The execution process of the classic K-means algorithm is divided into the following steps:

Step 1: The value of user input parameter K [5] , which is the number of initial clustering centers and is generally obtained from given data samples based on empirical observation. The algorithm randomly generates K clustering centers m 1 , m 2 , ⋯ , m k , represent clusters c 1 , c 2 , ⋯ , c k .

Step 2: To calculate the Euclidean distance from each sample point x_i in data set D to K clustering centers [6] , and put the samples into the cluster c i ( i = 1 , 2 , ⋯ , k ) where the nearest clustering center is located. D = { x i | x i ∈ R m , i = 1 , 2 , ⋯ , n } . Euclidean distance represents the similarity degree between the sample point and the cluster center. The smaller the distance, the higher the similarity degree. The calculation formula is shown in formula (1).

D i s t ( x i , m j ) = ( x i − m j ) T ( x i − m j ) (1)

i ∈ { 1 , 2 , ⋯ , n } , j ∈ { 1 , 2 , ⋯ , k } .

Step 3: To calculate the mean value of all sample points in each cluster, and update all clustering centers in step 1 with the obtained mean value.

Step 4: Repeat step 1 and step 2 until the clustering center obtained two times in a row is no longer changed, then ending the clustering.

The traditional K-means clustering algorithm is simple in thought and easy to implement, which is one of the widely studied and applied clustering algorithms. However, random selection of the initial clustering center also causes unstable clustering results and clustering efficiency, as well as local optimal problems [7] .

The fast global K-means algorithm is an improvement on the traditional K-means algorithm. By considering global data, the initial clustering center is found to reduce the sensitivity of the algorithm to outliers and noise [14] [15] . The algorithm flow chart is shown in Figure 1.

The calculation formula of b_n is shown in formula (2). Where N is the total number of samples, d k − 1 j is the minimum distance between sample point x_j and k initial clustering centers, and x_n is the sample points except the clustering center.

b n = ∑ j = 1 N max ( d k − 1 j − ‖ x n − x j ‖ 2 , 0 ) (2)

This algorithm can effectively solve the random problem of the initial clustering center [8] , and can effectively reduce the clustering times and thus shorten the clustering time. However, in the selection process of clustering center, repeated distance calculation is required for each sample point, which increases the time complexity of initial value selection.

The global K-means algorithm mentioned in literature [9] replaces the maximum relative distance b_n from the existing clustering center in the Fast Global K-means algorithm with the maximum absolute distance ‖ x n − x j ‖ 2 between two points. During the selection of the initial cluster center, d_j is only calculated as the distance between the pre-selected cluster center x_n and other sample points x_j, and d_j is summed up. Finally, the point with minimum accumulation value is selected as the clustering center.

This method reduces the computational steps when the initial cluster center is selected and reduces the time complexity of the algorithm to some extent. However, the influence of the selected initial clustering center on the next initial clustering center is ignored, which reduces the constraint conditions of initial value selection and improves the randomness.

In practical applications, each cluster center will be a certain distance away, and the next cluster center must be outside a certain neighborhood of the known cluster center. According to formula (2), there must be no point that maximizes b_n in a certain neighborhood of the known initial cluster center. Therefore, it is not necessary to calculate the initial cluster center search for the sample points in the neighborhood. Under the circumstance that the distribution of the whole class of samples is unknown, the size of the neighborhood is largely affected by the number of clustering centers k.

Suppose the first initial cluster center m₁ is located at the middle point of the sample, and sample x_max is the farthest sample point from m₁, and the distance is d_mmax. In the extreme case, K initial clustering centers are evenly distributed on the line segment formed by x_max and m₁, and the vertex of the line segment is two initial clustering centers, so the distance between each two initial clustering centers is d r = d m max / ( k − 1 ) . After comprehensive consideration, sufficient sample points are ensured to serve as the next initial clustering center after each initial clustering center is determined, and the time complexity of the algorithm is minimized. In this paper, R is selected as formula (3)

R = d m max / ( 2 ∗ ( k − 1 ) ) (3)

where k is the number of clustering centers, d_mmax is the maximum distance between all sample points and the first initial clustering center (i.e., sample median).

Taking the selection of the second initial clustering center as an example, calculate the distance d_m between all samples in the initial sample set D { x 1 , x 2 , ⋯ , x n } and the first initial clustering center m₁. D 1 { x 1 , x 2 , ⋯ , x m } , the set composed of d_m sample points, is selected. From D₁, each sample x_n is respectively selected as the second clustering center. According to formula (2), b_n is calculated to determine the second initial clustering center m₂.

Then, the distance between all sample points in D₁ and m₂ is calculated respectively, and the points whose distance is greater than the minimum radius R are formed into the set D₂. m 3 , m 4 , ⋯ , m k can be obtained according to the above methods.

In the current researches on K-means clustering algorithm, most of them conduct clustering based on Euclidean distance, but Euclidean distance is only applicable to clustering of spherical structure, and the correlation between variables and the difference in importance of each variable are not considered when processing data [10] . It has some defects in the application of high correlation data and image fuzzy segmentation. Mahalanobis distance is a method of calculating distance similarity proposed by P. C. Mahalanobis, an Indian statistician. Can be used to calculate both follow the same distribution and its covariance matrix of the Σ degree of difference between random variables. When the covariance matrix Σ matrix for the unit, the Mahalanobis distance can be converted into Euclidean distance. The Mahalanobis distance formula is shown in formula (4).

