Output-Only Identification and Spectral Monitoring of Coupled Feedback Networks with Known Time-Varying Actuation

Jihwan Woo

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Output-Only Identification and Spectral Monitoring of Coupled Feedback Networks with Known Time-Varying Actuation

Jihwan Woo

arXiv:2608.25844v1Aug 26, 2026eess.SPq-fin.TR

TL;DR

The paper addresses output-only identification of coupled feedback when inputs, coupling, and clearing-window disturbances are latent. It uses known time-varying gains and a partial-reversal moment with persistent excitation to separate coupling from confounds, proving local identification through gain-regime rank conditions. The resulting estimand is a resolvent sensitivity for transmission screening and first-stage spectral recovery, while simulations and case-study evidence expose limitations of heuristic spectral inversion and multiple-testing behavior.

Problem
Output-only coupled-feedback identification lacks the reference, probing, measured-input, or external-excitation signals required by standard frameworks.
Method
The paper combines known time-varying actuation gains, persistent pre-window signal excitation, and a known partial-reversal moment to identify feedback responses while separating gain-invariant confounds.
Results
The estimand is a resolvent sensitivity for transmission screening and first-stage spectral recovery; in the case study, the Hynix-to-Samsung interaction has z = −2.82.
Takeaways & Limitations
Known gain variation can replace external excitation for the paper’s output-only identification setting, subject to rank, excitation, and exogeneity conditions.
Takeaways & Limitations
The first-order implementation estimates a direct-coupling approximation with O(∥L∥2) projection bias, while direct-edge and spectral recovery require a second-stage inversion.

Abstract

from arXiv · show

Coupled feedback networks are often monitored channel by channel even though cross-channel paths alter both stability margins and transmitted disturbances. We study identification of a structured feedback matrix L_t = Phi diag(gamma_t) in an output-only setting: no commanded, probing, or reference input exists -- only temporally separated outputs and the scheduling gains gamma_t are observed, while the coupling response Phi and the clearing-window inputs are not. Identification rests jointly on the persistent excitation of the observed pre-window output and on two structural features separating coupling from confounds: the known time variation of the gains, which modulates the closed-loop response in a predictable pattern, and a partial-reversal moment by which a known fraction of transient displacement is corrected in a subsequent window. We give a hierarchy of results: exact local identification of the coupling under a Jacobian rank condition on the gain regimes; a first-order interaction estimator whose identification strength is the minimum eigenvalue of the residualized interaction information matrix (provably unidentified under constant gains); and a characterization of the estimand as a resolvent sensitivity -- the right object for screening transmitted disturbances and a first-stage input to spectral-margin recovery -- with sqrt(T) asymptotics for the first-order estimator, a cross-identification theorem mapping each varying gain to exactly identified resolvent rows and columns, and bootstrap validity under consistent selection; the implemented heuristic's empirical coverage (90% at nominal 95%) quantifies the remaining gap. Simulations verify sharpness of the rank condition and quantify benchmark failures under confounding. A case study on leveraged-fund rebalancing feedback, where daily fund disclosures play the role of the known gains, illustrates the method on real data.

I. INTRODUCTION

The paper studies output-only identification of coupled feedback when no commanded or probing input is available. Known, time-varying gains and a signed partial-reversal moment supply structural variation that distinguishes coupling from confounds.

Motivation: Coupled loops can transmit disturbances through the resolvent, making per-loop stability audits optimistically incomplete when coupling is strong.The spectral radius of a nonnegative gain matrix is at least its largest diagonal entry, while off-diagonal paths transmit displacement across channels.
Research gap: Standard closed-loop, dynamic-network, LPV, and output-only methods do not directly cover latent inputs with no reference, probing signal, or external excitation.The paper contrasts its setting with frameworks requiring measured inputs, external excitation, or constant structures with changing shock variances.
Identification idea: Known gain variation modulates the closed-loop response predictably, while partial reversal creates a signed, gain-scaled signature of transient feedback.The gains are disclosed and may vary for reasons unrelated to disturbances; the reversal corrects a known fraction of displacement in a later window.
Contributions: The paper contributes exact local identification via gain-regime Jacobian rank, a reversal-based interaction estimator, and a resolvent-sensitivity interpretation for transmission screening.It also separates direct-edge recovery from the second-stage inverse problem of recovering the structural coupling matrix.
Related work: The setting is positioned as distinct from heteroskedasticity, bilinear, varying-coefficient, and modulation-based identification because its structure varies observably inside a feedback fixed point.The paper characterizes the known-gain, output-only, signed-reversal combination as an uncovered structural corner in the neighboring literatures.

