CN105635964A - Wireless sensor network node localization method based on K-medoids clustering - Google Patents

Abstract

The invention discloses a wireless sensor network node localization method based on K-medoids clustering. The method firstly uses a trilateration localization algorithm to obtain a plurality of localization results, and takes the primary localization results as initial samples for clustering analysis. Then a K-medoids clustering algorithm is used to perform clustering analysis on the primary localization results. The division of the best cluster can be obtained through iteration. Through analysis of the cluster member number, beacon nodes with the large error are found and removed. Finally the localization calculation is performed on the preferable beacon nodes by using a multilateral localization method which is modified by reference values. The localization method of the invention effectively reduces node localization error, and improves the localization precision of wireless sensor network nodes.

Description

Based on the wireless sensor network node locating method of K central point cluster

Technical field

The present invention relates to the wireless sensor network node locating method based on K central point cluster, belong to wireless sensor network node field of locating technology.

Background technology

Wireless sensor network (WirelessSensorNetworks, WSN) by have perception ability, computing power, wireless communication ability sensor node form, be widely used in the aspects such as military reconnaissance, production process monitoring, environmental monitoring. Self location of node of network is basis and the important support of sensor network application, the more specific location information of node of network is all needed, to utilize positional information to complete specific requirement in network communication and node cooperation in many Application Areass such as the route of position-based information, target monitoring and tracking.

Sensor network node locating method, with according to the estimation mechanism to node location, can be divided into (Range-based) based on range finding and the location algorithm (Range-free) two kinds without the need to range finding. Based on range finding location algorithm be by measured node between distance or special angle information, it may also be useful to trilateration, triangulation method or maximum likelihood estimation method estimate node apparent position; And the location algorithm without the need to range finding only relies on the connection of network to estimate the position coordinate of unknown node, its power consumption is smaller, but the accuracy of the node coordinate estimated is lower. Along with the raising to positioning accuracy request, the location algorithm based on range finding will have better development in node locating technique.

In the location algorithm based on range finding, conventional distance-finding method has based on time of arrival (toa) (or time difference) with based on received signal intensity instruction (receivedsignalstrengthindicator, RSSI) ranging technology. Do not need the hardware outside additionalamount based on the ranging technology of RSSI, its positioning precision relatively has again obvious advantage without the need to distance-finding method, so being the relatively more conventional method in wireless sensor network location based on the range finding of RSSI. But, in actual environment, signal is by the impact of multiple factor, and take off data can exist certain error, and these errors comprise the less systematic error of amplitude and the bigger thick error of amplitude, the existence of error directly affects positioning precision, thus the positioning precision causing algorithm reduces.

Summary of the invention

Technical problem to be solved by this invention is: provide a kind of wireless sensor network node locating method based on K central point cluster, the beaconing nodes that there is thick range finding error is screened, carry out with remaining beaconing nodes taking precious location Calculation by force, effectively improve positioning precision.

The present invention is for solving the problems of the technologies described above by the following technical solutions:

Based on the wireless sensor network node locating method of K central point cluster, utilize the positional information of each beaconing nodes unknown node to be positioned, comprise the steps:

Step 1, each beaconing nodes sends own location information to unknown node, after unknown node receives positional information, record Received signal strength intensity level, and utilize logarithm-normal distribution model that Received signal strength intensity level is converted into distance value, above-mentioned distance value appoints to get 3 be a combination, each is combinationally used three limit location algorithms, obtains the initial position message of unknown node;

Step 2, the initial position message utilizing K central point cluster algorithm step 1 to be obtained carries out cluster, obtains cluster result;

Step 3, t the cluster that contained element is minimum is found out from the cluster result that step 2 obtains, t is less than the cluster numbers that K central point cluster algorithm obtains, and finds out the beaconing nodes that in above-mentioned t cluster, each position information is corresponding, m the beaconing nodes that record occurrence number is maximum;

Step 4, removes m maximum for occurrence number beaconing nodes, and remaining beaconing nodes utilizes the polygon location algorithm improved unknown node is positioned calculating, and the reference value chosen in the polygon location algorithm of described improvement is:Wherein, the coordinate that (x, y) is unknown node, (x_i,y_i) it is the coordinate of remaining beaconing nodes, i=1 ..., F,F is the number of remaining beaconing nodes.

