CN110399686B - Parameter-independent aircraft flight trajectory clustering method based on contour coefficients - Google Patents
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Abstract
The invention discloses a parameter-independent aircraft flight trajectory clustering method based on a contour coefficient. Firstly, normalizing the space position coordinates of all tracks of a track set; then, track similarity, track distance matrix, degree matrix and Laplacian matrix are established based on the dynamic time bending distance and Gaussian kernel function; then, clustering the track characteristic subspaces cluster by cluster in a given interval by using a k-means algorithm; and finally, taking the average profile coefficient of the track set as an evaluation standard of the clustering quality, and determining the optimal clusters, the optimal cluster number and the maximum average profile coefficient. Compared with the existing method, the method has the advantages that: expert experience or domain knowledge is not needed, manual intervention is eliminated, and the objectivity of the clustering process and the result is strong; the method is not influenced by the difference of the value ranges of the horizontal abscissa, the horizontal ordinate and the altitude coordinate of the length and the speed of the flight track, and is suitable for various data formats; the workload of parameter adjustment of users is reduced, and the time cost is saved.
Description
Technical Field
The invention relates to a space-time trajectory clustering method, in particular to a parameter-independent aircraft flight trajectory clustering method based on contour coefficients.
Background
Most current flight trajectory clustering algorithms require one or even more input parameters, the clustering results of which are often severely dependent on these parameters, but determining reasonable parameters is difficult. There are generally three methods and bases for determining the parameters necessary for the algorithm. The first method relies on expert experience or domain knowledge, and has the disadvantage of being highly subjective and lacking in objective quantitative evaluation indexes, depending on expert experience. The second method relies on the data characteristics of the track samples, and has the defects of strong limitation and poor generalization capability, and when the track test samples are obviously different from the training sample characteristics, the clustering quality of an algorithm for setting parameters by using the training samples is inevitably reduced. The third method is regulated through multiple experiments, and the defects are two, namely, the multiple experiments obviously increase the time cost of a user; and secondly, the candidate value of the parameter is generally selected in a certain range and a change step length manually specified, and is often not the theoretical optimal value.
Disclosure of Invention
Because one or more input parameters exist in the traditional flight trajectory clustering method, in the process of determining the parameters, the clustering result is doped with subjective favorites of an expert, or the generalization capability of an algorithm is limited by the characteristics of experimental sample data, or a user has to perform experimental tuning and optimizing for multiple times. Aiming at the problems, the invention provides a parameter-independent aircraft flight trajectory clustering method based on contour coefficients.
Firstly, normalizing the space position coordinates of all tracks of a track set; then, track similarity, track distance matrix, degree matrix and Laplacian matrix are established based on the dynamic time bending distance and Gaussian kernel function; then, clustering the track characteristic subspaces cluster by cluster in a given interval by using a k-means algorithm; and finally, taking the average profile coefficient of the track set as an evaluation standard of the clustering quality, and determining the optimal clusters, the optimal cluster number and the maximum average profile coefficient.
The technical scheme adopted by the invention is as follows: a method for clustering flight trajectories of parameter-independent aircraft based on contour coefficients, which is characterized in that firstly, a flight path is defined: a characteristic of a certain aircraft recorded by the monitoring device at a certain moment;
defining a track: the historical trail of aircraft flight through over a period of time consists of a series of discrete tracks arranged in a time series order.
In general, a trajectory set can be expressed as:
TS＝{T 1 ,T 2 ,…,T i ,…,T n }
wherein: t (T) i The ith track, n is the total number of tracks;
track T i The data set in tracks is expressed as:
T i ＝{p i1 ,p i2 ,…,p ij ,…,p im }
wherein: p is p ij Representing the jth track in the ith track, wherein m is the total number of tracks;
each track p ij Defined as a 4-dimensional vector, i.e
p ij ＝{x,y,z,t}
Wherein: x, y, z, t respectivelyRepresenting track p ij Spatial position abscissa, spatial position ordinate, flying height and recording time.
The method comprises the following steps:
(1) Normalized spatial position coordinates
Normalizing the three-dimensional coordinates of the space position of each track in the track set TS one by one, and then calculating Euclidean distance between tracks; the calculation method of the height coordinate normalization is shown in the formula (1):
wherein: z min 、z max Respectively representing the minimum and maximum height coordinates, z of all tracks in the track set norm And representing the normalized track height coordinates.