M ( x i , x j ) = ( x i − x j ) T Σ − 1 ( x i − x j ) (4)

The x_i, x_j for two vectors of the same sample concentration, Σ as the covariance matrix of the sample, M(x_i, x_j) for the Mahalanobis distance between two samples.

Compared with the Euclidean distance, the Mahalanobis distance reflects the internal relationship between sample attributes [11] , can effectively describe the global relationship between two sample points, and contains more neighborhood information and spatial information [12] , which can play a better analysis effect in big data processing and image segmentation.

The K-means clustering algorithm usually uses the square sum of clustering error D to represent the clustering effect, which is the sum of the distance from each sample to K cluster centers, and is defined as the formula (5).

D = ∑ j = 1 K ∑ 1 N | x i − m j | 2 (5)

where, x_i represents the ith sample, and there are N samples, m_j represents the jth clustering center, with a total of K clustering centers. In order to facilitate the observation of values, this paper uses the average error L to represent the clustering effect, which is defined as the formula (6).

L = 1 N D (6)

For the same data set, the smaller the value of L is, the better the clustering effect is.

Steps of fast global K-means clustering algorithm based on neighborhood screening and Mahalanobis distance:

Input: K: The number of cluster clusters;

D: A data set containing n objects.

Output: Sets and categories of K clusters

Method:

(1) Calculate the median value of all samples as the initial cluster center of the first cluster, and set s = 1.

(2) Calculate the distance d_j from each sample point x_i to its clustering center m₁, taking d m m a x = max ( d j ) , j = 1 , 2 , ⋯ , n , and D 1 = D .

(3) Calculate the minimum radius R in set D_i, as shown in formula (3).

(4) Set s = s + 1, if s > k jumps to (7).

(5) As m 1 , m 2 , ⋯ , m s − 1 are the first s − 1 cluster center, to calculate the distance d s − 1 j from each sample x_j in the set D_s−1 to the cluster center m_s−1. The new data set D i { x 1 , x 2 , ⋯ , x m } is composed of d > R sample x_j.

(6) To calculate b_n, for example, formula (2), select the x_n with the largest b_n as the s-th cluster center m_s, and jump to (3).

(7) The Mahalanobis distance from x j ( j = 1 , 2 , ⋯ , n ) of all samples in set D to m i ( i = 1 , 2 , ⋯ , k ) of k cluster centers is calculated respectively, and the sample points are divided into clusters closest to the cluster center.

(8) Calculate the sample mean m i N in each cluster. If m i N = m i , jump to (9); otherwise, set m i = m i N , and jump to (7).

(9) Output the set, class number and average error of K clusters to end the clustering.

In this paper, the algorithm time and the mean value of error sum square are used as the evaluation criteria of clustering effect.

Clustering experiments were carried out on traditional K-means algorithm, Fast Global K-means algorithm (FGK-means), fast global K-means algorithm based on neighborhood screening (RFGK-means), and fast global K-means algorithm based on neighborhood screening and Markov distance (RMFGK-means), respectively. Set the number of clustering centers to 4, and the clustering effect is shown in Figure 2.

After 10 times of clustering simulation, the average value is obtained. The time and average error of each algorithm are shown in Table 1 (kept three decimal places).

There are 1600 pieces of data in Wine quality-red, and each piece of data has 12 characteristics, among which the data under the quality attribute can be used as sample labels. After removing the title and the last quality feature, 1599 pieces of data are used, and 11 feature data of each piece of data are normalized for clustering. The number of cluster centers was set to 6, and the average value was obtained after 10 cluster simulations. The clustering results were shown in Table 2 (kept three decimal places).

In this paper, the ratio of the number of correctly classified samples to the total number of samples was defined as the correct classification rate, which was used to test the clustering effect of RFGK-means and RMFGK-means.

According to the 6 qualities of Wine quality-red, the original samples are classified into classes D1 to D6, and the clustering results are classified into classes DA1 to DA6 respectively. Each data sample in DA1 is fitted with samples from D1 to D6 respectively, and the number of the same samples is recorded. The fitting results of RFGK-means are shown in Table 3. Finally, the sample quality classification set with the largest number of samples and no duplication with other category sets as the similar set of clustering results, and statistics the number of identical samples in similar sets. Similar set results of RFGK-means are shown in Table 4.

The fitting results and similar set results after RMFGK-means clustering are shown in Table 5 and Table 6 respectively.

By analyzing the above results, the clustering effects of RFGK-means and RMFGK-means are shown in Table 7 (kept three decimal places).

According to the above data, the correct classification rate of samples obtained by RMFGK-means clustering is higher than that obtained by RFGK-means clustering.

In the process of using traditional K-means for clustering, the clustering time and average error fluctuate greatly. Since the initial value is randomly selected, the clustering time is unstable, and the clustering effect is easy to fall into local optimal. The other three algorithms use the global method to find the initial clustering center, and can output the clustering center stably, so as to obtain stable clustering results. RFGK-means and RMFGK-means are faster than FGK-means in the selection of initial clustering center. Mahalanobis distance is used to take into account the global distribution of data, instead of Euclidean distance, which can improve the accuracy of clustering results in real data sets.