Sports and Fairness

Today FIFA announced a decision to prohibit World Cup matches beyond altitudes of 2500 meters above sea level. The reason, ostensibly, is that the advantage offered to teams who regularly play at such altitudes is unfair.

Makes me wonder why FIFA doesn't ban nations with more than 50 million citizens from entering. With their larger pools of potential football players from which to draw their teams, these nations gain an unfair advantage, right?

Alternatively, FIFA could mandate that national teams are drawn at random. Just an idea.

Determining the Transition Matrix

Back when I was introduced to Markov chains in Robert Messer's Linear Algebra, the prime real-world examples of what Markov chains might model were interacting populations in a biosphere. Oddly though, there was no mention of how one might determine the transition matrix based on observations of actual populations. Nor can I seem to find a simple explanation of this online. So here is my own calculation.

We wish to determine the Markov Chain transition matrix based on an observed set of transitions where the dynamics are unknown. The update function is
where p is the probability distribution, q is the old probability distribution, and M is the transition matrix. In order to find the optimal transition matrix, we minimize the variance
where d is the dimensionality and E is the size of the statistical ensemble over time and space to which p and q belong. The index of the ensemble e is suppressed. We find the differential
Setting the differential to zero we get
which, assuming M_aj is known for all j ≠ b gives the solution
Our strategy then is to iteratively increase the size of the ensemble of observations and compute the transition matrix for each iteration. For j ≠ b we use the value M_aj from the previous iteration. For the initial matrix there are many possible choices. We will use

To be continued...

I Wish I Could do This

Link.

I've tried all the Melbourne Shuffle instructional videos out there, but I just can't quite get the hang of it. What I need is an instructional video by Leroy of Stampa med Leroy fame. He has a nice, relaxed style and takes things slow so even a layman like me can keep up.

Names for Cars

The automobile industry is obviously in trouble and needs my help. Nissan Xterra? Subaru Impreza? Need I go on?

Here are my suggestions:

Cherenkov
Doppler
Landau
Moebius
Sobolev
Tycho
Camus
Gödel
Dante
Dyson
Fourier
Napier
Schafer

Someone should pay me money for this.