to provide a comprehensive view of variance through multiple components. Although the principal components do preserve the overall structure of variance, the focus is on maximizing variance captured rather than simply maintaining it.