
Ratios as constraints
Hello,
I have a problem of which I do not really see how to tackle it and
whether it can be solved. Its a real life problem, this is just a
simplifed version.
Lets assume I observe three variables (A, B, C) which consists of
two elements: a true component and a systematic component.
A= A_true + A_systematic
B= B_true + B_systematic
C= C_true + C_systematic
Lets assume the the systematic part can only take three values (1, 0,
1).
I know as a restriction the ratios of the true values with respect to
each other but not the true values themselves.
Rab = A_true / B_true
Rbc = B_true / C_true
Rac = A_true / C_true
I only observe A, B, and C, so I can calculate observerd ratios
ObsRab = A / B
ObsRbc = B / C
ObsRac = A / C
Is there something I can say about the systematic part of each
variable?
For example, if each of the ObsR is equal to their corresponding R,
then I know that the systematic part for all three variables must have
the same value, i.e either 1 or 0 or 1. Otherwise one of the ObsR
wouldn't equal the R.
So, given I observe ObsRab > Rab and the other two are equal, then
only certain values for the respective systematic components are
admissable. How can I calculate that/infer/ generalize it? I tried a
brute force approach, but writing all combantions out, that doesn't
seem practical.
Furthermore, given I can increase the number of variables and, hence,
the number of Rab, Rac, Rad, ... that serve as restrictions, wouldn't
that reduce the nuber of admissable combination for the systematic
component of the variables. In effect these are inequality
restrictions. So is there number of variables and by that restrictions
which allow me to identify the systematic component for all variables?
(still assuming the systematic componente can only the three values, I
think otherwise it would be impossible)
I dont't think I am the first to have such a problem, in fact, it
looks like something that might be pretty common for some
applications. But I don't know where to look (that also explains the
broad subject). Or maybe the problem is not solvable at all, i.e. the
knowledge of the true ration doesn't help? Event that insight would
help a lot.
Thanks for any comment!
Best,