Fuzzy C-means (FCM) Clustering algorithm

Algorithm principle Allows the same data to belong to several different classes. The algorithm (developed by Dunn in 1973 and improved by Bezdek in 1981) is often used for pattern recognition, based on the minimization of The following target functions: , where M is a real number greater than 1, Uij is XI belongs to the class J membership degree, xi I measure to the D-dimensional data, CJ is a class J Clustering Center, | | *|| Represents any measurement data and clustering The similarity of the center. Fuzzy partitioning is using the following two-type update iterations to make the above objective function very small: , When the iteration stop,0< <1 is the iteration termination parameter, K is the number of iterations. The process converges to a minimum value of JM. The algorithm consists of the following steps: Initialize Membership matrix U=[uij], U (0) K-Round iteration: Computing Center vector C (k) =[CJ] with U (k) Update Membership Matrix U (k), U (k+1) If | | U (k+1)-u (k) | | < then STOP; Otherwise return to step 2. Description As mentioned earlier, the data belongs to a class is determined by the membership function, which is the embodiment of the algorithm's fuzzy behavior. In this algorithm, a matrix expression consisting of elements between 0 and 1 is used. The affiliation of the object to the category. To give a one-dimensional example, given a particular data set, the map is distributed as follows: It is easy to distinguish two kinds of data from the graph, which is expressed as ' A ' and ' B ' respectively. Using the aforementioned K-means algorithm, each data is associated with a specific centroid, membership function As shown below: Using FCM algorithm, the same data does not belong to a single classification, but can appear in the middle. In this example, the membership function becomes smoother, indicating that each data May belong to several classifications. In the above image, the red dot indicates that the data is more likely to belong to category B, rather than a, and the value of ' m ' 0.2 indicates the degree to which the data is subordinate to a. Now, without a diagram, we introduce A matrix whose elements are taken from the membership function: (a) (b) The number of rows and columns of a matrix depends on the number of data and categories, the exact number of rows representing the number of data, the number of columns representing the number of categories, and the matrix elements represented as UIJ. In the example above, we consider the example of K-means (a) and FCM (b), and we can see that the sparse is always binary in the first example (a), indicating that each data can only belong to one Classification, other properties are represented as follows: References J. C. Dunn (1973): "A Fuzzy Relative of the ISODATA Process and its use in detecting Compact well-separated Clusters" , Journal of Cybernetics 3:32-57 J. C. Bezdek (1981): "Pattern recognition with Fuzzy Objective Function Algoritms", Plen Um Press, New York Tariq Rashid: "Clustering" Http://www.cs.bris.ac.uk/home/tr1690/documentation/fuzzy_clustering_initial_report/node11.html Hans-joachim Mucha and Hizir Sofyan: "Nonhierarchical Clustering" Http://www.quantlet.com/mdstat/scripts/xag/html/xaghtmlframe149.html

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