Bioinspired evolutionary metaheuristic based on COVID spread for discovering numerical association rules

Bioinspired evolutionary metaheuristic based on COVID spread for discovering numerical association rules. C. Herruzo-Lodeiro Department of Languages and Computer Systems, University of Seville Seville, Spain [email protected] F. Rodríguez-Díaz Data Science and Big Data Lab, Pablo de Olavide University Seville, Spain [email protected] A. Troncoso Data Science and Big Data Lab, Pablo de Olavide University Seville, Spain [email protected] M. Martínez-Ballesteros∗ Department of Languages and Computer Systems, University of Seville Seville, Spain [email protected] Abstract The social impact and global health crisis caused by the coronavirus since late 2019 led to the development of a novel bio-inspired algorithm. This algorithm simulates the behavior and spread of the virus, known as the Coronavirus Optimization Algorithm. It provides several advantages over similar approaches and serves as a basis for generalizing pattern or association identification from numerical datasets. In this study, essential updates and modifications are proposed to adapt the CVOA algorithm for mining numerical association rules. These changes involve adjustments to the encoding of individuals and the infection/mutation process. Additionally, parameter values are updated, and a new fitness function is proposed to be maximized. The main objective is to obtain highquality numerical association rules for any dataset regardless of the number and range of attributes in the dataset. The implemented algorithm is compared to others designed for mining quantitative association rules in order to validate the results. For this reason, different datasets from the BUFA repository are used, confirming that Coronavirus Optimization Algorithm is a promising option for discovering interesting association rules within numerical datasets. CCS Concepts •Information systems → Association rules;•Theory of computation →Evolutionary algorithms. Keywords Evolutionary algorithms, numerical association rules, bioinspired metaheuristic, COVID ACM Reference Format: C. Herruzo-Lodeiro, F. Rodríguez-Díaz, A. Troncoso, and M. MartínezBallesteros. 2025. Bioinspired evolutionary metaheuristic based on COVID spread for discovering numerical association rules.. In The 40th ACM/SIGAPP ∗Corresponding author This work is licensed under a Creative Commons Attribution 4.0 International License. SAC ’25, March 31-April 4, 2025, Catania, Italy ©2025 Copyright held by the owner/author(s). ACM ISBN 979-8-4007-0629-5/25/03 https://doi.org/10.1145/3672608.3707787 Symposium on Applied Computing (SAC ’25), March 31-April 4, 2025, Catania, Italy. ACM, New York, NY, USA, 8 pages. https://doi.org/10.1145/3672608. 3707787 1 Introduction After more than 4 years since the World Health Organization (WHO) declared coronavirus disease 2019 (COVID-19) a global pandemic, new infections and deaths caused by the virus known as severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) are still being recorded around the world. Currently, the number of confirmed COVID-19 cases reported to the WHO is 768,237,788, including 6,951,677 deaths [ 24 ]. Although the WHO declared the end of COVID-19 as an international public health emergency on May 2023, this does not mean that it has ceased to be a threat to global health or a global public health priority [ 17 ]. For this reason, it is necessary to continue collecting datasets that allow us to analyze and understand the spread of this and potential future diseases. Metaheuristics are approximation algorithms, high-level strategies that allow exploration of the solution space through a wide variety of methods. Population-based metaheuristics mimic the behavior of living organisms in nature and aim to solve optimization problems by searching for the best result within the solution space. The search process involves, on one hand, measuring each agent’s proximity to the best solution through the fitness function, and on the other hand, performing the mechanisms of exploration, to collect information, and exploitation, which uses that information to explore favorable areas of the search space in detail. Bio-inspired metaheuristics, drawing from biological phenomena and natural processes, have been widely developed and studied over the past few decades [4]. Due to the rapid spread and the significant health crisis caused by the SARS-CoV-2 virus, the Coronavirus Optimization Algorithm (CVOA) [ 13 ] was developed as an optimization algorithm based on the behavior and spread model of COVID-19. This algorithm offers several advantages over similar strategies. One of them is that, given the abundance of data and statistics due to the global impact and importance of the disease, there is no need for arbitrary initialization of parameters, as values such as infection and mortality rates are already known. Viruses spread by infecting individuals, who can, in turn, infect others, die, or recover. The concept 138 SAC ’25, March 31-April 4, 2025, Catania, Italy C. Herruzo-Lodeiro et al. of “super-spreaders” also exists, referring to individuals who infect a larger number of people, thereby intensifying the search space. The number of initially infected individuals increases exponentially over several iterations but eventually starts to decline until no individuals remain, which is a key advantage of the algorithm as it eliminates the need to define a stopping criterion. The CVOA algorithm can be combined with other Artificial Intelligence techniques, such as association rules (AR), which aim to extract correlations, patterns, or associations between the attributes of a dataset. When association rules are derived from datasets with numerical attributes, they are referred to as numerical association rules (NAR) [ 2 ]. This work proposes a new algorithm based on CVOA to obtain NAR. Specifically, this algorithm, henceforth referred to as CVNAR, introduces the necessary modifications to, on one hand, obtain high-quality numerical association rules and, on the other, generalize the solution and the search for such rules for any dataset, number of attributes, and range of numerical values. The paper is structured into the following sections. Section 2 presents an analysis of the current state of the art of numerical association rules. Section 3 describes the adjustments made to the CVOA algorithm, adapting it for use in NAR mining to achieve the defined objectives. The results obtained are presented and analyzed in Section 4. Finally, Section 5 summarizes the key conclusions drawn from the results and discusses the future work. 2 Related works In 1993, Agrawal et al. [ 1 ] first proposed the use of AR to identify relationships between variables in datasets. The most popular AR algorithms are Apriori, Eclat, and FP-growth. These algorithms handle binary and categorical attributes, but the real world also consists of numerical attributes. To address these challenges, NARs, or quantitative association rules, are introduced as a means of handling numerical data more effectively. Initially, problem-solving with NAR relies on discretization, using partitioning and combination routines, clustering, and fuzzy logic. Additionally, other methods such as optimization and distribution have been proposed to further enhance the process [ 23 ]. These three main approaches are extensively discussed throughout the literature, and a wide range of NAR algorithms have been developed based on them, as demonstrated in [7] and [8]. Although discretization methods can result in information loss and reduce the quality of the rules, they remain one of the most popular approaches for addressing numerical association rule mining problems due to their simplicity and flexibility. Discretization can be performed using techniques such as fuzzification, clustering, or partitioning and combination. Clustering techniques group numerical columns into categories based on value similarity, using methods like hierarchical, densitybased, and grid-based clustering. For instance, ARCS [ 9 ] clusters association rules with bitwise operations, while MQAR [ 26 ] uses a dense grid frequent pattern tree to generate non-redundant NARs by clustering subspaces. More recently, GCQAR [ 15 ] combines modularity-based graph clustering with NAR to uncover relationships within cohesive subgraphs. Similarly, partitioning techniques are widely used in discretization. For example, Srikant and Agrawal [ 22 ] proposed an algorithm for handling numeric attributes in NARM, using equi-depth discretization to partition attributes and generate frequent itemsets based on minimum support. Later approaches, such as Rastogi’s algorithm [ 19 ], leverage predefined intervals to discretize numerical attributes into segments. More recently, Song and Ge [ 21 ] introduced NAR-Discovery, a divide-and-conquer algorithm that efficiently partitions attributes into buckets to mine NAR. Finally, fuzzification offers a flexible discretization method where interval boundaries are not strictly defined. For instance, OFARM [ 27 ] optimized fuzzy set partition points by using multiple objective functions and a two-level iteration process to generate frequent itemsets. Many studies focus on optimization methods for solving problems in the context of NAR to address the disadvantages of algorithms based on discretization methods. These methods use heuristic algorithms inspired by natural phenomena, such as animal behavior and biological processes. They are particularly noteworthy for their ability to detect relationships and patterns within large datasets, without the need to define thresholds or perform discretization steps. These methods can handle both numerical and categorical data and are robust against noise and missing information. However, discretization methods can face challenges with convergence, often leading to the discovery of local optima, along with high computational complexity and significant resource requirements. Optimization methods consist of two phases: first, all sets of frequent items are identified, and then all relevant association rules are extracted. They are divided into two categories: bio-inspired