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9. Attenuation
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Attenuation is a byproduct of Elo’s
handling of cumulative averaging: There are several
combination processes. One
might simply average the results from N
and No, obtaining a new average rating for the entire sample
No + N, but this preserves fully the rating contribution of
the earlier samples and produces,
if real changes in ability have occurred, a false statement of the
current rating. Formula [8.4], which uses the entire sample No + N, does
not fully preserve the contribution of earlier samples because No
is maintained as a constant from one event to the next, but it is more
accurate as an average than Elo’s blending process.
The sample weight of the original rating in the blending process,
(No - N) /
No , clearly depends on the size of the new sample, which must never
exceed the size of the
original. The process does produce attenuated samples, as depicted
in the graph of the
previous section. In the Elo System attenuation does not occur until the transition to
established ratings, but there is no logical reason why it should not
occur from the outset of the rating process. Attenuation can be
achieved directly by an attenuating factor a,
such that 0 < a
< 1, which is applied to a sequence of change formulas for cumulative
averaging. It will be
helpful first to have an accurate description of cumulative averaging.
The process requires that separate records
be maintained for the average itself, in
this case the cumulative rating, and for the sample on which it is
based, which is also cumulative. This
gives the two formulas [9.1]
Ri+1 = (RiNi + RN) / Ni+1 and [9.2]
Ni+1 =
Ni + N . We use a simple notation in which subscripted variables refer to
cumulative values, while unsubscripted variables refer to
incremental values, i.e., a new performance rating and its sample size.
The next step in the cumulative process would be [9.3]
Ri+2 = (Ri+1Ni+1 +
RN) / Ni+2 and [9.4]
Ni+2 =
Ni+1 + N . The unsubscripted values, R and N, have most likely
changed since they refer to a new performance.
The whole process begins with a single performance rating,
as [9.5]
R0 =
R , and [9.6]
N0 =
N . Note that cumulative results are no different from those that would
be obtained by ordinary averaging, given the same data.
The difference is that in cumulative averaging we do not have to
write out the the results of all the previous steps.
Now we apply the factor a
to obtain attenuated ratings. [9.7] Ri+1 = (aRiNi + RN) / Ni+1 and [9.8]
Ni+1 =
aNi + N for i =
0, 1, 2, 3, . . . . Though difficult to describe succinctly, the method is
clearly superior
to those that rely on a sampling constant.
It is based on true cumulative averaging and can be applied from
the outset of the averaging process.
Its primary disadvantage, apart from the fact that it is still a
sequential process, is that a record of sample size must be
maintained in addition to the
rating. As an interesting sidelight, if we write the expansions for a series
of attenuated sample sizes, [9.9]
No =
N ,
N1 =
aNo + N ,
N2
= a2No
+ aN1 + N ,
N3
= a3No
+ a2N1
+ aN2 + N , and so forth, we get a geometric series.
Assuming N to be
constant, [9.10] lim Ni = N / (1 - a) as i goes to infinity.
Now if [9.11]
a =
(No – N) / No , the weight of the old sample in Elo’s blending process, then the limit is the sampling constant No. Furthermore, if [9.12] a = No / ( No + N) , the weight of the old sample in formula [8.2], then the limit is No + N. |