## 4 Experimental Results
### Statistical Analysis  
We applied the Student's \(t\)-test to compare FQS _vs._ FJS, FQS _vs._ HOR and FQS _vs._ QS in 19 varied pattern lengths using randomized files and three practical files. The testing results are shown in \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Dataset**} & \multirow{2}{*}{\(m\)} & \multicolumn{2}{c}{**QS**} & \multicolumn{2}{c}{**FQS**} & \multirow{2}{*}{\(p_{\text{-value}}\)} \\ \cline{3-3} \cline{5-6}  & & **Mean** & **STD** & & **Mean** & **STD** \\ \hline Bible & 10 & 2,509,581 & 349,838 & 2,233,411 & 124,671 & **0.0029** \\ Bible & 100 & 762,316 & 75,952 & 646,298 & 54,656 & **<0.01** \\ Bible & 500 & 436,243 & 69,142 & 366,246 & 52,643 & **0.0010** \\ Bible & 1,000 & 371,849 & 47,702 & 311,520 & 45,886 & **0.0002** \\ _E. coli_ & 10 & 1,595,760 & 345,988 & 1,197,866 & 265,086 & **0.0002** \\ _E. coli_ & 100 & 1,634,972 & 548,912 & 657,987 & 128,279 & **<0.01** \\ _E. coli_ & 500 & 1,563,532 & 435,567 & 541,158 & 75,234 & **<0.01** \\ _E. coli_ & 1,000 & 1,777,232 & 505,260 & 538,972 & 87,332 & **<0.01** \\ World192 & 10 & 314,182 & 44,298 & 307,453 & 30,050 & 0.5777 \\ World192 & 100 & 75,189 & 11,321 & 70,636 & 11,953 & 0.2238 \\ World192 & 500 & 33,607 & 6,834 & 30,483 & 6,272 & 0.1403 \\ World192 & 1,000 & 26,898 & 4,649 & 23,800 & 3,675 & **0.0250** \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The number of symbol comparisons used by QS and FQS. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Dataset**} & \multirow{2}{*}{\(m\)} & \multicolumn{2}{c}{**QS**} & \multicolumn{2}{c}{**FQS**} & \multirow{2}{*}{\(p_{\text{-value}}\)} \\ \cline{3-3} \cline{5-6}  & & **Mean** & **STD** & & **Mean** & **STD** \\ \hline Bible & 10 & 2,091,864 & 88,154 & 1,981,448 & 170,460 & **0.0156** \\ Bible & 100 & 668,353 & 53,648 & 638,690 & 54,324 & 0.0904 \\ Bible & 500 & 395,943 & 45,934 & 361,286 & 52,932 & **0.0332** \\ Bible & 1,000 & 336,126 & 48,988 & 304,023 & 46,194 & **0.0395** \\ _E. coli_ & 10 & 1,173,807 & 274,324 & 1,060,892 & 235,219 & 0.1706 \\ _E. coli_ & 100 & 1,220,728 & 410,989 & 603,276 & 119,927 & **<0.01** \\ _E. coli_ & 500 & 1,167,201 & 326,151 & 497,990 & 69,480 & **<0.01** \\ _E. coli_ & 1,000 & 1,326,734 & 378,141 & 495,055 & 79,672 & **<0.01** \\ World192 & 10 & 285,185 & 29,089 & 265,368 & 29,591 & **0.0392** \\ World192 & 100 & 71,759 & 11,011 & 70,235 & 12,103 & 0.6794 \\ World192 & 500 & 31,265 & 6,038 & 29,869 & 6,252 & 0.4768 \\ World192 & 1,000 & 24,299 & 4,146 & 22,650 & 3,672 & 0.1910 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The number of pattern shifts used by QS and FQS. Figure 13. The \(x\)-axis represents the 19 varied pattern lengths (\(m\)); the \(y\)-axis is arranged accordingly with the alphabet size \(|\Sigma|\) (11 total cases). From the figure, we can find that the statistical testing results match with the previous performances well. In most cases, FQS performed better than FJS, HOR and QS. Figure 13. The Student \(t\)-test for: (**a**) FQS _vs._ FJS (**b**) FQS vs. HOR and (**c**) FQS _vs._ QS in the condition of execution time _versus_ pattern length, \(m\). “B” denotes cases when FQS has a statistically significant better performance over the comparison algorithm; “S” denotes cases where FQS and the comparison algorithm have statistically the same performance; “W” denotes cases where FQS has a statistically significant worse performance than the comparison algorithm. Let us examine Figure 13a in the figure. There are 209 (19 \(\times\) 11) total comparison cases: FQS performed better than FJS in 156 cases with statistically significant differences; they had the same performance in 40 cases; FJS had better performance in 13 cases.