Sunday, November 14, 2004

Sussing out Markov Chains

But first, the Club Champs report

4:45am was way too early. Had to have the 2 hour pre/post-midday nap to compensate. But I did get to experience the delights of Eastern Creek International Raceway Be There at approx. 7:00am for our Club Champs. It was only 51 km (13 laps) so it wasn't really long enough to extend beyond a crit. I remember the days when it used to be 100 km around Bargo. Now that was a good course.

Of eight A grade starters I managed a fairly modest fourth, mainly because I'd done several do or die attacks in the last half in an attempt to get away. Not today - the Others were plenty strong enough to chase me down, although the last attack with 2 km to go caused a few problems. Josh Marden was chiefly responsible for closing the gap and then he won the sprint so I can't complain about that! It's the Anthony Rappo Memorial out there next week and that'll be a bit more fun with teammates to work with and stuff.

We absolutely flew coming back, averaging 40+ along the freeway with a bit of a tailwind until we stopped in Leichhardt for een koffie of twee. No beer though - it was only 9:30am when we got back!

Sleep followed, then looking at houses that I can't afford in the Western Slurbs Courier, food shopping, i.e. very little of interest apart from an ice cream...mmmm...and then (ik denk) Sag Gosht.

The Axis of Weevil problem revisited

I knew I had forgotten something in the general coarse of the week (it has been very coarse). Dad and I discussed remodelling of the Axis of Weevil problem in our kitchen, which may involve remodelling our kitchen. I note that it could get complicated by the Axis of White Ants that has eaten a goodly portion of our back door frame.

The competing Axes may bring each other down in a cloud of wood dust and quick oats but the only way to determine this is by an accurate mathematical model. Such a model would have to know how to differentiate an inverse exponential function on the back of a bus ticket and also look suitably alluring in the kitchen. That's what models are for.

Before we get onto the combined model, it's first necessary to sort out the Axis of Weevil problem in a multi-step process. Step 1 involves throwing away the partial differential equations that were alluded to in an earlier blog. They are wrong, wrong, wrong, Jana. Wrong. PDE's have very little to do with Markov Chains, which can be used to model birth-death-catastrophe processes. In this case the birth refers to the birth of the weevils; death refers to when we kill them or they die of natural causes like being eaten by Lucy's deceased guinea pig Smiley (R.I.P.); and a catastrophe is when they take over the kitchen and eat everything, including the awful muesli that mum likes.

A Markov Chain requires using matrices and integration and stuff. It doesn't necessarily have to converge to a steady state of constant weevil population, although it could, if asked nicely. The input values have to be made up out of thin air. Something like 5 and 7, for the number of moths we kill and that are born from an egg on a mountaintop each day. We can plug this into our equation, which is left as an exercise for the reader to suss out, and that will give us the weevil population on day 2.

Of course this becomes rather tedious after a few days, as you have to put in different death and birth values. Instead, what we can do is make up some arbitrary matrix to describe how many weevils we will kill on average and how many are born, again on average. They key word is on average, except that it's more than one word. The solution to the equation can be found with the Viterbi algorithm, which again is left as an exercise for the reader. I won't bore you with such trivial details.

In layman's terms, the weevil moth population will vary a bit from day to day, depending on some different stuff.

The tricky part is when we combine this problem with the Axis of White Ants. We can rename this combined system the Axis of the Willing because the weevils are willing to take over the kitchen and the white ants are willing to devour our house from the top down. The combined problem becomes - in technical mathematical terms - a great big bloody mess that you wouldn't even want to think about solving unless you have access to NASA's superduperparallelwithonelattheendcomputer that they used to send Voyager to Uranus. I've never been to Uranus, even though our physics teacher always used to refer to perturbations in its orbit.

In other words, a little knowledge is a dangerous thing.

No comments: