Analyze Video is always run when Media Center imports a video file, or even looks at it, the option in the context menu is only offered to re-run it because we extended the data it gathers back in MC30 I think it was.
As of now, you should never see a video file in MC itself that does not have this data, unless it has been in the database for a while already. There is no way to even turn it off, as its just the basic file information gathering that takes under a second per file.
We don't directly track if a file has been analyzed, because its just going to happen to all files. If you want to know if a file needs re-analysis, because its been in the database for a while, you can check one of the new fields, like [Video Codec]
The only optional feature here is detecting black bars, and if you want to check if that has run, there is the [Video Black Bar Analysis Report], which you can check for emptiness, or also check what things it detected.