# Numerical Association Rule Mining from a Defined Schema Using the VMO Algorithm
## 5 Results and Discussion
### Preliminaries for the Experiments  
Population meta-heuristics and all evolutionary algorithms require preliminary experimentation to obtain a good configuration of their parameters. We evaluated the sensitivity of some parameters, such as the size of the population and the number of evaluations. The tests were developed with three real databases from UCI Repositories and five synthetic ones. We prepared five synthetic datasets, which are specified in Table 2; all used the same rule schema: between two and four numerical attributes on the condition side and one class attribute in the conclusion. To configure the experiment, we used the rule schemes defined in Table 3, from which we could generate rules of two to four numerical attributes. In the synthetic datasets, we used a single defined rule scheme of two to four numerical attributes in the LHS and a class in the RHS. We inserted the experimental configuration into the code of the algorithm, so the number of iterations was defined in the vector \(it=\) (100, 200, 500, 1000, 5000, and 10,000). QM_VMO required two parameters in order to set the population size: \(P\) (initial mesh size) \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Data Set** & **No. Records** & **Att. Quant.** & **Att. Qual.** \\ \hline iris & 150 & 4 & 1 \\ abalone & 4177 & 7 & 2 \\ basketball & 96 & 5 & 0 \\ \hline synthetic dataset & & & \\ \hline gen5k & 5000 & 4 & 1 \\ gen10k & 10,000 & 4 & 1 \\ gen20k & 20,000 & 4 & 1 \\ gen50k & 50,000 & 4 & 1 \\ gen100k & 100,000 & 4 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summary of the experimental datasets. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Dataset** & **Side of Rule** & **Rule Template** \\ \hline iris & Antecedent & sepal_length in [\(l_{1}\),\(u_{1}\)] and sepal_width in [\(l_{2}\),\(u_{2}\)] and petal_length in [\(l_{3}\),\(u_{3}\)] and petal_width in [\(l_{4}\),\(u_{4}\)] and class = [\(c_{1}\),\(c_{2}\),\(c_{3}\)] \\ abalone & Antecedent & length in [\(l_{1}\),\(u_{1}\)] and diameter in [\(l_{2}\),\(u_{2}\)] and height in [\(l_{3}\),\(u_{3}\)] and whole_weight in [\(l_{4}\),\(u_{4}\)] and shuked_weight in [\(l_{5}\),\(u_{5}\)] and viscera_weight in [\(l_{6}\),\(u_{6}\)] and shell_weight in [\(l_{7}\),\(u_{7}\)] and rings = [\(c_{1}\),\(c_{2}\),\(c_{3}\)] \\ basketball & Antecedent & all attributes \\  & Consequent & all attributes \\ basketball & Antecedent & all attributes \\  & Consequent & all attributes \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Experimental rule templates. and \(T\) (mesh expansion size); \(P=\) (8, 12, 24, 50 and 100) in all cases, the total expansion size used for this study was \(T=\frac{3}{2}\cdot P\), and the number of closest neighbors was \(k=5\). Table 4 shows the ordering with the fit function achieved by the algorithm in each of the configurations; thus, the population was in the rows. In contrast, the columns show the iterations tested. We proceeded with Equation (21) to obtain the values of column \"\(R\)\"; the results correspond to a statistical measure of variation and are interpreted according to an ascending order (lower indicates better quality). In general, performance can be affected by the number of rows present in the database. To address this, we analyzed the behavior of our algorithm based on the size of the database. To do this, we will followed the same ordering structure as in the previous test, but this time, instead of iterations, we varied the dataset. We set the iterations to six hundred. In Table 5, the minimum \(R\) is located in row 3, which corresponds to a population size of 24, thus supporting our chosen population. \(x_{i}\) denotes the rankings assigned in different iterations, and \(d_{j}\) is the ranking obtained in each dataset. Equation (21) is a utility measure based on the ordering of the evaluation function of the algorithm, where \(x_{i}\) denotes the order assigned for each element of \(X\), and \(STDEV(X_{j})\) is the standard deviation for each evaluated population measure \(X_{j}\).