A digital-based aerosol fire extinguishing monitoring system

Abstract

The present invention discloses a digital aerosol fire extinguishing monitoring system, which includes a data acquisition module, a distance metric definition module, a local factor design module, a clustering processing module, a parameter optimization module, and an aerosol fire extinguishing monitoring module. The present invention belongs to the field of data processing, and specifically refers to a digital aerosol fire extinguishing monitoring system. This solution introduces the definition of a positive definite matrix for distance metric, taking into account the correlation of each sensor; introduces Euclidean distance attenuation and time weight to design local factors, effectively suppressing single-point noise and instantaneous anomalies in aerosol fire extinguishing monitoring; based on the reward and punishment mechanism of quadratic membership update, rewards data in the core area and suppresses outliers, further strengthening the identification of true abnormal monitoring; thereby significantly improving the accuracy of fire extinguishing monitoring; based on the exploratory global update mechanism and the exploitative local update mechanism, meeting the dual requirements of fast response and optimized accuracy; thereby improving the fire extinguishing monitoring effect.

Description

Technical Field

[0001] The present invention relates to the field of data processing, and specifically refers to a digital aerosol fire extinguishing monitoring system. Background Art

[0002] An aerosol fire extinguishing monitoring system is a comprehensive system for real-time sensing, analyzing, and feedbacking the distribution and concentration of aerosol fire extinguishing agents in a protected area. However, in general aerosol fire extinguishing monitoring systems, it is difficult to fairly compare parameters with different correlations, which is prone to false detection or missed detection, lacks the collaborative utilization of information between adjacent sensors, and fails to effectively process sudden instantaneous interferences, thereby resulting in poor accuracy of fire extinguishing monitoring; in general aerosol fire extinguishing monitoring systems, there are problems such as improper setting of hyperparameters, and it is difficult to balance rapid response and optimization accuracy during the optimization process, thereby resulting in poor fire extinguishing monitoring effects. Summary of the Invention

[0003] In view of the above situation, to overcome the defects of the prior art, the present invention provides a digital aerosol fire extinguishing monitoring system. Regarding the problems in general aerosol fire extinguishing monitoring systems, such as difficult fair comparison of parameters with different correlations, prone to false detection or missed detection, lack of collaborative utilization of information between adjacent sensors, and failure to effectively process sudden instantaneous interferences, resulting in poor accuracy of fire extinguishing monitoring, this solution introduces a positive definite matrix to define distance metrics and takes into account the correlations of each sensor; introduces Euclidean distance attenuation and time weights to design local factors, effectively suppressing single-point noise and instantaneous anomalies in aerosol fire extinguishing monitoring; based on a reward and punishment mechanism updated by quadratic membership, rewards data in the core area and suppresses outliers, further strengthening the identification of true abnormal monitoring; thereby significantly improving the accuracy of fire extinguishing monitoring; regarding the problems in general aerosol fire extinguishing monitoring systems, such as improper setting of hyperparameters, and difficult to balance rapid response and optimization accuracy during the optimization process, resulting in poor fire extinguishing monitoring effects, this solution is based on an exploratory global update mechanism and an exploitative local update mechanism, comprehensively balancing global and local searches, meeting the dual requirements for rapid response and optimization accuracy; thereby improving the fire extinguishing monitoring effect.

[0004] The technical solution adopted by the present invention is as follows: A digital aerosol fire extinguishing monitoring system provided by the present invention includes a data acquisition module, a distance metric definition module, a local factor design module, a clustering processing module, a parameter optimization module, and an aerosol fire extinguishing monitoring module;

[0005] The data acquisition module acquires historical aerosol fire extinguishing monitoring data and monitoring status;

[0006] The distance metric definition module introduces a positive definite matrix to define distance metrics;

[0007] The local factor design module introduces time weights to design local factors;

[0008] The clustering processing module iteratively updates the membership degree, clustering center, and positive definite matrix based on the defined objective function to achieve clustering of the monitoring data;

[0009] The parameter optimization module optimizes the hyperparameters of the clustering processing model;

[0010] The aerosol fire extinguishing monitoring module conducts aerosol fire extinguishing monitoring on the real-time collected data based on the clustering results.

[0011] Furthermore, the data acquisition module uses the monitoring status as a data label, which is only selected as a cluster label; performs feature engineering processing on the collected data to obtain a fire extinguishing monitoring data set.

