10 – M4 L2b 13 Principal Components V3

Great. So, we found this new and special set of basis vectors, which are the principal components. What now? Well, let’s look at what we have. We have these new basis dimensions. What are the significance of these? How do we interpret them? What do they represent? Well, we’ve been talking about vectors in geometric space so far. So, we picture the principal components as directions in space, and they are that. But when we have a data table, the dimensions have another meaning as well. Each dimension represents a feature in a dataset, one type of measurement. For example, if we’re talking about stock returns, each dimension represents the returns of an individual company. The new basis dimensions, the PCs, are combinations of the original dimensions. Therefore, they have funny units, some amount of returns of company A plus some other amount of returns of company B. These new units may or may not correspond to any quantity that makes sense in the real world. We say they may or may not be interpretable. Whether or not we think the new basis dimensions represent some real-world quantity and whether or not we care depends on the problem we’re trying to solve.