A.3.3.PI:5.1 - Recover a treatment forecast from component masses or a short history

Let a_n and b_n be nonnegative masses in kilograms after n treatment cycles. Each cycle leaves half of the first substance and a quarter of the second, with no new material:

a_(n+1) = a_n/2
b_(n+1) = b_n/4
y_n = a_n + b_n

The instrument reports only y_n. States (a_0,b_0) = (1,0) and (0,1) both report y_0 = 1 kg but give next totals 1/2 kg and 1/4 kg. The aggregate has lost the composition needed for a unique next total.

One repair retains the two masses and updates them separately. Another uses two successive readings without measurement error. From y_0 = a_0 + b_0 and y_1 = a_0/2 + b_0/4,

a_0 = 4*y_1 - y_0
b_0 = 2*y_0 - 4*y_1
y_2 = (3/4)*y_1 - (1/8)*y_0

The recovered masses must be nonnegative, requiring y_0/4 <= y_1 <= y_0/2. For y_0 = 1 kg and y_1 = 0.35 kg, the masses are 0.4 kg and 0.6 kg, and y_2 = 0.1375 kg.

The recurrence applies at later cycles under the same law. A rolling pair (previous total, current total) can replace the hidden component values for predicting future totals. On receiving a new total, retain it and the former current total. If no new measurement arrives, the recurrence can propagate the pair as a conditional model prediction.

With only y_0, nonnegativity yields the bound

y_0/4^n <= y_n <= y_0/2^n, for integer n >= 0.

For y_0 = 1 kg, the next total is at most 0.5 kg. This settles a requirement at most 0.6 kg without learning the composition. A limit of 0.4 kg remains unresolved by that bound.

Now suppose the two readings each have absolute error at most epsilon kilograms while the retention law is fixed. The reconstructed a_0 can err by at most 5epsilon, and b_0 by at most 6epsilon. The formula for y_2 has error at most (7/8)*epsilon from those two reading errors: add the absolute contributions (3/4)*epsilon and (1/8)*epsilon. At epsilon = 0.01 kg this gives 0.00875 kg. A marginal threshold decision must account for that interval; a recovered negative mass calls for checking the measurement uncertainty and the model assumptions.

If both substances instead have the same fixed retention factor r, the total obeys y_(n+1) = r*y_n. Composition is then unnecessary for predicting totals under that law. The needed information changes with the operations and the question.