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PSI-SINDy: Post-Selection Inference for Sparse Identification of Nonlinear Dynamics

Ashraful Islam, Shuichi Nishino, Tomohiro Shiraishi, Ichiro Takeuchi

arXiv:2610.11486v1stat.MLcs.LG

TL;DR

Testing dynamical terms selected from noisy data is difficult because reusing the same trajectory can introduce selection bias, while SINDy also has design measurement error and shared response–design noise. PSI-SINDy uses four independent data-thinned views and bridges an auxiliary target with finite-sample-valid inference to a primary target with asymptotically valid inference. In primary simulations, its Type I error and coverage were compatible with nominal levels, while naive reuse inflated rejection.

  • Problem

    Assessing selected dynamical terms with the same noisy trajectory used to select them can introduce selection bias, complicating tests and confidence intervals.

    Source: §1 · p. 1

  • Method

    PSI-SINDy thins one trajectory into four independent views for selection, testing response, corrected design, and instrument, and connects auxiliary and primary inferential targets.

    Source: §1 · p. 2

  • Results

    In primary simulations, PSI-SINDy rejection rates were 4.41%–5.39% at the true selected BLA and 95% intervals had 94.61%–95.59% coverage; naive rejection rates were 7.36%–8.69%.

    Source: §4.1 · p. 10, §5 · p. 11

  • Takeaways & Limitations

    The procedure gives finite-sample exact conditional inference for the auxiliary realized IV coefficient and asymptotically valid inference for the primary selected-model BLA coefficient.

    Source: §5 · p. 11

    The primary BLA guarantee is asymptotic and established for scalar trajectories, fixed polynomial libraries, and independent Gaussian noise of known variance; broader settings and finite-sample calibration remain open.

    Source: §5 · p. 11

Abstract

from arXiv · show

Sparse identification of nonlinear dynamics (SINDy) is a data-driven framework for discovering governing dynamics from time-series data by identifying a sparse subset of candidate dynamical terms from a prespecified library. In this work, we develop a statistical inference framework for quantifying the reliability of dynamical terms selected by SINDy through hypothesis tests and confidence intervals. A key difficulty is that using the same noisy trajectory for both selecting dynamical terms and assessing their statistical significance can introduce selection bias. Post-selection inference provides a principled framework for addressing such bias, and we propose PSI-SINDy, a post-selection inference method tailored to SINDy. Direct application of existing post-selection inference techniques is challenging because SINDy involves measurement error in the candidate terms and shared noise between the response and design. To address these challenges, PSI-SINDy uses data thinning to decompose a single observed trajectory into four mutually independent views with distinct roles in selection and inference. This construction enables inference for selected dynamical terms while accounting not only for selection bias but also for measurement-error and shared noise effects. We establish the theoretical validity of PSI-SINDy under stated conditions and evaluate its performance through numerical experiments on simulated and experimental dynamical-system data.

1 INTRODUCTION

SINDy identifies interpretable dynamical models by selecting sparse combinations of candidate terms, but testing terms selected from the same noisy trajectory can introduce selection bias. PSI-SINDy uses four independent data-thinned views to address selection and noise-related challenges, with finite-sample validity for an auxiliary target and asymptotic validity for the primary target under stated conditions.

1 INTRODUCTION

  • Reusing a noisy trajectory for both term selection and inference can introduce selection bias because the same data determine hypotheses and supply evidence for testing them.This is also described as data double dipping.Source: §1 · p. 1
  • Four mutually independent views assign separate roles to term selection, testing response, corrected design, and instrument.Their separation addresses selection bias, shared response–design noise, and measurement error in the design.Source: §1 · p. 2
  • PSI-SINDy provides finite-sample-valid conditional inference for an auxiliary realized target and asymptotically valid inference for the primary noise-free target through a theoretical bridge.The two targets generally differ because corrected dynamical terms retain random measurement-error fluctuations.Source: §1 · p. 2
  • The paper evaluates Type I error, confidence-interval coverage and width, and the effect of accounting for uncertainty between auxiliary and noise-free targets in simulated and experimental settings.The described settings include larger candidate libraries and multistate systems.Source: §1 · p. 2
  • Existing SINDy uncertainty methods do not establish selected-model tests and confidence intervals while jointly accounting for design measurement error and shared response–design noise.PSI-SINDy addresses this joint problem for the selected noise-free approximation without requiring correct support recovery.Source: §1 · p. 3

2 PROBLEM SETUP

The setup formulates weak-form SINDy inference for selected terms from noisy scalar time-series data. It specifies three challenges the inference must address jointly and distinguishes a primary noise-free target from an auxiliary target that supports exact conditional inference under assumptions.

