Crossed tables can show the results of correlations between categories given by titles. 2 differents categories appear as a square if with the same number of categories in each side of quadrangle. In this square the results appear as a triangle or triangular matrix, especially if the correlations betweens are commutative. Categories can ben related to each other. So it is often necessary to self-cross and study what is called self-correlation matrix. Of course a reduciotnist perspective will prefer, only pure partition given by just main diagonal of matrix, because this makes matrix calculus more easy. But confusion and social matrix can consider that categories are significant but correlations more complicated. As a result other cells than main diagonal are not empty. Perfect discrimating categories would show only one line of correlation.
Now trying to study some phenomena at some complex level of definition like trying to 'geometrically' enclose a multilevel phenomena, at a given significant level of scale (be it for example, micro, meso and macro) and consider 4 perspectives at the level or covering a piece of scale. These 4 perspectives will statistically be approached by 4 matrices of correlations of categories used and categories will be tested on 4 autocorrelation matrix. The disposal of the frame may be imagined as a plate of 4 sectors and the pathway organizing the exploration or sequence of tests or inference. There is also an entry and a way out, which fit well with the idea that it is some supposed relevant level of scale which is explored coming from a micro down and a macro going up. This of course because we suggest to explore complex things at each level like an hypercube. The same way a cube has 4 lateral squares, plus one down and one up. It could be less or more but the number of 4 seems to be able to be inspiring, for geometrical and formal reasons. Also what apply directly may not be a simple matrix calculus:, this depends on categories and selected problems. At least there is a systemic management that can be softwared more automatically easilly, instead of having to reapply same procedures separatelly for many specific definitions.
For a practical example: imagine you want to set the medical treatment (just at the mid scale of the patient). You will be interested in reviewing systematically medicines: the biochemical class of the treatment, its organic effects, its pathological indications and the prognosis effects. Be for a given medicine with complex effects, type of treatment or for a group, a class and so on. You put crossed tables successively. Auto or self-correlated matrices will mean, successively the fundamental biochemical or metbolical relations, the relations to organ, the similarities between pathologies (or groups of pathologies) and the issues. Weakest links often make the definition and many authorized medical drugs have same effect or non individually specifiable effect. Hability to show positive and negative effects provide suggest therapeutic as well as adverse effects. Lack of data, because there are groups of effects, drugs and so on, may be approached by inner matrices, partial calculus, suggested by cells in correspondance, or practically compared to a theoretical estimate.