Preferably, the calculation formula of logarithm described in step 1-normal distribution model is:Wherein, PL (d) is the received signal power through distance d, PL (d₀) it is reference range d₀Corresponding received signal power, �� is path loss index, X_��For average is the Gaussian distributed random variable of 0, �� is standard deviation.

Preferably, described reference range d₀Get 1m.

Preferably, the detailed process of described step 2 is: (1), using the initial clustering sample of initial position message as K central point cluster, setting clusters number is K, random from initial sample selects K position as initial center point; (2) remaining position is added from the class representated by its nearest central point; (3) from remaining position, choose one at random replace any one central point in K initial center point; (4) calculate in (3) cost replaced, if cost is less than 0, then replace; (5) (3) and (4) are repeated, until when the cost replaced is greater than 0, cluster terminates.

Preferably, the calculation formula of described cost S is:Wherein, �� is initial clustering sample, ��_jFor jth class �� being clustered in K class, j=1 ..., K, c_jFor class ��_jCentral point, z_lFor current class ��_jIn except central point c_jOutside other optional positions.

The present invention adopts above technical scheme compared with prior art, has following technique effect:

1, localization method of the present invention, the defect of thick error effect it is subject to for the wireless sensor network node location algorithm found range based on RSSI, introduce K-central point cluster algorithm, utilize the positional information of the positional information location unknown node of beaconing nodes, first unknown node is carried out Primary Location; Then, utilize K central point cluster algorithm that Primary Location result is carried out cluster.

2, localization method of the present invention, according to cluster result, the beaconing nodes that there is bigger range finding error is screened, utilize the polygon location algorithm improved that unknown node is positioned calculating, improve the reference value of polygon location algorithm, reduce the interference of thick error and random noise, it is to increase wireless sensor network node positioning precision.

Accompanying drawing explanation

Fig. 1 is the schema of the present invention based on the wireless sensor network node locating method of K central point cluster.

Fig. 2 is the exemplary plot of K central point cluster algorithm of the present invention.

Fig. 3 be simulation example 100m*100m of the present invention region in the distribution plan of 100 wireless sensor nodes.

Fig. 4 is that beaconing nodes number is to the influence curve figure of positioning error.

Fig. 5 is that different thick error is to the influence curve figure of positioning result.

Fig. 6 is the positioning error distribution plan obtained after utilizing existing polygon algorithm simulating.

Fig. 7 is the positioning error distribution plan obtained after utilizing existing OTWC algorithm simulating.

Fig. 8 is the positioning error distribution plan obtained after utilizing the inventive method to emulate.

Embodiment

Being described below in detail embodiments of the present invention, the example of described enforcement mode is shown in the drawings, and wherein same or similar label represents same or similar element or has element that is identical or similar functions from start to finish. It is exemplary below by the enforcement mode being described with reference to the drawings, only for explaining the present invention, and limitation of the present invention can not be interpreted as.

In order to improve location algorithm precision further, the present invention mainly considers to reduce thick range finding error and the impact of locating effect is introduced K-central point cluster algorithm, improvement location algorithm (multilaterallocalizationalgorithmbasedK-medoidsclusterin ganalysis, KCML) based on K-central point cluster is proposed. First this algorithm utilizes three location, limit to obtain first positioning result, then, first positioning result is carried out cluster analysis, according to clustering result, find out and remove the beaconing nodes that there is thick range finding error, finally, the beaconing nodes utilizing distance measurement value that remaining error is less corresponding positions calculating. Comparing by carrying out emulation with existing method, the inventive method improves in positioning precision.

As shown in Figure 1, the detailed process of KCML algorithm flow of the present invention is:

1) beaconing nodes broadcast self information: node ID, own location information. Unknown node is receiving after the information of beaconing nodes, and record Received signal strength intensity level RSSI, is converted into distance value by signal strength values, and information is stored in the set self maintained.

2) unknown node uses the data of its own beacon node set to position calculating, utilizes three limit location algorithms to obtain M positioning result, and it can be used as the initial sample of cluster.

3) determine to need number K and the initial cluster center of classification, and obtain final cluster result by iteration, cluster result W₁,W₂,��,W_K, member's number is respectively n₁,n₂,��,n_K��

4) find out t the cluster subset that contained element is minimum, find out the beaconing nodes that in these subsets, each position point is corresponding, and record m the beaconing nodes that occurrence number is more, be the beaconing nodes that there is thick range finding error.

