Unsupervised Clustering Algorithm Based on Normalized Mahalanobis Distances

Some of the well-known fuzzy clustering algorithms are based on Euclidean distance function, which can only be used to detect spherical structural clusters. Gustafson-Kessel clustering algorithm and Gath-Geva clustering algorithm were developed to detect non-spherical structural clusters. However, the former needs added constraint of fuzzy covariance matrix, the later can only be used for the data with multivariate Gaussian distribution. Three improved Fuzzy C-Means algorithm based on different Mahalanobis distance, called FCM-M, FCM-CM and FCM-SM were proposed by our previous works, In this paper, an improved Fuzzy C-Means algorithm based on a Normalized Mahalanobis distance (FCM-NM) by taking a new threshold value and a new convergent process is proposed The experimental results of two real data sets show that our proposed new algorithm has the better performance. Key-Words: Unsupervised Clustering Algorithm, FCM-NM algorithm, GK-algorithm; GG-algorithm; Normalized Mahalanobis Distances 1 Motivation and Preface Fuzzy clustering is a branch in clustering analysis and it is widely used in the pattern recognition field. The well-known ones, such as Bezdek’s Fuzzy C-Means (FCM) and Li et al’s Fuzzy Weighted C-Means (FWCM) [1,2], are based on Euclidean distance. These fuzzy clustering algorithms can only be used to detect the data classes with the same super spherical shapes. . To overcome the drawback due to Euclidean distance, we could try to extend the distance measure to Mahalanobis distance (MD). However, Krishnapuram and Kim (1999) [3] pointed out that the Mahalanobis distance can not be used directly in clustering algorithm. Gustafson-Kessel (GK) clustering algorithm [4] and Gath-Geva (GG) clustering algorithm [5] were developed to detect non-spherical structural clusters. In GK-algorithm, a modified Mahalanobis distance with preserved volume was used. However, the added fuzzy covariance matrices in their distance measure were not directly derived from the objective function. In GG algorithm, the Gaussian distance can only be used for the data with multivariate normal distribution. In our three previous works, to add a regulating factor of Each covariance matrix to each class in the objective function, and deleted the constraint of the determinants of covariance matrices in the GK algorithm, the Fuzzy C-Means algorithm based on adaptive Mahalanobis distances, common Mahalanobis distance and standardized Mahalanobis distance, respectively (FCM-M, FCM-CM, and FCM-SM), [8-12,16] were proposed, and then, the fuzzy covariance matrices in the Mahalanobis distance can be directly derived by minimizing the objective function. In this paper, not only replacing the common covariance matrix with the correlation matrix in the objective function in the FCM-CM algorithm but also replacing the threshold D in equation (43) of FCM-CM algorithm with the determinant value of the crisp correlation matrix,R, and then, a new fuzzy clustering method, called the Fuzzy C-Means algorithm based on normalized Mahalanobis distance (FCM-NM), is proposed. Proceedings of the 9th WSEAS Int. Conference on APPLIED COMPUTER and APPLIED COMPUTATIONAL SCIENCE ISSN: 1790-5117 180 ISBN: 978-960-474-173-1 2 Literature Review Clustering technique plays an important role in data analysis and interpretation. It groups data into clusters so that the data objects within a cluster have high similarity in comparison to one another, but are very dissimilar to those data objects in other clusters. 2.1 GK Algorithm FCM can only work well for spherical shaped clusters. In the objective function the distances between data points to the centers of the clusters are calculated by Euclidian distances. To overcome the above drawback, we could try to extend the distance measure to Mahalanobis distance (MD). However, Krishnapuram and Kim (1999) [2] pointed out that the Mahalanobis distance can not be used directly in clustering algorithm. Gustafson and Kessel (1979) extended the Euclidian distances of the standard FCM by employing an adaptive norm, in order to detect clusters of different geometrical shape without changing the clusters’ sizes in one data set. The objective function of GK algorithm is given in Equation (1),(2),(3) and (4).