optimization methods and physics-based optimization methods. Depending on the optimization objectives, they can also be classified into single-objective and multi-objective approaches. Bio-inspired methods can be categorized as follows: • Evolution-based algorithms: These algorithms mimic the ability of living organisms to adapt to their environment. They leverage search methods inspired by natural selection and genetics. • Swarm-intelligence-based algorithms: These can be further divided into two sub-optimization methods: – Particle Swarm Optimization: An algorithm based on animal behaviors, such as the collective movement of bird flocks or fish schools, and designed for population-based optimization of non-linear functions. – Wolf Search Algorithm: A bio-inspired heuristic optimization algorithm that replicates how wolves search for food and survive by avoiding predators. Several evolution-based methods have been developed for NARM. GENAR [ 14 ], for example, leverages genetic algorithms to discover rules with numerical attributes, while Yan et al. proposed EARMGA [ 25 ], which incorporates genetic operators and a fitness function that eliminates the need for predefined support thresholds, handling both categorical and quantitative attributes effectively. In contrast to evolutionary approaches, particle swarm optimization offers a different strategy. For instance, Beiranvand et al. introduced MOPAR [ 3 ], a multi-objective PSO algorithm that extracts 139 CVNAR: Conavirus Optimization Algorithm to discover Numerical Association Rules SAC ’25, March 31-April 4, 2025, Catania, Italy NAR by optimizing three key objectives: confidence, comprehensibility, and interestingness. MOPAR also redefines the traditional swarm components to more effectively handle numerical attributes. Recently, Moleshi et al. proposed HGP-QAR [ 16 ], a hybrid algorithm combining genetic algorithms and particle swarm optimization to enhance the efficiency of NARM. This algorithm leverages GA’s selection, crossover, and mutation with PSO’s velocity and position updates for more efficient exploration of the solution space. The fitness function is based on confidence, interestingness, and comprehensibility. On the other hand, physics-based optimization methods simulate the behavior and properties of matter or follow the laws of physics. One such method is the Gravitational Search Algorithm (GSA), an optimization algorithm inspired by the laws of gravity [ 18 ]. In this approach, agents behave as objects, and their performance is measured by their mass. After a thorough review of the literature, it is noteworthy that there are still many challenges to be addressed in developing NAR methods to discover patterns in real-world datasets. In this paper, we propose the CVNAR algorithm, which leverages optimization techniques and builds on the CVOA algorithm to address common limitations, including information loss, convergence problems, the requirement for user-defined thresholds, and falling into local minima. To address these challenges, several novel features are proposed, which will be detailed in the following sections. 3 Description of the CVNAR algorithm The CVNAR algorithm aims to find high-quality NAR based on the CVOA algorithm. The goal is to discover a generalized optimal solution for any numerical dataset, regardless of the number of attributes or their range of values. The following sections describe and detail the various phases of the algorithm’s methodology. First is the initialization phase, which is responsible for generating the initial population of individuals. In this case, the population consists of a single subject, known as Patient Zero (PZ). Simulating the behavior of the coronavirus, PZ represents the first infected individual and is generated randomly. Once the initial population is created, the iterative evolution phase begins. This phase involves applying a series of steps until the stopping condition is met. In the CVNAR algorithm, the stopping condition is satisfied when a maximum number of iterations is reached or when no infected individuals remain. The steps corresponding to this phase are: Step 1. Calculate the fitness function for each individual in the infected list. If the fitness function does not return a result, that is, it returns an undefined or unrepresentable value, the individual is removed from the infected list and moved to the deceased list. Step 2. Maintain a global list and update it in each iteration with the ten individuals having the best fitness function. The number of top solutions or individuals stored by the algorithm is a configurable parameter and can be adjusted in each run. Step 3. Spread the disease. There are various scenarios that can occur depending on the individual, which directly affect the number of new infected individuals and, consequently, the spread of the disease: (1) An infected individual will die with a certain probability ( 𝑃_𝐷𝐼𝐸 ). In this case, the individual will not infect new individuals. (2) Individuals who do not die may become ordinary spreaders, meaning they infect new individuals at a normal