That Makes it Easy

I just had my mind blown.  In my summer course, Random Processes, I had a homework problem regarding estimation theory and determining 2nd order statistics about linear combinations of jointly Gaussian random variables. Part of the problem is to determine the Covariance of these linear combinations. As worked away I had an incredible realization: Covariance behaves like an inner product!

Check it out, it satisfies the properties of an inner produce on the probability space:
1) It's bilinear
\[
Cov\lbrack\gamma A + \delta B,Z\rbrack = \gamma Cov\lbrack A,Z\rbrack +\delta Cov\lbrack B,Z\rbrack\]
2) It's symmetric
\[
Cov\lbrack A,B\rbrack = Cov\lbrack B,A\rbrack\]
3) It's positive semi-definite (or what I think is better said non-negative definite, but that's for another post)
\[
Var\lbrack A\rbrack =Cov\lbrack A,A\rbrack\ge 0\]
Where $Cov\lbrack A,A\rbrack = 0$ implies that $A$ is a constant random variable.

So it's not exactly an inner product--there isn't the existence of a single zero. Rather, all constant random variables behave like zero. Apparently, this defines a Quotient Space--a vector space with an subspace $N$ that forms an equivalence class with $0$--and Covariance is an inner product over such a space.

Anyway, I looked this up on Wikipedia and, sure enough, there is a little section titled "Relationship to inner products" (which pointed me towards the article on quotient spaces). I guess it's not any huge discovery. I wasn't expecting that. But I am a little peeved that Variance and Covariance were never taught this way, or that this cool perspective was never even mentioned. I feel as though I pushed my understanding and intuition of Covariance way ahead by seeing this little change of face, and it would have been a huge help when taking probability courses in the past.

Grow Your Own

A little over a year ago, as part of a gift for my mother, I flew up to Oregon and installed a hops garden at my parents home in Tumalo.  I knew that they had some spare tipi poles lying around which I could use to create a unique frame for some hop vines.

After picking a good spot on their three acres I set to work, removing a small juniper and more than a few sizable rocks from the area.  The tipi (an 18 footer) went up and the rhizomes went in the ground, complete with a gravity fed soaker hose!  I can't quite remember at this point, but I think there were 16 plants--4 Chinook, 5 Centennial, and 7 Cascade--planted between each pair of poles, with two lengths of twine running up the each triangular face of the tipi.  Last season things went alright.  The plants only grew to about 6 feet and 3 Cascade plants never even broke ground.  I'd chalk it up to planting a little late in an already short growing season.

This season, things are looking much better.  We replaced 3 Cascades that didn't make it and did a little fertilizing.  This year the soaker is running 24 hours a day, which is okay in such sandy soil and the plants seem to love it.  The result?  The biggest hop cone you've ever seen!

Clearly there is more growing to be done.  But the sides are filling in quite nicely and it's creating a neat space inside.  Hopefully this season the entire thing will fill in, but definitely in a couple years, once all the plants really mature, this will be quite the cool spot to grab some shade and enjoy an Arnold Palmer or a cool beer, if you're into that sort of thing.

From a brewing standpoint, this is also quite exciting.  I'll need to teach my parents when and how to harvest, but I think I can look forward to an impressive shipment of fresh hops to play around with this fall.  The only problem is that once the plants get to the top they will just start tangling around and create a big, knotted mess.  While this will work wonders to create shade in the middle of the day it may not be so easy to determine which hops are which during harvest.  The solution?  Say "fuck it" and just brew away!  Chinook, Centennial and Cascade can all be used as late addition hops and work well in APAs and IPAs, so I think blindly adding hops by the fistful in the last 10 minutes of the boil could yield some great, one-time-only big hoppy beers.  Talk about seasonal brewing.