Dick Tuthill wrote...
 
<snip>
 
> Two points: the "jumpy" phenomenon described above is
> completely analogous to what happens with an insular schedule
> such as the MAAC's. The paucity of connecting data make this
> method, like the others, inapplicable to judging the MAAC
> teams against the establishment. Thus I do not think it can
> be used with any more reliability or credibility than any of
> the others unless these "erratic" effects are thoroughly
> studied.
 
Actually, with some (considerable?) more effort, the KRACH could estimate
not only the mean value of each team's rating, but also the relative
uncertainty in those estimates -- a so-called "likelihood interval." Actually,
one could do it for an individual team without too much difficulty if one
assumed all its opponents ratings were fixed at their estimated values,
then treat that individual team's rating as a variable. I'd try messing
with that except, because USCHO is down I'm not able access anybody's
complete record for the season. When USCHO recovers I might try playing with
that. My guess is that the uncertainties in the MAAC teams' ratings are
probably too small to consider promoting any of them from the bottom 12 to the
top 12. But at the moment that's sheer speculation on my part.
 
> The second point is that I recall Ken sending out an
> early ranking a few years ago which he may have modified to
> take out or smooth these effects. (I will go into the
> archives to research this point when I get a chance.) That
> may have been where I remember the initial weighting factors
> from. Believe me, I may have recollected inaccurately, but I
> wasn't making that up.
 
I believe what you're thinking of is that he introduces into each team's
record a tie against a hypothetical perfectly average team. He does this
because the logistic analysis introduces problems for undefeated teams and
for winless teams. It would like to assign absolute zero ratings to
winless teams and infinite values to undefeated teams. That's OK in principle,
but it wreaks havoc on the computer. Ken's hypothetical tie against a
perfectly average team avoids these computer overflows, but otherwise has very
little affect on different teams' relative ratings. Especially by the end
of the season when every team has played a lot of real games. In any case,
this minor fudge is totally unbiased.
 
> A final question about KRACH: since the mathematics are not
> given, and since we know (only because Ken tells us as it is
> not on the top of our heads) that the problem is non-linear,
> do we know that the solution is single valued? Or is it
> possible that the error residual has multiple local minima
> which can be converged upon? And if we know that there is
> only a single solution, then how do we know that? From first
> principles, or from experience only? These are questions that
> all need to be answered before any official weight is given
> to another system in something as important as NCAA bids. The
> "erratic" effects to me are the most troubling.
 
Not a problem. They're nonlinear, but perfectly well behaved. Here's the
equations you need to solve:
 
Sum-over-team-i's-games [ ri/(ri+ropp) ] = Wi
 
where ri is team i's rating, ropp is the rating of each of team i's
opponents, and Wi is the number of team i's wins. Ken's fudge factor is to give
each team 1/2 a win against a team whose rating (ropp for that game) is
defined to be 100, say.
 
> Lastly, no apology necessary, Bob.
 
Tell that to Wayne. He's got me on his persona-non-grata,
gotta-screen-their-posts list.
 
Bob "HOCKEY-L Leper" Stagat
 
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