spread rate, or “super-spreaders”, who will propagate the disease at a higher rate. The probability of being a super-spreader (𝑃_𝑆𝑈 𝑃𝐸𝑅𝑆𝑃𝑅𝐸𝐴𝐷𝐸𝑅) will determine the rate of spread. (3) Finally, there is a probability that the individual will travel, represented by 𝑃_𝑇𝑅𝐴𝑉 𝐸𝐿 , which causes the solutions created through infection to change based on the travel distance, denoted as 𝑇𝑅𝐴𝑉 𝐸𝐿_𝐷𝐼𝑆𝑇𝐴𝑁𝐶𝐸 . The same individual can be both a “super-spreade” and a traveler. Step 4. Manage and update three individual lists at each iteration after the previous step is completed: (1) Deceased. The individuals added to this list will no longer be used. (2) Recovered. For simplicity, infected individuals will move to the recovered list if the isolation probability is satisfied ( 𝑃_𝐼𝑆𝑂𝐿𝐴𝑇𝐼𝑂𝑁 ). There is a probability of reinfection, which means that a recovered individual can return to the infected list (𝑃_𝑅𝐸𝐼 𝑁 𝐹𝐸𝐶𝑇 𝐼𝑂𝑁 ). (3) Newly infected. Refers to the group of individuals infected in the previous step. The goal of the CVNAR algorithm is to identify the individuals with the highest fitness function value in the entire population. Initially, PZ is considered as the best solution found. The list grows with each iteration as new individuals become infected, until the predefined number of top solutions is reached. From that point on, during each iteration where the disease spreads, the algorithm checks whether any infected individual has a better fitness value than the optimal stored solutions, replacing them if necessary. Finally, as previously mentioned, the size of the best solutions list in CVNAR is determined by an input parameter of the algorithm. 3.1 Individual codification Association rules can be defined as implications of the form “if 𝑋 then 𝑌 ”, considering that the occurrence of 𝑋 in a transaction implies the occurrence of 𝑌 , where 𝑋 is defined as the antecedent and 𝑌 as the consequent [ 1 ]. In the population, each individual represents a rule, and each of these rules undergoes an evolutionary mutation process. At the end of the process, the individual with the best fitness function is designated as the best rule [10]. When the dataset is to be analyzed numerical, it is necessary to perform a discretization of the attribute domain into intervals, defining an upper limit and a lower limit for each of them. This can lead to a potential loss of information; for this reason, the CVOA algorithm learns the width of the intervals in each iteration [ 20 ]. This decision-making process regarding the interval width is explained in detail in Section 3.2.4. The encoding employed for individuals that are candidates to be solutions of the CVNAR algorithm is described below, considering that: •the attributes of the dataset are numerical, • the range of values for each attribute belongs to the set of real numbers, 140 SAC ’25, March 31-April 4, 2025, Catania, Italy C. Herruzo-Lodeiro et al. Figure 1: Individual codification. • it is not fixed whether each attribute belongs to the antecedent, the consequent, or does not belong to the rule. Therefore, given a dataset with 𝑚 attributes and 𝑛 instances, an individual consists of 2 ×𝑚 elements, where each element 𝑘 is an integer between [ 1 ,𝑛 − 1 ] . Each attribute will have a lower limit, corresponding to the odd positions in the list, and an upper limit, corresponding to the even positions. The value of both is an integer, 𝑘𝑖∈ [ 1 ,𝑛 − 1 ] for the lower limit and 𝑘𝑠∈ [ 1 ,𝑛 −𝑘𝑖] for the upper limit. The structure of the individual consists of three parts: • Encoded value of the limits. This refers to the random value generated between [ 1 ,𝑛 − 1 ] for the lower limit 𝑘𝑖 and the random value generated between [ 1 ,𝑛 −𝑘𝑖] for the upper limit 𝑘𝑠. • Decoded value of the limits. First, all the values from the dataset for the attribute are taken and sorted in ascending order. The decoded value of the lower limit corresponds to the value found at position 𝑘𝑖 , and the decoded value of the upper limit corresponds to the value found at position 𝑘𝑠. • Attribute type. Each attribute of the individual can be of one type: –0 when the attribute does not belong to the rule, –1 when the attribute belongs to the antecedent, –2 when the attribute belongs to the consequent. Figure 1 graphically shows the encoding of an individual and an example of a coded NAR. In particular, the rule 𝑎1∈ [ 98 , 98 . 5 ] ∧ 𝑎2∈ [ 1 , 1 ]=⇒𝑎3∈ [ 61 , 89 ] is represented. It can be observed that the attributes 𝑎1 and 𝑎2 belong to the antecedent, as 𝑡1 and 𝑡2 have the value 1. Similarly, the attribute 𝑎3 belongs to the consequent because 𝑡3 has the value 2. For attribute 𝑎1, the decoded interval values are 98 as the lower bound and 98.5 as the upper bound. These values correspond to the positions 𝑘𝑖1= 110 and 𝑘𝑠1= 0in the dataset, where 𝑘𝑖1 and 𝑘𝑠1 represent the k-th indices of the attribute values when sorted in ascending order. This ordering ensures that the interval boundaries are accurately identified within the context of the dataset. Similarly, for attribute 𝑎2 , the decoded interval values are both 1, meaning the lower and upper bounds of the interval coincide. These values correspond to the k-th positions 𝑘𝑖2= 44 and 𝑘𝑠2= 57, respectively, when the values of attribute 𝑎2 are sorted from smallest to largest. Finally, for attribute 𝑎3 , the decoded interval values are 61 for the lower bound and 89 for the upper bound. These values correspond to the positions 𝑘𝑖3= 4and 𝑘𝑠3= 0in the dataset when attribute 𝑎3is ordered in ascending order. 