= Cycle-gain correlation
<Cycle-gain correlation> describes statistical dependence between the multipliers achieved in different cycles of a <chemical multiplication reaction>. Shared reagent conditions, preparation or <measurement calibration> can make cycle gains move together. For positive multipliers $K_j$, total gain is $G=\prod_jK_j$ and
$$
\operatorname{Var}(\ln G)=\sum_{i,j}\operatorname{Cov}(\ln K_i,\ln K_j).
$$
This is an exact identity for finite log-<variances>; approximating the <standard deviation> of $\ln G$ as relative <standard uncertainty> of $G$ additionally requires small relative variation. It is an application of the correlated-input framework to the repeated-cycle gain model.<Evaluation of measurement data — Guide to the expression of uncertainty in measurement by JCGM>[Section 5.2, covariance contributions]{ref}
An original shared-condition model separates batch and cycle effects:
$$
\ln K_j=\mu_j+B+\varepsilon_j,\qquad
\operatorname{Var}(B)=\tau^2,\quad\operatorname{Var}(\varepsilon_j)=\sigma^2,
$$
where $B$ and the centered independent $\varepsilon_j$ are mutually independent. Then each log-gain has <variance> $\tau^2+\sigma^2$, distinct cycles have <covariance> $\tau^2$, and
$$
\rho=\frac{\tau^2}{\tau^2+\sigma^2},\qquad
\operatorname{Var}(\ln G)=n^2\tau^2+n\sigma^2.
$$
The common contribution accumulates with $n^2$ in <variance>, while independent contributions accumulate with $n$.
For six cycles, $\tau=0.01$ and $\sigma=0.02$, the pairwise log-gain <correlation coefficient> is0.2and the final log-gain <standard deviation> is $\sqrt{0.006}\approx0.0775$. Treating those same individual log-gain <variances> as independent would give $\sqrt{6(0.01^2+0.02^2)}\approx0.0548$. A per-cycle uncertainty value without its dependence structure cannot determine <cycle-gain uncertainty>.
Across replicated sequences, observing each cycle's gain permits estimation of cross-cycle <covariances>. Observing only the final product at one fixed cycle count generally cannot identify how its spread divides between shared and independent terms. Matched intermediate measurements and controlled changes of reagents or temperature provide more information, but their measurement errors must be included. Correlation alone does not identify the physical cause, and the positive common-effect model does not exclude other dependence patterns such as depletion across cycles.
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