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This paper shows the meaning of Pearson residuals as an indicator of statistical independence in a multi-way contingency table. While information granules of statistical independence of two variables can be viewed as determinants of 2 ?? 2- submatrices, those of multivariate cases consist of linear sum of residuals for odds ratios, which can be viewed as an extention of determinants in 2 ?? 2 matrices.
Granular computing (GrC) is a recent label, roughly speaking, jointly coined by Lin and Zadeh in 1996 to denote an emerging technology that is based on the computing/mathematical theory of an ancient concept of granulation. In this paper, we present the rdquofinalrdquo GrC model that simplify the earlier version.
Current state of granular computing (GrC) is summarized. The historical development of infinitesimal granules is reviewed for future lessons. A category theory based GrC model, that nearly realizes (we believe it will realize) all classical examples of GrC, is re-examined; all early models are derived. Some possible future directions are proposed.
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