2 PROBLEM SETUP

  • The setup considers noisy scalar time-series observations of an underlying continuously differentiable, noise-free trajectory, with the noise variance treated as known in the theoretical development.The variance may be estimated from an independent dataset in practice.Source: §2.1 · p. 3

2.1 Weak-form SINDy and model selection

  • Weak-form SINDy approximates the noise-free weak response with a sparse combination of candidate-function columns, using integration against smooth weights to avoid numerical differentiation.Sparse regression selects candidate functions from the prespecified library using a selection view.Source: §2.1 · pp. 3–4

2.2 Statistical challenges for inference after SINDy

  • Inference must jointly address data-dependent selection, dependence between response and design errors, and measurement error and possible mean bias in the noisy design.Reusing selection data while treating the selected set as fixed can invalidate tests and confidence intervals.Source: §2.2 · p. 4

2.3 Inferential targets and hypotheses

  • For a selected model, the primary target is its noise-free weak-regression BLA coefficient; an auxiliary realized IV coefficient admits exact conditional inference and provides an asymptotic bridge to the primary target.The targets generally differ in finite samples.Source: §2.3 · p. 4
  • Testing whether a selected-model coefficient is zero does not by itself establish that the corresponding term is absent from the underlying governing equation.The null hypothesis concerns the coefficient within the selected model.Source: §2.3 · p. 5

3 PSI-SINDY

PSI-SINDy separates selection from inference using four independent views of one noisy trajectory, then links an exactly inferable auxiliary IV coefficient to the primary selected-model BLA. The resulting BLA inference is asymptotically valid under stated conditions.

3.1 Four-view Gaussian data thinning

  • Four mutually independent views assign separate roles to selection, testing response, corrected design, and instrument, separating selection from inference and the response from noisy design information.Conditioning on selection, corrected design, and instrument leaves the testing response with independent Gaussian noise.Source: §3 · p. 6, §3.1 · p. 6

3 PSI-SINDY

  • The framework targets post-selection inference for dynamical terms from a noisy trajectory, first establishing exact inference for an auxiliary IV target and then an asymptotic bridge to selected BLA inference.The primary BLA is a best linear approximation within the selected model, not necessarily a governing-equation coefficient.Source: §3 · p. 6, §3.3 · p. 7

3.2 Exact conditional inference for the realized IV coefficient

  • With known measurement-noise variance and a nonzero residualized instrument direction, the realized IV target admits finite-sample exact conditional tests and confidence intervals.When that direction is zero, the IV coefficient is unidentified and no IV test or interval is reported.Source: §3.2 · pp. 6–7

3.3 Relating the IV and BLA coefficients

  • An asymptotic IV–BLA bridge accounts for random errors in the corrected design and instrument; its gap contributes uncertainty at the response-error scale when its limiting variance is positive.When that variance is zero, the gap is negligible at that scale.Source: §3.3 · p. 7

3.4 Asymptotic inference for the selected BLA

  • The selected-BLA interval combines response-error uncertainty with the IV–BLA gap scale, which PSI-SINDy estimates using a parametric bootstrap around a pilot trajectory.Bootstrap draws fix the selected support and pilot response while regenerating the corrected design and instrument.Source: §3.4 · p. 8
  • The selected-BLA p-value converges to Uniform(0, 1) conditional on the selected support under the stated asymptotic conditions.The BLA result assumes dense observations of a scalar trajectory, a fixed polynomial library, and fixed weak weights.Source: §3.4 · p. 8

4 EXPERIMENTS

The experiments evaluate calibration, power, coverage, and interval width at nominal level 0.05, using simulated and experimental dynamical-system data. Results depend on the target, adjustment, and noise setting.

