47 research outputs found
About the maximal rank of 3-tensors over the real and the complex number field
High dimensional array data, tensor data, is becoming important in recent
days. Then maximal rank of tensors is important in theory and applications. In
this paper we consider the maximal rank of 3 tensors. It can be attacked from
various viewpoints, however, we trace the method of Atkinson-Stephens(1979) and
Atkinson-Lloyd(1980). They treated the problem in the complex field, and we
will present various bounds over the real field by proving several lemmas and
propositions, which is real counterparts of their results.Comment: 13 pages, no figure v2: correction and improvemen
SVD-based principal component analysis of geochemical data
Principal Component Analysis (PCA) was used for the mapping of geochemical data. A testing data matrix was prepared from the chemical and physical analyses of the coals altered by thermal and oxidation effects. PCA based on Singular Value Decomposition (SVD) of the standardized (centered and scaled by the standard deviation) data matrix revealed three principal components explaining 85.2 % of the variance. Combining the scatter and components weights plots with knowledge of the composition of tested samples, the coal samples were divided into seven groups depending on the degree of their oxidation and thermal alteration.
The PCA findings were verified by other multivariate methods. The relationships among geochemical variables were successfully confirmed by Factor Analysis (FA). The data structure was also described by the Average Group dendrogram using Euclidean distance. The found sample clusters were not defined so clearly as in the case of PCA. It can be explained by the PCA filtration of the data noise