Matching ontologies with kernel principle component analysis and evolutionary algorithm

2.1 Ontology and OM

An ontology is formally represented as a 3-tuple O = ⟨ C , P , ℐ ⟩ [8], comprising the sets of concepts ( C ), properties ( P ), and instances ( ℐ ), which correspond to objects in the real world. Their union ℰ = C ∪ P ∪ ℐ is called ontology entities. Due to the absence of uniform design standards, ontologies often suffer from the heterogeneity problem [12], and OM addresses this issue by identifying correspondences between semantically similar entities [8].

The task of matching two ontologies can be framed as a binary classification problem that aims to find a classifier h : ℰ 1 × ℰ 2 → { 0 , 1 } . In this scenario, a correspondence is specified by a 5-tuple ⟨ i d , e 1 , e 2 , r , c ⟩ [20], where i d is the ID of the correspondence, r ∈ { ≡ , ≤ , ⊥ } indicates the relational context between e 1 and e 2 , and c signifies the confidence level of the correspondence. The set of such correspondences is called ontology alignment. This alignment can be illustrated as a binary matrix: each row and column represents an entity from the respective ontologies, and matrix entries ( i , j ) denote whether entities i and j are mapped (1) or not (0). Evaluation metrics such as recall, precision, and f -measure [21] require the comparison of this alignment against a ground truth to assess its accuracy.

2.2 SF selection and combination for OM

Due to the challenges posed by the high-dimensional search space and complex interactions between SFs in OM, specialized FS methods are essential [12] to improve the quality of matching results. These methods aim to enhance matching accuracy by removing irrelevant or redundant SFs while retaining semantically significant ones, which can be divided into three categories, i.e., embedding-based, filter-based, and wrapper-based FS methods [22]. Todorov [23] developed an embedding-based FS method to filter irrelevant SFs with support vector machines (SVM) [24]. However, the performance of this approach is constrained by SVM limitations in high-dimensional or varied contexts. Belhadi et al. [25] used the wrapper-based FS method to identify the SF subset, which can improve the quality of the chosen SF. They further expanded their work by introducing a pattern mining solution for OM [26], focusing on frequent ontology patterns to refine alignment without the need for exhaustive SF analysis. Later, Yap and Kim [27] used a filter-based FS method to determine the SF subset. They mainly focus on individual SF’s confidence, but did not consider scenarios involving complex SF dependencies. In contrast with the other two FS methods, the filter-based approach distinctly stands out due to its intrinsic evaluation mechanism. It evaluates the relevance of features rooted in the data’s inherent properties, eliminating the need for external learning algorithms. This inherent efficiency results in faster processing speeds and improved scalability. Given that OM frequently operates in real-time or on-line contexts, the rapidity and adaptability make it particularly apt for such tasks.

Moreover, how to combine the selected SF is critical to the quality of the matching results [8]. In recent years, there has been a growing focus on employing EAs for SF combination in OM [28]. Among these, Genetics for ontology alignments stands out as a genetic algorithm (GA)-based approach focused on optimizing aggregation weights for SFs [29]. Traditional EAs, however, face challenges such as slow convergence and premature optimization. To mitigate these issues, Acampora et al. proposed a hybrid GA that incorporates local search strategies [30]. Although most EA-based approaches concentrate on parameter optimization, including weight aggregation and threshold setting, they often overlook the practical challenges of domain-specific requirements and time constraints. Martinez-Gil and Chaves-González addressed this gap by introducing a symbolic regression-based method using genetic programming (GP) to optimize SF ensemble structures [31], which leverages GP’s genotype to explore and select optimal ensemble structures.

Despite significant progress in SF selection and combination for OM, existing solutions still suffer from critical drawbacks that compromise their efficacy and broad utility. First, these methods concentrate mainly on SF selection or SF combination, neglecting the other. This one-sided focus inevitably undermines the overall accuracy of the resulting ontology alignments. Second, current FS approaches do not adequately account for the complex, non-linear interrelations between various SFs in OM scenarios, lowering the quality of the selected feature set. Finally, the existing EA-based SF combination methods lack a comprehensive evaluation mechanism, which hinders their suitability for deployment in real-world applications. To overcome these drawbacks, we introduce a novel kPCA-EA framework that automatically selects and combines SFs to enhance the accuracy of OM results.

