9. Attenuation

 
  


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.
     The proper method to combine Rp with Ro should attenuate the earlier performances in favor of the later ones . . . [E1,  8.25].

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.