(2) Calculating dynamic time warping distance
The calculation method of the dynamic time bending distance between every two tracks is shown in the formula (2):
wherein: DTW (T) i ,T j ) Representing the trajectory T i And track T j A dynamic time warp distance between; m is m i And m j Respectively represent the tracks T i And track T j The number of tracks of the track; dist (p) i1 ,p j1 ) Representing the trajectory T i Is the first track p of (2) i1 And trajectory T j Is the first track p of (2) j1 Euclidean distance between them; rest (T) i ) Representing the trajectory T i Removing the first track p i1 The remaining trace, rest (T j ) Representing the trajectory T j Removing the first track p j1 The remaining trace after that.
(3) Constructing an average dynamic time warping distance matrix
The average dynamic time warping distance matrix is denoted as R, matrix element R ij Representation ofTrack T i And trajectory T j The average dynamic time bending distance between the two is calculated as shown in the formula (3):
wherein: DTW (T) i ,T j ) Representing the trajectory T i And track T j A dynamic time warp distance between; m is m i And m j Respectively represent the tracks T i And track T j Is a track number of the track(s).
(4) Calculating the bandwidth of a Gaussian kernel function
The bandwidth calculation method of the Gaussian kernel function is shown in the formula (4):
wherein: beta is the bandwidth of the Gaussian kernel function, sigma 2 Is the variance of the average dynamic time warping distance matrix R, μ is the mean of the average dynamic time warping distance matrix R, parameter λ= (μ/σ) 2 F (λ) is a function with respect to the parameter λ, and the function f (λ) and the parameter λ satisfy the relationship shown in the formula (5):
wherein: e is a natural constant.
(5) Constructing a similarity matrix
The similarity matrix is denoted as S, and the matrix elements S ij Representing the trajectory T i And trajectory T j The similarity between the two is calculated as shown in the formula (6):
wherein: r is (r) ij Representing the trajectory T i And trajectory T j Average betweenDynamic time warping distance, β is the bandwidth of the gaussian kernel function, e is a natural constant;
the similarity matrix S is a symmetrical square matrix, and matrix elements S ij Are normalized.
(6) Reconstructing distance matrix
The distance matrix is denoted as R ', and the matrix element R' ij Representing the trajectory T i And trajectory T j The distance between the two is calculated as shown in the formula (7):
r ij ′＝1-s ij (7)
wherein: s is(s) ij Representing the trajectory T i And trajectory T j Similarity between them.
(7) Degree of structuring matrix
The degree matrix is a diagonal matrix, denoted as D, the diagonal element D ii The calculation method of (2) is shown as the formula (8):
wherein: s is(s) ij Representing the trajectory T i And trajectory T j Similarity between them.
(8) Constructing a Laplace matrix
The Laplace matrix is marked as L, and the calculation method is shown as a formula (9):
wherein: d is a degree matrix, S is a similarity matrix;
the elements of the laplace matrix L are normalized.
(9) Initialization parameters
a. Initializing a minimum cluster number k min =2, maximum cluster number k max >>2；
b. Initializing the number of candidate clusters k=k min The value range of the candidate cluster number k is k min ≤k≤k max ；
c. Initializing a maximum average profile coefficient s max ＝-1。
(10) Judging whether the search is completed
Judging whether the number k of candidate clusters is less than or equal to the maximum number k of clusters max The method comprises the steps of carrying out a first treatment on the surface of the If yes, jumping to the step (11); if not, step (17) is skipped.
(11) Constructing feature subspaces
And solving the first k minimum eigenvalues and corresponding eigenvectors of the Laplace matrix L, and constructing an eigenvoice by using the solved k eigenvectors.
(12) K-means clustering
Clustering the feature subspaces by using a k-means clustering algorithm to obtain a clustering result C, which is expressed as:
C＝{C 1 ,C 2 ,…,C i ,…,C k }
wherein: c (C) i For the ith cluster, k is the cluster number.
