1.1. Our motivation

Outlier detection, particularly in large datasets, can be incredibly challenging due to the complexity of the data. As datasets grow, they often contain a mix of both useful and irrelevant features, making it difficult to accurately identify patterns that truly stand out. These irrelevant characteristics can negatively affect the precision and efficiency of outlier detection methods, leading to misleading or erroneous results. Another major challenge is accurately identifying the nearest neighbors, especially in high-dimensional data. Many existing outlier detection methods struggle with this, and if the nearest neighbors are chosen incorrectly, it can cause the entire clustering process to break down, leading to incorrect identification of outliers. Moreover, many current approaches don’t compute the Z-score and the degree of each point simultaneously, which is essential for understanding how far an observation deviates from the norm. Without this, it is harder to determine the true significance of potential outliers. Given these challenges, our work is motivated by a clear need to improve outlier detection by addressing the issue of irrelevant data, ensuring more accurate neighbor selection, and providing a more comprehensive analysis of each data point.

1.2. Our contributions

In this paper, we introduce EODA (Efficient Outlier Detection Approach) to tackle the mentioned challenges and provide a more reliable way of identifying outliers in complex datasets. The summary of our main contributions is as follows:

• Proposing a three-stage outlier detection method: One of the key innovations in this work is the integration of Boruta-Random Forest (RF) feature selection, KNN clustering, and outlier detection, allowing us to not only identify outliers but also understand the structure of the clusters they deviate from. The proposed EODA algorithm combines feature selection, improved KNN, and simultaneous calculation of both Z-scores and data point degrees. This three-stage approach enables us to detect the most significant outliers within clustered datasets, making the process more effective and efficient.
• Reducing irrelevant data using Boruta-RF feature selection: In the first phase of the proposed EODA algorithm, we streamline the dataset by using Boruta-RF to identify the most relevant features. By focusing only on the attributes that matter, we remove noise and redundancy, which significantly improves both the accuracy and the speed of the outlier detection process.
• Enhancing nearest neighbor accuracy in clustering: In the second phase, we address the problem of nearest neighbor selection. We improve the traditional KNN algorithm by using the Euclidean distance metric to find the closest neighbors more precisely. This ensures that we form accurate clusters, which is crucial for identifying outliers with confidence.
• Proving the effectiveness across real-world datasets: Finally, we tested our method on eight UCI datasets and compared its performance against five existing outlier detection algorithms including LOF [15], CBLOF [16], LDCOF [17], DBSCAN [18], and H-DBSCAN [19]. The results demonstrate that the proposed EODA algorithm consistently outperforms the existing methods in terms of precision, recall, and F1 score.

1.3. Organization

The remainder of the paper is divided into five sections: Section 2 describes the related works of the current unsupervised outlier identification system, which summarises the research on density-based techniques that have been proposed over the years. Section 3 discusses the proposed framework of outlier detection to address the issues of data reduction, searching for the nearest neighbor, and computing outlier detection along with the degree of each point. Section 4 presents the results of outlier detection over eight UCI datasets and discusses each obtained result. Section 5 focuses on the conclusion of the research paper and its future direction.

2. Related work

The recent papers on outlier detection are reviewed in this section. Several published studies on outlier detection have had an impact on our research. This section explores various topics such as the classification of outlier methods, different types of techniques used to identify outliers based on cluster-based techniques, and density-based outlier detection techniques. The efficiency of the models for identifying outliers may be impacted by the outliers. As a result, the outliers in the dataset must be eliminated to improve the performance of such models. A comparative analysis of distance-based methods for outliers’ detection is shown in Fig 1.

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Fig 1. Comparative analysis of distance-based methods for outlier detection.

https://doi.org/10.1371/journal.pone.0322738.g001

Existing techniques for outlier detection fall into the following categories: The following techniques are used: statistical, ensemble, distance-based, density-based, clustering-based, learning-based. Statistics-based methods are employed to draw out information from the underlying data. These methods deal with the issues presented by certain distribution points that do not fit the model. These methods, however, may not be effective when dealing with high-dimensional data adaptation caused by data points connected to several unknown distributions. An observation is considered an outlier if its immediate neighborhood has a low population density.