4.1 Type I error, power, coverage, and interval width

  • 4.41%–5.39% PSI-SINDy rejection at the true selected BLA contrasts with 7.36%–8.69% for Naive and 1.61%–2.02% for Bonferroni; PSI-SINDy intervals cover 94.61%–95.59%.As M rises from 200 to 3200, median BLA interval width falls from 0.3846 to 0.0957.Source: §4.1 · p. 10
  • 6.75% to 64.85% is PSI-SINDy's reference-null power range across the signal grid, versus 2.85% to 47.10% for Bonferroni; the ordering reverses in a supplementary comparison.The authors caution that neither comparison establishes superiority at matched conditional Type I error.Source: §4.1 · p. 10

Comparison with coefficient-bootstrap methods.

  • Among 2,000 datasets in which PSI-SINDy selects x2, rejection rates were 4.70% for PSI-SINDy, 8.10% for the longer-smoothing ensemble, and 20.48% for the residual adapter, for the specified adapters and structural target.The latter two Monte Carlo intervals exclude 5%; the five-point ensemble had no observed rejections but about 15 times PSI-SINDy’s median interval width.Source: Comparison with coefficient-bootstrap… · p. 10

4.2 Effect of measurement-error adjustments

  • Omitting the gap scale increases rejection for all three coefficients at both noise levels; removing feature correction raises high-noise intercept rejection from 5.53% to 11.06%.The latter ablation leaves linear and quadratic tests unchanged because the included intercept absorbs the constant quadratic-feature correction for this basis.Source: §4.2 · p. 10, §4.3 · p. 11
  • In the sine-drift study, true-coefficient rejection was 12.18% for BLA and 5.15% for IV inference; BLA failure coincided with a larger bridge remainder.Near-nominal IV calibration therefore did not ensure finite-sample BLA calibration in this setting.Source: §4.2 · p. 10

4.3 Larger libraries and multistate systems

  • Larger-library inference remained informative within sparse selected models, but the experiments do not establish growing-dimensional validity; gap adjustment improved multistate coverage without uniformly attaining nominal coverage.Unregularized selection substantially widened intervals in a separate study.Source: §4.3 · p. 10

4.4 Supercapacitor discharge data

  • For eight supercapacitor discharge traces, equal-device selected-coefficient BLA coverage was 95.23% under independent noise and 81.86% under correlated noise.The measured-trace application is outside the theoretical guarantees, and these simulations demonstrate sensitivity to dependence rather than validate measured-trace coverage.Source: §4.4 · p. 11

5 CONCLUSION

The paper proposes PSI-SINDy for inference after selecting dynamical terms from a noisy trajectory. It combines finite-sample exact inference for an auxiliary IV target with asymptotically valid inference for the selected-model BLA under stated conditions.

5 CONCLUSION

  • Four-view data thinning and measurement-error correction address selection bias, shared response–design noise, and design measurement error; primary simulations report nominal-compatible Type I error and coverage, while naive response reuse inflates rejection.Omitting the IV–BLA gap adjustment also increases rejection in the matched ablation.Source: §5 · p. 11
  • The selected-model BLA guarantee is asymptotic and currently established for scalar trajectories, fixed polynomial libraries, and independent Gaussian noise of known variance.Finite-sample BLA undercoverage occurs in some supplementary settings; extensions to multistate systems, broader libraries, dependent noise, and estimated noise variance remain open.Source: §5 · p. 11

AI Use Statement

  • The authors report using generative AI for manuscript organization, references, language editing, technical passages, analyses, code, and verification scripts, and state that they reviewed the work and take responsibility for the final manuscript.They also report checking cited references against original sources.Source: AI Use Statement · pp. 11–12

APPENDICES

The appendices formalize the finite-grid weak formulation and provide proofs for PSI-SINDy’s corrected-view construction, conditional IV inference, and weak-design limits.