[0012] Furthermore, the distance metric definition module defines a distance metric value, expressed as: ; where is the i-th data vector; is the k-th clustering center vector; is the positive definite matrix used in the k-th region; T is the matrix transpose; is a user-defined distance metric.

[0013] Furthermore, the local factor design module utilizes neighborhood information by designing local factors; introduces an additional time weight factor ; The local factor is expressed as: ; ; where is the local factor; both g and i are data indices; is the Euclidean distance between the i-th data and the g-th data; is the membership degree of the g-th data belonging to the k-th cluster; is the updated membership degree; is the g-th data vector; is the neighbor set of the i-th data; and are the sampling times of the data points; is the time scale parameter.

[0014] Furthermore, the clustering processing module clusters the fire extinguishing monitoring data set, specifically including the following:

[0015] Objective function definition unit, expressed as: ; where J is the objective function; c is the number of clusters; n is the number of data points; m is the fuzzy index; is the balance parameter; initializes the clustering center and membership degree; is the membership degree of the i-th data belonging to the k-th cluster; Is the updated ;

[0016] Iterative optimization unit; update the membership degree, and update the membership degree by using the weighted distance between the current data and the cluster center and the local factor; expressed as: ; update the cluster center, and update the cluster center by weighted average; expressed as: ;

[0017] Positive definite matrix update unit; construct the covariance matrix, and the covariance matrix is expressed as: ; positive definite matrix is updated to: ; where; is the silhouette coefficient of the k-th cluster; is the proportionality constant; M is the dimension of the data vector; det(·) is the determinant;

[0018] Quadratic update mechanism setting unit; when the monitoring data significantly tends to the core area, adopt the reward mechanism; adopt suppression for the outlier data; thus realizing the quadratic update of the membership degree; after dividing each cluster area, when , it is determined to fall in the core area, ; otherwise it is determined to fall in the outlier area, ; where, is the membership degree of the i-th data point belonging to the k-th cluster; is the membership degree after quadratic update; is the local information determination suppression rate; the local information determination suppression rate is expressed as: ; ; where, is the local membership degree; is whether the j-th data point belonged to the k-th cluster in the previous iteration, if so, then , otherwise ; is the Euclidean distance between the i-th data point and the j-th data point; and are respectively the weighted distance metric value and the weighted local factor value between the data point in the k-th cluster and the cluster center; and are respectively the weighted distance metric value and the weighted local factor value within the neighborhood of the i-th data point;

[0019] Iterative determination unit; if the objective function converges, the iteration ends and the clustering result is obtained; if the maximum number of iterations is reached, parameter optimization is performed based on the parameter optimization module; otherwise, continue to update the cluster center and assign data points.

[0020] Further, the parameter optimization module optimizes the hyperparameters in the clustering process, specifically including the following:

[0021] Initialization unit; establish an optimization space based on the hyperparameters in the clustering process and initialize the optimization population;

[0022] Fitness function definition unit; the fitness function is expressed as: ; where and are different cluster centers, p and q are cluster center indices;

[0023] Exploration-based global update unit; perform a large-step search globally, expressed as: ; ; where is the position of the optimized individual after the first update; is the position of the optimized individual before the update; W is the dynamic weight; RB is the search range; S is the decay rate; t is the current optimization iteration; is the maximum number of optimizations; RS is the minimum search range; is a random number between 0 and 1; is the initial movement threshold; is the flight operator;

[0024] Utilization formula local update unit; perform a refined search based on the current optimal solution, expressed as:

[0025] ; where is the position after the second update; is the behavior weight; rand is a random number between 0 and 1; is a random number between 0 and 1, independent of r1; is the second movement threshold; bf is the benefit factor; is the position of the optimal solution;

[0026] Optimization determination unit; set the fitness threshold. If the maximum number of optimizations is reached or the fitness value of the optimal solution is higher than the fitness threshold, the optimization ends, and the clustering parameters are set based on the position of the optimal solution to obtain the clustering result.

[0027] Further, the aerosol fire extinguishing monitoring module, based on the clustering result, uses the label of the cluster center as the cluster label; real-time collects aerosol fire extinguishing monitoring data, distributes it based on the clustering processing module, and uses the corresponding cluster label of the aerosol fire extinguishing monitoring data as the monitoring result.

