Table 4 Feature selection metrics.

Average Error

The average error across all optimization iterations, providing a general assessment of the algorithm’s performance.

\(\text {Average Error} = \frac{1}{N} \sum _{i=1}^{N} E_i\)

Average Select Size

The average number of features selected during the optimization process, indicating the algorithm’s ability to reduce data dimensionality.

\(\text {Average Select Size} = \frac{1}{N} \sum _{i=1}^{N} |F_i|\)

Average Fitness

The mean fitness score achieved across all optimization runs, reflecting the overall quality of selected feature subsets.

\(\text {Average Fitness} = \frac{1}{N} \sum _{i=1}^{N} \text {Fitness}(F_i)\)

Best Fitness

The lowest error achieved during the optimization process, representing the optimal subset of selected features.

\(\text {Best Fitness} =\min \left\{ \text {Fitness}(F_1), \text {Fitness}(F_2),..,\text {Fitness}(F_N) \right\}\)

Worst Fitness

The highest error observed during the optimization process, providing a baseline for algorithm reliability.

\(\text {Worst Fitness} = \max \left\{ \text {Fitness}(F_1), \text {Fitness}(F_2),.., \text {Fitness}(F_N) \right\}\)

Standard Deviation of Fitness

The variability in fitness scores across multiple optimization runs, indicating the stability of the feature selection process.

\(SD = \sqrt{\frac{1}{N} \sum _{i=1}^{N} \left( \text {Fitness}(F_i) - \text {Average Fitness}\right) ^2}\)

Feature Stability Analysis

Consistency of selected features across multiple optimization runs, typically measured using the Jaccard index.

\(J(A, B) = \frac{|A \cap B|}{|A \cup B|}\)

Alternatively, the average pairwise Jaccard index:\(\text {Stability} = \frac{2}{N (N-1)} \sum _{i=1}^{N-1} \sum _{j=i+1}^{N} J(F_i, F_j)\)