A.1 Continuous-time dynamics and finite-grid approximation

  • Smooth weights supported inside the observation interval make the integration-by-parts boundary term zero; finite-grid sums approximate the corresponding weak quantities.The derivation separately bounds quadrature error and defines the selected approximation on the finite grid.Source: §A.1 · p. 15

A FINITE-GRID WEAK-FORM FORMULATION

  • The selected best linear approximation is defined using finite-grid quantities, so its finite-grid interpretation does not require zero continuous-approximation or quadrature error.The appendices distinguish approximation by the selected library from numerical quadrature.Source: §A.1 · p. 15

A.2 Normal equations and residualization within the selected model

  • Within a selected model, each coefficient is adjusted for the other selected terms, and its value can change when that adjustment set changes.The appendix derives this representation by residualizing the response and regressor against nuisance columns; the selected design is assumed to have full column rank.Source: §A.2 · p. 16

B PSI-SINDY CONSTRUCTION AND PROOFS

  • The proofs establish independent measurement views and unbiased corrected features, derive auxiliary IV inference, connect it to the BLA, and use pilot and bootstrap consistency for the primary BLA result.The proof sequence also establishes the weak-design limits and differentiates the nuisance projector for the IV–BLA bridge.Source: B PSI-SINDY CONSTRUCTION AND PROOFS · p. 16

B.1 Independent views and polynomial correction

  • The corrected design and instrument are independent, each has expectation equal to the noise-free design, and both still fluctuate around it.For polynomial features, the correction has expectation u^d under Gaussian measurement error; the exact guarantees assume the thinning variance is known, and substituting an estimate is not covered.Source: §B.1 · pp. 17–18

B.2 Exact conditional IV inference

  • Conditional on the inference information, the standardized IV error is N(0, 1), yielding exact conditional coverage when the IV denominator is nonzero.The argument requires testing noise independent of that information; exact coverage conditional on selection additionally relies on identification.Source: §B.2 · p. 18

B.3 Sufficient conditions for the weak-design limit

  • Under the stated smoothness and fixed-degree conditions, the corrected weak-design quantities have Gaussian limits, while full rank remains a separate identification condition.The appendix derives the limits using a Lyapunov central limit theorem and establishes that whitening is well defined for sufficiently large grids when the limiting matrix is positive definite.Source: §B.3 · p. 19

B.4 IV–BLA expansion and bridge proof

This section develops the IV–BLA bridge and bootstrap consistency arguments, then presents conditional coverage, calibration, and simultaneous inference procedures. It also specifies comparator error control and simulation assessment protocols, accounting for selection conditioning and allowing a nonzero limiting approximation residual.

B.4 IV–BLA expansion and bridge proof

  • The IV–BLA bridge is centered at the finite-grid target and yields the stated first-order limit and asymptotic identification under a positive limiting denominator.Its expansion separates instrument, nuisance-adjustment, and corrected-design uncertainty; the latter two can be correlated, and their covariance enters the limiting variance.Source: §B.4 · pp. 20–21

B.5 Pilot consistency and the bootstrap gap scale

  • Under the pilot and rank conditions, the bootstrap gap scale is consistent, including when the limiting scale is zero, and this holds conditional on any positive-probability selection event.The local cubic pilot assumes a four-times continuously differentiable trajectory and bandwidth hM→0 with MhM/log M→∞; filtering must have acceptance probability tending to one and Bboot→∞.Source: §B.5 · pp. 22–24

B.6 Selected-BLA coverage and simultaneous inference

  • Conditional on selection, the exact pivot is standard normal and the resulting selected-BLA intervals have asymptotic coverage, with p-values valid despite vanishing-probability failures.The proof does not require finite-sample second moments of the ratio or independence between the estimated scale and numerator.Source: §B.6 · p. 24
  • Bonferroni simultaneous inference controls FWER at α for identified IV targets and asymptotically for fixed-support BLA targets, without requiring independence across coordinates.Simultaneous IV coverage applies only to available intervals; BLA intervals include the fallback on failures.Source: §B.6 · p. 25
  • For true nulls, the first rejection has p-value at most α/m0, yielding familywise error at most α for identified IV targets and α + o(1) for a fixed BLA family.Coverage may be pooled across a fixed finite collection of eligible supports if conditions hold for each support and intervals use their support-specific targets.Source: §B.6 · p. 25

B.7 Inference and bootstrap algorithms

  • Algorithm 1 separates selection, IV inference, and BLA inference across independent views, while Algorithm 2 estimates the IV–BLA gap scale by bootstrap.Algorithm 1 returns coefficientwise intervals and tests; simultaneous inference requires a separate adjustment.Source: §B.7 · pp. 25–26