5), after removing m more beaconing nodes of occurrence number, the polygon location algorithm improved is utilized to position calculating.

Wireless signal propagation model typically has freeboard model, log-distance path loss model model and logarithm-normal distribution model. In the applied environment of reality, it is subject to the impact of the factors such as multipath, diffraction, obstacle due to signal, radio signal propagation paths loss difference to some extent compared with theoretical value. Thus producing following logarithm-normal distribution model on this basis, available (1) formula represents receiving end and the path loss model launching end:

$PL\left(d\right)=PL\left(d_0\right)+10*\mathrm{\α}log\frac{d}{d_0}+X_{\mathrm{\σ}}---\left(1\right)$

Wherein, PL (d) is the received signal power through distance d, PL (d₀) it is reference range d₀Corresponding received signal power (dB), d₀Generally getting 1m, �� is path loss index, its scope generally between 2 to 4, X_��For average is the Gaussian distributed random variable of 0, the scope of its standard deviation sigma is 4��10, d is true distance.

The basic thought of K central point cluster algorithm is: assume to there is n object, first K that wishes to obtain all objects divide, choosing the center (i.e. representative object) of K object as class at random, other remaining objects are according to adding corresponding nearest class to the distance of representative object; Then, utilize non-representative object to replace representative object by the mode of iteration, and utilize cost function to assess replacement cost, seek optimal classification scheme, to improve cluster quality. Repeatedly carrying out iteration, until meeting certain stop condition, drawing final division.

In fact K central point clustering method is a combinatorial optimization algorithm, it is possible to be described as following mathematical model, objective function:

$S={\min }_{z_l}{\Σ}_{j=1,j\∈\mathrm{\Ω}}^K{\Σ}_{l\∈{\mathrm{\Ω}}_j}\left|\right|c_j-z_l\left|\right|---\left(2\right)$

In formula, set omega is whole data set, subset ��_jRepresent a class, c_jRepresent central point, z_lRepresent class ��_jIn data sample except central point, | | c_j-z_l| | represent sample point z_lWith central point c_jDistance, formula (2) meets constraint condition:

��₁�Ȧ�₂�ȡ��Ȧ�_K=��, ��₁�ɦ�₂�ɡ��ɦ�_K=�� (3)

In optimizing process, c_jIt is constantly change, also can cause subset �� simultaneously_jChange, it is possible to think c_jIt is controlled variable, ��_jFor state variables. Traditional method of calculation (such as PAM algorithm) are all samples of traversal, constantly select new central point, if objective function decline, then replace, and repartition class. The optimization problem that target function type (2) is divided into two layers can be equivalent to:

$S={\min }_{c_l}\underset{z_l}{\min }{\Σ}_{j=1,j\∈\mathrm{\Ω}}^K{\Σ}_{l\∈{\mathrm{\Ω}}_j}\left|\right|c_j-z_l\left|\right|---\left(4\right)$

Wherein, internal layer optimization problem is aimed at given central point c_jDetermine the z that distance central point is nearest_l, and carry out the division of respective class; And outer optimization problem is equivalent to for given class ��_jDetermine best central point. Clustering is completed when objective function reaches optimum.

Such as, 15 sampled datas being divided into 4 classes, as shown in Figure 2, the "��" in dotted line circle represents and belongs to same class, "+" represent the point of the cluster centre in each class, if a class only exists data, its cluster centre is data itself.

When utilizing Received signal strength intensity unknown node to be positioned, the distance between node is obtained according to transmission model, unknown node filters out M the beaconing nodes nearest with oneself, using every 3 nodes wherein as a combination, utilize each combination location unknown node, the positioning result that each combination obtains is averaged, as finally locating coordinate. But, the mode directly getting average can not remove the impact of thick error of finding range, it is easy to causes error to increase. Use cluster algorithm to remove in the process of the present invention slightly poor, improve locating effect.

Use and first to be determined initial sample point during cluster algorithm, be three limit positioning results in the present invention. If a certain unknown node obtains M distance value through range finding, it is a combination by every for this M range information 3, each group is used three limit location algorithms, obtains altogetherIndividual positioning result, is initial position sample points.

