Principal Components Regression, Pt. The Standard Method In this note, we discuss principal components regression and some of the issues with it:

First you explain it to your great-grandmother; then to you grandmother; then to your mother; then to your spouse; finally, to your daughter who is a mathematician. Each time the next person is less of a layman.

Here is how the conversation might go. I heard you are studying "Pee-See-Ay". I wonder what that is Ah, it's just a method of summarizing some data. Look, we have some wine bottles standing here on the table.

We can describe each wine by its colour, by how strong it is, by how old it is, and so on see this very nice visualization of wine properties taken from here.

We can compose a whole list of different characteristics of each wine in our cellar. But many of them will measure related properties and so will be redundant.

If so, we should be able to summarize each wine with fewer characteristics! This is what PCA does. So this PCA thing checks what characteristics are redundant and discards them?

No, PCA is not selecting some characteristics and discarding the others. Instead, it constructs some new characteristics that turn out to summarize our list of wines well. Of course these new characteristics are constructed using the old ones; for example, a new characteristic might be computed as wine age minus wine acidity level or some other combination like that we call them linear combinations.

In fact, PCA finds the best possible characteristics, the ones that summarize the list of wines as well as only possible among all conceivable linear combinations. This is why it is so useful. Hmmm, this certainly sounds good, but I am not sure I understand. What do you actually mean when you say that these new PCA characteristics "summarize" the list of wines?

I guess I can give two different answers to this question. First answer is that you are looking for some wine properties characteristics that strongly differ across wines. Indeed, imagine that you come up with a property that is the same for most of the wines.

This would not be very useful, wouldn't it? Wines are very different, but your new property makes them all look the same!

This would certainly be a bad summary. Instead, PCA looks for properties that show as much variation across wines as possible. The second answer is that you look for the properties that would allow you to predict, or "reconstruct", the original wine characteristics.

Again, imagine that you come up with a property that has no relation to the original characteristics; if you use only this new property, there is no way you could reconstruct the original ones! This, again, would be a bad summary.

So PCA looks for properties that allow to reconstruct the original characteristics as well as possible. Surprisingly, it turns out that these two aims are equivalent and so PCA can kill two birds with one stone.

But darling, these two "goals" of PCA sound so different! Why would they be equivalent? Perhaps I should make a little drawing takes a napkin and starts scribbling. Let us pick two wine characteristics, perhaps wine darkness and alcohol content -- I don't know if they are correlated, but let's imagine that they are.

Here is what a scatter plot of different wines could look like: Each dot in this "wine cloud" shows one particular wine. A new property can be constructed by drawing a line through the center of this wine cloud and projecting all points onto this line.

Now look here very carefully -- here is how these projections look like for different lines red dots are projections of the blue dots: As I said before, PCA will find the "best" line according to two different criteria of what is the "best".

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One of the main results from a principal component analysis, factor analysis, or a linear discriminant analysis is a set of eigenvectors that are called components, factors, or linear discriminant functions.

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Principal Component Analysis