B.8 Calibration conditional on feature inclusion

  • Conditional on positive feature inclusion, IV p-values are uniform when every eligible support is identified, or superuniform if inference is unavailable; BLA intervals attain limiting coverage under fixed-support conditions.No positive lower bound on the inclusion probability is required. For pooled reference-null experiments, the specified value must equal the current true BLA on every possible support; pooled empirical calibration alone cannot establish calibration in a rare support.Source: §B.8 · pp. 26–27

B.9 Comparator calibration and model-count Bonferroni inference

  • Fixed-support comparator statistics have Gaussian or asymptotically normal laws under stated assumptions, and model-count Bonferroni control averages over selection rather than controlling support-conditional BLA error.The asymptotic bound gives neither support-conditional BLA control nor a power ordering relative to PSI-SINDy.Source: §B.9 · pp. 28–29

C SIMULATION PROTOCOLS AND RESULTS

  • The appendix documents the primary selected-BLA study, supporting realized-IV calibration, a separate coefficient-bootstrap comparison, and bridge-approximation diagnostics.The primary, support-conditioned, and additional simulation studies use separate cohorts.Source: C SIMULATION PROTOCOLS AND RESULTS · p. 29

C.1 Primary simulation protocol

  • Across 30,437 selected primary focal cases, all were identified, had positive estimated gap scales, and retained every requested bootstrap draw.The primary study generated 5,000 datasets in each of eight cells spanning four observation counts and four positive signals.Source: §C.1 · pp. 29–30

C.2 Assessment denominators and Monte Carlo uncertainty

  • Assessment distinguishes conditional from joint rejection rates, counts unavailable BLA intervals in coverage when targets are assessable, and reports pointwise 95% Wilson intervals for Monte Carlo error.Replications, not coefficients within a replication, are the independent sampling units.Source: §C.2 · pp. 30–31

C.3 Primary numerical results

Primary numerical results assess calibration at both inferential targets and report selected-BLA power. The results also qualify their scope: pooled assessments do not establish precise calibration for rare supports.

C.3 Primary numerical results

  • All pointwise Monte Carlo intervals for IV and BLA coverage in the four primary cells contain 95%, while the guarantees are established separately.Table 2 reports primary calibration conditional on selection and median coefficient-interval widths.Source: §C.3 · p. 31, Table 2 · p. 31
  • Primary PSI-minus-Bonferroni paired power differences are 3.90, 6.41, 12.66, and 17.75 percentage points, respectively.These differences use selected denominators; Table 3 also reports joint selection-and-rejection frequencies.Source: §C.3 · p. 31, Table 3 · p. 31
  • The pooled results predominantly assess supports containing the linear term and do not establish precise calibration for rare supports.Source: §C.3 · pp. 31–32

C.4 Comparison with coefficient-bootstrap methods

The supplementary analyses compare coefficient-bootstrap adapters, assess calibration across selected supports, and examine the IV–BLA bridge. They find setting-dependent bootstrap trade-offs and improving primary-regime bridge diagnostics, while noting limits on generalization and interpretation.

C.4 Comparison with coefficient-bootstrap methods

  • PSI-SINDy’s observed rejection is compatible with the nominal 5% level, and its intervals are substantially shorter than conservative SG5 ensemble intervals.The longer-SG ensemble and matched residual adapter produce narrower intervals but undercover in this structural-null setting.Source: Table 4 · p. 32, §C.4 · p. 33
  • The bootstrap comparison uses 1,382 paired datasets where PSI selects x2, and evaluates structural-null rejection, coverage, and median interval width.It is a comparison of specified application adapters at M = 800, σX = 0.06, and γ = δ = 0.Source: Table 4 · p. 32, §C.4 · p. 32
  • Performance varies across settings; the comparison neither establishes a uniform ordering nor tests cited methods’ theoretical guarantees outside their assumptions.At M = 200 and σX = 0.12, longer-SG rejection and median width are 5.00% and 0.5707, versus 3.45% and 0.8086 for PSI-SINDy.Source: §C.4 · p. 33

D.1 Support-conditioned calibration and familywise error

  • Across prespecified exact-support groups, selected-BLA coverage is 92.92%–94.70% for polynomial and 94.50%–95.65% for sine-augmented settings.From M = 200 to 3200, median widths fall from 0.11091 to 0.02769 for polynomial and from 0.07656 to 0.01908 for sine-augmented groups; sine-augmented familywise rejection is 6.98% at M = 200.Source: §D.1 · p. 33, Table 5 · p. 34