III. PROBLEM FORMULATION

The model observes temporally separated outputs and disclosed gains while coupling, clearing-window displacement, and inputs remain latent. A fixed-point feedback mechanism, partial reversal, persistent excitation, and gain exogeneity define the identification setting.

A. Observation model: At each period, the observed variables are the pre-window output, post-window output, and actuation-gain vector.The clearing window lies between the two observations.
A. Observation model: Each channel acts proportionally to its own gain and full-period signal, while the latent displacement feeds back through the coupling response.The self-reference creates the fixed point and couples actions across channels.
A. Observation model: Collecting latent displacement terms yields the fixed-point relation (I−L_t)r2,t = L_tr1,t + v_t, with stability requiring ρ(L_t) < 1.The resolvent expansion represents repeated feedback rounds, with cross-channel transmission entering at the first round.
A. Observation model: The post-window mechanism reverses a known share θ of transient displacement while carrying a constant confound.The reversal supplies a signed response associated with the feedback path.
A. Observation model: The unknowns are constant coupling, constant confounding loadings, and noise second moments; gains and the reversal share are known or externally calibrated.Unknown θ creates a first-order global scale ambiguity unless calibrated or normalized.
A. Observation model: Gain exogeneity requires gains to be deterministic or independent of disturbances and requires the confound not to vary with disclosed gains.This is the exclusion restriction separating gain-tracking feedback from gain-invariant confounding.
A. Observation model: Identification also requires stationarity, ergodicity, finite fourth moments, conditional orthogonality, and persistent excitation of the pre-window output.Gain variation alone is insufficient under the stated population identification results.

B. The moment, derived

The reversal moment produces an observable gain-indexed projection linking post-window output to pre-window output. With the pre-window regressor, it identifies the resolvent response up to known reversal scale and additive confounding.

B. The moment, derived: The post-window output is formed from the reversal mechanism and its projection on the pre-window output is analyzed conditional on gain regimes.The projection uses conditional second moments of the pre-window output and clearing-window noise.
B. The moment, derived: Using the pre-window regressor, the reversal moment identifies M_t up to known scale θ and an additive confound exactly, without a small-gain approximation.This exact separation is specific to the pre-window moment.
B. The moment, derived: With the full-period output as regressor, dividing the projection ingredients yields a different exact estimand.The full-period formulation incorporates the feedback transformation into the projection.
B. The moment, derived: The full-period confound becomes gain-dependent through C(I+M_t)^−1, so the clean constant-confound separation does not carry over.The discrepancy is O(∥M∥) under bounded C, and small clearing-window variance alone does not restore the clean theorem.

IV. IDENTIFIABILITY

Identifiability depends on variation and conditioning in the gain design, together with signal excitation. Linearized identification fails under constant gains, while exact local identification is governed by the rank of a stacked Jacobian across regimes.

A. Interaction estimator: The first-order interaction regression uses standardized observed gain paths as effect modifiers and treats the confound as an intercept.The centered gain paths define the interaction design used by the linearized estimator.
A. Interaction estimator: The linearized coefficient map identifies each pair (c_ij, θϕ_ij) if and only if the corresponding gain varies over time.If the gain is constant, observationally equivalent confound-coupling pairs form a line.
A. Interaction estimator: The pooled interaction design additionally requires λ_min(Q) > 0, with Q determined by the signal covariance and gain-path Gram matrix.Residualization makes the interaction block orthogonal to intercept and signal main effects under the stated exogeneity conditions.
A. Interaction estimator: Signal correlation governs how gain-design conditioning inflates variance, with factor-dominated signals making the conditioning of G decisive.For uncorrelated signals, gain-path correlation is harmless in the stated bound.
A. Interaction estimator: A correlated staircase gain design with condition number ∼247 nearly doubles the signed detector’s false-alarm rate relative to independent switching with condition number ∼12.The comparison holds equal per-channel variances.
IV. IDENTIFIABILITY: Exact local identification of (C, Φ) is sufficient and necessary at regular points when the stacked differenced Jacobian across gain regimes has full column rank.At least two distinct regimes are necessary, and coinciding regimes reproduce the constant-gain continuum.
IV. IDENTIFIABILITY: Linearizing the exact Jacobian recovers the per-column condition that each gain path must vary in at least one regime.The proof eliminates C by differencing regime-specific observation blocks.
IV. IDENTIFIABILITY: Unknown reversal scale θ is a first-order global scale ambiguity, while exact resolvent powers make spectral statements depend on its calibration.The level of Φ is identified once θ is calibrated or normalized.

