The home stretch - having established that we want to experiment with some sort of heaviside function, we take a moment to check that they have the right properties, then do some fitting.
One restriction we had on F is that it be an odd function; F(-x) = -F(x). It isn't completely obvious to me that the heaviside functions would have this property, I take a moment to demonstrate this.
We've established that F(x) = (2H(x) - 1) / ( 2H(1) - 1 ). The denominator is a constant for any fixed function H, so we can ignore that and look at the numerator alone - is it odd when we substitute?
2H(x) - 1
= 2 ( 1/2 erfc( -Nx ) ) - 1
= erfc( -Nx ) - 1
= 2 - erfc( Nx ) - 1
= 1 - erfc( Nx)
= 1 - 2 ( 1/2 erfc( Nx ) )
= -1 ( 2( 1/2 erfc(Nx) - 1 )
= -1 ( 2H(-x) - 1 )2H(x) - 1
= 2 ( 1 / ( 1 + exp(-Nx))) - 1
= ( 2 / ( 1 + exp(-Nx))) - 1
= ( 2 - ( 1 + exp(-Nx))) / ( 1 + exp(-Nx))
= ( 1 - exp(-Nx) ) / ( 1 + exp(-Nx) )
= (exp(Nx) (1 - exp(-Nx)) ) / ( exp(Nx)(1 + exp(-Nx)))
= (exp(Nx) - 1 ) / ( exp(Nx) + 1 )
= -1 ( 1 - exp(Nx) ) / ( 1 + exp(Nx) )
= -1 ( 2 ( 1 / (1 + exp(Nx))) - 1 )
= -1 ( 2H(-x) - 1 )
So I now have to different flavors of F to consider. The game at this point is to try different values of N to see how well each fits. If we get lucky, one or the other of these curves will be a close matched with our calculated datapoints.
Of course, we have to find N, and I tried two different approaches. In one approach, I simply let excel figure out what N best fit all of my datapoints. In the other, I let excel best fit my first datapoint - equivalent to trying to pick N using the first derivative.
In each case, the equations were very closely matched to the data. The erfc based function in both cases is closer than the best fit exp variation, which in turn is better than using the derivative to approximate the exp coefficient. The best fit curves come out of the gate a bit too quickly, then cross and fall behind the true curve. The best fit variations give very good matches in the early part of the range (up to an .800 winning percentage at 9.7 R/G) but the maximum error is ultimately larger.
September 11, 2004 7:09 PM
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