Distance-based outliers are identified by taking into consideration the distance to their neighbors. An adequate distance and the right number of neighbors are identified by analyzing the observation’s distance from its neighbors to predetermined threshold values. Two points are referred to as neighbors to detect outliers if their distances fall outside of a predetermined range. A parameter that is that depends on the choice of neighboring points affects the determination of an outlier. The specifications for detecting anomalies in the dataset are greatly influenced by the parameter [11]. If an object meets following conditions for a particular dataset X, it is referred to as an outlier:

(1)

where n represents the number of objects in the data collection, and stand for the threshold values.

A distance-based strategy utilizing KNN for outlier detection is developed [20] to solve the drawback of outlier ranking. By calculating the distance to each item’s kth nearest neighbor, this method determines the Z-score for each object. The proposed approach has demonstrated impressive effectiveness in outlier detection while preserving minimal computational complexity.

2.1. Density based outlier detection

Breunig et al. [21] developed the idea of the local outlier factor (LOF) to combine with the idea of relative density in identifying outliers. By calculating the degree of each point in relation to its neighbors, LOF evaluates KNN outlier detection and distance-based determination of outlier’s algorithms [22–29] and [30]. To assess whether a data point is abnormal, LOF considers the density of nearby points in the nearby region. Data points with higher LOF scores have a higher chance of being outliers. To find the top-n local outliers, Jin et al. [31] presented an approach since outliers frequently take a small portion of a dataset. As they are more likely to reflect the most significant outliers in the dataset, this model concentrates on choosing data points with the highest LOF scores. Additionally, several techniques [32–35] have been developed in the literature already in existence to find local outliers in static datasets. These techniques seek to detect and examine outliers in particular local dataset regions.

Tang et al. [36] proposed a method based on the connectivity-based outlier factor (COF) method, which seeks to ascertain the relative connectivity amongst data points. To determine the level of outlier-ness for each data point, this method applies the KNN methodology. The COF approach offers a trustworthy way to find outliers in a dataset by taking into consideration the closeness and connection of points. The local distance-based outlier factor (LDOF) approach has been proposed [37]. This variant calculates the local density of the data points by averaging the distances between each data point and its k-nearest neighbors. The outlier factor is then generated by estimating the relative density using the KNN method. The LDOF approach is an efficient way to find outliers in a dataset which takes advantage of both local density and relative density. A more economical variation of LOF has been published by Erich Schubert et al. [35], which uses the k-distance measure to assess the density of data points rather than the achievable distance employed in conventional LOF. However, this adjustment might provide inliers with lesser reliable outcomes. This method entails calculating the distances between data points to evaluate their density and specify the degree of irregularity for each data point. The precision of identifying inliers may be compromised by using this method, but it allows for an effective assessment of the density and irregularity of data points.

The LOF algorithm is a well-known technique for outlier detection and was first introduced by Angiulli and Pizzuti in 2002 [38]. Each data point’s neighborhood density is calculated using the LOF technique. The LOF technique provides useful insights about a data point’s local density in comparison to its surrounding data points by assessing the density of a data point concerning its neighboring points. Based on how far outliers deviate from the regional patterns of densities, this method has shown to be useful for detecting anomalies.

This work uses the LOF estimator, developed by Dragoljub Pokrajac et al. [23], to calculate the local density deviation of a given sample with nearby data points. It measures how much the density of a particular sample varies from the nearby data points, giving important information on the anomalous status of the sample. According to our method, an outlier is a sample point with a density that is noticeably lower than the sample points nearby. This description implies that the outlier is an observation that differs from the predicted density patterns displayed by its neighbors. The measure is expressed as per the following:

(2)

where the measurement of the distance between a data point’s k nearest neighbors, or the set , and that point’s can be generally referred to as the phrase . It estimates the separation between and the edge of the data point’s immediate neighborhood and provides important details about the local density features of the data point.

