# Multi-Objective Optimization for High-Dimensional Maximal Frequent Itemset Mining
## 4 Case Studies
### Supermarket Data Test  
The transaction set of a business supermarket contained 65,435 marketing records, which were equivalent to 18,548 transactions. One transaction corresponds to the list of items that the same consumer purchased. The dimension of commodities (5547) is the number of categories of all commodities. Compared with the accident transaction sets, the dimension of commodities was higher, and the feasibility searching of high-dimensional data was an objective proposed by the algorithm. The test results are introduced in the following texts.  
Support of the 1-itemset of the supermarket data is shown in Figure 3 (Support1D), which is generally low and relatively sparse. If the support threshold is 0.001, then 360 frequent 1-itemsets, which account for 6.49% of the dimension of commodities (total number) could be intercepted. The support distribution of these frequent 1-itemsets is  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **100** & **200** & **300** & **400** & **500** \\ \hline
10\% & 13.916395 s & 26.744861 s & 25.966303 s & 33.145599 s & 46.008133 s \\ \hline
20\% & 488.286400 s & 771.563331 s & 872.430373 s & 919.101463 s & 979.192209 s \\ \hline
30\% & 8292.354528 s & \textgreater{}2.3 h & \textgreater{}2.3 h & \textgreater{}2.3 h & \textgreater{}2.3 h \\ \hline \hline \end{tabular}
\end{table}
Table 1: FP-growth for the extracted accidents.  
Table 2 shows that the volume of the transaction set is decreased by deleting the infrequent 1-itemset. The number of transactions decreased from 18,548 to 16,701, the average transaction length decreased from 3.53 to 2.92, and the maximum transaction length decreased from 49 to 25. In addition, the valid frequent 1-itemset decreased from 5547 to 360. In other words, the dimension of the solution space decreased from 5547 to 360, and the searching space significantly decreased. In the 100 s of the algorithm operation time, more than 9 s was consumed for preprocessing, in which the transaction set had to be visited twice. The first visit gained the overall volume of the transaction set, including the number of transactions and support of each dimension. The second visit aimed to reduce the transaction set, including the reduction in the number and length of transactions. A total of 1010 maximal frequent itemsets with support higher than the threshold were searched after the operation of the algorithm. The supports of these maximal frequent itemsets would be lower than the threshold by adding any item. Most of these itemsets are 1-itemsets, and the rest is a little of the 2-itemsets, 3-itemsets and 4-itemsets. Association rules can be directly and intuitively reflected from these itemsets because the number of items in the itemset is small. Some interesting frequent itemsets are listed in Table 3, including the top 1, 2, and 3-itemsets.  
Additionally, some interesting 2-itemsets have relatively high supports which are not listed in Table 3. These 2-itemsets are daikon-ginger itemsets, bamboo shoots-spare ribs itemsets, tomato-egg itemsets, spring onions-tofu itemsets, and carrots-lettuce itemsets. These commodity combinations are frequently purchased together by consumers, thereby reflecting the dietary culture of China and the Fujian cuisine culture of southeast China.