Proposal of Clustering Approach Based on Structural Mechanics: An Application of Multi-Dimensional Truss

We deal with an approach of clustering based on structural mechanics. Rupture of multi-dimensional truss due to a universal repulsive force is adopted as the process of clustering. The structural behavior of multi-dimensional truss is formulated. The feasibility of the proposed approach is examined and demonstrated by a number of calculation examples. Keywords–Clustering; Multi-Dimension; Truss; Tidal force. I. I NTRODUCTION Clustering is one of the important processes for data management, especially in the case of pattern identification and recognition [1]. Several methods of clustering have been proposed [2][3]. There are typical approaches such as hierarchical algorithms e.g., the group average method and the Ward method and partitioning algorithms e.g., the k-means method and its families. Another approach such as based on PCA (Principal Compoment Analysis) has also been developed and studied [4]. These approaches having being developed so far are basically based on some mathematical or geometrical viewpoints. The authors see that these approaches are somewhat artificial, in the sense that the clustering processes are controlled by one or more mathematical parameters that are intentionally determined. In the current study, we deal with an approach of clustering based on structural mechanics. Taking account of the mechanical characteristics of the target data set, a clustering of somewhat natural manner is considered to be possible. One of the significant problem is that structural systems dealt with in structural mechanics are two or three-dimensional entities, but the data set to be clustered can be an entity of higher dimensional space. In the case of truss structural system, however, it is possible to formulate the structural mechanical characteristics such as the stiffness matrix even in the case of a truss structure of four or higher dimensional space. In this article, we regard the data elements to be clustered as the truss nodes. We develop the formulation of stiffness matrix of truss structure of general dimension. Rupture of the truss structure due to universal repulsive force is calculated; the obtained separated parts are recognized as the clusters. In Section II, a general formulation of the nodal stiffness matrix of multi-dimensional truss is introduced. The developed clustering procedure is explained in Section III. Some preliminary example calculation results are demonstrated and discussed in Section IV and Section V gives the conclusion and future work. II. M ULTI -DIMENSIONAL TRUSS A truss structure consists of a number of truss nodes and truss members connecting them. We denote the truss nodal positions vector asX = [x1 , · · · , xN ] , where xn is a Ddimensional vector corresponding to the nth element of the data set to be clustered. Truss member connections are denoted as C = {c1, · · · , cM}, wherecm = {cm0, cm1} and cm0 and cm1 correspond to the two truss nodes connected by the mth truss member. A. Geometrical Relation We denote the truss member lengths vector as L = [l1, · · · , lM ] . Each of the member lengths is given as the Euclidean distance between the corresponding nodes expressed as lm = [ (xcm1 − xcm0) (xcm1 − xcm0) ](1/2) (1) For all of the truss member lengths and the truss nodal positions, (1) can be collected and expressed in the following form: L = L(X) (2) The total differential of (2) is given as