5.1 SF combination problem

Given an SF set Sℱ = { SF 1 , SF 2 , … , SF n } , let M i be the similarity matrix corresponding to SF i for i = 1 , … , n , the SF combination problem is to find optimal aggregation weights W = { w 1 , w 2 , … , w n } and a threshold T ∈ [ 0 , 1 ] that will maximize the alignment quality. The process involves the following steps:

• Aggregate the similarity matrices M i using the weighted sum strategy to form a new aggregated similarity matrix M agg :

  (3) M agg = ∑ i = 1 n w i ⋅ M i .

• Convert M agg into a binary matrix M bin using the threshold T :

  (4) M bin ( i , j ) = 1 , if M agg ( i , j ) > T , 0 , otherwise .

5.2 Algorithm overview

EA is a global optimization algorithm renowned for its versatility and robustness, and its inherent capability to explore complex solution spaces makes it particularly suitable for the linear regression problem to combine SFs [28]. In this work, we empirically use the Gray code [33], a well-established binary encoding scheme, to represent both the aggregating weights and a threshold within each individual. Algorithm 1 outlines the pseudo-code of EA for SF combination. Initially, the population P is initialized and evaluated, and the elite individual indiv elite is determined by selecting the individual exhibiting the highest fitness value. In each generation, single-point crossover [34] and bit mutation [35] are used to create a new population P ″ , and the next generation’s population is determined by roulette wheel selection [36]. At the end of each generation, indiv elite is updated, which replaces the individual with the lowest fitness values. The algorithm terminates when reaching the maximum generation MaxGen, and outputs indiv elite as the final result.

The incorporation of EA in our model uniquely leverages its global optimization prowess, specifically tailored to the nuanced demands of SF combination in OM. Using Gray code for binary encoding, our approach distinctively captures the granularity of aggregating weights and threshold nuances within the evolutionary process. This methodological innovation facilitates a precision-oriented optimization, enabling the EA to adeptly balance and refine the ensemble of SFs. Our model’s effectiveness is further amplified through the strategic deployment of evolutionary operations, which can enhance both the diversity and quality of SF combinations. This leads to an optimized aggregation that is not just theoretically robust but practically superior, as evidenced by the enhanced matching accuracy.

5.3 New fitness function

Conventional evaluation of alignment quality relies on traditional metrics such as recall, precision, and f -measure [37]. However, these metrics require the availability of expert-validated groundtruth-matching results, which are unavailable in practice. To overcome this limitation, we introduce a more flexible evaluation framework grounded in three novel metrics that serve as approximations of these conventional measures. In particular, we assume that all entity pairs have a priori probability of constituting a valid mapping, and to formalize this assumption, we introduce two virtual SFs SF ⊤ and SF ⊥ . SF ⊤ unconditionally maps all entity pairs, whereas SF ⊥ abstains from mapping any. Given these premises, we define the probability P ( δ i ) of a particular entity mapping δ i being a true-positive correspondence as follows:

where num SF represents the total number of SFs used, and SF k ( δ i ) equals 1 if the k -th SF designates δ i as positive based on a threshold determined by the EA, and 0 otherwise. In particular, conf k is the confidence value of SF k , which is estimated by analyzing the number of logic conflict mappings within its corresponding alignment. Given an SF SF k and its corresponding alignment A SF , an inconsistent subset of alignment A inconsist can be determined in the following steps:

• Sort the entity correspondences in A SF in descending order of their SF values;

• Add the correspondences with SF values one by one into a temp alignment subset A ′ , where any correspondences being added do not violate the locality consistency principle [38] with the ones already in A ′ ;

• Add any mappings that conflict with the ones in A ′ in the second step to A inconsist .

where ∣ ⋅ ∣ is the number of elements in a set.

On this basis, we define precision as the probability that an entity pair δ i is true-positive:

where ∣ A ∣ is the number of entity mappings in the alignment A . Similarly, we define recall as the probability for an entity pair to be judged as true-positive:

The f -measure is defined as the harmony mean of pre and rec :

Our proposed evaluation metrics provide an effective alternative to traditional metrics such as precision, recall, and f -measure, tailored for EA-based SF combination. At the heart is a consensus model, using an ensemble of SFs to compute P ( δ i ) for a true-positive mapping δ i . Distinguishing our approach are two “virtual” SFs that act as statistical controls, preventing convergence to misleading values. Furthermore, we incorporate confidence values for each SF, enhancing the accuracy of aggregation by weighting SFs based on their logical consistency. This approach reduces the sensitivity to individual SF performance, resulting in a refined evaluative metric for ontology alignments.