[0028] The beneficial effects achieved by the present invention using the above solution are as follows:

[0029] (1) Aiming at the problems existing in general aerosol fire extinguishing monitoring systems, such as the difficulty in fairly comparing parameters with different correlations, prone to false detection or missed detection, lack of collaborative utilization of information between adjacent sensors, and ineffective handling of sudden instantaneous interference, which lead to poor accuracy of fire extinguishing monitoring. This solution introduces the definition of distance metric using positive definite matrices and takes into account the correlations of each sensor; introduces Euclidean distance attenuation and time weights to design local factors, effectively suppressing single-point noise and instantaneous anomalies in aerosol fire extinguishing monitoring; based on the reward and punishment mechanism updated by quadratic membership, rewards data in the core area and suppresses outliers, further strengthening the identification of true anomalies; thereby significantly improving the accuracy of fire extinguishing monitoring.

[0030] (2) Aiming at the problems existing in general aerosol fire extinguishing monitoring systems, such as improper setting of hyperparameters, and the optimization process is difficult to balance rapid response and optimization accuracy, which leads to poor fire extinguishing monitoring effect. This solution is based on an exploratory global update mechanism and an exploitative local update mechanism, comprehensively balancing global and local searches to meet the dual requirements of rapid response and optimization accuracy; thereby improving the fire extinguishing monitoring effect. Description of the Drawings

[0031] Figure 1 It is a schematic flow chart of an aerosol fire extinguishing monitoring system based on digitization provided by the present invention;

[0032] Figure 2 It is a schematic flow chart of the clustering processing module.

[0033] The drawings are used to provide a further understanding of the present invention and constitute a part of the specification. Together with the embodiments of the present invention, they are used to explain the present invention and do not constitute a limitation to the present invention. Detailed Embodiments

[0034] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments; based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative work belong to the scope of protection of the present invention.

[0035] In the description of the present invention, it should be understood that the terms "upper", "lower", "front", "rear", "left", "right", "top", "bottom", "inner", "outer", etc. indicating the orientation or positional relationship are based on the orientation or positional relationship shown in the drawings, and are only for the convenience of describing the present invention and simplifying the description, rather than indicating or implying that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation, and therefore should not be construed as a limitation to the present invention.

[0036] Example 1, refer toFigure 1 , a digital aerosol fire extinguishing monitoring system provided by the present invention includes a data acquisition module, a distance metric definition module, a local factor design module, a clustering processing module, a parameter optimization module, and an aerosol fire extinguishing monitoring module;

[0037] The data acquisition module acquires historical aerosol fire extinguishing monitoring data and monitoring status;

[0038] The distance metric definition module introduces a positive definite matrix to define the distance metric;

[0039] The local factor design module introduces a time weight to design the local factor;

[0040] The clustering processing module iteratively updates the membership degree, cluster center, and positive definite matrix based on the defined objective function to achieve clustering of the monitoring data;

[0041] The parameter optimization module optimizes the hyperparameters of the clustering processing model;

[0042] The aerosol fire extinguishing monitoring module performs aerosol fire extinguishing monitoring on the real-time collected data based on the clustering results.

[0043] Example 2, refer to Figure 1 , based on the above example, the data acquisition module uses the monitoring status as a data label, which is only used as a cluster label selection; performs feature engineering processing on the collected data to obtain a fire extinguishing monitoring data set; the historical aerosol fire extinguishing monitoring data includes basic environmental parameters, aerosol-related data, optical parameter data, sampling time, and auxiliary data; the basic environmental parameters include temperature, humidity, and atmospheric pressure; the aerosol-related data includes aerosol concentration and particle size distribution; the optical parameter data includes the density, distribution, and thermal radiation of the smoke; the auxiliary data includes wind speed, wind direction, and other pollutant concentrations; the monitoring status includes normal, mildly abnormal, moderately abnormal, and severely abnormal.

[0044] Example 3, refer to Figure 1 , based on the above example, in the aerosol fire extinguishing monitoring, there is a cross-influence in the monitoring data of different sensors. To effectively reflect the true differences between the data, a distance metric value is defined, expressed as: ; where is the i-th data vector; is the k-th cluster center vector; is the positive definite matrix used in the k-th region; T is the matrix transpose; is the custom distance metric; using the positive definite matrix Incorporating prior information into the covariance structure can more accurately characterize the correlation and scale differences among various monitoring data, which helps to accurately detect abnormal aerosol distributions.

