## Mathematical Description

What is the general mathematical description of necessary conditions?

What is the mathematical description of a single necessary condition?

What is the mathematical description of multiple necessary conditions?

What is the general mathematical description of necessary conditions?

The general mathematical description of necessary conditions is Y ≤ fi(Xi), where Y is the outcome, Xi is the condition, and fi(Xi) is the ceiling line for Xi. NCA assumes that the ceiling line is non-decreasing. This allows making statements like "Xi ≥ Xic is necessary for Y = Yc", where C is a point (Xic ,Yc) on the ceiling line.

What is the mathematical description of a single necessary condition?

With one condition, NCA puts a line on the data and this line is the ceiling line f1(X1), such that X1 ≥ X1c is necessary for Y = Yc. The maximum possible Y = yc for a given value X1 = x1,  is  yc = f1(x1).

What is the mathematical description of multiple necessary conditions?

With two conditions (X1 and X2), there is a three dimensional space (X1,X2,Y) and  NCA puts a blanket on the data (ceiling surface). The projection of the ceiling surface on the X1,Y plane is the ceiling line f1(X1) for X1. The projection of the ceiling surface on the X2,Y plane is the ceiling line f2(X2) for X2.  Both X1 ≥ X1c  AND X2 ≥ X2c are necessary for Y = Yc. The maximum possible Y = yc for given values X1 = x1,  and X2= x2 is yc = min {f1(x1), f2(x2)}.

With more than two conditions there is a multidimensional ceiling with projections fi(Xi). The mathematical descriptions of a necessary AND configuration with several conditions are: Xi ≥ Xic for Y = Yand the maximum possible  Y = yc for given values of Xi = xi  is  yc =  min { fi(Xi) }, where Y is the outcome, Xi  is the i-th condition, fi(Xi)  is the i-th ceiling line and C is a point on the ceiling line.