D.2 Bridge approximation and bootstrap scale accuracy

  • In the primary quadratic cohort, the median normalized remainder falls from 0.0607 to 0.0156 as M rises from 200 to 3200, while the scale ratio approaches one.The median estimated-to-oracle scale ratio changes from 1.0195 to 1.0002; bridge diagnostics do not alone establish tail accuracy, absence of bias, or nominal coverage.Source: §D.2 · p. 34, Figure 2 · p. 35
  • Both highlighted undercoverage cases have larger normalized remainders, but these diagnostics identify approximation limitations rather than a unique cause of undercoverage.The pendulum’s median scale ratio exceeds one, and its equationwise calculation does not assert the scalar asymptotic theorem for multistate observations.Source: §D.2 · p. 34, Table 7 · p. 35

D.3 Component ablations and sensitivity analyses

The section compares PSI-SINDy components and alternative procedures through paired ablations, comparator sensitivity, and support-specific calibration checks. The findings show both specific calibration departures and limits on what the comparisons establish.

D.3 Component ablations and sensitivity analyses

  • Removing feature correction raised intercept rejection by 5.53 percentage points [2.73,8.33] at σX = 0.12, while removing the gap scale raised it by 2.09 points [1.11,3.08] at σX = 0.06.The matched comparisons shared observations, selected support, views, and bootstrap randomness; in the stated {1, x, x2} basis, feature correction changed only the quadratic feature by an intercept-absorbed constant shift.Source: §D.3 · pp. 35–36, Table 8 · p. 36
  • The separate component study compared variants that alter feature correction, response–design independence, the instrument, weighting, or model-count adjustment, with arms assessed at their own targets.It was distinct from the matched ablation, and no pointwise Monte Carlo interval for full-method BLA coverage lay entirely below 95%.Source: §D.3 · p. 36, Table 9 · p. 37
  • At γ = 0.10, Bonferroni had 165/177 reference-null rejections (93.22% [88.52,96.08]) versus PSI-SINDy’s 84/177 (47.46% [40.23,54.79]).The supplementary comparator reused 26,000 datasets and changed multiple settings, so the comparison does not establish equal conditional Type I error or isolate individual setting effects.Source: §D.3 · p. 36
  • Six of 110 support–coordinate groups had pointwise Monte Carlo coverage intervals entirely below 95%, including the polynomial M = 200 slope and five additional groups.The scan was descriptive and not multiplicity-adjusted; for example, polynomial M = 1600, S = {1, 2} intercept coverage was 755/810, or 93.21% [91.27,94.75].Source: §D.3 · p. 37

E INFERENCE WITH LARGER CANDIDATE LIBRARIES

The larger-library study compares weak and integrated ridge-STLSQ selection and inference as the scalar polynomial library grows. Quartic recovery remains difficult in some settings, while selected-term coverage is near nominal in the reported large-sample cells; the accompanying construction settings also describe multistate assessment boundaries.

E INFERENCE WITH LARGER CANDIDATE LIBRARIES

  • The paired study varies M and polynomial-library size P across null, quadratic, and quartic dynamics, using 500 datasets per dynamics/M combination and both fixed selectors.The scalar library is {1, x, …, xP−1}; increasing P also raises polynomial degree, and the study does not establish a growing-P theorem or uniform weak-identification robustness for the BLA target.Source: E INFERENCE WITH LARGER CANDIDATE… · p. 37
  • The integrated selector uses a cumulative-integral design from a smoothed first view, while the matched weak-design variant uses the corrected weak first-view design and response.The integrated variant applies Hermite correction using propagated marginal smoothing variance, which does not remove bias in the smoothed mean; selection considers a fixed ridge grid and has no cardinality cap.Source: E INFERENCE WITH LARGER CANDIDATE… · pp. 37–38
  • At M = 3200, integrated-selector quartic recovery was 83.2%, 82.8%, and 81.8% at P = 10, 15, and 20, respectively; focal BLA coverage was 94.5%, 94.5%, and 94.4%.At P = 20, coverage was 387/410 with Wilson interval [91.7%, 96.2%] and median interval width 0.1068; weak-design recovery in these cells was 76.0%, 75.8%, and 75.0%. Figure 3 compares recovery, focal coverage, and median interval width; Table 10 reports the integrated-selector grids.Source: E INFERENCE WITH LARGER CANDIDATE… · p. 38, Figure 3 · p. 39, Table 10 · p. 39
  • The integrated selector was not uniformly better: at M = 3200 and P = 20, quadratic recovery was 98.0% for the weak selector and 97.6% for the integrated selector.Quartic P = 4 is omitted because recovery is zero and the focal term x4 is absent from the library, making focal inference undefined.Source: Figure 3 · p. 39, Table 10 · p. 39, E INFERENCE WITH LARGER CANDIDATE… · p. 40

