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Role-Specific Predictive Geometries for Nonstationary Multivariate Graph-Signal Forecasting

Yanbo Chen, Anamitra Makur

arXiv:2609.06519v1cs.LG

TL;DR

Nonstationary multivariate graph signals can contain equilibrium and transient dynamics that need not share one cross-feature geometry. The paper models directed Long and Short predictive relations separately and evaluates their contribution beyond a graph-temporal backbone. Controlled experiments recover planted relations, while real-data comparisons report gains over the backbone on multiple benchmarks.

  • Problem

    A shared learned graph or product geometry can mediate several dynamical effects, although equilibrium restoration and transient propagation act on different coordinates in nonstationary processes.

    Source: §I · p. 1

  • Method

    The method assigns matrix-valued Long responses to estimated equilibrium coordinates and graph-filtered Short responses to lagged differences, with edges evaluated conditionally beyond a graph-temporal backbone.

    Source: §I · p. 1

  • Results

    Controlled experiments recover every planted active Long and Short relation and both roles in every Dual realization; real-data comparisons show gains over G-VARMA on multiple benchmarks, including all 25 fold–horizon cells on the five-fold financial task.

    Source: §VII · p. 7

  • Takeaways & Limitations

    The reported results indicate that separating predictive roles can add forecast value beyond graph-temporal baselines within the evaluated settings.

    Source: §VII · p. 7

    The real-data forecasting benefit varies across graph-temporal regimes, so its value is dataset dependent.

    Source: F. Limitations · p. 7

Abstract

from arXiv · show

Forecasting multivariate graph signals is challenging when node-level trajectories are nonstationary but stable relations persist across nodes and features. In an error-correction representation, long-run equilibrium restoration and short-run transient propagation represent different predictive roles and need not share a common cross-feature geometry. We introduce role-specific predictive geometries in which directed Long relations act on estimated equilibrium coordinates, whereas directed Short relations act on lagged differences. Matrix-valued Long responses mix equilibrium coordinates before graph propagation, while Short responses use graph-filtered transient designs; a direct multi-horizon estimator couples forecast corrections across adjacent horizons. Temporal cross-fitting and Frisch-Waugh-Lovell partialling-out give selected edges a conditional predictive interpretation relative to a graph-temporal backbone. The Long operator remains right-factorized through the equilibrium subspace and therefore annihilates source common-trend directions. Controlled experiments recover all planted Long relations (20/20), all planted Short relations (20/20), and both role families in every Dual realization (10/10). Across four real-world benchmarks, the proposed predictor improves on the G-VARMA backbone in three datasets, with all 25 fold-horizon comparisons favorable on the five-fold financial benchmark.

I. INTRODUCTION

The introduction argues that a shared graph geometry can be restrictive for nonstationary multivariate signals, because equilibrium restoration and transient propagation act on different coordinates. It proposes separate directed Long and Short predictive relations and evaluates their conditional contribution beyond a graph-temporal backbone.

A. Graph Forecasting

  • Graph filters share parameters across nodes, providing structure when an unrestricted vector autoregression would be too large to estimate.Graph VARMA, time–vertex, product-graph, and neural models use differing graph-temporal representations.Source: §I · p. 1

I. INTRODUCTION

  • Long relations map estimated equilibrium coordinates through matrix-valued responses, while Short relations map lagged differences through graph filters; their directed supports may differ.The Long response can mix equilibrium coordinates before graph propagation.Source: §I · p. 1
  • Temporal cross-fitting and FWL partialling-out make each selected edge a conditional forecast contribution beyond the graph-temporal backbone, not a measure of intervention-level causality.The Long operator's right factorization through the equilibrium basis prevents direct action on source common-trend directions.Source: §I · p. 1
  • Across real-data comparisons, the proposed forecaster shows incremental value over the graph-temporal backbone on SP500 OHLC and Natural Gas.The complete benchmark comparison spans SP500 OHLC and three EIA datasets.Source: §I · p. 2

B. Multifeature and Product-Graph Models

  • A product feature graph supplies one cross-feature geometry, whereas role-specific geometry permits separate directed relations for equilibrium and transient coordinates while retaining the node shift.A shared geometry is economical when the same feature relation suits all modeled dynamics.Source: B. Multifeature and Product-Graph Models · p. 2

C. Cointegration and Error Correction

The paper separates equilibrium restoration from transient propagation in an error-correction model, then defines learned edges as conditional predictive contributions beyond a graph-temporal nuisance model. The forecasting task predicts each feature at multiple future horizons from its available history.

