2.1 Quadratic Regularization Clustering Algorithm
Among the fuzzy clustering algorithms, the fuzzy c-means clustering algorithm is the most classical algorithm(Dubey, Y.K., & Mushrif, M.M.,2016). FCM algorithm adjusts the fuzzy factor 
, regards the nonlinear parameters 
as the regularization of c-means, and iteratively updates the clustering center and membership matrix until the best clustering center is obtained. Different from FCM, quadratic regularization clustering algorithm takes maximum entropy clustering (MEC) as an example (Karayiannis, N.B.,1994; Li,R.,& Mukaidono,M.,1995) and quadratic function 
as a new nonlinear term. Give a dataset 
consist of 
data, where 
denotes the dimension of the data in the dataset, and 
denotes the number of samples in the dataset. The number of clusters in the dataset can be expressed as 
. The quadratic regularization clustering algorithm can be expressed mathematically as the following functions:
(1)where
represents the clustering centroid,
represents the membership matrix,
indicates that the sample point
belongs to the fuzzy membership value of the
clustering centroid, and
represents the regularization parameter.
By using Lagrange optimization and iteratively updating the objective function, it is easy to derive the updated clustering centroid equation as follows:
(2)(3)The formula (3) can be obtained from formula (1). Therefore:
Each 
is minimized by 
individually:
(4)Suppose 
, 
can be calculated by formula (4) to satisfy 
. Let:
when
, let
. According to this algorithm, the membership matrix
can be obtained.