## 2 Numerical Association Rule Mining and Swarm Intelligence Algorithms
### Swarm Intelligence-Based Algorithms  
Swarm intelligence [35] represents a computational paradigm inspired by the collective behavior of simple entities in nature. These algorithms simulate the complex group behaviors that emerge from local interactions among simple individuals. Typical algorithms include Differential Evolution (DE) [36], Particle Swarm Optimization (PSO) [37], and Bat Algorithms (BA) [38]. The core concept of swarm intelligence algorithms is to guide the collective behavior of numerous simple agents (referred to as \"agents\" or \"particles\") through basic rules to address complex optimization problems.  
Swarm intelligence algorithms focus on the emergent behaviors of multiple interacting agents following simple rules. While each agent may be considered non-intelligent on its own, the collective system can exhibit self-organizing behavior, functioning akin to a form of collective intelligence.  
Typically, swarm intelligence algorithms consist of the following key components:  
* **Agents:** Agents are the basic units of the algorithm, moving within the search space and interacting based on the state of the environment and other agents.
* **Search Space:** This is the solution space explored by the algorithm, within which agents move to find optimal solutions.
* **Variation Operators:** These operators are used to update the state of the agents, which can be rule-based or equation-based.
* **Evaluation Function:** Used to evaluate the fitness of the agents, determining which agents survive to the next iteration.
* **Self-Organization:** Interactions among agents lead to self-organizing behavior, guiding the population toward the optimization objective.  
By simulating the interactions among simple individuals, swarm intelligence algorithms demonstrate powerful capabilities in solving complex problems. Their characteristics of self-organization, parallelism, and adaptability make them effective tools for addressing various optimization challenges. A general pseudocode for swarm intelligence algorithms is provided in Algorithm 1.  
```
1:procedureEnhancedSwarmIntelligence
2: Initialization:
3: Set the search space \(S\) (the area in which agents search for solutions)
4: Initialize the population of agents agents (each agent has an initial position within \(S\))
5: Define the fitness evaluation function \(f(\cdot)\) (assesses the quality of a solution)
6:while termination condition is not met do
7:for each agent agent\({}_{i}\) in agents do
8: Compute the fitness value \(f({\bf position}_{i})\) of the agent agent\({}_{i}\)'s position
9:endfor
10:for each agent agent\({}_{i}\) in agents do
11: Update the agent's position position\({}_{i}\) using variation operators
12: Consider the influence of neighboring agents neighbors\({}_{i}\) when updating position\({}_{i}\)
13:endfor
14: Select the most fit agents based on their fitness values for the next iteration
15: Record the best solution found so far and its corresponding fitness value
16:endwhile
17: Output the best solution and its fitness value
18:endprocedure
```  
**Algorithm 1** General Pseudocode for Swarm Intelligence Algorithms