3.2 General process of CVNAR This section describes the phases of the CVNAR algorithm, detailing the generation of the initial population and the execution of the infection process, highlighting the proposed fitness functions, and the selection of individuals based on the chosen fitness function. 3.2.1 Generation of the initial population. The initial population of CVNAR algorithm consists of a single individual, generated randomly. Randomness encompasses the number of attributes present in the rule, the type of each attribute, and their respective intervals. Regarding the attribute limits, a random integer 𝑘𝑖∈ [ 1 ,𝑛 − 1 ] is generated for the lower limit, and another integer 𝑘𝑠∈ [ 1 ,𝑛 −𝑘𝑖] is generated for the upper limit. The values from the entire dataset for each attribute are sorted in ascending order, and finally, the value at position 𝑘𝑖 is selected for the lower limit, while the value at position 𝑘𝑠 is selected for the upper limit. For the type, a random number between 0, 1, and 2 is generated for each attribute. To ensure that the generated individual represents a robust rule, the following constraints are considered: • The upper limit must be greater than the lower limit. If this condition is not met, the random generation process mentioned earlier is repeated. • The value of the intervals for each attribute must fall within the range of values in the dataset. • The number of attributes belonging to the antecedent must be greater than or equal to 1. • The number of attributes belonging to the consequent must be greater than or equal to 1. • If the number of attributes is greater than 2, the number of attributes corresponding to the antecedent must be greater than the number of attributes belonging to the consequent. 3.2.2 Fitness function. Association rules represent quantifiable patterns that reveal dependencies between attributes in a dataset. Typically, the number of extracted associations is high, then, it becomes crucial to sort or filter them based on specific measures of 141 CVNAR: Conavirus Optimization Algorithm to discover Numerical Association Rules SAC ’25, March 31-April 4, 2025, Catania, Italy relevance [ 12 ]. Quality measures used in this paper are described as follows: • Support(X ⇒ Y): Percentage of instances in the dataset that satisfy 𝑋and 𝑌. 𝑆𝑢𝑝(𝑋⇒𝑌)=(𝑃(𝑋∪𝑌)) (|𝐷|) where 𝐷 is the total number of instances o transactions of the dataset. • Confidence(X ⇒ Y): Probability that instances satisfying 𝑋also satisfy 𝑌. 𝐶𝑜𝑛𝑓 (𝑋⇒𝑌)=𝑆𝑢𝑝 (𝑋⇒𝑌) 𝑆𝑢𝑝 (𝑋) • Accuracy(X ⇒ Y): Degree of coincidence of the data obtained with the real data. 𝐴𝑐𝑐(𝑋⇒𝑌)=𝑆𝑢𝑝 (𝑋⇒𝑌) 𝑆𝑢𝑝 (¬𝑋⇒¬𝑌) • Leverage(X ⇒ Y): Proportion of cases covered by both 𝑋 and 𝑌 compared to what would be expected if 𝑋 and 𝑌 were independent. Lev(X ⇒Y)=Sup(X ⇒Y)-Sup(X)Sup(Y) • Certainty Factor(X ⇒ Y): Probability that 𝑌 is present in an instance when considering only those instances where 𝑋 appears. Si Conf(X ⇒Y) >Sup(Y): FC(X⇒Y) = 𝐶𝑜𝑛𝑓 (𝑋⇒𝑌)−𝑆𝑢𝑝 (𝑌) 1−𝑆𝑢𝑝 (𝑌) If Conf(X⇒Y) ≤Sup(Y): FC(X⇒Y) = 𝐶𝑜𝑛𝑓 (𝑋⇒𝑌)−𝑆𝑢𝑝 (𝑌) 𝑆𝑢𝑝 (𝑌) The fitness function is used to identify the best individuals. Two distinct objective functions are proposed for maximization. Both functions are composed of different measures of interest: Function 1 =𝐴𝑐𝑐(𝑋⇒𝑌) + 𝐶𝑜𝑛𝑓 (𝑋⇒𝑌) + 𝐿𝑒𝑣(𝑋⇒𝑌)(1) Function 2 =𝐹𝐶(𝑋⇒𝑌) + 𝐶𝑜𝑛𝑓 (𝑋⇒𝑌) + 𝑆𝑢𝑝(𝑋⇒𝑌)(2) These functions aim to obtain high-quality numerical association rules by combining measures of interest to address the disadvantages presented by some of them. Leverage and accuracy, in some cases, are insufficient, as they only measure co-occurrences and do not consider implication. Therefore, it may be beneficial to combine them with confidence, which measures the reliability of the rule. On the other hand, Certainty Factor has the advantage of taking into account the support of both the antecedent and the consequent, as well as the implication. Given this context, a comparative analysis is conducted to identify the most effective measure for generating interesting numerical association rules. 3.2.3 Individual selection. As previously mentioned, the CVNAR algorithm generates a list of the best individuals, specifically those with the highest fitness function values. Initially, the only stored element is PZ, and the list increases as new individuals become infected until it reaches the defined size. In each iteration, it is checked whether any of the individuals in the infected list has a higher fitness value than those in the list of best individuals. If so, the infected individual replaces the one with the lower fitness value. The size of the list of best individuals is an input parameter of the CVNAR algorithm and can be adjusted as needed. 