V. THE ESTIMAND IS A RESOLVENT SENSITIVITY

The interaction estimator targets a gain-distribution-weighted sensitivity of the closed-loop resolvent, rather than the direct coupling entry alone. This resolvent object captures transmitted disturbances, but entrywise detection can confuse indirect reachability with direct edges.

Beyond first order, the estimator approximates a gain-weighted projection of resolvent sensitivity rather than ϕij itself.With discrete gain regimes, the coefficient is a secant rather than a derivative, affecting second-stage inversion.
Each sensitivity entry aggregates all network walks from j to i that pass through the perturbed gain, including indirect paths.Thus nonzero sensitivity can occur without a direct edge whenever a directed path connects the pair.
The resolvent sensitivity is appropriate for transmission alarms and spectral-margin monitoring, but not direct-edge recovery without a second-stage inverse problem.A signed entrywise detector can flag reachable zero entries above nominal size, while unreachable entries retain nominal size.

A. Which gains identify which entries

Varying gains identify a cross-shaped subset of the resolvent: each varying node with an outgoing path reveals its full row and column. The complementary block remains unresolved at first order, with higher-order identification still open.

Which gains identify which entries: For each varying gain with positive operating value and local support, the observable derivative identifies the corresponding resolvent column and row when the node has an outgoing path.The identified set includes every entry whose sender or receiver is a varying node with an outgoing path.
Which gains identify which entries: At first order, varying gain l exposes only column l of Φ, leaving entries whose row and column indices are both nonvarying unidentified.The theorem therefore guarantees a point-identified cross pattern, not complete recovery of Φ or M.
Which gains identify which entries: The derivative with respect to gain l factors into the resolvent column m·l and row r·l, enabling exact cross-identification from rank-one structure.The diagonal relation x(1+x)=γl pll fixes the nonnegative factorization scale.
Which gains identify which entries: A sink node with no outgoing feedback reveals only that its sensitivity vanishes, while nonmoving gains leave the corresponding nonvarying block dark at first order.Partial gain variation is nevertheless sufficient to identify all transmission entries with a varying sender or receiver under local support.
Which gains identify which entries: Higher-order joint gain variation may identify the dark block through nonlinear dependence, but the paper leaves that possibility unresolved.All derivatives expose products of cross-pattern entries, so any recovery must exploit their nonlinear dependence on the full coupling matrix.

VI. ESTIMATION, ALGORITHMS, AND MONITORING

The paper estimates first-order gain interactions after residualizing main effects and diagnoses identification through the residualized interaction information matrix. Its asymptotic theory is fixed-design and targets a pseudo-true parameter, whose projection bias persists with sample size.

Interaction estimator: The estimator separates constant confounds from gain-tracking responses, using the known negative reversal direction for signed coupling tests.The interaction coefficient satisfies dij = −θ ϕij sγj + O(∥L∥2).
Scope qualifications: The implementation is first-order and same-index, whereas the exact model contains all n^2 cross-interactions and requires a nonlinear second stage for exact recovery.The full interaction regression reduced the reachable-zero flag rate from 0.39 to 0.27 at unchanged size on unreachable zeros.
Estimation theory: Under mixing, moment, stability, and deterministic-design conditions, the interaction estimator has sqrt(T) asymptotics with HAC or design-preserving block-bootstrap covariance estimation.The central limit theorem is formulated for a triangular score array centered at date-specific means.
Algorithms: Algorithm 1 standardizes varying gains, reports λmin( Q̂ ) as identification strength, and uses OLS with robust or dependence-aware uncertainty estimates.Columns with negligible gain variation are declared unidentified and dropped.
Algorithms: The first-order estimate is assembled into L̂t and the resolvent M̂t=(I−L̂t)^−1−I for subsequent monitoring.The procedure applies a nonnegative projection to the assembled feedback matrix.
Estimation theory: Consistency targets the pseudo-true coefficient, while its structural first-order gap is an O(supt ∥Φ diag(γt)∥2) projection bias that does not shrink with T.The bias grows as λmin(Q) approaches zero and is a modeling error rather than a statistical one.
Design corollaries: Tripling T over {250, 750, 2250, 6750} reduced RMSE by factors 1.75/1.74/1.68, close to the theoretical sqrt(3)≈1.73 rate.Under random gain designs, unconditional convergence stalls at a design-variability floor, motivating inference conditional on the realized gain path.

