Small-Loop Error Rates and Experimental Design

Conceptual map

III-05.01Rectangle expansions
III-05.02Reversal and mirroring
III-05.03Two-scale calculations
III-05.04Designable policy directions
III-05.05Local power

1. From a geometric identity to a finite experiment

A finite loop contains the desired area effect together with higher-order dynamics, numerical integration error, and sampling noise. Symmetry and multiple scales separate these components only under a stated expansion.

Let $C(\varepsilon)$ be an oriented loop contrast for a fixed pair of policy directions. Assume a local expansion

\[C(\varepsilon )=\varepsilon ^{2}K+\varepsilon ^{3}R+\varepsilon ^{4}S+O(\varepsilon ^{5}).\](1)

The estimand $$K$$ is the area-normalized local bracket or target-curvature term. Define $Q(\varepsilon)=C(\varepsilon)/\varepsilon ^{2}$ . A single finite scale estimates $$K$$ with an order- $\varepsilon$ bias under (1).

2. Mirroring and two-scale correction

Proposition 1 · Bias cancellation

Suppose a mirrored design has expansion $C^{m}(\varepsilon)=\varepsilon ^{2}K-\varepsilon ^{3}R+\varepsilon ^{4}S+O(\varepsilon ^{5})$ . Then

\[\begin{aligned}\bar{Q}(\varepsilon )=[C(\varepsilon )+C^{m}(\varepsilon )]/(2\varepsilon ^{2})=K+\varepsilon ^{2}S+O(\varepsilon ^{3}), \\ Q^{R}(\varepsilon )=[4\bar{Q}(\varepsilon /2)-\bar{Q}(\varepsilon )]/3=K+O(\varepsilon ^{3}).\end{aligned}\](2)

Proof. Add the two expansions to cancel the cubic coefficient and divide by $2\varepsilon ^{2}$ . Evaluate the resulting expression at $\varepsilon$ and $\varepsilon/2$ . The weighted difference in (2) cancels the $\varepsilon ^{2}S$ term. ∎

The cancellation follows from the assumed symmetry of the two physical designs. Label reversal, time reversal, and geometric mirroring need separate derivations because they can transform nuisance terms differently.

3. Worked error decomposition

Take $C(\varepsilon)=2\varepsilon ^{2}+3\varepsilon ^{3}+5\varepsilon ^{4}$ and $C^{m}(\varepsilon)=2\varepsilon ^{2}-3\varepsilon ^{3}+5\varepsilon ^{4}$ . At $\varepsilon=0.1$ , the raw estimator is $$Q=2.35$$ and the mirrored estimator is $\bar{Q}=2.05$ . At $\varepsilon=0.05$ , $\bar{Q}=2.0125$ . The two-scale expression gives exactly 2 for this fourth-order polynomial.

Now add observation noise with variance $\sigma ^{2}/n$ to each loop contrast. Dividing by $\varepsilon ^{2}$ produces variance proportional to $\sigma ^{2}/(n\varepsilon ^{4})$ . Shrinking $\varepsilon$ reduces truncation bias and amplifies sampling error. A design therefore needs a scale range where the estimated slope is stable and the signal remains above its resolution limit.

Failure case · A visually small loop

A small $\varepsilon$ does not certify the asymptotic regime. Discontinuous policies, active constraints, branch changes, solver tolerances, and treatment noncompliance can dominate the $\varepsilon ^{2}$ term. The scale plot must include achieved policy paths and report closure alongside the normalized contrast.

4. Design and inference ledger

A randomized loop experiment assigns orientation, mirror status, and scale. The outcome contrast should be formed from the randomization unit specified by the policy. Standard errors must respect clustering induced by common paths, organizations, or time blocks. Numerical integration error is evaluated separately by solver refinement on the same assigned paths.

Choose policy directions and verify local feasibility in both orientations.
Randomize path variants at the intervention unit and record achieved paths.
Estimate raw, mirrored, and two-scale contrasts with a shared covariance calculation.
Plot normalized estimates against $\varepsilon$ and report the estimated error slope.
Declare the smallest scale whose sampling and solver uncertainty permit the target contrast to be resolved.

5. Exercises and sources

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Derive the bias orders in (2) when the remainder in (1) is uniform over the tested directions.
Minimize the leading mean-squared error $a^{2}\varepsilon ^{4}+\sigma ^{2}/(n\varepsilon ^{4})$ and interpret the resulting scale.
Construct a mirror operation that also changes $$K$$ and show why averaging then targets a different object.

Partial solutions

1. Mirroring removes the $\varepsilon ^{3}$ term before division by $\varepsilon ^{2}$ , leaving an $\varepsilon ^{2}$ bias. The weights 4/3 and $$-1/3$$ remove that term across the two scales. 2. Differentiation gives $4a^{2}\varepsilon ^{3}-4\sigma ^{2}/(n\varepsilon ^{5})=0$ . Thus $\varepsilon ^{8}=\sigma ^{2}/(a^{2}n)$ and $\varepsilon$ is proportional to $n^{-1/8}$ .

Ernst Hairer, Syvert Nørsett, and Gerhard Wanner, Solving Ordinary Differential Equations I, Chapters II–III.Local truncation error and extrapolation methods.
Douglas C. Montgomery, Design and Analysis of Experiments, Chapters 5–6.Factorial contrasts and blocking principles.
III-04 · Curvature, Transport, and Holonomy.Geometric object estimated by the finite loop.

6. Audit checkpoint

Report the expansion, symmetry action, loop scales, achieved paths, closure errors, randomization unit, covariance estimator, solver tolerance, and resolution limit. A finite contrast equals the local target plus the displayed remainder terms.

7. Scope boundary

The chapter covers local rectangular-loop designs and bias accounting. Global path optimization and nonlinear optimal control require separate tools.