6.1 Experiment design

To assess the efficacy of the kPCA-EA approach, we used the Benchmark and Conference datasets provided by the Ontology Alignment Evaluation Initiative (OAEI)^[1]. The OAEI Benchmark dataset^[2] is meticulously designed for a broad spectrum of heterogeneous OM tasks, offering test cases that vary in complexity, size, and structure:

• Test cases 101–104 encompass nearly identical ontology pairs, posing simpler matching challenges.

• Test cases 201–210 include ontologies with lexical heterogeneities that require advanced matching approaches.

• Test cases 221–247 present ontologies with linguistic variations, adding layers of complexity to matching tasks.

• Test cases 248–262 are the most challenging, consisting of ontologies with pronounced structural heterogeneities.

The kPCA-EA configuration for SF selection is as follows:

• Syntax-based SF candidates: Levenshtein distance [39], Jaro distance [40], N-Gram [41], Dice coefficient-based distance [42], and SMOA distance [43];

• Linguistic-based SF candidates: Wu & Palmer distance [44], Path-based distance [45], Leakcock & Chodorow distance [46], and Resnik distance [47];

• Structure-based SF candidates: Similarity flood (SF) distance [48], Neighbor-concept-based distance [49], Instance-based distance [30], and RDF2Vec [50];

• The SF subset siz: 4.

• Population size size p : 60;

• Maximum generation MaxGen: 2000;

• Crossover rate rate c : 0.6;

• Mutation rate rate m : 0.05.

6.2 Sensitivity analysis

The SF subset size is a critical determinant of the efficacy of the kPCA-EA method, significantly affecting both the precision of OM and the computational efficiency. An overly extensive SF subset can encumber the SF combination stage, leading to a prolonged run time that detracts from the method’s efficiency. Conversely, a too narrow SF subset may fail to capture essential interactions among features, inadequately addressing the nuanced heterogeneities between ontological entities. Thus, identifying an optimal balance in the size of the SF subset is crucial to maximize the kPCA-EA’s capabilities, enabling a synergistic interplay among SFs to effectively navigate and reconcile entity discrepancies.

Figure 2 presents a compelling trade-off between the f -measure and the run time against the size of SF subset. With the size of the subset increasing from 2 to 6, a pronounced increase in run time is observed, indicating that larger subsets exert greater computational demand. In parallel, the f -measure, which serves as a barometer for subset accuracy, ascends with increasing subset size, reaching its zenith at a subset size of 4. Advancing beyond this optimal size, the f -measure stabilizes or experiences a marginal decline, suggesting that an excessively large SF subset may not only fail to elevate performance but could also introduce conflicts among SFs, thus undermining efficacy. Consequently, a subset size of 4 is pinpointed as the ideal configuration, striking a harmonious balance between a commendable f -measure and a manageable computational duration, thereby endorsing both the efficiency and the precision of the OM endeavor.

Figure 2

Sensitivity testing on SF subset size.

6.3 Results and discussions

In the experiment, kPCA-EA was compared against OAEI’s participants, with results sourced directly from the official OAEI website.^[4] The selection of these comparative methods was strategically made to cover a wide range of OM approaches, from basic string-based techniques to sophisticated machine learning algorithms, aiming for a thorough benchmark against leading-edge solutions. These methods were specifically chosen for their proven excellence in navigating the multifaceted challenges of OM as outlined by OAEI, including dealing with structural heterogeneity and lexical variances. This diverse set provides a solid foundation for evaluating the adaptability and efficiency of kPCA-EA. Furthermore, their computational efficiency was a key factor in our selection, establishing a standard against which kPCA-EA’s ability to achieve efficient, high-quality ontology alignments could be measured. Our comparative analysis, grounded in the transparent and rigorous evaluation criteria of OAEI, not only demonstrates kPCA-EA’s enhanced accuracy in terms of precision, recall, and f -measure but also underscores its suitability for a broad spectrum of real-world OM challenges, thereby underscoring the innovative impact of our approach within the domain.

