PCA is the process of finding the principal components. What exactly are these?

We can consider that a component is a normalized linear combination of the features (James, 2012). The first principal component in a dataset is the linear combination that captures the maximum variance in the data. A second component is created by selecting another linear combination that maximizes the variance with the constraint that its direction is perpendicular to the first component. The subsequent components (equal to the number of variables) would follow this same rule.

A couple of things here. This definition describes the **linear combination**, which is one of the key assumptions in PCA. If you ever try and apply PCA to a dataset of variables having a low correlation, you will likely end up with a meaningless analysis. Another key assumption is that the mean and variance for a variable are sufficient statistics. What this tells us is that the data should fit a normal...