An assessment of criteria of fit in Patterson search
The problem of reliably detecting a known set ofn vectors of weight wi embedded in a heavily overlapped Patterson function Pi is investigated by a Monte-Carlo simulation based on searches of computer-generated random number sequences. Several formulations of the criterion of fit were compared. All were found to improve when the criterion was based on a subset of m “worst fitting” vectors as judged by a low value of (Pi/wi). The best criteria were Σi = 1m (wiPi)/Σi =1m wi2, withm ≈ (0.4−0.5)n, Σi= 1/m wi, withm ≈ 0.3n, and Σi=1m (wiPi) withm ≈ 0.7n. In each case the detectability of the embedded vectors wi increases with increasing Σ(w) in relation to Σ(N), the standard deviation of the overlaid noise. A related simulation of a Patterson search for non-crystallographic symmetry shows that for a given size of the non-crystallographically symmetric region, the detectability increases with the order (2-fold, 6-fold, 12-fold) of the symmetry.
Volume 134, 2022
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