3.2.4 Infection/Mutation. The infection process of an individual involves performing a mutation on the infected subject, which means that the values of the intervals and the type of each attribute encoding the individual are modified. On one hand, the number of new infections will depend on whether the individual is a regular spreader or a “super-spreader”. On the other hand, the number of affected intervals will depend on 𝑇𝑅𝐴𝑉 𝐸𝐿_𝐷𝐼𝑆𝑇𝐴𝑁𝐶𝐸 . The distance is a random value between 1 and the total number of intervals. When referring to infecting an individual, both the interval and the type of attribute are considered. As mentioned above, as many intervals will be infected, as indicated by the randomly generated value 𝑇𝑅𝐴𝑉 𝐸𝐿_𝐷𝐼𝑆𝑇𝐴𝑁𝐶𝐸 . A different index is maintained for each type of infection, allowing for the mutation of the interval of the second attribute and the type of the third for the same individual. The mutated indexed for each type are stored and not reused. The process is as follows: • Infection of an interval attribute: It consists of increasing or decreasing the amplitude of the interval by a certain percentage (25%, 50%, 75%). A variable 𝑑=|𝑎𝑓−𝑎𝑖| is defined, which represents the difference between the new amplitude and the initial one, and a random number 𝑃 is generated between 0 and 2. If 𝑃 = 0, 𝑑 = 0.25 x 𝑎𝑖 ; if 𝑃 = 1, 𝑑 = 0.5 x 𝑎𝑖 and if 𝑃 = 2; 𝑑 = 0.75 x 𝑎𝑖 . The decision to increase or decrease the interval is also random. In case of increasing, 𝑟𝑜𝑢𝑛𝑑(𝑑)/ 2 (rounded down) is subtracted from each limit of the interval, and in case of decreasing, 𝑟𝑜𝑢𝑛𝑑(𝑑)/ 2(rounded up) is added. If the resulting value is less than 0, it is assigned 0. • Infection of a type attribute: It consists of changing the type of an attribute from 0 to 1 or 2; from 1 to 0 or 2 and from 2 to 0 or 1. If the type of the infected attribute does not meet the following conditions: – The number of attributes belonging to the antecedent must be greater than or equal to 1. – The number of attributes belonging to the consequent must be greater than or equal to 1. – If the total number of attributes is greater than 2, the number of attributes corresponding to the antecedent must be greater than the number of attributes belonging to the consequent. Then, the types of all attributes of the individual are infected following the same procedure as during the generation of the initial population. 3.3 Computational complexity of CVNAR The computational complexity of CVNAR can be expressed as 𝑂(𝐼·𝑃·𝐹(𝑁, 𝑀)) , where 𝐼 represents the number of iterations, 𝑃 is the average population size per iteration, and 𝐹(𝑁, 𝑀) is the computational cost of evaluating the fitness function for each individual in the population. The fitness function 𝐹(𝑁, 𝑀) evaluates the quality of each individual, NAR in this case, derived from the dataset. This involves computing quality measures such as support, confidence, which require scanning the dataset to verify the rule conditions. For a dataset with 𝑁 instances and 𝑀 attributes, 142 SAC ’25, March 31-April 4, 2025, Catania, Italy C. Herruzo-Lodeiro et al. 𝐹(𝑁, 𝑀)=𝑂(𝑁·𝑀) , where 𝑁 and 𝑀 determine the linear complexity of encoding and evaluating rule conditions. To optimize runtime and enhance the scalability of CVNAR for larger datasets, parallelizing the fitness evaluations and dynamically controlling the population size (e.g., adjusting infection rates or isolation probabilities) can reduce execution time without compromising performance [11]. 4 Experimentation and Results This section outlines the results obtained from the execution of the CVNAR algorithm, including the execution parameters, datasets, and evaluation metrics used. 4.1 Execution parameters The parameters used in the CVNAR algorithm and the value of each are detailed below. These parameters are extracted from various sources, one of the most imporant being the WHO. (1) Mortality rate: This is the probability that an infected individual will die. It is calculated by dividing the number of confirmed deaths by the number of confirmed cases. According to the data available in [ 6 ], a value of 𝑃_𝐷𝐼𝐸 = 0.06 is assigned. (2) Probability of being super-spreader. It is the probability that an individual spreads the disease at a higher rate. It is considered that 10% of the population are super-spreader individuals, therefore, 𝑃_𝑆𝑈 𝑃𝐸𝑅𝑆𝑃𝑅𝐸𝐴𝐷𝐸𝑅 = 0.1. Two additional parameters related to the disease’s propagation rate are taken into account, 𝑂𝑅𝐷𝐼𝑁𝐴𝑅𝑌_𝑅𝐴𝑇 𝐸 and 𝑆𝑈 𝑃𝐸𝑅𝑆𝑃𝑅𝐸𝐴𝐷𝐸𝑅_𝑅𝐴𝑇𝐸 . Both are random numbers and refer to the number of people that can infect. An ordinary spreader will infect between 0 and 5 people, while a superspreader will infect between 6 and 15. (3) Travel probability: According to [ 5 ], it is considered that 10% of the population can travel to any location during a week and infect other people, therefore, 𝑃_𝑇 𝑅𝐴𝑉 𝐸𝐿 = 0.1. (4) Re-infection probability: The likelihood of re-infection with SARS-CoV-2 is estimated to occur in less than 1% of previously confirmed cases. Therefore, 𝑃_𝑅𝐸𝐼𝑁 𝐹𝐸𝐶𝑇𝐼𝑂𝑁 = 0.001. (5) Isolation probability: A high value is assigned, as it helps reduce the exponential growth of infection. The selected value is 𝑃_𝐼𝑆𝑂𝐿𝐴𝑇𝐼𝑂𝑁 = 0.7. (6) Algorithm iterations: The total number of iterations is represented by 𝑃𝐴𝑁𝐷𝐸𝑀𝐼𝐶_𝐷𝑈 𝑅𝐴𝑇𝐼𝑂𝑁 , which is set to 20. The number of iterations without social distancing measures is indicated by 𝑆𝑂𝐶𝐼𝐴𝐿_𝐷𝐼𝑆𝑇𝐴𝑁𝐶𝐼𝑁𝐺 , with values ranging from 7 to 12, inclusive. Table 1 summarizes the key parameters and their values used in the CVNAR algorithm, organized according to the main phases and steps outlined in Section 3.2. 