I'm not sure about the tremendous value of understanding PCA inside out - do you want to calculate it manually? Common sense is sufficient to apply it.
I don't need to get down to the nitty gritty of it.. but i think i have already.. I realize its another way to reduce complexity and to score the data in relation to loadings/components.. So how would you take into account the backwardation, and contango with crude oil into this.. obviously the convience yeild is negative in contango, and it pays to carry the physical when the curve is very steep.. what consideration would be had for this? And would we not be trying to place relative value trades by spreading with calenders and flys.. This would isolate slope to a degree?
The more interesting application is trading the curvature. But, isn't that the same as looking at the modality in a forward curve, then saying I will sell the peak/buy the bottom of that curve by spreading between those months. I am not sure where the additional value of PCA is, but this is from a lack of understanding. As for the slope, you can see where the glide path of a slope can turn steeper or flatter. However, trading that would result in very small (but maybe, more reliable?) moves.
Well basically your pricing it against it's driver/component instead of just it's price.. this way you can bring things back to a different unit of measure and look at an individual contracts relative price to the "driver" rather then just what it visual looks like against the rest of the curve.. Your looking for dislocations relative to a factor.. Obviously like in another thread another factor is the containment of the degree of contango based on the ability to arb the physical against the forward.. as it gets steeper it gets more and more enticing to buy the front , take delivery and sell the forward.. .
Yes, the factors are supposed to be uncorrelated, indeed... I would say that there is no contradiction. Normally curves are steep when short-dated rates fall (that might not be true any more). Your "level" principal component (the first one) doesn't necessarily correspond to the short rate, so it should all be internally consistent. I generally always used to look at the two domains completely separately. Connecting the "dots", so to speak, is a job for rates-specific term structure models, rather than generic PCA. So I suppose I am sorta dodging the question by saying that I don't really know how to intuitively interpret the relationship/difference between the two. I do know there are supposed to be ways to do it, but, to be honest, I've never found any of these methods to be particularly satisfactory.