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Learning source weights for target-position estimation: comparing fixed weights and an input-dependent rule

Jun KobayashiiD

PDFDOI

In-depth summary

Claude Opus 5.5 · xhigh effort · 8 Nov 2026 · from the full text of version 2 (6 Oct 2026) · CC BY 4.0

In simulation, under reduced accuracy or temporally correlated errors, recalibration or fixed-weight learning captured most gains; a disagreement-dependent rule beat fixed weights mainly under occasional large errors (4.05% lower MSE).

Inverse-variance weights from an earlier calibration can give an unreliable input too much influence once measurement errors change, and individual variances do not capture shared errors. Using only simulated measurements and noisy references, the study compares four ways to improve estimates: recalibrating error moments, learning one fixed weight set, a disagreement-dependent weighting rule, and keeping a longer observation history.

Key results

  1. ShownWhen measurement accuracy dropped or errors became temporally correlated, most of the gain over weights from an earlier calibration came from re-estimating error moments or learning one fixed weight set. Input-dependent weighting added little on top.

    Input-dependent rule vs supplied-MSE weighting: mean MSE 57.33% lower under reduced current accuracy and 9.34% lower under temporal correlation. Learned fixed and training-recalibrated weights obtained most of these reductions. Under independent errors the reduction was 0.36% (100 runs per condition). · 3.1 Updating an earlier calibration; 4.1 What learning added beyond recalibration

  2. ShownWith occasional large measurement errors, the nine-parameter input-dependent rule beat learned fixed weights in every run. In the other conditions its extra benefit was below 0.5%, and under temporal correlation fixed weights did marginally better.

    Large errors: mean MSE 2.7140 × 10^−5 vs 2.8285 × 10^−5 m^2 (4.05% lower), lower in all 100 runs; paired difference 1.145 × 10^−6 m^2 (95% interval [1.102, 1.187] × 10^−6). Independent errors: 0.37% lower (rule better in 84 runs). Reduced accuracy: 0.47% lower (65 runs). Temporal correlation: 0.098% higher (rule better in 40 runs). · 3.2 Additional benefit of input-dependent…; Abstract

  3. ShownUnder large errors, the rule gave less weight to current measurements that carried an added offset, even though the offset labels were never available to it. This is descriptive evidence of selective weighting, not of causal inference.

    Mean weight on the current measurement: 0.268 without offsets vs 0.175 with offsets. Clean-minus-offset difference 0.093 (95% interval [0.091, 0.096]), positive in all 100 runs; mean within-run relative reduction 34.7% of the clean weight. · 3.2 Additional benefit of input-dependent…; 2.5 Outcomes, sensitivity, and verification; 4.2 Disagreement, large measurement errors, and…

  4. ShownOver the whole sequence, estimators that keep the full measurement history usually beat both two-input rules. At the first evaluated measurement, however, the input-dependent rule beat them under large errors.

    History averaging had lower mean MSE than the input-dependent rule under independent, large-error and temporal-correlation conditions, but not under reduced current accuracy (later SD 0.03 m vs 0.01 m). History GLS had the lowest mean MSE under reduced accuracy and temporal correlation. Post hoc, at the first evaluated point, the rule beat fixed weights, averaging and GLS in all 100 large-error runs. · 3.3 History controls and measurement sensitivity; 4.2 Disagreement, large measurement errors, and…; Appendix A. Complete method comparisons

  5. SuggestedLearning cannot detect a biased coordinate standard or correlated reference errors, an identifiability limit that the exploratory sensitivity runs illustrate. Lower error against the references therefore need not mean better physical accuracy.

    A standard bias shifts corrected measurements and training references together. Across all nine settings the large-error benefit stayed positive on average, but its percentage was smaller with 5- and 10-mm bias and in the combined setting. At ρ = 0.9 the difference-based noise estimate and the mean reference's error variance differ by a ratio of 19 (12 runs per setting). · 3.3 History controls and measurement sensitivity; 4.3 Measurement uncertainty and biological…; Appendix B. Sensitivity to measurement…

Methods, data and reproducibility

Fixed 3D targets got 23 simulated measurements per trial; two-input estimators combined each new one with the mean of the first three using supplied, recalibrated, learned fixed, or nine-parameter input-dependent weights (Adam then L-BFGS, minimizing mean squared error against noisy reference means), versus history mean, median and generalized least squares (GLS); 100 runs in each of four error conditions.