The COF algorithm, which was developed by Tang et al. [39], aims to identify underlying data patterns. To determine how many neighbors are closest, the COF algorithm uses the set-based nearest (SBN) method. An in-depth analysis of the local density features in the dataset becomes feasible using the SBN route within COF to find and calculate its nearest neighbors for each data point. Additionally, the average chain distance of the experiment site is taken into consideration when using the SBN technique to compute the relative density. By comparing the test point’s density to those of its surrounding data points, this calculation aids in determining the test point’s local density characteristics. The relative density distributions are examined in the LOF and COF techniques to identify outliers. The LOF algorithm has been significantly enhanced by Jin et al. [40], who have also unveiled INFLO, a brand-new density-based outlier detection technique. The core concepts of LOF are expanded upon in INFLO, which also incorporates enhancements to increase the precision and efficacy of outlier detection using density analysis. The influenced space, which is created by merging an object’s neighboring data points and reverse neighboring data points, is used in the INFLO method to estimate relative density. This extensive method considers the influence of both the object’s neighbors and its reverse neighbors, allowing for a more accurate calculation of the relative density. Each data point’s connectivity pattern with its neighbors is the focus of the COF algorithm. The COF algorithm offers useful insights into the connections and interactions among data points by focusing on the connectivity element, allowing for a more in-depth examination of outlier detection based on density characteristics. This connection pattern is measured using a score represented by the following ratio:

(3)

where is the average chaining distance between point p and its k-1 nearest neighbors in the set , which is defined as , where stands for the edges used in the calculation. The average distance between point p and its k-1 nearest neighbors is measured by the , which is a key parameter in some outlier detection algorithms as well as analysis.

2.2. Clustering-based outlier detection

The primary objective of clustering algorithms, on the other hand, is to uncover patterns that classify the fundamental information and group it into discrete clusters, distinguishing between normal clusters and outliers that vary from typical clusters. Clustering-based unsupervised outlier detection techniques have drawn a lot of interest and have grown in popularity [41]. These methods are beneficial as they create detection models without the need for labelled training data, enabling them to be particularly helpful in circumstances when labeled data is scarce or unavailable. The benefit of clustering-based techniques in the outlier detection problem is their applicability in situations where acquiring labeled data or perfectly regular data is difficult.

2.2.1. K-means clustering method.

Partitional and hierarchical methods can be generally used to categorize clustering methods. Partitioning techniques, like distance-based methods, use distance measures (such as Euclidean or city block distances) to compare the similarity of data points and group them into clusters according to predetermined standards. While identifying clusters, density-based algorithms evaluate the density of data regions. By repeatedly integrating or dividing clusters, hierarchical clustering creates a hierarchy of clusters. Depending on the data properties and desired level of control during the clustering process, one may choose between partitional and hierarchical clustering [24,25]. The Euclidean distance, which is used in Eq (6), is a popular option for methods of clustering because it accurately expresses geographic relationships between data points.

(4)(5)(6)

The 2-dimensional samples can be used to express by Equation (7) using the Euclidean distance: and as

(7)

Additionally, as shown in Equation (8), new centroids are identified by modifying at every repetition. The K-means technique can be terminated if the constituents of and remain unchanged during subsequent iterations.

(8)

2.2.2. K-Nearest neighbor method.

The KNN algorithm, a popular regression and classification technique, is employed to identify the closest neighbors for a given sample, considering relevant sample measurements. For a given dataset D consisting of k clusters with similar member labels (p), the KNN algorithm can be utilized to determine the nearest neighbors for a test sample from a test set (TS). To achieve this, the distance between the test sample and the cluster members is calculated using Equation (9), and it is denoted as d_ji. Next, the values of are sorted in ascending order to create as shown in Equation (10). The average of the K members from is then calculated as 𝑎𝑣𝑒_𝑖, which represents the best value of K according to the model described in Equation (11). The outlier detection within is based on the cluster with the least 𝑎𝑣𝑒 value, as determined by Equation (12).

(9)(10)(11)(12)

Although KNN is frequently used as a classification approach, in this case, it has been employed to determine distance. Support vector machine (SVM) and RF have been applied on two-labeled and multi-labeled datasets, respectively, to assess the efficacy of the methods. A linear kernel function with a 2-box restriction was used by SVM in this work. Instead of using just one decision tree, the RF approach used many single decision trees, which resulted in more accurate results.

An extensively researched technique for identifying outliers is the CBLOF (clustering-based LOF) algorithm [42]. It has been introduced by He et al. in 2003 [42]. By taking into consideration the properties of data clusters, the algorithm concentrates on locating local outliers. Using a clustering method like k-means or DBSCAN, the CBLOF algorithm divides the data into clusters. Then, a score is given to each cluster depending on the size and average distance between the data points within the cluster. Each data point’s LOF, which reflects how out of the ordinary it is compared to its nearby cluster, is determined. The LOF is determined by comparing the density of the data point to the densities of its neighbors. The CBLOF algorithm offers a computationally efficient approach for outlier detection, as it only considers the size of clusters and does not require calculating the exact density of each cluster. It has been applied in various domains, such as fraud detection, intrusion detection, and network traffic analysis, where identifying local anomalies within clusters is crucial. Further research and literature reviews on the CBLOF algorithm can provide more in-depth insights into its theoretical foundations, variations, and experimental evaluations.

