By first applying some thought to the boundary terms, we can greatly narrow down the space of functions we need to consider.
A team that allows runs as frequently as it scores them should be expected to finish right around .500. A team the scores RS and allows RA should have the opposite record to that of a team which scores RA and allows RS. A team that shuts out its opponents should win every game.
We can simplify the algebra by rewriting these ideas referenced to the average. Rather than looking for winning percentage, instead define W = Wpct - .500. Thus we can write
W = f(RS,RA)
where f is the elusive predictor, and its properties taken from above are
f(RS,RA) = - f(RA,RS) f(RS,RS) = 0 f(RS,0) = .5
The second equation follows from the first, as a special case, but offers a big hint on how to proceed. We need to rotate from RS, RA to "natural units", where one dimension runs along that axis where W = 0. A simple rotation (technically a rotation and a reflection, to keep the signs consistent with the natural language) by 45 degrees should do it. The factor 2^-0.5 comes from the trigonometry of that rotation.
X = (RS+RA)*2^-0.5 Y = (RS-RA)*2^-0.5
So X is simply an expression for the run scoring environment, with a scaling factor. Y is the run differential with a similar factor. We can now think about fixing X, and studying how W varies with Y.
Observe that X is unchanged when RS and RA are reversed, but Y changes sign. We also need W to change sign[ W(X,a) = - W(X,-a) ] with Y, and can deduce that W is an odd function of Y.
In other words, when we consider a Taylor expansion of W, we need only consider odd powers of Y - the even coefficients are known to be zero.
June 25, 2004 11:00 PM
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