(13) Calculating average profile coefficients
a. Track T i The average distance from other tracks in the same cluster is shown in formula (10):
wherein: r's' ij Representing the inner track T of the same cluster i And trajectory T j Distance between C i Representing the trajectory T i And trajectory T j Clusters of the same genus; c i I represents cluster C i Total number of tracks in the box.
b. Track T i To not include track T i The minimum average distance of all clusters of (a) is shown in formula (11):
wherein: r's' ij Representing the trajectory T i To and trace T i Tracks T of different clusters j Distance between each otherSeparation, C j Representing not including track T i Is a cluster of (a); c j I represents cluster C j Total number of tracks in the box.
c. Track T i Is defined as:
d. the average value of the contour coefficients of all tracks in the track set is calculated, and the calculation method is shown as a formula (13):
wherein: s is(s) avg For the average profile coefficient of the entire trajectory set, s (T i ) Is the track T i Is n is the total number of tracks.
(14) Judging whether to obtain better clustering result
Using the average profile coefficient of the track set as the evaluation of the track clustering result; determining whether to average the profile coefficient s avg >Maximum average profile coefficient s max The method comprises the steps of carrying out a first treatment on the surface of the If yes, jumping to the step (15); if not, step (16) is skipped.
(15) Preserving the clustering result
Let the maximum average contour coefficient s max Mean contour coefficient s of this sub-cluster avg Optimum cluster number k opt Number of secondary candidate clusters k, best clustering result C opt The result of this clustering C.
(16) Continue searching
The number k of candidate clusters is increased by 1, and then the step (10) is skipped.
(17) Outputting the best clustering result
Output maximum average profile coefficient s max Optimum cluster number k opt And best clustering result C opt 。
The beneficial effects of the invention are as follows: the present invention has three advantages over existing methods. Firstly, the method does not need expert experience or domain knowledge, eliminates human intervention, and has strong objectivity of clustering process and result. Secondly, the method is not influenced by the length of the flight track, the speed and the difference of the value ranges of the horizontal abscissa and the vertical ordinate and the altitude coordinate, and is suitable for various data formats. Third, the method reduces the workload of parameter adjustment of users and saves time cost.
Drawings
FIG. 1 is a flow chart of a method of aircraft flight trajectory clustering;
fig. 2 is a graph of average profile factor versus number of candidate clusters.
Detailed Description
The invention is further described below with reference to the drawings and examples.
Examples:
the method is illustrated by selecting 365 real aircraft approach flight trajectories for 48 hours in total at 28L runway of the san Francisco International airport.
A trace is shown in table 1:
TABLE 1 track data
| Sequence number | Abscissa of the circle | Ordinate of the ordinate | Height coordinates | Time |
| 1 | 18483 | -74632 | 3996 | 0 |
| 2 | 18225 | -73838 | 3963 | 5 |
| 3 | 17959 | -73048 | 3930 | 10 |
| 4 | 17696 | -72265 | 3895 | 14 |
| 5 | 17426 | -71486 | 3858 | 19 |
| 6 | 17148 | -70711 | 3820 | 23 |
| 7 | 16864 | -69937 | 3781 | 28 |
| 8 | 16585 | -69165 | 3741 | 33 |
| 9 | 16317 | -68395 | 3700 | 37 |
| 10 | 16052 | -67627 | 3659 | 42 |
| …… | …… | …… | …… | …… |
| 126 | -10768 | -10962 | 82 | 578 |
| 127 | -11037 | -10817 | 65 | 583 |
| 128 | -11302 | -10672 | 47 | 587 |
| 129 | -11561 | -10526 | 29 | 592 |
| 130 | -11817 | -10381 | 11 | 597 |
The complete flow of the algorithm is shown in fig. 1, and the specific method comprises the following steps:
(1) Normalized spatial position coordinates
Because the horizontal transverse coordinate, the vertical coordinate and the height coordinate of the track space position are different in units, the value range of the horizontal transverse coordinate, the vertical coordinate and the height coordinate may have larger difference, and if the Euclidean distance between tracks is directly calculated without processing, the influence degree of the coordinate with a small numerical range on the track space distance can be reduced. Therefore, the three-dimensional coordinates need to be normalized one by one, and the Euclidean distance between tracks is calculated. For specific methods of normalization see formula (1).
After normalizing the three-dimensional coordinates according to formula (1), the trajectory data is shown in table 2.
