I have computed PC1 and PC2 wts on future contracts derived from cumulative log differences. How can I use them to get back the theoretical price of each contract using those 2 pcs? Thanks in advance.
I found a link and I have to repeat: I don't think that PCA helps you to find a price ... it helps to model the movements of prices but not their values. You get something like a factor model ... this does not directly give you a price ... maybe you also want to have a look at this link where PCA is applied to the oil market.
Yes you can, how depends fully on your required accuracy and also whether PC1 and PC2 are sufficient in explanatory power of the log differences of your futures contract.
Also, make sure you understand the signs of the eigenvalues (sign of the PC) can be different from one experiment to the next as they are arbitrary (the values are obviously not). Here some comments on that which I found when I tried to find supporting documents:
The following describes in a somewhat theoretical way how to PCA-Reverse but they also bring up couple neat examples. You are not presented here with a off-the-shelf R code toolbox that gets you to your results in the next 5 minutes but I think anyone using PCA should actually first understand the related math and stats behind it first. I am sure you will be able to easily reverse the PCA post reading this:
Edit: I was actually so intrigued by the above paper's examples that I currently play with PCA and the PCA-Reverse in regards to image manipulation. Sorry that comment is not quant finance related but just wanted to share my excitement with math that sometimes overcomes me when playing with some interesting stuff.