4.2 Datasets The datasets used to validate the performance of the CVNAR algorithm are sourced from the Bilkent University Function Approximation Repository (BUFA) repository. Table 2 shows the name, the Table 1: Summary of parameters in the CVNAR algorithm. Phase (Step) Parameter Value Spread the disease: Infection /Mutation (Step 3) 𝑂𝑅𝐷𝐼𝑁𝐴𝑅𝑌_𝑅𝐴𝑇 𝐸 Random [0, 5] 𝑆𝑈 𝑃𝐸𝑅𝑆𝑃𝑅𝐸𝐴𝐷𝐸𝑅_𝑅𝐴𝑇𝐸 Random [6, 15] 𝑃_𝑆𝑈 𝑃𝐸𝑅𝑆𝑃𝑅𝐸𝐴𝐷𝐸𝑅 0.1 𝑃_𝑇𝑅𝐴𝑉 𝐸𝐿 0.1 𝑇𝑅𝐴𝑉 𝐸𝐿_𝐷𝐼𝑆𝑇𝐴𝑁𝐶𝐸 Random [1, total intervals] Manage and update lists (Step 4) 𝑃_𝐷𝐼𝐸 0.06 𝑃_𝑅𝐸𝐼𝑁 𝐹𝐸𝐶𝑇𝐼𝑂𝑁 0.001 𝑃_𝐼𝑆𝑂𝐿𝐴𝑇𝐼𝑂𝑁 0.7 Global iterations 𝑃𝐴𝑁𝐷𝐸𝑀𝐼𝐶_𝐷𝑈 𝑅𝐴𝑇𝐼𝑂𝑁 20 iterations 𝑆𝑂𝐶𝐼𝐴𝐿_𝐷𝐼𝑆𝑇𝐴𝑁𝐶𝐼𝑁𝐺 7-12 iterations number of instances, and the number of features for each of the 5 datasets used in this paper. Table 2: Datasets from the BUFA repository. Dataset Instances Features Basketball (BK) 96 5 Bodyfat (FA) 252 18 Bolts (BL) 40 8 Pollution (PO) 60 16 Quake (QU) 2178 4 4.3 Quality measures The metrics or measurements of interest selected in this paper to evaluate the results obtained are those described in Section 3.2.2. Additionally, we include lift as an additional metric, which captures the correlation between the antecedent and the consequent. A lift value greater than 1 indicates that the rule provides valuable information when 𝑋and 𝑌appear together. Support, confidence and accuracy can take values between 0 and 1. The goal is for these values to be as high as possible, with 1 being the best result. Leverage can take values from -1 to 1, with values greater than 0 considered favorable; values close to 1 indicate high-quality association rules, while values below 0 suggests strong independence between 𝑋 and 𝑌 . Certainty factor also ranges from -1 to 1, where 1 indicates that the rule is completely certain and precise. 4.4 Results This section presents in Section 4.4.1 the results of applying the CVNAR algorithm to the datasets from Section 4.2, optimizing the objective functions outlined in Section 3.2.2 to determine which function performs better. Then, the best function is compared with other existing algorithm to obtain NAR in Sections 4.4.2. The stopping criterion is set to 20 iterations, and the 10 best solutions are being saved. 4.4.1 Objective function comparison. As described in Section 3.2.2, two different objective functions are defined in Equations 1 and 2 respectively, to maximize and solve the NAR optimization problem. 143 CVNAR: Conavirus Optimization Algorithm to discover Numerical Association Rules SAC ’25, March 31-April 4, 2025, Catania, Italy Table 3: Objective function comparison for datasets from the BUFA repository. Av. Conf (%) Av. Lift Av. Lev Av. Acc (%) Av. Sup (%) Av. FC BK Function 1100.00 96.00 0.0103 100.00 1.04 1.00 Function 2 100.00 1.01 0.0073 50.41 48.54 1.00 BL Function 1100.00 5.00 0.1600 100.00 20.00 1.00 Function 2 100.00 1.13 0.0472 55.00 43.50 1.00 FAFunction 1 100.00 173.52 0.0039 99.12 0.39 1.00 Function 2 100.00 1.45 0.0115 32.53 5.03 1.00 PO Function 133.33 7.16 0.0061 97.66 0.66 0.33 Function 2 100.00 1.38 0.0375 42.00 18.00 1.00 QU Function 1100.00 4.06 0.0003 81.67 19.49 0.80 Function 2 99.71 1.10 0.0138 23.48 13.82 0.96 Total Av. Function 1 86.66 57.14 0.0361 95.69 8.32 0.83 Function 2 99.94 1.21 0.0235 44.36 25.78 0.99 Table 3 shows the values of each quality metric for the different datasets and objective functions. The analysis of the results shows that for most datasets, the measurement values for both objective functions are within the quality threshold. Analyzing each metric, it can be observed that the second objective function achieves higher confidence than the first, although in both cases the values are high and close to 100 %. Confidence is not suitable for determining which objective function is more appropriate, as it does not consider the support of the consequent and, consequently, cannot detect negative dependencies. However, this is addressed by quality measures such as lift, leverage, or accuracy, which take into account the relationship between the antecedent and the consequent. Although the values of these measures are higher for objective function 1, it cannot be concluded that the association rules obtained are more interesting than those from objective function 2. This is because for the second function, the lift is consistently greater than 1, the leverage is always positive, and the accuracy values are close to 100 %. Furthermore, lift, leverage, and accuracy are not symmetric measures, and in some cases, this is insufficient as they fail to account for the direction of the implication. Lastly, the certainty factor considers not only the support of both the antecedent and consequent but also the direction