[0045] Example 4, refer to Figure 1 , this example is based on the above example. In the local factor design module, in the monitoring network and in the regional sensor network, adjacent monitoring data usually has a high correlation; by designing local factors, the influence of individual noises is weakened using neighborhood information; an additional time weight factor is introduced to reduce the local perturbation caused by instantaneous anomalies; the local factor is expressed as: ; ; where is the local factor; both g and i are data indices; is the Euclidean distance between the i-th data and the g-th data; is the membership degree of the g-th data belonging to the k-th cluster; is the updated membership degree; is the g-th data vector; is the neighbor set of the i-th data; and are the sampling times of the data points; is the time scale parameter.

[0046] Example 5, refer to Figure 1 and Figure 2 , this example is based on the above example. The clustering processing module clusters the fire monitoring data set, which specifically includes the following:

[0047] Objective function definition unit, expressed as: ; where J is the objective function; c is the number of clusters; n is the number of data points; m is the fuzzy index; is the balance parameter; initialize the cluster centers and membership degrees; is the membership degree of the i-th data belonging to the k-th cluster; is the updated ;

[0048] Iterative optimization unit; for each cluster region, grouping and anomaly detection of the monitoring data are achieved through iterative updates; update the membership degrees, and update the membership degrees of the monitoring data using the weighted distance between the current data and the cluster centers and the local factors to reflect the dependence degree of the data points on each cluster; expressed as: ; update the cluster centers, update the cluster centers by weighted averaging, and synthesize the influence of individual data points and their neighborhood data, so that the cluster centers better fit the overall characteristics within the actual region; expressed as: ;

[0049] Positive definite matrix update unit; construct a covariance matrix to capture the distribution characteristics of each monitoring data, and then use its inverse matrix to construct a positive definite matrix; covariance matrix It is expressed as: ; Positive definite matrix Is updated to: ; Where; Is the silhouette coefficient of the k-th cluster; Is a proportionality constant; M is the dimension of the data vector; det(·) is the determinant;

[0050] Quadratic update mechanism setting unit; there are often external environmental interferences and noise data in aerosol fire monitoring, and effective weakening of abnormal data is achieved through inhibitory competitive learning; when the monitoring data significantly tends to the core area of the fire, a reward mechanism is adopted; on the contrary, for outlier data, inhibition is adopted; thus realizing the quadratic update of membership degree; after dividing each clustering area, different update strategies are adopted according to whether the sample falls in the core area or the outlier area. When , it is determined to fall in the core area, ; On the contrary, it is determined to fall in the outlier area, ; Where, Is the membership degree of the i-th data point belonging to the k-th cluster; Is the membership degree after quadratic update; Is the inhibition rate determined by local information; the inhibition rate determined by local information is expressed as: ; ; Where, Is the local membership degree; Is whether the j-th data point belonged to the k-th cluster in the previous iteration. If so, , otherwise ; Is the Euclidean distance between the i-th data point and the j-th data point; And Are respectively the weighted distance metric value and the weighted local factor value between the data point in the k-th cluster and the cluster center; And Are respectively the weighted distance metric value and the weighted local factor value within the neighborhood of the i-th data point;

[0051] Iterative decision-making unit; if the objective function converges, the iteration ends and the clustering result is obtained; if the maximum number of iterations is reached, parameter optimization is performed based on the parameter optimization module; otherwise, continue to update the cluster center and assign data points.

[0052] By performing the above operations, for the general aerosol fire extinguishing monitoring system, it is difficult to fairly compare parameters with different correlations, which is prone to false detection or missed detection, lacks the collaborative utilization of information between adjacent sensors, and fails to effectively handle sudden instantaneous interference, thus resulting in poor accuracy of fire extinguishing monitoring. This solution introduces the definition of distance metric using positive definite matrices and takes into account the correlations of each sensor; introduces Euclidean distance attenuation and time weights to design local factors, effectively suppressing single-point noise and instantaneous anomalies in aerosol fire extinguishing monitoring; based on the reward and punishment mechanism updated by quadratic membership degree, rewards data in the core area and suppresses outliers, further strengthening the identification of true anomalies; and thus significantly improves the accuracy of fire extinguishing monitoring.

