= Cycle-gain uncertainty
<Cycle-gain uncertainty> describes uncertainty in the product of the multipliers achieved over repeated <chemical multiplication reaction> cycles. Its size depends on <cycle-gain correlation>: environmental or reagent effects shared across cycles behave differently from independent cycle variation.
For positive multipliers $K_j$, write $G=\prod_{j=1}^nK_j$ and $\ell_j=\ln K_j$. Then $\ln G=\sum_j\ell_j$, so
$$
\operatorname{Var}(\ln G)=\sum_{i=1}^n\sum_{j=1}^n\operatorname{Cov}(\ell_i,\ell_j).
$$
For small relative errors, $u(G)/G$ is approximately the <standard deviation> of $\ln G$. This applies the correlated-input <uncertainty propagation> framework to the product model.<Evaluation of measurement data — Guide to the expression of uncertainty in measurement by JCGM>[§5.2, covariance contributions and sensitivity coefficients.]{ref}
An original intermediate-correlation model sets $\operatorname{Cov}(\ell_i,\ell_j)=\sigma^2\rho^{|i-j|}$. Grouping equal separations gives
$$
v=\operatorname{Var}(\ln G)=\sigma^2\left[n+2\sum_{d=1}^{n-1}(n-d)\rho^d\right].
$$
For $n=4$, $\sigma=0.02$ and $\rho=0.5$, the bracket is8.25, giving relative <standard uncertainty> approximately0.0574. Independent cycles would give0.040, while completely shared log-error would give0.080. Specifying only the uncertainty of each individual cycle leaves total-gain uncertainty undetermined.
If the log-multipliers are jointly normally distributed, $G$ has a <log-normal distribution>. With $\mu_G=\operatorname{E}(\ln G)$,
$$
\operatorname{median}(G)=e^{\mu_G},\qquad
\operatorname{E}(G)=e^{\mu_G+v/2},\qquad
\frac{\operatorname{sd}(G)}{\operatorname{E}(G)}=\sqrt{e^v-1}.
$$
These exact relations under the stipulated distribution distinguish the mean gain from its median when variation is substantial.
Estimate correlations from a justified shared-input model or appropriately replicated sequences. Common calibration, reagent variation and cycle-dependent depletion may contribute. Adding more ideal cycles can increase nominal gain while making the final <measurement uncertainty> more sensitive to those shared inputs.
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