C. Cointegration and Error Correction

  • In a VECM, equilibrium coordinates from levels support level corrections, while lagged differences propagate transients; this is an algebraic role distinction, not a split by forecast horizon.Either role can affect every recursively generated horizon.Source: C. Cointegration and Error Correction · p. 2

D. Learned Predictive Topology

  • A conditional predictive edge asks whether a source role adds out-of-fold forecast information after response and design are projected away from a graph-temporal nuisance model.This differs from marginal similarity and from the node shift used for spatial propagation.Source: D. Learned Predictive Topology · p. 2

A. Signals and Graph-Temporal Backbone

  • The task is to predict every feature's graph signal at horizons 1 through H from the available history, using a common node shift.The within-feature graph error-correction backbone separates equilibrium restoration from transient filtering but has no cross-feature role relation.Source: A. Signals and Graph-Temporal Backbone · pp. 2–3

B. Long Equilibrium Geometry

The Long operator maps a source feature's estimated equilibrium coordinates to a target correction through a matrix-valued response and graph propagation. The Short relation instead acts on observed differences, so the two directed feature geometries need not coincide.

B. Long Equilibrium Geometry

  • A directed Long relation maps source equilibrium coordinates through a matrix-valued response that can rotate and mix them before graph propagation.The deployed map retains a right factor through the equilibrium basis, preventing direct loading on source common-trend directions.Source: B. Long Equilibrium Geometry · p. 3

C. Short Transient Geometry

  • A Short relation propagates the source feature's observed difference toward a target feature's forecast correction.Because Long and Short relations act on different coordinates, their supports are not constrained to coincide.Source: C. Short Transient Geometry · p. 3

D. Unified and Direct Multi-Horizon Prediction

The paper combines role-specific corrections with a direct multi-horizon predictor and selects specifications using chronological validation. Its estimation procedures preserve temporal separation, condition candidate edges on a graph-temporal backbone, and encourage—but do not force—similarity across adjacent-horizon responses.

D. Unified and Direct Multi-Horizon Prediction

  • The real-data predictor stacks directed Short fields into a design and jointly estimates coefficients across horizons on top of the backbone forecast.The frozen specification uses P = 2, KS = {0, 1}, retains every ordered feature pair, and performs no support thresholding.Source: D. Unified and Direct Multi-Horizon… · p. 3
  • The direct form adds role corrections to the graph-temporal baseline forecast rather than recursively feeding predicted levels and differences back into later steps.Predicted levels and differences are fed back in the recursive multi-step form.Source: D. Unified and Direct Multi-Horizon… · p. 3

A. Temporal Separation

  • Chronological splits keep normalization, graph quantities, role designs, candidate rankings, and regularization choices on the fitting side before evaluation on the next segment.SP500 uses five expanding-window outer folds; each EIA study uses fixed 60%/20%/20% train, validation, and test blocks.Source: A. Temporal Separation · p. 3

B. Conditional Edge Learning

  • Temporal cross-fitting produces out-of-fold nuisance residuals, and FWL partialling-out removes the nuisance span from both response and candidate design before edge evaluation.Training-only scores rank Long and Short candidate groups separately; a chronological validation segment selects support size.Source: B. Conditional Edge Learning · p. 3

C. Joint Multi-Horizon Role Response

  • The joint multi-horizon objective penalizes coefficient magnitude and encourages adjacent horizons to have similar role responses without forcing equality.The objective is a convex quadratic solved directly.Source: C. Joint Multi-Horizon Role Response · p. 4

D. Computation and Interpretation

The section separates role-specific predictive relations from node propagation and establishes structural properties of conditional edge scores and the deployed Long operator.