[0053] Example Six. Refer to Figure 1 , based on the above example, the parameter optimization module optimizes the hyperparameters in the clustering process, specifically including the following:

[0054] Initialization unit; establish an optimization space based on the hyperparameters in the clustering process and initialize the optimization population;

[0055] Fitness function definition unit; the fitness function is expressed as: ; where and are different clustering centers, p and q are clustering center indices;

[0056] Exploration-style global update unit; perform a large-step search within the global scope, expressed as: ; ; where is the position of the optimized individual after the initial update; is the position of the optimized individual before the update; W is the dynamic weight; RB is the search range; S is the attenuation rate; t is the current optimization number; is the maximum optimization number; RS is the minimum search range; is a random number from 0 to 1; is the initial movement threshold; is flight operator;

[0057] Utilization formula local update unit; perform a refined search based on the current optimal solution, expressed as:

[0058] ; where is the position after the secondary update; is the behavior weight; rand is a random number from 0 to 1; is a random number from 0 to 1, independent of r1; is the secondary movement threshold; bf is the benefit factor; is the position of the optimal solution;

[0059] The optimization determination unit sets a fitness threshold. If the maximum number of optimization times is reached or the fitness value of the optimal solution is higher than the fitness threshold, the optimization ends. The clustering parameters are set based on the position of the optimal solution to obtain the clustering result.

[0060] By performing the above operations, for the problem that the general aerosol fire extinguishing monitoring system has improper hyperparameter settings, and it is difficult to balance rapid response and optimization accuracy in the optimization process, resulting in poor fire extinguishing monitoring effects. This solution is based on the exploratory global update mechanism and the exploitative local update mechanism, comprehensively balancing global and local searches, meeting the dual requirements for rapid response and optimization accuracy; thereby improving the fire extinguishing monitoring effect.

[0061] Example Seven, refer to Figure 1 , based on the above example, the aerosol fire extinguishing monitoring module takes the label of the clustering center as the cluster label based on the clustering result; real-time collects aerosol fire extinguishing monitoring data, and distributes it based on the clustering processing module, and takes the corresponding cluster label of the aerosol fire extinguishing monitoring data as the monitoring result; if the monitoring result is moderately abnormal, the sampling frequency is increased; if it is severely abnormal, a warning process is performed.

[0062] It should be noted that in this article, relational terms such as first and second are only used to distinguish one entity or operation from another entity or operation, and do not necessarily require or imply any actual relationship or order between these entities or operations. Moreover, the term "comprises", "comprising" or any other variation thereof is intended to cover a non-exclusive inclusion, such that a process, method, article or device comprising a series of elements includes not only those elements, but also other elements not expressly listed, or elements inherent to such process, method, article or device.

[0063] Although the embodiments of the present invention have been shown and described, for those of ordinary skill in the art, it can be understood that various changes, modifications, substitutions and variations can be made to these embodiments without departing from the principles and spirit of the present invention.

[0064] The above describes the present invention and its embodiments. This description is not restrictive, and what is shown in the drawings is only one of the embodiments of the present invention, and the actual structure is not limited thereto. All in all, if those of ordinary skill in the art are inspired by it and design similar structural ways and embodiments without creative efforts without departing from the purpose of the present invention, they should all fall within the protection scope of the present invention.

