A.3.3:5.6 - When the observed total is not enough to predict

A team wants to predict how much of two removable substances will remain after treatment. For this worked model, let a_n and b_n be their nonnegative remaining masses in kilograms after n cycles. Assume that each cycle leaves half of the first substance and a quarter of the second, with no new material added:

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

The instrument reports only their total, y_n = a_n + b_n. The model’s state is (a_n, b_n); its observation relation maps that pair to the total. The states (1, 0) and (0, 1) both give y_0 = 1 kg, but their next totals are 1/2 kg and 1/4 kg. No deterministic law using only the current total can reproduce the next total for every admitted state.

Retain the two masses when they are available. If only totals are observed, two successive exact readings recover the composition in this model: solve a_0 + b_0 = y_0 and a_0/2 + b_0/4 = y_1, giving a_0 = 4y_1 - y_0 and b_0 = 2y_0 - 4y_1. These masses must be nonnegative. Substitution into the next-cycle law gives y_2 = (3/4)y_1 - (1/8)y_0; the same recurrence applies at every later cycle. The prediction now uses one previous total as well as the current one. Equivalently, take that pair of totals as the predictive state.

If only the initial total is available, the model still gives a range: for integer n ≥ 0, y_0/4^n ≤ y_n ≤ y_0/2^n. Each extreme is attained by putting all the initial mass in one substance. Use this range when it answers the working question; otherwise obtain information about the composition. If the two substances instead have the same known retention factor r with 0 ≤ r ≤ 1, the total alone obeys y_(n+1) = r y_n. Thus whether aggregation preserves the needed law depends on the modeled operations.

The calculation assumes exact readings and fixed retention factors. Applying it to treatment data requires accounting for measurement error and establishing the retention law over the intended operating range. Lin and Lu, §§2.1–2.2 explain the broader state/observation and model-reduction problem; the two-substance case here supplies an elementary construction.