STATISTICAL-THEORY OF SPOT OVERLAP FOR N-DIMENSIONAL SEPARATIONS

A theory by Roach describing the overlap of circles in a two-dimensional plane is generalized to an n-dimensional space, where n is the number of orthogonal axes. The theory is proposed as a model of overlap in an n-dimensional separation. Extensive computer simulations show that the theory describes overlap very well in three-dimensional spaces and modestly well in four-dimensional spaces, when the number of components is large. Perturbations from edge effects somewhat limit the theory's application to four-dimensional spaces. The theory shows that the maximum number of spots per unit capacity and the maximum number of any kind of multiplet per unit capacity both decrease geometrically with increasing n. The theory also shows that the maximum fraction of any kind of multiplet is independent of n. Equations are derived to relate the saturation of an n-dimensional space to the one-dimensional saturation of any axis and the one-dimensional peak capacities of the other n - 1 axes. Other equations relate the saturation of an n-dimensional separation to the saturation of an (n + 1)-dimensional separation and the one-dimensional peak capacity of the (n + 1)th dimension. With these last equations, the internal consistency of the theory is justified. A critique finally is presented, which explains why the theory does not apply to a single dimension of separation.