F.1 Construction and simulation settings

  • For multistate experiments, corrected monomial features have the corresponding noise-free monomial as their expectation, but the construction does not assert a multistate version of the scalar dense-grid BLA theorem.Inference is assessed within individual equations, not simultaneously across equations; the pendulum’s second equation lies outside its finite polynomial library, so its target is the selected BLA.Source: §F.1 · pp. 40–41

F.2 Metrics and assessment denominators

The assessment distinguishes selected-coefficient calibration, interval availability, and rejection behavior using selected-support denominators. The reported multistate results include calibration departures despite complete bootstrap retention.

F.2 Metrics and assessment denominators

  • All 328,110 recorded selected-coordinate assessments returned finite intervals, with no invalid, unassessable, unavailable-IV, or whole-line cases.The reported rejection and coverage rates use datasets selecting the term as their denominator; never-selected terms have undefined conditional rates.Source: §F.2 · p. 41
  • No bootstrap rank failure or invalid fit occurred, yet pendulum xy2 undercoverage remained; the authors attribute it to a large linearization remainder rather than discarded bootstrap draws.Every selected coordinate retained every requested draw.Source: §F.2 · p. 41
  • For pendulum xy2 at M = 1600 and σX = 0.05, selected-BLA coverage was 85.82% among 275 selections, versus 60.36% without the gap scale.Higher coverage than the no-gap variant does not establish nominal calibration.Source: §F.2 · p. 41, Table 11 · p. 42
  • Across six Lorenz cells, second-equation intercept rejection at the true selected BLA ranged from 6.00% to 9.05%, with all six pointwise Wilson intervals above 5%.For the misspecified pendulum equation, x rejection ranged from 6.50% to 11.50%, again with all six intervals above 5%.Source: §F.2 · p. 41, Table 11 · p. 42

G SENSITIVITY ANALYSES AND INFERENCE AVAILABILITY

The sensitivity analyses vary procedure settings, supplied noise scale, model form, signal strength, and sample-size/noise conditions. They report isolated calibration departures and explicitly limit the conclusions to descriptive grid comparisons rather than uniform robustness.

G SENSITIVITY ANALYSES AND INFERENCE AVAILABILITY

  • The procedure-setting grid changed one setting at a time, and no pointwise Monte Carlo interval for full-method BLA coverage lay entirely below 95%.The grid included unequal view allocations as well as changes to threshold, K, and weak radius.Source: G SENSITIVITY ANALYSES AND INFERENCE… · p. 42
  • With supplied noise-scale ratio 0.5, x3 had the largest reported BLA rejection rate, 15.51% [14.32,16.79] (n = 3301); its realized-IV rejection rate was 15.24% [14.05,16.50].An incorrect scale affects thinning, feature correction, selection, and both uncertainty components, rather than merely resizing the final interval.Source: G SENSITIVITY ANALYSES AND INFERENCE… · p. 42
  • In the sine signal sweep, x2 rejection was 12.18% [11.05,13.40] at the largest sine signal, while its realized-IV check was 5.15% [4.42,6.00].Near-nominal auxiliary-IV calibration does not ensure finite-sample selected-BLA calibration; seven calibration-grid coordinate cells had BLA rejection intervals above 5%.Source: G SENSITIVITY ANALYSES AND INFERENCE… · p. 43
  • At M = 200 and σX = 0.03, x2 BLA rejection was 6.14% [5.33,7.06] (n = 2950), and its IV rejection was 5.86% [5.07,6.77].This was one grid coordinate; the pointwise comparisons were unadjusted and do not by themselves establish systematic miscalibration.Source: G SENSITIVITY ANALYSES AND INFERENCE… · p. 43
  • The scalar and unregularized larger-P grids reported no invalid coordinates, unavailable IV targets, undefined selected BLA targets, or whole-line intervals.These grids do not establish uniform robustness to weak identification; multistate retention diagnostics also showed no discarded draws.Source: G SENSITIVITY ANALYSES AND INFERENCE… · p. 43