Table 1 delineates the f -measure scores achieved by kPCA-EA in comparison with its counterparts for all test cases examined. To statistically substantiate the comparative performance, we used the Wilcoxon rank sum test [51] at a significance threshold of 0.05. In the table, the symbols “(+)/(‒)/(=)” appended to the outcomes of each competing OM method signify whether kPCA-EA outperformed, underperformed, or achieved similar results to those methods, in a statistically significant manner. In both Benchmark and Conference datasets, kPCA-EA consistently achieves higher precision, f -measure, and recall, highlighting its superior matching accuracy. To further validate the method’s advantages, a comparison with non-EAs was conducted. The results, sourced from 30 independent runs, show that kPCA-EA maintains a perfect precision score of 1.00, outperforming competitors in f -measure and recall with minimal standard deviation. These findings indicate kPCA-EA’s robustness and reliability in diverse ontological landscapes, making it a versatile tool for OM. The comprehensive performance evaluation also sheds light on the unique benefits of using an EA-based approach in OM. kPCA-EA’s superior f -measure and recall scores suggest that the EA component significantly contributes to the method’s ability to effectively combine and optimize SFs. In contrast, non-EAs, while effective in certain scenarios, may not provide the same level of adaptability and optimization capabilities as EA-based methods. This comparison not only highlights kPCA-EA’s strengths but also its potential limitations, offering insights into specific contexts where traditional or alternative approaches might be more suitable. Overall, this balanced comparison provides a clearer understanding of kPCA-EA’s applicability and showcases its potential in a variety of complex real-world OM scenarios.

The experimental results offer compelling evidence supporting the claim that the success of kPCA-EA is largely due to the novel framework we designed for automatic SF construction. The data showcase kPCA-EA’s superiority over an array of OAEI competitors, with it consistently achieving higher precision, f -measure, and recall, which are crucial for OM as they collectively measure how accurately and completely the system can identify true matches between entities across ontologies. The framework we introduced plays a critical role in achieving this level of performance due to several factors:

• Two-stage sequential process: Our framework breaks down the SF construction into two strategic stages: selection and combination. By treating these stages sequentially, the framework ensures that each stage is optimized independently before moving on to the next, allowing for a more refined and targeted approach to SF construction;

• Optimization of SF subset selection: The first stage involves selecting the most pertinent SF subset from a possibly vast feature space. The results demonstrate that kPCA-EA can discern the most informative features, which is evidenced by the high precision scores–signifying that when kPCA-EA predicts a match, it is almost always correct;

• Effective combination of selected SFs: The second stage focuses on the combination of the selected SFs. The superior f -measure and recall scores indicate that kPCA-EA is not only accurate, but also consistently identifies a high proportion of true matches (recall) and balances this with precision ( f -measure). This suggests that the SFs are combined in such a way that they complement each other, leading to a more holistic representation of entity similarity;

• Consistency and stability: The minimal standard deviations across multiple runs emphasize the reliability and reproducibility of the framework. This consistency is critical in OM, where varying results can drastically affect the system’s trustworthiness;

• Adaptability across ontological domains: The method’s robust performance across both Benchmark and Conference datasets indicates its adaptability. This adaptability can be attributed to the framework’s ability to handle the unique semantic complexities inherent in different ontologies, optimizing SFs accordingly.

In addition, Table 1 also presents a detailed comparison of the average run time for kPCA-EA and other OAEI participants over 30 independent runs. The experimental results reveal that kPCA-EA not only demonstrates superior performance in terms of precision, f -measure, and recall but also exhibits remarkable efficiency, with significantly lower average run time in both Benchmark and Conference datasets compared to the competing methods. Specifically, in the Benchmark dataset, kPCA-EA’s average run time stands at 103 s with a minimal standard deviation of 2 s showcasing its capability to deliver high-quality OM results more rapidly than several other participants. Notably, methods such as Lily, Wikimatch, and Mamba report substantially higher run times, ranging from 2,211 to 3,244 s, which underscores the computational efficiency of kPCA-EA in handling complex OM tasks without compromising the accuracy or the quality of the results. Similarly, in the Conference category, kPCA-EA maintains its computational advantage with an average run time of 204 s and a standard deviation of 5 s, further highlighting its efficiency. Compared to other methods such as Wikimatch and Mamba, which exhibit run times of 2,765 and 2,174 s, respectively, kPCA-EA presents a more time-efficient solution without sacrificing performance metrics.

The experimental results demonstrate the superiority of the kPCA-EA method in the field of OM, showcasing its consistently high precision, f -measure, and recall metrics when compared to a broad spectrum of existing methods. This performance is not limited to specific test cases or competitors but extends across various ontological frameworks, underscoring the method’s robustness and adaptability. The inclusion of an EA component significantly enhances kPCA-EA’s ability to effectively construct SF, offering a level of adaptability and optimization that traditional and non-EAs struggle to achieve. Moreover, kPCA-EA’s computational efficiency further distinguishes it from other methods. It achieves superior matching accuracy and quality at a fraction of the computational cost, making it an efficient solution for complex OM tasks.

6.4 Further analysis