Table 2 track data after normalization
(2) Calculating dynamic time warping distance
In the process of solving the distance between tracks, tracks of different tracks are matched according to a certain rule, and then the sum of Euclidean distances of the matched tracks is used as the distance between the tracks. In reality, most of flight tracks are different in length, and due to the differences of machine types and flight performances, the flight speeds of the aircrafts are different, so that the track densities of different tracks are different. Therefore, the present invention uses the characteristic of dynamic time warping distance to represent the distance between tracks. First, the dynamic time warping distance does not require the length of the two tracks compared, i.e. the lengths of the two tracks may be either equal or unequal. And secondly, on the premise of ensuring that the track sequence is unchanged, the method based on the dynamic time bending distance completes local scaling of the time dimension by repeating part of tracks, so that the optimal track pairing relation is generated by taking the minimum Euclidean distance of two tracks as a target.
According to the method of the formula (2), the dynamic time bending distance between every two tracks is calculated in sequence.
(3) Constructing an average dynamic time warping distance matrix
If the dynamic time bending distance is directly used as the measurement of the distance between two tracks, the obtained dynamic time bending distance is definitely larger than that of two short tracks far away from each other because the two long tracks close to each other contain more tracks. This is contrary to the goal of clustering. The invention divides the dynamic time bending distance of the two tracks by the average track number of the two tracks, and takes the average dynamic time bending distance as the measurement of the distance of the two tracks, and the calculation method is shown in the formula (3).
And (3) sequentially calculating the average dynamic time bending distance between every two tracks in the track set according to the method of the formula (3) to obtain an average dynamic time bending distance matrix R. It is a 365 x 365 symmetric matrix with some of the data shown below.
(4) Calculating the bandwidth of a Gaussian kernel function
The smaller the distance between the tracks, the higher the similarity; conversely, the lower the similarity. The Gaussian kernel function is adopted for converting the average dynamic time bending distance matrix into the similarity matrix, and the difference of the tracks can be highlighted by the Gaussian kernel function, so that the similarity of the tracks with long space distance is obviously reduced, and the clustering quality is improved.
The result of the gaussian kernel function is sensitive to the bandwidth parameters, but with the calculation methods of formula (4) and formula (5), no domain knowledge or trial-and-error experiments are required, and only the characteristics of the data sample set itself are relied on. From the formulas (4) and (5), the variance σ is obtained 2 =0.0189, μ= 0.2256, λ= 2.6978. Because of 0.01<λ<100, f (λ) = 0.5353, β=0.0352.
(5) Constructing a similarity matrix
The similarity matrix S is obtained from the average dynamic time warping distance matrix R according to the calculation method of equation (6). The similarity matrix S is a symmetric matrix of 365×365, and the matrix elements are normalized, and part of the data is shown below.
(6) Reconstructing distance matrix
The greater the similarity of the two tracks, the smaller the distance; conversely, the smaller the similarity, the greater the distance. Since the elements of the similarity matrix S are normalized, the distance matrix R' is obtained from the similarity matrix S according to the calculation method of equation (7). The distance matrix R' is a 365 x 365 symmetric matrix, part of which is shown below.
(7) Degree of structuring matrix
The degree matrix D is constructed according to equation (8). The degree matrix D is a 365×365 diagonal matrix, and a part of the data is as follows.
(8) Constructing a Laplace matrix
The laplace matrix L is constructed according to equation (9). The laplace matrix L is a 365 x 365 matrix, with normalized elements, and a portion of the data is shown below.
(9) Initialization parameters
a. Initializing a minimum cluster number k min =2, maximum cluster number k max >>2；
b. Initializing the number of candidate clusters k=k min The value range of the candidate cluster number k is k min ≤k≤k max ；
c. Initializing a maximum average profile coefficient s max ＝-1。
(10) Judging whether the search is completed
Judging whether the number k of candidate clusters is less than or equal to the maximum number k of clusters max The method comprises the steps of carrying out a first treatment on the surface of the If yes, jumping to the step (11); if not, step (17) is skipped.
(11) Constructing feature subspaces
And solving the first k minimum eigenvalues and corresponding eigenvectors of the Laplace matrix L, and constructing an eigenvoice by using the solved k eigenvectors. For example, when the cluster number is 5, the feature subspace is as shown in table 3.
TABLE 3 feature subspace
(12) K-means clustering
And (4) calling a k-means clustering algorithm to cluster the feature subspaces, and obtaining a clustering result shown in a table 4.
