# A New String Edit Distance and Applications
## 3 Results and Discussion  
An overarching goal of this work was to develop a metric that was inversely proportional to sequence relative frequency in MPS data analysis output. A demonstration of the effect of the RFL distance algorithm using the result from Table 2 is shown in Figure 2. \begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline  & & & & & **Mean** & **Total** & **Unique** \\
**Locus** & **Motif** & **Median** & **Min** & **Max** & **on-LUS** & **Panets** & **Parents** \\
**Back** & **at Locus** & **at Locus** & **at Locus** & **at Locus** & **at Locus** \\
**with** & **at Locus** & **at Locus** & **at Locus** & **at Locus** & **at Locus** \\ \hline CSF1 PO & TCTA & 11 & 4 & 15 & 0.0219 (163) & 1159 & 17 \\ D10S1248 & GGAA & 14 & 8 & 19 & 0.0291 (190) & 1182 & 13 \\ D12S391 & TAGA & 12 & 7 & 19 & 0.0279 (115) & 1287 & 84 \\ D12S391 & CAGA & 6 & 3 & 10 & 0.0085 (67) & 1287 & 84 \\ D13S317 & TATC & 12 & 7 & 16 & 0.0183 (105) & 1219 & 30 \\ D13S317 & AATC & 2 & 2 & 3 & 0.0021 (56) & 803 & 10 \\ D165S39 & GATA & 11 & 5 & 14 & 0.0214 (123) & 1207 & 17 \\ D18S51 & AGAA & 15 & 9 & 24 & 0.0249 (125) & 1242 & 27 \\ D195433 & TCCT & 13 & 7 & 18 & 0.0218 (133) & 1194 & 22 \\ D151656 & TATC & 13 & 9 & 17 & 0.0332 (141) & 1270 & 30 \\ D151656 & AC & 6 & 5 & 6 & 0.0059 (26) & 1270 & 30 \\ D21S11 & TATC & 12 & 7 & 15 & 0.0224 (96) & 1264 & 79 \\ D21S11 & TGTC & 6 & 4 & 8 & 0.0021 (21) & 1264 & 79 \\ D22S1045 & ATT & 12 & 5 & 16 & 0.0316 (233) & 1179 & 13 \\ D2S1338 & GGAA & 13 & 8 & 17 & 0.0221 (80) & 1269 & 69 \\ D251338 & GGCA & 7 & 3 & 9 & 0.0033 (34) & 1269 & 69 \\ D25441 & CTAT & 11 & 7 & 14 & 0.0188 (114) & 1205 & 24 \\ D351358 & CTAT & 13 & 5 & 17 & 0.0309 (143) & 1238 & 26 \\ D351358 & CTGT & 2 & 2 & 4 & 0.0035 (85) & 987 & 19 \\ D55818 & ATCT & 12 & 7 & 15 & 0.0257 (119) & 1237 & 31 \\ D78820 & CTAT & 10 & 6 & 13 & 0.0163 (96) & 1229 & 30 \\ D81179 & CTAT & 12 & 8 & 15 & 0.0251 (115) & 1237 & 34 \\ D881179 & CTGT & 2 & 2 & 3 & 0.0013 (51) & 20 & 6 \\ FGA & GAAA & 14 & 9 & 19 & 0.0211 (109) & 1247 & 32 \\ Penta D & GAAAA & 11 & 5 & 17 & 0.0051 (47) & 1236 & 18 \\ Penta E & TTTIC & 12 & 5 & 25 & 0.0114 (89) & 1240 & 25 \\ TH01 & AATG & 7 & 5 & 11 & 0.008 (63) & 1165 & 12 \\ TPOX & AATG & 9 & 5 & 13 & 0.0113 (90) & 1140 & 14 \\ VWA & ATAG & 12 & 3 & 16 & 0.0247 (146) & 1242 & 32 \\ VWA & ACAG & 4 & 3 & 6 & 0.0023 (50) & 1242 & 32 \\ VWA & GATG & 3 & 3 & 4 & 0.0019 (8) & 81 & 2 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: High-stutter motifs at CODIS-20 and PENTA loci, based on 661 individuals, using a threshold of 0.167 Allele Coverage Ratio (ACR)—the ratio of the second-highest count to the highest—to determine homo– vs. heterozygosity in identifying true alleles. For each motif, LUS summary statistics are taken across all alleles. Back stutter rate is the average of all single-stutter proportions resulted from amplification and sequencing, averaged across all loci where parents with that motif were present. The last two columns describe how many alleles and unique alleles with the relevant motif at each locus were present in the data. This analysis is described further in Section 3.1. Figure 2: Proportion by distance plots at a CSF locus with unit edit costs. The RFL distance in (**b**) fixes the non-monotonicity of the classical Levenshtein in (**a**). The non-monotonic plot in Figure 2 has a peak in frequency at a distance of 4 from the true parent sequence in the file, corresponding to stutter of an entire tetranucleotide motif being a more common occurrence than deleting 3 or 5 characters individually. The RFL distance counts this stutter with a cost of 1, thus giving monotonicity. In loci with motifs of other lengths, the frequency peak appears at that particular length. In loci exhibiting multiple motifs, there are increases in frequency at both lengths. There are smaller spikes at a distance of twice the motif length (a distance of 8 in Figure 2), corresponding to double stutter, but they are much less noticeable.