Paper
Analytical Description of Boundary Effect in the 1D Kohonen Scheme for Constructing Adaptive Grids
- Authors:
- D. S. Khmel; A. V. Voytishek
- Abstract
- Kohonen self-organizing maps (SOM) have a lot of fruitful applications. In the classical monograph Kohonen T., Self-Organizing Maps (Third edition), New York: Springer-Verlag, 2001 the following fields of SOM applications are presented: signal processing, control theory, models of the biological brain function, experimental physics, chemistry and medicine, financial analysis, etc. One of the main applications of SOM is “automated” stochastic iterative numerical algorithm for constructing adaptive grids. Moreover, this algorithm can be treated as the most natural mathematical model of SOM. The algorithm starts with introducing arbitrary initial positions of points (nodes) inside some domain. On every iterative step a sample value of the stochastic variable, which is distributed with respect to given probability distribution density (this density defines the demanded arrangement of nodes in the domain), is realized numerically. The closest (“winner”) point to this sample value defines the learning coefficient (or neighborhood function) which influences on shift of every node. The special choice of the learning coefficient allows getting satisfactory arrangement of points after several iterations. There are difficulties in analytical description of the Kohonen scheme. This description is rather useful for theoretical investigation of the self-organizing algorithm (convergence, estimating of errors, etc.). In monograph the mentioned above, T.Kohonen suggested the following “continuous” approach. Under some principal simplifications (1D case, simplified form of the learning coefficient, assumption about ordering of the initial distribution of nodes, uniform distribution for “attraction nodes”) the recurrent formulas for node’s shifts are replaced by the system of differential equations for “most probable” positions of nodes. This system has no general analytical solution, and only numerical experiments can be used for investigation of asymptotic positions of nodes. In this paper we have proposed direct use of the formulas for node’s shift to get analytical recurrent expressions for the most probable positions of nodes under the Kohonen’s simplifications. We also showed that our approach helps to investigate some special effects of the self-organizing algorithm, in particular, the “boundary effect” which defines undesirable noticeable distances between the boundary nodes and the boundary of the domain. In addition we considered the possibilities for weakening of Kohonen’s restrictions: in particular, we have constructed recurrent formulas for special practical learning coefficient.
- Keywords
- Kohonen Self-Organizing Maps; Recurrent Formulas; Boundary Effect; Learning Radius
- StartPage
- 68
- EndPage
- 74
- Doi