![]() |
8. Attenuation
|
|
|
|
We have seen that attenuation in the Elo
System is achieved by means of an arbitrary constant No,
which may be viewed as a limit for sample size in recursive averaging.
Attenuation can be achieved directly though the application of an
attenuation factor f, such that 0 < f < 1. A
formula that captures the recursive nature of attenuated weighted
sampling for sequential ratings would then be where [8.2] |Ni
= f |Ni-1 + Ni Similar to Elo's notation, Ri is a performance rating based on Ni games. Recursive values are preceded by a vertical line. The constant No is replaced by the variable |Ni, and the new rating Rn by |Ri. The process begins with the original performance rating, which is to say, [8.3] |R0 = R0 and [8.4] |N0 = N0 . At each stage in the rating process, the sampling weight is decreased with multiplication by f and increased with addition of Ni. Thus, [8.5] |N1 = f N0 + N1 , [8.6] |N2 = f 2N0 + f N1 + N2 , and so forth. This is a geometric series. Assuming Ni to be a constant N, then |Ni converges to the value [8.7] lim = N / (1 - f). The process is self-correcting. For example, if we start with some value other than N but otherwise maintain Ni at a constant N, the series still converges to the indicated value. Interestingly, if we set f to 1-N/No, the weight of the old rating in Elo's blending process [7.4] , we find the limit to be No. From [8.1] and [8.2] it follows that [8.8] DR = |Ri - |Ri-1 = (Ri - |Ri-1) (Ni / |Ni ) , which is essentially a more subscripted version of [7.6]. The one difference, aside from differences in notation, is the substitution of the variable |Ni for the constant No. From this we see that an arbitrary constant like No is not necessary for attenuation. Formula [8.8] could actually be applied from the outset of the rating process, obviating the distinction between performance and established ratings. A drawback of [8.8] is that the variable |Ni must be stored along with the current rating. If an arbitrary constant is preferred, a better candidate than No would be the next to last value in the recursive sequence, which may be designated Nk. Reverting to Elo's notation, we have [8.9] Rn = [RoNk + RN] / (Nk + N) . The advantage over Elo's blending process is that the ratio of the original sample weight to the new sample weight, Nk / N no longer depends on the difference between sampling weight and sample weight, as in the ratio (No - N) / N . In the latter, N cannot exceed No. |