TABLE 4 clustering results
| Track | Cluster number |
| 1 | 3 |
| 2 | 1 |
| 3 | 2 |
| 4 | 5 |
| 5 | 2 |
| 6 | 1 |
| 7 | 4 |
| 8 | 5 |
| 9 | 1 |
| 10 | 3 |
| …… | …… |
| 361 | 3 |
| 362 | 4 |
| 363 | 2 |
| 364 | 5 |
| 365 | 3 |
(13) Calculating average profile coefficients
The invention uses the contour coefficient as the evaluation of the flight trajectory clustering result. The profile coefficient of each track takes a value between-1 and 1, and the larger the value is, the more compact the cluster of the track is, and the more distant from other clusters, namely, the better the cluster quality is. To measure cluster quality on the complete set of trajectories, the average profile coefficients of the set of trajectories are calculated according to equations (10), (11), (12), and (13).
Intuitively, increasing the number of clusters appears to help reduce the average distance of tracks within each cluster, as there is a chance that more dense clusters will be formed, with tracks in the clusters being more similar. However, too many clusters are divided due to marginal effect, which results in a decrease in the effect of decreasing the average distance of tracks within the clusters. Multiple experiments confirm that as the number of candidate clusters increases, the average profile factor always tends to increase first and then decrease, as shown in fig. 2. Thus, the average profile coefficient s is used avg Finding the optimal cluster number k with respect to the inflection point of the curve of the candidate cluster number k opt And a maximum average profile coefficient s max 。
(14) Judging whether to obtain better clustering result
Using the average profile coefficient of the track set as the evaluation of the track clustering result; determining whether to average the profile coefficient s avg >Maximum average profile coefficient s max The method comprises the steps of carrying out a first treatment on the surface of the If yes, jumping to the step (15); if not, step (16) is skipped.
(15) Preserving the clustering result
Let the maximum average contour coefficient s max Mean contour coefficient s of this sub-cluster avg Optimum cluster number k opt Number of secondary candidate clusters k, best clustering result C opt The result of this clustering C.
(16) Continue searching
The number k of candidate clusters is increased by 1, and then the step (10) is skipped.
(17) Outputting the best clustering result
Output maximum average profile coefficient s max Optimum cluster number k opt And best clustering result C opt 。
Claims (1)
1. A method for clustering flight trajectories of parameter-independent aircraft based on contour coefficients is characterized in that firstly,
defining a track: a characteristic of a certain aircraft recorded by the monitoring device at a certain moment;
defining a track: in a period of time, the history trace of the aircraft flight is composed of a series of discrete tracks arranged according to a time sequence;
in general, a trajectory set can be expressed as:
TS＝{T 1 ,T 2 ,…,T i ,…,T n }
wherein: t (T) i The ith track, n is the total number of tracks;
track T i The data set in tracks is expressed as:
T i ＝{p i1 ,p i2 ,…,p ij ,…,p im }
wherein: p is p ij Representing the ith tracej tracks, m being the total number of tracks;
each track p ij Defined as a 4-dimensional vector, i.e
p ij ＝{x,y,z,t}
Wherein: x, y, z, t each represents a track p ij A spatial position abscissa, a spatial position ordinate, a flying height and a recording time;
the method comprises the following steps:
(1) Normalized spatial position coordinates
Normalizing the three-dimensional coordinates of the space position of each track in the track set TS one by one, and then calculating Euclidean distance between tracks; the calculation method of the height coordinate normalization is shown in the formula (1):
wherein: z min 、z max Respectively representing the minimum and maximum height coordinates, z of all tracks in the track set norm Representing normalized track height coordinates;
(2) Calculating dynamic time warping distance
The calculation method of the dynamic time bending distance between every two tracks is shown in the formula (2):
wherein: DTW (T) i ,T j ) Representing the trajectory T i And track T j A dynamic time warp distance between; dist (p) i1 ,p j1 ) Representing the trajectory T i Is the first track p of (2) i1 And trajectory T j Is the first track p of (2) j1 Euclidean distance between them; rest (T) i ) Representing the trajectory T i Removing the first track p i1 The remaining trace, rest (T j ) Representing the trajectory T j Removing the first track p j1 A remaining trace after the step;
(3) Constructing an average dynamic time warping distance matrix
The average dynamic time warping distance matrix is denoted as R, matrix element R ij Representing the trajectory T i And trajectory T j The average dynamic time bending distance between the two is calculated as shown in the formula (3):
