# Multi-Objective Optimization for High-Dimensional Maximal Frequent Itemset Mining
## 4 Case Studies
### Accident Transaction Set Test  
As stated previously, the accident transaction set covered 340,183 transactions and 468 dimensions. The transaction length of the accident transaction set has uniform distributions, a high quantity of transactions and high dimensions. It is representative and can be used as the testing objects. The proposed algorithm cannot be compared with existing traditional algorithms in optimization, which targets the maximal frequent itemset because of two reasons. First, traditional algorithms are generally unworkable with high-dimensional big data. Second, existing algorithms have hardly used a maximal frequent itemset as the searching objective. Thus, the number of frequent itemsets that were identified by the algorithm in each iteration under the constraint of appropriate _support_ thresholds, as well as the proportion of maximal frequent itemsets, were only investigated in this study. On this basis, the feasibility of the proposed algorithm was analyzed. The CPU time for 100 generations of algorithm operation was 23,931 s (more than 6 h). Various results of algorithm operation are shown in Figure 2. The \(x\) axis in Figure 1(a) shows the number list of 468 items, and the \(y\) axis is the _support_ of the corresponding item used as the 1-itemset. Clearly, the probability distribution of the occurrence of a single item in the transaction set was relatively uniform, and the maximum _support_ was even close to 1. This finding indicates that this item occurs in almost every transaction, and it is the major cause of accidents if used in traffic accident association analysis. For the purpose of the experiment, approximately 5% of the items were intercepted as the frequent 1-itemset. Therefore, it was relatively appropriate to set the _support_ threshold as 0.5, which can intercept 24 frequent 1-itemsets. All non-frequent 1-itemsets were deleted when generating the reduced transaction sets, and the average length (number of items) of all objects was decreased from the original 33.8 to 12.3. The number of transactions in the reduced transaction set was not decreased because the _support_ of the highest single item in the target transaction set was almost close to 1. If the algorithm was applied to other transaction sets (e.g., the supermarket data in the following text), then the quantity of transactions significantly decreased, which reduced the complexity of the algorithm in the assessment of each evolutionary individual. All frequent 1-itemsets were combined in pairwise, and a total of 24 \(\times\) 23/2 = 276, 2-itemsets were generated. The histogram of _support_ is shown in Figure 2b. The number of frequent itemsets with _support_ higher than the threshold was 168. The algorithm randomly increased items based on these 168 frequent 2-itemsets to generate the initial population for evolution. The population scale was also set as equal to the number of frequent 2-itemsets, which was 168. The number of frequent itemsets that are searched in each generation of the evolutionary process is shown in Figure 2c. Among them, the maximal and non-maximal frequent itemsets are available. The solution space is very large (2468), and the number of all frequent itemsets cannot be verified because this dataset is high dimensional. In this experiment, the evolution was performed for 100 generations to show the feasibility. Clearly, the number of frequent itemsets searched in each generation generally decreased with the increase in evolution generations. The number of all frequent itemsets that was obtained with the increase in evolution generations is shown in Figure 2d. The curves in Figure 2d are integrals of the function of Figure 2c at zero. After 100 iterations, at least approximately 500 frequent itemsets--including 168 maximal frequent itemsets--were gained. In fact, 216 maximal frequent itemsets exist when the threshold is set to 0.5. In other words, 77% of maximal frequent itemsets were obtained through 100 generations of evolutions.