Informatica

Multidimensional scaling (MDS) is a widely used technique for mapping data from a high-dimensional to a lower-dimensional space and for visualizing data. Recently, a new method, known as Geometric MDS, has been developed to minimize the MDS stress function by an iterative procedure, where coordinates of a particular point of the projected space are moved to the new position defined analytically. Such a change in position is easily interpreted geometrically. Moreover, the coordinates of points of the projected space may be recalculated simultaneously, i.e. in parallel, independently of each other. This paper has several objectives. Two implementations of Geometric MDS are suggested and analysed experimentally. The parallel implementation of Geometric MDS is developed for multithreaded multi-core processors. The sequential implementation is optimized for computational speed, enabling it to solve large data problems. It is compared with the SMACOF version of MDS. Python codes for both Geometric MDS and SMACOF are presented to highlight the differences between the two implementations. The comparison was carried out on several aspects: the comparative performance of Geometric MDS and SMACOF depending on the projection dimension, data size and computation time. Geometric MDS usually finds lower stress when the dimensionality of the projected space is smaller.

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.

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.

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.