wherein: DTW (T) i ,T j ) Representing the trajectory T i And track T j A dynamic time warp distance between; m is m i And m j Respectively represent the tracks T i And track T j The number of tracks of the track;
(4) Calculating the bandwidth of a Gaussian kernel function
The bandwidth calculation method of the Gaussian kernel function is shown in the formula (4):
wherein: beta is the bandwidth of the Gaussian kernel function, sigma 2 Is the variance of the average dynamic time warping distance matrix R, μ is the mean of the average dynamic time warping distance matrix R, parameter λ= (μ/σ) 2 F (λ) is a function with respect to the parameter λ, and the function f (λ) and the parameter λ satisfy the relationship shown in the formula (5):
wherein: e is a natural constant;
(5) Constructing a similarity matrix
The similarity matrix is denoted as S, and the matrix elements S ij Representing the trajectory T i And trajectory T j The similarity between the two is calculated as shown in the formula (6):
wherein: r is (r) ij Representing the trajectory T i And trajectory T j The average dynamic time warping distance between beta is the bandwidth of the Gaussian kernel function, and e is a natural constant;
the similarity matrix S is a symmetrical square matrix, and matrix elements S ij All normalized;
(6) Reconstructing distance matrix
The distance matrix is denoted as R ', and the matrix element R' ij Representing the trajectory T i And trajectory T j The distance between the two is calculated as shown in the formula (7):
r ij ′＝1-s ij (7)
wherein: s is(s) ij Representing the trajectory T i And trajectory T j Similarity between;
(7) Degree of structuring matrix
The degree matrix is a diagonal matrix, denoted as D, the diagonal element D ii The calculation method of (2) is shown as the formula (8):
wherein: s is(s) ij Representing the trajectory T i And trajectory T j Similarity between;
(8) Constructing a Laplace matrix
The Laplace matrix is marked as L, and the calculation method is shown as a formula (9):
wherein: d is a degree matrix, S is a similarity matrix;
the elements of the Laplace matrix L are normalized;
(9) Initialization parameters
a. Initializing a minimum cluster number k min =2, maximum cluster number k max >>2；
b. Initializing the number of candidate clusters k=k min The value range of the candidate cluster number k is k min ≤k≤k max ；
c. Initializing a maximum average profile coefficient s max ＝-1；
(10) Judging whether the search is completed
Judging whether the number k of candidate clusters is less than or equal to the maximum number k of clusters max ；
If yes, jumping to the step (11); if not, jumping to the step (17);
(11) Constructing feature subspaces
Solving the first k minimum eigenvalues and corresponding eigenvectors of the Laplace matrix L, and constructing an eigenvoice by using the obtained k eigenvectors;
(12) K-means clustering
Clustering the feature subspaces by using a k-means clustering algorithm to obtain a clustering result C, which is expressed as:
C＝{C 1 ,C 2 ,…,C i ,…,C k }
wherein: c (C) i The i-th cluster, k is the cluster number;
(13) Calculating average profile coefficients
a. Track T i The average distance from other tracks in the same cluster is shown in formula (10):
wherein: r's' ij Representing the inner track T of the same cluster i And trajectory T j Distance between C i Representing the trajectory T i And trajectory T j Clusters of the same genus; c i I represents cluster C i The total number of tracks in the box;
b. track T i To not include track T i The minimum average distance of all clusters of (a) is shown in formula (11):
wherein: r's' ij Representing the trajectory T i To and trace T i Tracks T of different clusters j Distance between C j Representing not including track T i Is a cluster of (a); c j I represents cluster C j The total number of tracks in the box;
c. track T i Is defined as:
d. the average value of the contour coefficients of all tracks in the track set is calculated, and the calculation method is shown as a formula (13):
wherein: s is(s) avg For the average profile coefficient of the entire trajectory set, s (T i ) Is the track T i Is the total number of tracks;
(14) Judging whether to obtain better clustering result
Using the average profile coefficient of the track set as the evaluation of the track clustering result;
determining whether to average the profile coefficient s avg >Maximum average profile coefficient s max ；
If yes, jumping to the step (15); if not, jumping to the step (16);
(15) Preserving the clustering result
Let the maximum average contour coefficient s max Mean contour coefficient s of this sub-cluster avg Optimum cluster number k opt Number of secondary candidate clusters k, best clustering result C opt =this secondary clustering result C;
(16) Continue searching
The number k of the candidate clusters is increased by 1, and then the step (10) is skipped;
(17) Outputting the best clustering result
Output maximum average profile coefficient s max Optimum cluster number k opt And best clustering result C opt 。
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