Effective repetition count across independent calibration blocks

Effective repetition count across independent calibration blocks accounts for dependence confined to each acquisition block rather than shared by every stored trace. Suppose block $j$ contains $n_j$ empty-interval indicator random variables, all with common probability $0<p<1$ and within-block correlation coefficient $\rho \ge 0$. Assume different blocks are independent. With $N={\sum }_j{n}_j$, summing within-block covariances givesMatching this to the variance of an average of independent indicators yieldsFor $K$ equal blocks of size $m$, this reduces to $N_{\mathrm{eff}}=Km/[1+(m-1)\rho ]$. The denominator involves the block size, not the total $Km$; the single-block expression in effective repetition count for detector intervals with shared drift must not be applied as though the same drift state affected every block.

For an original comparison, fix $N=120$ and $\rho =0.05$. One block gives $N_{\mathrm{eff}}=120/6.95\approx 17.27$. Six independent blocks of twenty give $N_{\mathrm{eff}}=120/1.95\approx 61.54$. Six very uneven blocks of sizes $(100,4,4,4,4,4)$ instead giveAll three designs store the same number of traces. The difference comes from how often the same fluctuating condition is reused. For fixed $N$, $K$, and positive $\rho$, minimizing ${\sum }_j{n}_j^2$ favors balanced blocks; this motivates optimal allocation of detector repetitions across calibration blocks when block-preparation costs also matter.

Independence must come from the physical acquisition model. Starting another file does not erase a shared clock, reference calibration, or slowly varying environmental influence. NIST's uncertainty law retains shared-input covariance explicitly.^[ref] If drift persists between nominal blocks, those cross-block terms must be added and the displayed formula changes. A cluster bootstrap likewise requires blocks that are suitable independent resampling units.

Optimal allocation of detector repetitions across calibration blocks chooses block sizes using the dependence and acquisition-cost model, not just the desired number of stored traces. For fixed total $N$, number of independent blocks $K$, common success probability, and positive within-block correlation $\rho$, effective repetition count across independent calibration blocks givesMaximizing it is equivalent to minimizing ${\sum }_j{n}_j^2$. If two integer block sizes differ by at least two, transferring one repetition from the larger to the smaller decreases that sum by $2(n_i-n_j-1)>0$. Therefore the optimal integer sizes differ by at most one.

For an original fixed-count example, $N=23$ and $K=5$ give balanced sizes $(5,5,5,4,4)$. At $\rho =0.1$, their effective count is $529/31.4\approx 16.85$. Sizes $(11,3,3,3,3)$ give only $529/36.4\approx 14.53$ under the same assumptions.

More independent blocks can help, but preparing one may consume time. In a continuous planning model let each block cost setup time $h>0$, each repetition cost $\tau >0$, and each block contain $n$ repetitions. With total time $T$, the effective count becomesFor $0<\rho <1$, differentiation gives the interior choice $n_{\ast }=\sqrt{h(1-\rho )/(\tau \rho )}$. Integer block counts, minimum sizes, and hardware restrictions must subsequently be imposed.

For chosen $h=9\mathrm{s}$, $\tau =1\mathrm{s}$, $\rho =0.1$ and $T=180\mathrm{s}$, the continuous optimum $n_{\ast }=9$ is feasible with ten blocks and gives $N_{\mathrm{eff}}=50$. Single-repetition blocks give 18, while two blocks of 81 repetitions also give 18. These are model-based comparisons, not measured apparatus timings.

Unequal block noise or different drift correlations require unequal-noise allocation across calibration blocks. Independence between prepared blocks must also be justified by the reference and environmental model; NIST explicitly retains shared-input covariance.^[ref]

Unequal-noise allocation across calibration blocks distributes repetitions using block-specific variability while keeping the intended estimator fixed. Suppose independent blocks supply means ${\overline{Y}}_j$ and the target estimate is $\hat{\theta }={\sum }_j{w}_j{\overline{Y}}_j$ with specified weights $w_j$. Let one observation in block $j$ have variance ${\sigma }_j^2$ and equicorrelation $0\le {\rho }_j<1$ within that block. ThenThe first sum is a block-specific floor; only the second decreases with repetition count. Changing the estimator weights merely to improve this calculation could change the estimand, so they remain fixed here.

For continuous positive allocations with positive costs $c_j>0$, budget ${\sum }_j{c}_j{n}_j=C$ and $a_j=w_j^2{\sigma }_j^2(1-{\rho }_j)>0$, a Lagrange-multiplier calculation givesFor equal observation costs, repetitions are proportional to $|w_j|{\sigma }_j\sqrt{1-{\rho }_j}$, not to the variance itself. Minimum counts and integer rounding must subsequently respect the actual acquisition constraints.

For an original two-block example, take $w_1=w_2=1/2$, equal costs, $C=100$, independent readings within each block, and standard deviations $1$ and $3$ in the output unit. The optimal counts are $25$ and $75$, giving variance $0.04$ instead of $0.05$ from fifty each. If both within-block correlations are instead $0.1$, the same proportions give variance $0.286$, while equal counts give $0.295$. The common-error floor $0.25$ now dominates and cannot be eliminated by taking more observations inside those same blocks.

This extends optimal allocation of detector repetitions across calibration blocks, whose balanced solution used homogeneous noise. For binary detector indicators, the per-reading variance is $p_j(1-p_j)$; differing probabilities require a clearly specified weighted target. The planning variances and correlations must come from an appropriate measurement calibration model, not merely from counting files as independent blocks. NIST's uncertainty propagation retains shared-input covariance for exactly this reason.^[ref]