Pub. online:1 Jan 2009Type:Research ArticleOpen Access
Volume 20, Issue 2 (2009), pp. 235–254
Most of real-life data are not often truly high-dimensional. The data points just lie on a low-dimensional manifold embedded in a high-dimensional space. Nonlinear manifold learning methods automatically discover the low-dimensional nonlinear manifold in a high-dimensional data space and then embed the data points into a low-dimensional embedding space, preserving the underlying structure in the data. In this paper, we have used the locally linear embedding method on purpose to unravel a manifold. In order to quantitatively estimate the topology preservation of a manifold after unfolding it in a low-dimensional space, some quantitative numerical measure must be used. There are lots of different measures of topology preservation. We have investigated three measures: Spearman's rho, Konig's measure (KM), and mean relative rank errors (MRRE). After investigating different manifolds, it turned out that only KM and MRRE gave proper results of manifold topology preservation in all the cases. The main reason is that Spearman's rho considers distances between all the pairs of points from the analysed data set, while KM and MRRE evaluate a limited number of neighbours of each point from the analysed data set.
Pub. online:1 Jan 1995Type:Research ArticleOpen Access
Volume 6, Issue 2 (1995), pp. 225–232
An algorithm for the sequential analysis of multivariate data structure is presented. The algorithm is based on the sequential nonlinear mapping of L-dimensional vectors from the L-hyperspace into a lower-dimensional (two-dimensional) vectors such that the inner structure of distances among the vectors is preserved. Expressions for the sequential nonlinear mapping are obtained. The mapping error function is chosen. Theoretical minimum amount of the very beginning simultaneously mapped vectors is obtained.
Pub. online:1 Jan 1993Type:Research ArticleOpen Access
Volume 4, Issues 1-2 (1993), pp. 81–93
An algorithm for the sequential analysis of multivariate data is, presented along with some experimental results. The algorithm is based upon the sequential nonlinear mapping of L-dimensional vectors from the L-hiperspace into a lower-dimensional (two-dimensional) vectors such that the inner structure of distances between the vectors is preserved. Expressions for the sequential nonlinear mapping are obtained. The sequential nonlinear mapping is applied to sequential c1usterization of random processes and creation of an essentially new method for sequential detection of many abrupt or slow changes in several unknown states of dynamic systems.