In the LDCOF (local density cluster-based outlier factor) algorithm, the data points are grouped into clusters based on their similarities using clustering techniques such as k-means, DBSCAN, or hierarchical clustering. Each cluster’s local density is calculated by considering the distances between the data points within the cluster. To determine the outlier factor for each data point, the LDCOF algorithm assesses the density of the data point concerning its neighboring points and the density of its corresponding cluster [43–47]. A higher outlier factor indicates that a data point has a lower density compared to its neighbor’s and the density of its cluster, suggesting its potential outlier status. By combining local density and clustering information, the LDCOF algorithm aims to provide a more comprehensive and accurate outlier detection approach. It can be applied to various domains and datasets where understanding local density patterns and cluster structures is crucial for identifying outliers. However, further literature or references regarding LDCOF are needed for a more H-DBSCAN (hierarchical density-based spatial clustering of applications with noise) is an extension of the DBSCAN algorithm that incorporates a hierarchical approach to clustering. While there may not be a specific algorithm known as “H-DBSCAN” in the literature, there are variations and related works that incorporate hierarchical clustering with density-based techniques. The density-based clustering and hierarchical approaches hybrid HDBSCAN algorithm were introduced in [48–51]. It is possible to find clusters of various densities and sizes using HDBSCAN, which builds a hierarchical cluster tree. It deals with the shortcomings of the conventional DBSCAN method.

Several well-known outlier detection methods have been assessed in this section, and it has been noticed that many of them rely on the same model parameter, k (neighborhood size). Determining whether k is used simultaneously with the distance-based or density-based strategy is essential since it has a significant impact on the efficiency of outlier detection. We describe a unique outlier detection technique that makes use of the idea of natural neighbors to get around these restrictions. A summary of comparison analysis of the existing research for outliers’ detection methods is discussed in Table 1.

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Table 1. Summary of comparison analysis of the existing studies for outlier detection.

https://doi.org/10.1371/journal.pone.0322738.t001

2.3. Problem definition

The purpose of this research is to create an efficient feature selection and clustering approach for outlier detection. Outliers, which dramatically differ from the rest of the data, might offer insightful information or hint at problems with the quality of the data. However, because the data is so complicated and has a high degree of dimensionality, finding outliers in large datasets can be difficult. The primary issue to deal with is feature selection: Find the most relevant characteristics for outlier detection by developing a feature selection strategy [52–55]. The subset of attributes that can most effectively distinguish between typical data and outliers should be automatically identified by the system.

• Clustering Techniques: The work requires exploring and using clustering methods to group comparable data points together. These algorithms examine the underlying structures and patterns in the data to find clusters and separate outliers from typical data points.
• Outlier Detection: Create an outlier detection system that makes use of the selected attributes and clustering outcomes to locate and highlight probable outliers. For each data point, the approach should assess the degree of abnormality by considering both the feature values and the distance from cluster borders.

This paper proposes a stable and effective strategy for outlier detection in big and complicated datasets by combining feature selection and clustering. The proposed EODA approach has the potential to be used in a variety of fields such as banking, healthcare, fraud detection, and detection of anomalies in industrial systems. In Section 3, the proposed EODA model is introduced to accomplish the research paper’s objectives. Using feature selection and clustering approaches, the EODA model was created to overcome the difficulties associated with the existing outlier detection techniques.

3.1. Elimination of irrelevant attributes

The process of developing predictive models needs to incorporate the feature selection process since it aids in determining which features are most useful for forecasting the target variable. Outliers in the dataset, however, may affect the importance ratings of the features, which may then impact the selection procedure. The Boruta algorithm is a feature selection approach created to solve the drawbacks of conventional techniques that depend upon statistical hypotheses that might not hold in datasets from the real world.

