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As it turns out to be, this is straightforward. For any
scoring matrix E for which S_{E}(d) is strictly monotonic we can invert
S_{E}(d). Our new score is computed using the inverse, i.e.
d^{*} = S_{E}^{1}(w/n)

(6) 
where w is the actual score obtained from a pair
of aligned sequences
.

(7) 
For n sufficiently large, w/n converges to its expected
value S_{E}(d) and d^{*} converges to d. Since S_{E}(d) is a sum
of exponentials in d, it is generally not possible to invert it
algebraically. The computation of the inverse will be done
numerically.
Let
be the k^{th} moment of the scoring matrix E.
The central moments of w/n are:
E[w/n] = S_{E}(d)

(8) 

(9) 

(10) 
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Chantal Korostensky
19990714