It's a pretty blurry thing, I believe. You are already 'analyzing the blockchain' when you're following some coins you sent to new recipients. You could also count the number of addresses that have more than 1 BTC in them. But you could also go and use sophisticated algorithms in order to find out certain distributions or determine to what probability some coins are connected to a certain address.
Neat, this seems like kinda of a decryption tactic, where you analyze letter frequency to break the encryption. Is this something similar in the block chain, where you analyze transaction frequencies to find irregularities?