C. Monitoring statistics and dynamic threshold

Monitoring combines cycle margin, resolvent transmission, and persistence alarms. Partial overnight correction tightens the dynamic stability boundary below the within-period threshold, while inference and sequential multiplicity remain explicit design concerns.

Monitoring statistics and dynamic threshold: Cycle margin uses ρ(L̂t), transmission uses off-diagonal resolvent entries M̂ij,t, and persistence chains uncorrected displacement across periods.Transmission measures displacement imported by channel i per unit of channel j’s innovation.
Monitoring statistics and dynamic threshold: The persistence chain applies partial correction to carried displacement before adding the next innovation through the resolvent.Its matching eigenvalue is 1−θ/(1−λ).
Monitoring statistics and dynamic threshold: At θ=0.88, the operative dynamic boundary is ρ(L)<0.56 rather than the within-period boundary 1.For real spectra, the Perron root binds under the stated conditions.
Monitoring statistics and dynamic threshold: The threshold is a frozen-L, scalar-θ, real-spectrum heuristic when those conditions fail, so its scope is narrower than general time-varying monitoring.The algorithm packages alarms using a consecutive-exceedance rule.
Uncertainty quantification: Bootstrap inference targets the pseudo-true dynamic spectral functional, with 90% empirical coverage at nominal 95% and pointwise validity while uniform validity remains open.The experimental SVD-slab point estimator carries no inferential claim in the stated algorithm.
Monitoring statistics and dynamic threshold: For γj>0, the product ujvj ranks channels by elasticity of the spectral margin to their gains.This gives the monitor a principled ranking of which varying channel drives an approaching spectral alarm.
Uncertainty quantification: Entrywise screening over n^2 edges requires familywise or false-discovery control, and Krun trades average run length against detection delay.Both calibrations are part of the sequential design.

D. Uncertainty quantification for spectral alarms

Spectral-alarm uncertainty propagates through coefficient estimation, selection, inversion, and eigenvalue calculation. Bootstrap validity requires consistent selection and regularity conditions, while the implemented heuristic achieves 90% coverage at nominal 95%.

From coefficients to spectral radius: Spectral uncertainty passes from HC covariance on interaction coefficients through entry estimates to the Perron-root functional.The pipeline uses first-order eigenvalue perturbation for an irreducible nonnegative matrix, with correlated entry errors handled by the corresponding quadratic form.
The plug-in trap: Naive plug-in spectral estimates are severely biased because indirect paths and projected zero-entry noise accumulate in the spectral radius.With true ρ = 0.30, the naive plug-in gives E[ˆρ] = 0.73, while signed sparsification gives 0.72.
The plug-in trap: The fixed-point inversion reduces, but does not eliminate, spectral bias: E[ˆρ] = 0.47 overall, 0.42 on the t-detected support, and +0.07 with oracle support.The inversion recovers Φ with mean absolute error 0.042, but delta-method coverage is only 61% on the oracle-support result.
Bootstrap calibration: 90% coverage at nominal 95% is obtained by reversed-percentile moving-block bootstrap intervals, versus 45–61% for delta intervals and 55% for naive percentile intervals.The bootstrap reruns interaction regression, signed selection, inversion, and eigenvalue calculation over dates.
Bootstrap calibration: Asymptotic bootstrap validity requires thresholded, support-consistent selection, a locally unique nonsingular fixed point, a simple Perron root, and a conditional bootstrap CLT.Under these conditions, basic dependent-multiplier bootstrap intervals attain nominal asymptotic coverage.
Scope and limitations: The implemented fixed-α screening and pairwise resampling heuristic differs from the theorem and leaves uniform local-to-zero and detection-delay theory open.Several sensitivities near the threshold cause selection randomness and coverage below nominal at moderate T.

VII. SYNTHETIC VALIDATION

Synthetic validation confirms the rank-condition boundary and shows that identification quality depends on joint gain design. The interaction estimator is robust to constant signed confounds, while benchmark methods fail under mismatched identifying variation.