Claims

1. A digital aerosol fire extinguishing monitoring system, characterized in that: The system includes a data acquisition module, a distance metric definition module, a local factor design module, a clustering processing module, a parameter optimization module, and an aerosol fire extinguishing monitoring module; The data acquisition module acquires historical aerosol fire extinguishing monitoring data and monitoring status; The distance metric definition module introduces a positive definite matrix to define the distance metric; The local factor design module introduces a time weight to design the local factor; The clustering processing module iteratively updates the membership degree, cluster center, and positive definite matrix based on the defined objective function to achieve clustering of the monitoring data; The parameter optimization module optimizes the hyperparameters of the clustering processing model; The aerosol fire extinguishing monitoring module performs aerosol fire extinguishing monitoring on the real-time collected data based on the clustering results; The data acquisition module uses the monitoring status as a data label, which is only used for selecting the cluster label; performs feature engineering processing on the collected data to obtain the fire extinguishing monitoring data set; The local factor design module utilizes neighborhood information by designing local factors; an additional time weight factor is introduced. ; The local factor is expressed as: ; ; where is the local factor; both g and i are data indices; is the Euclidean distance between the i-th data and the g-th data; is the membership degree of the g-th data belonging to the k-th cluster; is the updated membership degree; is the g-th data vector; is the neighbor set of the i-th data; and are the sampling times of the data points; is the time scale parameter; The clustering processing module clusters the fire extinguishing monitoring data set, which specifically includes the following: The objective function definition unit is expressed as: ; where J is the objective function; c is the number of clusters; n is the number of data points; m is the fuzzy exponent; is the balance parameter; initialize the cluster centers and membership degrees; is the membership degree of the i-th data belonging to the k-th cluster; is the updated ; Iterative optimization unit; update the membership degree, and update the membership degree by using the weighted distance between the current data and the cluster center and the local factor; expressed as: ; update the cluster center, and update the cluster center by the weighted average method; expressed as: ; Positive definite matrix update unit; construct a covariance matrix, the covariance matrix is expressed as: ; positive definite matrix is updated to: ; where; is the silhouette coefficient of the k-th cluster; is a proportionality constant; M is the dimension of the data vector; det(·) is the determinant; A secondary update mechanism setting unit; An iteration determination unit; if the objective function converges, the iteration ends and the clustering result is obtained; if the maximum number of iterations is reached, parameter optimization is performed based on the parameter optimization module; otherwise, the cluster center is updated and the data points are assigned; The parameter optimization module optimizes the hyperparameters in the clustering process, which specifically includes the following: An initialization unit; establishes an optimization space based on the hyperparameters in the clustering process and initializes the optimization population; Fitness function definition unit; fitness function Expressed as: ; where and are different cluster centers, and p and q are cluster center indices; Exploratory global update unit; performs a large-step search within the global scope, expressed as: ; ; where is the optimized individual position after the initial update; is the optimized individual position before the update; W is the dynamic weight; RB is the search range; S is the attenuation rate; t is the current optimization iteration; is the maximum optimization iteration; RS is the minimum search range; is a random number between 0 and 1; is the initial movement threshold; is the flight operator; A local update unit using the formula; performs refined search based on the current optimal solution, expressed as: ; among them, is the position after the second update; is the behavior weight; rand is a random number from 0 to 1; is a random number from 0 to 1, independent of r1; is the second movement threshold; bf is the benefit factor; is the position of the optimal solution; An optimization determination unit; sets a fitness threshold, if the maximum number of optimization times is reached or the fitness value of the optimal solution is higher than the fitness threshold, the optimization ends, sets the clustering parameters based on the position of the optimal solution, and obtains the clustering result; The aerosol fire extinguishing monitoring module, based on the clustering results, uses the label of the cluster center as the cluster label; real-time collects aerosol fire extinguishing monitoring data, distributes based on the clustering processing module, and uses the corresponding cluster label of the aerosol fire extinguishing monitoring data as the monitoring result.

2. The aerosol fire extinguishing monitoring system based on digitization according to claim 1, characterized in that: The distance metric definition module defines the distance metric value, expressed as: ; where is the i-th data vector; is the k-th cluster center vector; is the positive definite matrix used in the k-th region; T is the matrix transpose; is a user-defined distance metric.

3. The aerosol fire extinguishing monitoring system based on digitization according to claim 2, characterized in that: The secondary update mechanism setting unit adopts a reward mechanism when the monitoring data significantly tends to the core area; it suppresses the outlier data; thereby realizing the secondary update of the membership degree; after dividing each clustering area, when , it is determined that it falls within the core area, ; Conversely, it is determined that it falls into the outlier region. ; where is the membership degree that the i-th data point belongs to the k-th cluster; is the membership degree after the second update; is the local information determination inhibition rate; it is expressed by the local information determination inhibition rate as: ; ; where is the local membership degree; indicates whether the j-th data point belongs to the k-th cluster in the previous iteration. If so, , otherwise ; is the Euclidean distance between the i-th data point and the j-th data point; and are respectively the weighted distance metric value and the weighted local factor value between the data points in the k-th cluster and the cluster center; and are respectively the weighted distance metric value and the weighted local factor value within the neighborhood of the i-th data point.

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