KCML location algorithm is mainly divided into three parts: first, it is determined that initially estimate location sets, and utilizes K central point cluster algorithm to carry out big error range information screening, initial location estimate is carried out cluster, is divided into K classification by n position data. Then, algorithm finds out the big distance value of range finding error according to the result of cluster. Finally, the polygon location algorithm recycling improvement after being removed by big error distance measurement value positions calculating. Its detailed process is as follows:

(1) first determine initially to estimate location sets, and set the number K needing classification. In n positional information, random selection K position is as initial central point.

(2) remaining n-K position point is added from the class representated by its nearest central point.

(3) a central point object c not selected is selected_j, choose the non-central position sample points z not selected at random_l��

(4) calculate position point z_lAs total cost S during central point.

(5) if non-central some z_lReplace central point c_jAfter total cost S < 0 of calculating, then use z_lReplace current central point object c_j, form the set of K new central point.

(6) repeating step (3) is to (5), until not changing, namely all S are greater than 0.

After above-mentioned steps completes iteration, so that it may to obtain the cluster result of K final class, such as the information such as number of the estimation position point in each class.

By analyzing cluster result, according to the number of the element that each subset contains, the distance measurement value that error identifying is bigger, it is possible to reduce range finding error to the impact of positioning precision. Its process is as follows:

(1) the final cluster result utilizing K central point cluster algorithm to try to achieve represents for W={W₁,W₂,��,W_K, each subset W_iIn element number be respectively n₁,n₂,��,n_K��

(2) { n is gathered₁,n₂,��,n_KIn the corresponding cluster subset location data amount check of element, get the subset min{n that front t element is minimum₁,n₂,��,n_KForm new set H={h₁,h₂,��,h_t, the element number that these subsets comprise is respectively q₁,q₂,��,q_t��

(3) in the subset that t containing element is minimum, each element represents an estimation position point, namely combines the unknown node coordinate obtained by one group of beaconing nodes, represents the number of element in set H with G:

$G={\mathrm{\&Sigma;}}_{a=1}^t{q}_a,a=1,...,t---\left(5\right)$

And each estimation position is calculated by three distance measurement values in a combination, so just can obtain front m the distance measurement value that occurrence number in this G sample value is more, be the distance measurement value that error is bigger. So just obtain the distance measurement value that unknown node error is bigger, also just know beaconing nodes corresponding with it.

It is generally acknowledged and a cluster subset comprises estimation position is more is precision located higher result, and high positioning result is obtained by distance measurement value accurately. Therefore the range finding error existed in the cluster subset that contained element is less also can be bigger, the cluster that element is few is selected to find the bigger range information of error, after beaconing nodes corresponding for this bigger distance value of m error is removed, assume that residue beaconing nodes number is F, the polygon location algorithm improved is utilized to position calculating, it is possible to reduce range finding error to the impact of positioning precision.

The polygon location algorithm of tradition is by any one ranging information as a reference, and other range informations subtract this reference value, and solving equation obtains node estimation position. In this approach, if the reference value chosen exists error, the result so calculated through aforesaid method can bring more big error. All measuring results after removing the m beaconing nodes that relatively big error is corresponding are averaging and construct equation by the present invention, subtract this average equation with observed value equation and obtain F equation, and then simultaneous equations solve. Can avoid like this reducing the impact of measuring error because choosing that reference value is improper and the error that causes.

Assuming that unknown node coordinate is set to (x, y), the coordinate of its neighbours' beaconing nodes is (x_i,y_i), i=1,2 ..., F, the distance value obtained by signal strength measurement is d_i, then this unknown node can be set up system of equations (6):

$\left\{\begin{array}{c}{(x-x_1)}^2+{(y-y_1)}^2=d_1^2\\ .\\ .\\ .\\ {(x-x_F)}^2+{(y-y_F)}^2=d_F^2\end{array}\right.---\left(6\right)$

The polygon location algorithm of tradition appoints to get an equation as with reference to value, all the other equations subtract this reference value in system of equations (6), if the reference value error chosen is relatively big, then the result obtained like this can exist relatively big error. In order to reduce the impact of range finding error, the average of corresponding parameter as reference value, is made by the present invention:

$\stackrel{\&OverBar;}{d}=\frac{1}{F}{\mathrm{\&Sigma;d}}_i---\left(7\right)$

Equation (8) can be constructed:

