## 6 Analysis of the Algorithm
### Experiments  
There are many other tools for detecting and analyzing the motifs, like the EMmethod [23], MEME [24], Gibbs [25], Compo [26], MochiView [27], PhyME [28], HeliCis [29] and WebMotif [30], among others. Each of them has their advantages and disadvantages.  
Table 1 shows the results on simulated datasets. From the table, we could find that the results of our algorithm for finding the motif on simulated datasets are satisfied. Our algorithm could find all the motifs from each sequence and get the consensus with an accuracy rate of 100%. If the dataset has a high mutation rate, we could increase the number of repetitions, so that the result on the datasets will be more accurate. We also recorded the total time cost for each test, which mainly depends on the number of test repetitions. In the experimental tables, the parameter, \(N\), represents the number of sequences, parameter \(M\) represents the length of motifs, parameter \(M\) is the maximum length of sequences and \(R\) is the number of iterations. GCR1 is a famous DNA binding protein, whose ability to bind DNA is dependent on the CTTCC sequence motif [31]. Several other popular data sets are also used in the experiments of our motif detection and its comparisons to the other methods.  
\begin{table}
\begin{tabular}{l c c c c c c}  & **N** & **M** & **L** & **R** & **Accuracy Rate** & **Time cost(s)** \\ Set 1 & 20 & 600 & 15 & 60 & 100 & 98 \\ Set 2 & 15 & 600 & 15 & 10 & 100 & 18 \\ Set 3 & 20 & 600 & 12 & 15 & 100 & 22 \\ Set 4 & 20 & 500 & 15 & 40 & 100 & 79 \\ \end{tabular}
\end{table}
Table 1: Results on simulated data.  
In the second experiment, we tested our algorithm on a real sequence set, which was obtained from SCPD. SCPD contains a large number of gene data and transcription factors of yeast. Sequences in the same set are all regulated by a common motif. We chose 1,000 bp as the length of the input sequences. In order to show the advantages of our algorithm, we also compared the result of our algorithm with the results of several other existing motif finding methods on the same dataset, such as Gibbs, MEME, Info-Gibbs and Consensus. Table 2 shows the details of the data set we used in the experiment.  
Table 3 shows the results of the five algorithms on biological data sets. Table 4 and Figure 2 give the average mismatch numbers of each algorithm. We choose four well-known motif-detecting softwares to make the comparisons. From Table 3 and 4, we see that the average mismatch numbers of our algorithm on data sets GCN4 and GCR1 are significantly lower than other four well-known methods. On the data sets Bas1,Rap1 Ebf1 and HSE-HTSF, our algorithms also shows satisfied performance compared to other methods.  
In addition, our algorithm also shows its high speed in computations compared to other four motif finding methods. Because the starting pattern of algorithms are represented by a string, so our algorithm can avoid some extra time consuming computations unlike Gibbs sampling and EM methods, such as computations of likelihoods. According to this feature, we use the consensus string of the voting operation obtained from the last iteration as a new starting pattern in program, and continue doing voting operations repeatedly until there is no further improvement. Experimental results show that if we set the number of iterations to be a large integer, the programs could give more accurate results within a reasonable time.  
\begin{table}
\begin{tabular}{c c c c c c}  & **Bas1** & **GCN4** & **GCR1** & **Rap1 Ebf1** & **HSE-HTSF** \\ N & 6 & 9 & 6 & 15 & 5 \\ L & 10 & 10 & 10 & 15 & 10 \\ \end{tabular}
\end{table}
Table 2: Number of Sequences and Motif Length.  
\begin{table}
\begin{tabular}{c c c c c c}  & **Bas1** & **GCN4** & **GCR1** & **Rap1 Ebf1** & **HSE-HTSF** \\ Our Algorithm & 10 & 8 & 4 & 45 & 5 \\ Gibbs & 8 & 51 & 5 & 202 & 7 \\ MEME & 8 & 15 & 10 & 32 & 3 \\ InfoGibbs & 9 & 21 & 5 & 46 & 9 \\ Consensus & 8 & 9 & 5 & 42 & 7 \\ \end{tabular}
\end{table}
Table 3: Total number of mismatch positions.