Identification conditions: Constant gains produce non-identification on every path because the interaction regressor is degenerate.This exactly matches the predicted rank violation and is detectable from the data.
Identification conditions: Independent regime-switching gains reduce the false-alarm rate from 0.22 to 0.12 relative to correlated staircase gains.The result links identification strength to joint gain design and Gram conditioning under strongly factor-driven signals.
Confounding: The level/Granger benchmark’s false-alarm rate changes from 0.04 to 0.53 when the confound sign changes, whereas the interaction estimator remains at 0.12.A constant confound cannot track the varying gain, regardless of its sign.
Benchmark comparison: The variance-ratio benchmark has RMSE 43.4 versus 0.28 for the proposed estimator and produces no admissible estimate on 22% of paths after favorable tuning.The benchmark is mismatched because the simulated structural shock variances are constant.

A. Operating characteristics of the rolling monitor

The rolling monitor’s transmission alarms expose a trade-off between false alarms and missed detections. Threshold and run-length choices determine this operating frontier and should be calibrated jointly for sequential monitoring.

Operating characteristics: At entrywise z = 2.576, the null-arm per-path false-alarm rate is 0.90 because 20 edges and 151 overlapping windows are tested.The transmission branch evaluates signed exceedances over consecutive window ends.
Operating characteristics: Sequential calibration of z and Krun is a design choice for targeting average run length, rather than an afterthought.Detection delay and missed detections vary jointly with false alarms across the tabulated frontier.
Operating characteristics: Raising the threshold and lengthening the run rule lowers the false-alarm rate to 0.10 but increases missed detections to 18/40.The loosest rule produces 0/40 missed detections, making the trade-off visible across the frontier.

VIII. CASE STUDY: LEVERAGED-FUND REBALANCING

The case study applies the output-only interaction detector to leveraged-fund rebalancing in Korean and U.S. panels, using disclosed gains as the identification resource. Korean results show a gain-scaled Hynix-to-Samsung signal, while U.S. panels provide null scale-placebo and staggered-panel evidence alongside explicit screening limitations.

Korea: The Korean complexes use disclosed daily fund information to construct actuation gains, entered with a one-day lag.The gains are built from fund shares, NAVs, leverage multiples, and trailing 20-day traded value.
Korea: z = −2.82 for the Hynix-to-Samsung launch-break cross-reversal slope, ranking 1/183 against ordered placebo pairs.Newey–West gives −2.72, the leave-one-day-out range is [−3.05, −2.64], and the ten-day block-bootstrap interval excludes zero.
United States: The U.S. staggered-panel interaction grid is null after multiplicity adjustment, with the AAPL→TSLA cell only a marginal t = −1.77 flag.The caption states that this is about what twelve statistics produce by chance at 5%.
Korea: The Korean interaction is −0.106 with t = −2.29, while the standalone gain main effect is insignificant.The pattern is described as consistent with gain scaling, though not as validation of the theorem because the application regressor is contaminated.
Interpretation: The baseline spectral radius is ρ = 0.610, driven entirely by the imported diagonal because the detected cross-channel configuration is triangular.The detected cross channel does not affect the spectral radius in this baseline, making transmission the operative alarm.
United States: All six treated directions in the MicroStrategy–Bitcoin–Coinbase scale placebo are null, with |z| ≤1.45, as are the placebos.The approximately symmetric configuration has liquidity-scaled capital comparable to Korea, including γMSTR up to 1.72.

IX. CONCLUSION

The conclusion presents known time-varying actuation and a partial-reversal moment as the basis for output-only identification, provided gain exogeneity and signal excitation hold. It emphasizes resolvent sensitivity for monitoring while distinguishing detectable constant-gain failure from indistinguishable gain-tracking confounding.

Conclusion: Known time-varying actuation, signal excitation, and conditional orthogonality identify coupled feedback networks from outputs alone under gain exogeneity.The gain path substitutes for references, probing signals, and designed precoders, while partial reversal yields a signed estimator robust to gain-invariant confounds.
Conclusion: Identification is governed by the rank and conditioning of the joint gain design, while the estimand is resolvent sensitivity for transmission screening and spectral-margin recovery.The conclusion separates the identification condition from the monitoring object.
Conclusion: Constant-gain non-identification is detectable from the data, whereas a reversal-signed gain-tracking confound is observationally indistinguishable.The latter is identified as the maintained assumption’s unavoidable cost.
Conclusion: The case study’s intraday-data limitation makes its structural readings convention-dependent because the regressor uses full-period returns rather than the clean pre-window return.The application is therefore described as contaminated screening, with structural readings carrying O(∥M∥) convention dependence.