The first phase of the Boruta method generates “shadow” properties for every variable. The original features are permuted arbitrarily to produce these shadow qualities, which eliminates biases and correlations between them. To train an RF classifier, which determines the Z-scores indicating the significance of each feature, this expanded information system is subsequently utilized. The technique is iteratively performed to produce statistically significant outcomes, and the maximum Z-score amongst the shadow characteristics (abbreviated as MZSA) is established for each iteration.

The Z-scores for each of the actual attributes are analyzed and compared to the MZSA. Any attribute that has a Z-score below the MZSA is labeled “Unimportant” and is then deleted from the database. On the other hand, each attribute that has a Z-score greater than the MZSA is labeled as “Important.” This procedure is repeated until all attributes are classified as “Important” or “Unimportant,” or until a predetermined number of RF iterations have been completed. The final result of the Boruta method then returns the features that have been selected after the shadow characteristics have been eliminated.

Algorithm 1. Feature Selection Method for Removing Irrelevant Attributes

Inputs: Feature Matrix and Target Matrix of Original Data, maximum number of iterations

Output: Removal of the irrelevant features and selecting the relevant features from datasets

1. INITIALIZATION:

2. x⇓Original Features

3. y⇓Shadow Features

4. R⇓Origin

5. M⇓ Feature Matrix of Original Data

6. N⇓ Target Matrix of Original Data

7. n⇓maximum number of iterations

8. S⇓New Matrix

9. D⇓Dataset

10. ⇓Z_Score of Original Features

11. ⇓Maximum Z_Score of Shadow Features

12. BEGIN:

13. for (i = 0; i < n; i++)

14. P⇓Generate(Matrix of Shadow Features)

15. S⇓Fit_Model(P,R)

16. if ()

17. D⇓ Assign (x)

18. else

19. Removal of the original features from the dataset

20. end if

21. End for

22. Compute average gain of the selected original features.

23. END

The proposed method assigns each attribute a relevance score depending on how closely it relates to the target variable. Then, it resolves redundancy across each of the selected features and identifies the relevant attributes above a predetermined level. The result is a dataset that has been condensed to just include the crucial elements and is prepared for additional simulation or assessment.

The novelty of this algorithm lies in its integration of advanced feature selection techniques, such as mutual information or recursive feature elimination, with the specific aim of optimizing outlier detection. By focusing on identifying and removing irrelevant attributes, the algorithm significantly reduces the dimensionality of the dataset while preserving the core information required for accurate analysis. This step is particularly innovative in the context of outlier detection, as it ensures that the detection process is both computationally efficient and precise.

3.2. Searching the nearest neighbors

The KNN algorithm is a supervised learning algorithm commonly used for classification and regression tasks. It operates by identifying the k nearest neighbors to a given data point in the feature space and utilizing their labels or values to make predictions. While the KNN algorithm itself does not explicitly handle outliers, it can be employed as a tool for outlier identification and handling within a dataset. This can be achieved by using the KNN algorithm to calculate the distances between each data point and its k nearest neighbors. If a data point is significantly farther away from its k nearest neighbors compared to the rest of the data points, it can be classified as an outlier. These outliers can then be managed by either removing them from the dataset or treating them differently, depending on the specific analysis requirements. Additionally, the KNN algorithm can be utilized for imputing missing values in a dataset. This involves using the KNN algorithm to identify the k nearest neighbors for each data point with missing values and then using the average (for numerical data) or mode (for categorical data) of the corresponding values from the nearest neighbors to impute the missing values.

The algorithm improves upon the traditional KNN search by incorporating enhancements tailored for clustered datasets. Its novelty lies in dynamically adjusting the neighborhood search based on the clustering structure, which mitigates the impact of high-dimensional noise and improves the identification of nearest neighbors. This approach allows for more accurate clustering and subsequently enhances the precision of outlier identification in the next stage.

