 Allocation of a deterministic bias budget among baseline constraints

Allocation of a deterministic bias budget among baseline constraints assigns how much residual signal-response and background-leakage error each constraint may contribute. Let the allowed total bias be $E>0$, and let coefficient bounds $B_j>0$ multiply response errors $r_j$. Choose nonnegative fractions $f_j$ with ${\sum }_j{f}_j\le 1$ and setIf every computed response obeys $|r_j|\le {\delta }_j$, the triangle inequality gives $|\mathrm{Bias}|\le {\sum }_j{B}_j{\delta }_j\le E$. The response-normalized tolerance for baseline-feasibility residuals checks those component limits. Positive tolerances require positive allocated fractions; a zero allocation demands exact cancellation in the stated model.

For an original example, take $E=12\mathrm{nV}$ and bounds of $2\mathrm{mV}$ for signal amplitude, $10\mathrm{mV}$ for constant background and $100V/s$ for linear drift. Allocate fractions $1/6$, $1/3$ and $1/2$. The respective bias shares are $2$, $4$ and $6\mathrm{nV}$, giving tolerancesEqual bias shares would instead allow $4\mathrm{nV}$ each, relaxing the signal-response tolerance to $2\times 10^{-6}$ while tightening the drift tolerance to $4\times 10^{-11}\mathrm{s}$. Both allocations guarantee the same total bound when their assumptions hold; their numerical and experimental demands differ.

An allocation should reflect attainable constraint accuracy and the cost of tightening each response. Cost-weighted allocation of baseline-response tolerances can formalize that tradeoff. Equal numerical tolerances across differently dimensioned conditions do not mean equal physical bias contributions. If a bound $B_j$ is uncertain, use a justified upper bound or include its uncertainty explicitly before calling the total a guarantee.

NIST's pulse-energy apparatus used before/after measurements for baseline and drift correction.^[ref] The bounds above are illustrative design choices. This budget controls deterministic leakage within the assumed background family. It is separate from measurement-noise variance, whose covariance contributions must still be propagated.^[ref]