  • Simulated 3D target positions in a cubic workspace (±0.05 m), measured by a primary sensor (SD 0.01 m), two reference instruments (SD 0.002 m) and a coordinate standard (SD 0.0005 m) · n = seven groups of 256 targets per run; 100 runs per condition, 400 runs total
  • Development sensitivity simulations: standard bias 1, 5 or 10 mm; reference correlation 0.5 or 0.9; reference-B SD doubled or quadrupled; one combined setting · n = 12 repetitions per setting and condition; groups of 128 targets
  • Generated measurements, fitted parameters, predictions and analysis outputs (Zenodo archive, version 0.2.0) · 10.5281/zenodo.23117348
  • Earlier materials excluded from the revised manuscript (Zenodo archive, version 0.1.0) · 10.5281/zenodo.20580788
Methods & tools
Adam, L-BFGS (strong-Wolfe line search), generalized least squares (GLS), NumPy (independent verification implementation), OpenAI Codex (AI assistance), Anthropic Claude Code (AI assistance)
Reagents & environment
The package README lists dependencies and reproduction instructions, but the text gives no versions. Code is MIT-licensed in release observable-v0.2.0, uses float64 arithmetic, and outputs are kept with SHA-256 manifests; VERIFICATION.md documents the checks.
Reproducibility gaps
The text prints no URLs. Data and code are cited only as Zenodo DOI 10.5281/zenodo.23117348 (v0.2.0) and as a GitHub v0.2.0 release, with no address given for the release.
Limitations (6)
  • The sensors, references and coordinate standard are generated models whose error levels were set, not measured. No real recordings, human performance or reaching endpoints were tested. · stated by the authors
  • Only a fixed target and translation-only calibration were tested. Not tested: moving targets, large errors in the initial measurements, persistent large errors, occlusion, image formation, packet delays, and scale or rotation errors. · stated by the authors
  • Learning cannot reveal a bias shared by the standard and the references, or correlated reference errors. Gains measured against noisy references therefore do not establish better physical accuracy. · stated by the authors
  • Training and evaluation used the same post-calibration error condition, so transfer to a new error regime is untested. Validation stopping does not show global convergence, and the comparison does not separate the effect of parameter count or model selection from the nonlinear features. · stated by the authors
  • The sensitivity analyses used 12 runs per setting and so carry larger Monte Carlo uncertainty. The confidence intervals are exploratory and not corrected for multiple comparisons, and no equivalence testing was done. · stated by the authors
  • The author designed both the contamination model (offsets only after three offset-free initial measurements) and the rule's disagreement-magnitude feature for this comparison, so the 4.05% gain may not hold under other error structures. · noted in this summary

Read this if you combine noisy measurements or cues using calibrated weights and want to know when recalibration, learned fixed weights, disagreement-dependent weighting or longer observation histories actually reduce error.

Citations
1

Abstract

Estimating an object's location requires combining uncertain measurements. Weights derived from an earlier calibration may become inappropriate when measurement errors change. We examined whether learning from noisy reference measurements improves position estimates, and whether weights computed from current inputs improve further than one learned set of weights used throughout evaluation. In simulations, we generated noisy position measurements of a target that remained fixed within each trial. Each two-input method combined the current measurement with the mean of three initial measurements. Measurement-accuracy reductions and added measurement offsets applied only after initialization. The estimator received no current or previous true target position. We repeated coordinate calibration, training, validation, and independent evaluation 100 times in each of four error conditions. Recalibration and fixed-weight learning explained most of the improvement over earlier supplied weights under reduced measurement accuracy and temporal correlation. With occasional large measurement errors, input-dependent weighting reduced mean squared error by 4.05% relative to learned fixed weights and assigned lower mean weights to measurements with added offsets. Its additional benefit was below 0.5% under independent errors and reduced current accuracy, while fixed weights had slightly lower mean error under temporal correlation. Methods using longer observation histories performed better in several conditions. Sensitivity analysis showed that shared coordinate bias and correlated reference errors limit conclusions about physical accuracy. The results show benefits and limits of recalibration, input-dependent weighting, and observation history within these measurement models.

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