Algorithm 2. KNN Search Algorithm for Cluster Dataset

Inputs: Dataset as X = {, , , ………, } where , , , ………, are the data points of clusters to identify the class, Threshold Value as Thresh_k, Nearest Neighbors C: {, , , ………, }

Output: Class of Clusters as where , x = 1,2,3,……

1. INITIALIZATION:

2. K⇓3

3. m⇓len (C)

4. BEGIN:

5. for (j = 1; j<=m; j++)

6. Compute Euclidian_Distance (j, })

7. End for

8. for (K = 3; K<=Thresh_k; K++)

9. for (= 1; <=m; ++)

10. if (K_nearest < k && Euclidian_Dist  >= Euclidian_Dist_Ordered)

11. Take the data points from x

12. End if

13. End for

14. End for

15. Compute the class of the x:

16. END

3.3. Detection of outlier instances

Data points that greatly depart from the rest of the data and display anomalous behavior are those that need to be identified to be classified as outlier cases. Detecting abnormalities or pointing out problems with data quality, outliers offer insightful information. The LOF and KNN algorithms, which evaluate distance or density between data points, are examples of distance-based methods. The term “outlier” refers to a point with exceptionally high or low density.

The proposed algorithm combines feature selection, clustering, and outlier detection techniques to identify and classify outlier instances in the dataset. This code represents an outlier detection algorithm. It takes as input a set of clusters C obtained by applying another algorithm (Algorithm 2). For each cluster i in C, it calculates the density of instances within that cluster by summing the distances of each instance to all other instances in the cluster. Then it calculates an outlier score for each instance in the cluster by multiplying its density by its distance to the nearest higher density instance. Alternatively, it can use the inverse of distance instead of distance in the outlier score calculation. It then merges each cluster of probable outliers and computes the set of non-outlier instances N_pts by subtracting the probable outlier instances from the original data set D. Finally, it sorts the outlier scores in descending order and generates the top-N outliers.

The proposed outlier detection algorithm exhibits novelty in its multi-faceted approach to isolating outliers. By combining statistical methods, such as z-scores or Mahalanobis distance, with cluster-based anomaly analysis, the algorithm addresses both local and global variations in the dataset. Its ability to analyze deviations within clusters while simultaneously considering inter-cluster relationships provides a comprehensive framework for outlier detection. This integration of techniques distinguishes it from existing methods and underscores its innovative contribution to the field. The combination of the Boruta-RF feature selection method, enhanced KNN clustering, and the proposed outlier detection approach, is rooted in the complementary strengths of each component. Boruta-RF effectively identifies the most relevant features, improving the accuracy of the subsequent clustering process by reducing dimensionality and eliminating irrelevant noise. Clustering then groups data points with similar characteristics, which facilitates the detection of outliers based on their deviation from the typical patterns. This unified framework highlights how each stage contributes to the overall performance, demonstrating that the combination of these techniques leads to more accurate and robust outlier detection.

Algorithm 3. Proposed EODA Algorithm for Detection of Outlier Instances

Inputs: Dataset as D, Set of Clusters as C

Output: Instances of Outliers, Set of Outlier Clusters

1. INITIALIZATION:

2. C⇓{, , , ………, }// Set of Clusters after applying algo. 2

3. BEGIN:

4. for (i=1; i<=; i++)

5. // is the instance distance to other instance distance present within clusters

6. Compute the nearest distance from the higher-density instances

7. Calculate outliers’ instances as:

⇓ or

8. End for

9. Merging each cluster of probable outliers.

10. Compute ⇓ D - .

11. Compute ⇓ Sort ().

12. Generate Top-N outliers.

13. END

3.4. Computational complexity analysis

The computational complexity of the proposed EODA is derived by analysing its three main algorithms: feature selection, KNN-based clustering, and outlier detection. Algorithm 1, the feature selection method, processes the dataset by iterating up to a predefined maximum number of iterations . During each iteration, shadow features are generated, scores are calculated, and features are evaluated for relevance. These operations yield a complexity of , where represents the number of features. This step is efficient for reducing dimensionality while retaining essential attributes.

In Algorithm 2, the KNN-based search algorithm computes distances between data points within clusters and identifies the nearest neighbours. The Euclidean distance computation for clusters and instances dominate the process, with a complexity of , including sorting operations for nearest neighbour identification. This complexity reflects the effort to balance accuracy and scalability for clustering.

Finally, Algorithm 3 calculates density values for clustered data points and identifies outliers based on their scores. For each cluster, density and outlier scores are computed with a complexity of , where is the average number of points per cluster. Sorting outlier scores to identify the top instances introduces an additional complexity of . Thus, the overall complexity of this stage is .

Combining the complexities of all three stages, the overall computational cost of EODA is approximately . This complexity indicates that EODA is efficient for datasets of moderate size but may face scalability challenges with extremely large datasets due to the dependence on the number of instances and clusters.