H.1 Measurements, preprocessing and inference

The discharge analysis uses averaged voltage traces and working-model coefficient intervals, while the discharge-template simulation examines how covariance misspecification affects interval coverage. Independent-noise coverage is compatible with 95%, whereas correlated noise produces undercoverage in all observed support–coordinate groups.

H.1 Measurements, preprocessing and inference

  • Eight supercapacitor devices were analyzed using time and voltage, but the monotone voltage trace cannot establish voltage as a complete physical state.The application is exploratory, and the ten-point averaging attenuates the approximately 23.25 Hz component without eliminating aliasing or establishing independent Gaussian errors.Source: §H.1 · p. 43
  • Ten-point averaging of 100 Hz observations produced M = 1600 observations, and the resulting intervals have a working-model interpretation.The stated normalization is a working scale after smoothing, dependence, and reuse of fitting observations; exact-IV and BLA theorems do not establish coverage for these measured data.Source: §H.1 · pp. 43–44, Table 13 · p. 44
  • Across the eight device fits, 25 coefficients were selected, but missing terms have no coefficient interval and their absence is not a structural-zero test.The intervals are pointwise rather than simultaneous across terms or devices, and the devices are not repetitions with a known common coefficient.Source: §H.1 · p. 44, Table 14 · p. 44

H.2 Effects of covariance misspecification

  • Equal-allocation views are independent when the working covariance matches the true measurement covariance, but estimating it from the same observations does not automatically restore conditional Gaussianity.Under misspecification, cross-view covariance is Σobs − Σwork, while each view's variance is Σobs + 3Σwork.Source: §H.2 · p. 45

H.3 Discharge-template simulation

  • Coverage in the discharge-template experiment is averaged over selected-coefficient intervals and is not simultaneous coverage or a guarantee for every selected support.The experiment uses eight fixed templates and 1,000 replications per device; its supplied scale is the true marginal standard deviation but does not supply temporal correlation.Source: §H.3 · p. 46, Table 15 · p. 46
  • Under correlated noise, all 25 support–coordinate groups had coverage-interval upper endpoints below 95% with either scale input.With independent noise, coverage estimates ranged from 94.2% to 96.1% using supplied scale and 94.2% to 96.0% using estimated scale; every pointwise Wilson interval contained 95%. The paired supplied-minus-estimated coverage difference under correlated noise was 15.85 percentage points, with a 95% Monte Carlo interval of [15.13,16.56].Source: §H.3 · p. 46, Table 15 · p. 46

I COMPUTATIONAL RESOURCES AND RUNTIME

The runtime benchmark measured the primary focal-coefficient implementation on 30 generated datasets per setting. Inference medians increased from 0.0088 to 0.2701 seconds across four reference-trajectory settings as both M and Bboot increased.

I COMPUTATIONAL RESOURCES AND RUNTIME

  • Each of five settings used 30 generated datasets, and the benchmark assessed inference for the quadratic term under the primary protocol.Warm-up attempts were excluded; timing used a separate random seed.Source: I COMPUTATIONAL RESOURCES AND RUNTIME · pp. 46–47
  • Pipeline timing includes selection and, when the quadratic term is selected, inference, while excluding final test and confidence-interval arithmetic.Inference time is included in pipeline time, so the two timing columns are not additive; the reported measurements are not a complexity bound or the runtime of the full experimental study.Source: I COMPUTATIONAL RESOURCES AND RUNTIME · p. 47, Table 16 · p. 47
  • Inference time increased from 0.0088 to 0.2701 seconds across the four reference-trajectory settings as both M and Bboot increased.All 114 selected cases returned valid inference and retained every requested bootstrap draw; 36 other datasets did not select the quadratic term.Source: I COMPUTATIONAL RESOURCES AND RUNTIME · p. 47
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