comprise lesser amounts of the total variability and lower in subsets that comprise nearly all of the variability in the data set. Variability can be characterized by using principal components analysis (PCA).
Consider, for example, a PCA using the first three elements (As, Sb, and Sn—elements “123”), which yields 104.564 as the total variation in the data. PCA provides the three linear combinations that decompose this variation of 104.564 into three linear combinations of the three elements in a sequential fashion: the first linear combination explains the most variation (76.892); the second, independent of the first, explains the next-most (19.512), and the third accounts for the remainder (8.16). The total variation in all seven elements is 136.944. Thus, this three-element subset accounts for (104.564/136.944) × 100%, or 76.3% of the total variation. The results of PCA on all 35 3-element subsets are shown Table H.1; they illustrate that subset “237” (Sb, Sn, and Cd) appears to be best for characterizing the total variability in the set, accounting for (114.503/136.944) × 100% = 83.6% of the variability. Subset “137” (As, Sn, and Cd) is almost as good at (113.274/136.944) × 100% = 83.0%.
PCA is then applied to all 35 possible four-element subsets; the one that accounts for the most variation, (131.562/136.944) × 100% = 96.1%, is subset “1237” (As, Sb, Sn, and Cd). Among the five-element subsets, subset “12357” (As, Sn, Sb, Cu, and Cd) explains the greatest proportion of the variance: (134.419/136.944) × 100% = 98.2%, or about 2.1% more than the subset without Cu. The five-element subset containing Bi instead of Cu is nearly as efficient: (133.554/136.944) × 100% = 97.5%. Finally, among the six-element subsets, “123457” (all but Ag) comes very close to explaining the variation using all seven elements: (136.411/136.944) × 100% = 99.6%. Measuring all elements except Bi is nearly as efficient, explaining (134.951/136.944) × 100% = 98.5% of total variation. The values obtained for each three-, four-, five-, six-, and seven-element subset PCA are found in Tables H.1, H.3, H.5, H.7, and H.9 below. The corresponding variances in order of increasing percentages are found in Tables H.2, H.4, H.6, and H.8.
This calculation may not directly relate to results obtained by simulating the false match probability as described above, but it does give some indication of the contribution of the different elements, and the results appear to be consistent with the impressions of the scientists who have been measuring bullets and making comparisons (Ref. 1-3).