## 5 Experimental Results
This notion of quorum can generate several spurious patterns and, thus, will make the filtering problem more challenging. We also investigate a different setup in which the quorum value reflects the number of sequences where the pattern is contained. This latter scenario produces similar results; however, the range of values for this quorum is bounded by the number of sequences, and this can limit the possibility to study the efficiency of the filtering process, while varying the quorum values. For these reasons, we choose to use the first definition of quorum. Then, we use the extracted patterns as input to our algorithm, presented in Section 4, in order to compute the underlying patterns \(\mathcal{U}\). In our experiments, each pattern, \(m\in\mathcal{M}\), extracted by Varun is with character classes and no gaps, where we set the quorum, \(q\), for the underlying patterns, to be the same as the quorum for the original patterns in the input, as by the definition seen above. The length, \(n\), of the string in the input is equal to the length of the concatenations reported above. The size of \(\mathcal{M}\) varies with respect to the quorum, \(q\), and for some configurations, can be larger than the sequence length (see Figure 1). All experiments were conducted on a common PC, and the filtering process requires on average less than 10 s.  
For both sets of patterns, \(\mathcal{M}\) and \(\mathcal{U}\), we compute some global statistics. Figure 1 shows the number of patterns, the sum of lengths of all patterns and their mean \(z\)-score for the first two protein families (\(Ni\), \(Fa\)). The other families share the same behavior (figure not shown). The \(z\)-score is computed employing the same formula reported in [12]. As expected, the number of underlying patterns is always much smaller than the number of the original patterns. A similar conclusion can be drawn for the  
Figure 1: Total number, sum of lengths and mean \(z\)-score of the patterns extracted using Varun and their corresponding underlying patterns, for the two protein families, \(Ni\) and \(Fa\). The dashed line in the total length diagrams indicates the total size of each family. Note that in (**a**) Mean \(Z\)-Score and (**b**) All diagrams, the ordinate is plotted on a logarithmic scale. (**a**) Nickel-Dependent hydrogenases (\(Ni\)); (**b**) Coagulation factors 5/8 type C domain (\(Fa\)). sum of lengths. More importantly, for small quorums, the total length of original patterns exceeds the length of sequences in both families, indicating that the original patterns are even larger than the set of sequences under examination. Moreover, as seen above, the sum of lengths of underlying patterns is always bounded by the length of sequences. These first two measures indicate that, not only the number, but also the total length of underlying patterns are much smaller than those of the original patterns. Therefore, the filtering process is space-efficient, as expected. Another important measure is the mean \(z\)-score of \(\mathcal{M}\) and \(\mathcal{U}\). The \(z\)-score of a pattern, \(p\), is a statistical measurement of the degree of over-representation of \(p\) with respect to the expected number of its occurrences. The mean \(z\)-score is thus a global measure able to capture the average quality of patterns in a set; however, it is much more computationally expensive than our pattern priority, and we just use it to validate our approach. In Figure 1, we can see that, for all quorums, the average \(z\)-score of underlying patterns is always greater than those of original patterns, and in most cases, the difference is one or two orders of magnitude. To summarize, this first test confirms that the number and span of underlying patterns is much more manageable than the original set and, also, that their average quality is improved. Once we have verified that the notion of underlying is a suitable filter, in a second series of experiments, we test the ability to retain meaningful patterns.