of the rule’s implication. In this case, the second objective function yields the best results. Considering the above, objective function 2 is selected as the fitness function. This is supported by its consistent performance across all datasets, yielding positive leverage, lift values greater than 1, and accuracy values close to 100%. These results indicate that objective function 2 provides more reliable and robust rules compared to objective function 1, particularly when considering the directionality and dependency relationships in the NAR. 4.4.2 Comparative with other NAR algorithms. After selecting the second objective function, we compare the results obtained with the CVNAR algorithm to those of several existing optimization methods presented in Section 2. This comparison is based on the analyses outlined in [ 16 ]. The datasets used for the comparative analysis are: Basketball (BK), Bodyfat (FA), Quake (QU), Bolts (BL) and Pollution (PO). The metrics analyzed include support and confidence, as shown in Tables 4, 5, 6, 7 and 8. Considering the definitions of support and confidence, when support is low and confidence is high, rules with a large number of attributes are generated, expressing very specific patterns of behavior in the data. On the other hand, when support is high, Table 4: Comparison between CVNAR and NAR algorithms for the BK dataset. Algorithm Av. Sup (%) Av. Conf (%) GENAR (Mata et al. 2001) 30.82 96.52 EARMGA (Yan et al. 2009) 2.70 100.00 MOPAR (Beiranvand et al. 2014) 30.76 95.00 HGP-QAR (Moleshi et al. 2019) 62.37 97.40 CVNAR (Function 2 in this paper) 48.54 100.00 Table 5: Comparison between CVNAR and other NAR algorithms for the FA dataset. Algorithm Av. Sup (%) Av. Conf (%) GENAR (Mata et al. 2001) 41.52 96.52 EARMGA (Yan et al. 2009) 4.97 100.00 MOPAR (Beiranvand et al. 2014) 22.95 81.00 HGP-QAR (Moleshi et al. 2019) 65.43 98.90 CVNAR (Function 2 in this paper) 5.03 100.00 Table 6: Comparison between CVNAR and other NAR algorithms for the QU dataset. Algorithm Av. Sup (%) Av. Conf (%) GENAR (Mata et al. 2001) 35.17 64.40 EARMGA (Yan et al. 2009) 3.40 100.00 MOPAR (Beiranvand et al. 2014) 31.97 89.00 HGP-QAR (Moleshi et al. 2019) 63.35 99.80 CVNAR (Function 2 in this paper) 13.82 99.71 Table 7: Comparison between CVNAR and other NAR algorithms for the BL dataset. Algorithm Av. Sup (%) Av. Conf (%) GENAR (Mata et al. 2001) 30.82 96.52 EARMGA (Yan et al. 2009) 11.43 100.00 MOPAR (Beiranvand et al. 2014) 10.72 88.91 HGP-QAR (Moleshi et al. 2019) 81.14 89.77 CVNAR (Function 2 in this paper) 43.50 100.00 less significant rules are obtained, which may result in the loss of these patterns. Furthermore, when a rule covers all records, it does not provide any meaningful information. Therefore, a high or low support value is not enough to assess the effectiveness of an algorithm. It can be observed that for the five datasets, the support of the rules generated by CVNAR is neither the highest nor the lowest but generally remains in the mid-range. However, the confidence is consistently higher, reaching 100% in most cases, except for QU, where it remains at 99.8%. Notably, EARMGA is the only algorithm that achieves a similar level of confidence, but its support is significantly lower in all datasets. Therefore, it can be concluded that the CVNAR algorithm allows us to obtain more reliable association rules. 144 SAC ’25, March 31-April 4, 2025, Catania, Italy C. Herruzo-Lodeiro et al. Table 8: Comparison between CVNAR and other NAR algorithms published for the PO dataset. Algorithm Av. Sup (%) Av. Conf (%) GENAR (Mata et al. 2001) 22.64 99.72 EARMGA (Yan et al. 2009) 5.36 99.90 MOPAR (Beiranvand et al. 2014) 52.14 23.02 HGP-QAR (Moleshi et al. 2019) 70.66 96.60 CVNAR (Function 2 in this paper) 18.00 100.00 5 Conclusions and future works The objective of this work was to propose a new algorithm, based on the CVOA algorithm, for discovering NAR. This algorithm is intended to be a general purpose algorithm and can be applied to any data set. To achieve this, a new individual enconding is proposed, randomly determining whether each attribute is included in the rule and specifying its type. Additionally, the infection process was modified, and two new objective functions were defined to enhance the optimization problem. The results demonstrate that the CVNAR algorithm can handle any numeric dataset by learning, in each iteration, both the intervals and the attributes assigned to the antecedent, consequent, or excluded from the rule. This learning process leads to the generation of more precise and reliable association rules. Furthermore, the CVNAR algorithm enables the extraction of patterns that can predict future behaviors from current data across various domains and fields. As future work, the definition of a new objective function can be considered, combining other relevant metrics to increase the value of the lowest metrics and addressing the optimization problem from a multi-objective prespective. Additionally, it would be advisable to reduce the algorithm’s execution time. 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