${(x-\frac{1}{F}{\mathrm{\&Sigma;x}}_i)}^2+{(y-\frac{1}{F}{\mathrm{\&Sigma;y}}_i)}^2={\stackrel{\&OverBar;}{d}}^2---\left(8\right)$

By poor to each equation in system of equations (6) and equation (8), can obtain:

$x_i^2-{(\frac{1}{F}\&Sigma;x_i)}^2-2x(x_i-\frac{1}{F}\&Sigma;x_i)+y_i^2-{(\frac{1}{F}\&Sigma;y_i)}^2-2y(y_i-\frac{1}{F}\&Sigma;y_i)=d_i^2-{\stackrel{\&OverBar;}{d}}^2---\left(9\right)$

Order:

$\left\{\begin{array}{c}\stackrel{\&OverBar;}{x_i}=x_i-\frac{1}{F}\&Sigma;x_i,\stackrel{\&OverBar;}{y_i}=y_i-\frac{1}{F}\&Sigma;y_i\\ {\widetilde{x_i}}^2=x_i^2-{(\frac{1}{F}\&Sigma;x_i)}^2,{\widetilde{y_i}}^2=y_i^2-{(\frac{1}{F}\&Sigma;y_i)}^2,{\widetilde{d_i}}^2=d_i^2-{\stackrel{\&OverBar;}{d}}^2\end{array}\right.---\left(10\right)$

Formula (9) can be rewritten the system of equations of an accepted way of doing sth (11) form:

$\left\{\begin{array}{c}{\widetilde{x_1}}^2-2\stackrel{\&OverBar;}{x_1}x+{\widetilde{y_1}}^2-2\stackrel{\&OverBar;}{y_1}y={\widetilde{d_1}}^2\\ .\\ .\\ .\\ {\widetilde{x_F}}^2-2\stackrel{\&OverBar;}{x_F}x+{\widetilde{y_F}}^2-2\stackrel{\&OverBar;}{y_F}y={\widetilde{d_F}}^2\end{array}\right.---\left(11\right)$

Calculate the coordinate (x, y) of unknown node, the form of the formula (12) formula (11) can write as:

$\left[\begin{array}{c}\stackrel{\&OverBar;}{x_1},\stackrel{\&OverBar;}{y_1}\\ .\\ .\\ .\\ \stackrel{\&OverBar;}{x_F},\stackrel{\&OverBar;}{x_F},\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}\frac{1}{2}({\widetilde{x_1}}^2+{\widetilde{y_1}}^2-{\widetilde{d_1}}^2)\\ .\\ .\\ .\\ \frac{1}{2}({\widetilde{x_F}}^2+{\widetilde{y_F}}^2-{\widetilde{d_F}}^2)\end{array}\right]---\left(12\right)$

Solving equation (12) can obtain the coordinate of unknown node.

In order to check the validity of location algorithm, the present invention adopts MATLAB software to emulate, and verifies the validity of innovatory algorithm of the present invention. Scene setting: stochastic distribution 100 sensor nodes in the region of 100m*100m, choosing path loss index �� is 4, if no special instructions, by the communication radius R of sensor node₀For 40m, beaconing nodes number Bn is 30, and standard deviation sigma is 4, neighbor node number N is 7, and it is 2 that big error removes quantity m. It is averaging for 50 times often organizing data simulation. As shown in Figure 3, in figure �� be beaconing nodes, �� is the actual position of unknown node to Node distribution.

As shown in Figure 4, in order to verify that positioning error is affected by beaconing nodes number, the beaconing nodes number in network is increased to 45 from 10. Along with the increase of beaconing nodes number in wireless sensor network, the positioning error of sensor node has certain reduction, and after beaconing nodes number reaches certain value, node locating precision changes no longer thereupon. This illustrates, only by increase beaconing nodes number, it is to increase it is inadequate that beaconing nodes density reduces positioning error. From the graph of errors figure, the positioning error of KCML algorithm is lower relative to first two algorithm, and positioning precision promotes.

As shown in Figure 5, in order to verify that inventive algorithm is to relatively big error distance measurement value screening effect, Clustering Model of the present invention and equal value model, Gauss's model are compared. Adding thick error in ranging information, thick poor ratio is 0%��10%, and thick poor size is 1.25*RSSI signal value. Inventive algorithm and mean filter method and gaussian filtering method being emulated, the ratio of thick error is progressively increased to 10% from 0. As can be seen from Figure 5, when slightly poor be 0 time, equal value model and Gauss's model are better than KCML model, and equal value model is subject to slightly poor impact, and thick poor ratio is more big, on positioning error impact more serious. KCML algorithm can effectively suppress slightly poor impact, when thick error increases, still can keep ideal locating effect, have good robustness.

In order to the positioning performance of direct visual comparison algorithm, what Fig. 6, Fig. 7, Fig. 8 showed is appoint the physical location and estimation position getting and once emulate interior joint. In figure �� and it is beaconing nodes, �� is the actual position of unknown node, and zero is estimation position, and line represents error. As can be seen from the figure, there is the bigger node of error in first two algorithm (Fig. 6, Fig. 7), and inventive algorithm (Fig. 8) error distribution all with, locating effect is best.

Above embodiment is only the technological thought that the present invention is described, can not limit protection scope of the present invention with this, and every technological thought proposed according to the present invention, any change done on technical scheme basis, all falls within protection domain of the present invention.

Claims (5)

1., based on the wireless sensor network node locating method of K central point cluster, utilize the positional information of each beaconing nodes unknown node to be positioned, it is characterised in that, comprise the steps:

Step 1, each beaconing nodes sends own location information to unknown node, after unknown node receives positional information, record Received signal strength intensity level, and utilize logarithm-normal distribution model that Received signal strength intensity level is converted into distance value, above-mentioned distance value appoints to get 3 be a combination, each is combinationally used three limit location algorithms, obtains the initial position message of unknown node;

Step 2, the initial position message utilizing K central point cluster algorithm step 1 to be obtained carries out cluster, obtains cluster result;

Step 3, t the cluster that contained element is minimum is found out from the cluster result that step 2 obtains, t is less than the cluster numbers that K central point cluster algorithm obtains, and finds out the beaconing nodes that in above-mentioned t cluster, each position information is corresponding, m the beaconing nodes that record occurrence number is maximum;

Step 4, removes m maximum for occurrence number beaconing nodes, and remaining beaconing nodes utilizes the polygon location algorithm improved unknown node is positioned calculating, and the reference value chosen in the polygon location algorithm of described improvement is:Wherein, the coordinate that (x, y) is unknown node, (x_i,y_i) it is the coordinate of remaining beaconing nodes, i=1 ..., F,F is the number of remaining beaconing nodes.

2. as claimed in claim 1 based on the wireless sensor network node locating method of K central point cluster, it is characterised in that, the calculation formula of logarithm described in step 1-normal distribution model is:

$PL\left(d\right)=PL\left(d_0\right)+10*\mathrm{\&alpha;}log\frac{d}{d_0}+X_{\mathrm{\&sigma;}},$

Wherein, PL (d) is the received signal power through distance d, PL (d₀) it is reference range d₀Corresponding received signal power, �� is path loss index, X_��For average is the Gaussian distributed random variable of 0, �� is standard deviation.

3. as claimed in claim 2 based on the wireless sensor network node locating method of K central point cluster, it is characterised in that, described reference range d₀Get 1m.

4. as claimed in claim 1 based on the wireless sensor network node locating method of K central point cluster, it is characterized in that, the detailed process of described step 2 is: (1) is using the initial clustering sample of initial position message as K central point cluster, setting clusters number is K, and from initial sample, random selection K position is as initial center point; (2) remaining position is added from the class representated by its nearest central point; (3) from remaining position, choose one at random replace any one central point in K initial center point; (4) calculate in (3) cost replaced, if cost is less than 0, then replace; (5) (3) and (4) are repeated, until when the cost replaced is greater than 0, cluster terminates.

5. as claimed in claim 4 based on the wireless sensor network node locating method of K central point cluster, it is characterised in that, the calculation formula of described cost S is:

$S={\min }_{c_j}\underset{z_l}{min}{\mathrm{\&Sigma;}}_{j=1,j\&Element;\mathrm{\&Omega;}}^K{\mathrm{\&Sigma;}}_{l\&Element;{\mathrm{\&Omega;}}_j}\left|\right|c_j-z_l\left|\right|,$

Wherein, �� is initial clustering sample, ��_jFor jth class �� being clustered in K class, j=1 ..., K, c_jFor class ��_jCentral point, z_lFor current class ��_jIn except central point c_jOutside other optional positions.