D. Computation and Interpretation

  • The real-data fits use 48 role-design columns for SP500 and 24 for each EIA dataset; five SP500 heads take 0.019 s to fit, excluding the backbone.The node graph, graph-temporal backbone, and cross-feature role geometry are distinct objects; only the last is interpreted as a role-specific predictive relation.Source: D. Computation and Interpretation · p. 4

V. STRUCTURAL PROPERTIES

  • An edge score measures incremental linear predictive content conditional on the nuisance span, not causal identification.The residualized coefficient equals the coefficient in the full regression, and temporal cross-fitting keeps a scored observation out of its nuisance-prediction fit.Source: §V · p. 4
  • The block Long extension annihilates source components outside estimated equilibrium coordinates while allowing response matrices to mix those coordinates.Its right factor remains fixed through the equilibrium basis, adding expressiveness within that channel rather than opening a path from the common-trend complement.Source: §V · p. 4
  • When qf > 1, matrix-valued Long responses can mix equilibrium coordinates, strictly extending scalar responses that only rescale a fixed carrier.For rank-deficient loading, the strict-extension statement applies on the effective column space; the scalar/polar-carrier response is nested in the matrix-valued form.Source: §V · p. 4
  • Support, deployed operators, and their norms are coordinate-consistent reporting objects, whereas unaligned entries of R are not.The coordinate-consistency condition is introduced for Vraw = VnewT with nonsingular T.Source: §V · p. 5
  • The direct multi-horizon objective is a convex quadratic with a unique solution when its block Hessian is positive definite, including selected ridge-regularized fits.Controlled experiments also verify finite one- through five-step forecasts, semisimple unit roots, and nonpermanent roots inside the unit circle in every realization.Source: §V · p. 5

A. Experimental Setup

Across controlled validation and four benchmark comparisons, this section evaluates role recovery and forecasting performance using chronological evaluation. The results show complete inclusion of planted relations and gains over G-VARMA on three datasets, with paired evidence on SP500.

A. Experimental Setup

B. Controlled Structural Validation

  • 20/20 planted Long and 20/20 planted Short relations are recovered, with both role families recovered in 10/10 Dual runs; recovery does not imply exact support.The recovered Short geometry is exact in the Dual realizations, while extra Long selections occur at frequencies between 0.2 and 0.4.Source: B. Controlled Structural Validation · p. 5, Figure 2 · p. 6
  • Median held-out forecast change versus the role-free backbone is +4.3041% in Long-only, +0.6498% in Short-only, and +3.6662% in Dual runs.Positive changes occur in 9, 7, and 8 of the 10 realizations, respectively.Source: B. Controlled Structural Validation · p. 5

C. Real-World Forecasting

  • Proposed beats G-VARMA on SP500 (0.0411 vs 0.0418), Natural Gas (0.6658 vs 0.6722), and State Electricity (0.2790 vs 0.2815), but is not best on every benchmark.TimeFilter has the lowest State Electricity rNMSE, while Gasoline is closely contested.Source: C. Real-World Forecasting · p. 5

D. Paired Forecast Evidence

  • SP500 pooled skill versus G-VARMA is 2.2942% (95% interval [1.5632%, 3.3466%], HAC p = 1.81×10−7); cumulative loss difference is positive over 99.3% of the path.The largest 10% of positive origin contributions account for 60.1% of the total positive contribution.Source: D. Paired Forecast Evidence · pp. 5–6

E. Forecast Behavior and Efficiency

Forecasting gains vary across real-data benchmarks, while the paper concludes that role-specific corrections can add forecast value beyond graph-temporal baselines.

E. Forecast Behavior and Efficiency

  • The role correction is small relative to the SP500 backbone forecast, while cumulative loss differences are positive across the chronological path.The solution time for all five selected SP500 heads is 0.019 s, reported as incremental cost excluding the backbone.Source: E. Forecast Behavior and Efficiency · p. 6
  • Proposed is best on SP500 OHLC and Natural Gas, TimeFilter is best on State Electricity, and Gasoline is closely contested.Fig. 3 reports absolute rNMSE over forecast horizons; Table III summarizes mean rNMSE over horizons.Source: Figure 3 · p. 7, Table III · p. 7

F. Limitations

  • Real-data forecasting benefit varies across graph-temporal regimes, so its value is dataset dependent.Source: F. Limitations · p. 7

VII. CONCLUSION

  • The paper concludes that separating predictive roles can add reliable forecast value beyond strong graph-temporal baselines.The reported financial-task improvement holds across all five forward folds and all 25 fold–horizon cells.Source: §VII · p. 7

A. Dense Short-role design

The real-data Short-role design builds graph-filtered lagged-difference features and fits horizon corrections jointly, using chronological selection and paired uncertainty assessment. The supplement also formalizes the conditional regression meaning of selected edge contributions.

A. Dense Short-role design

  • For each origin, ordered feature pair, lag, and graph power, the Short design uses graph-filtered lagged differences and stacks the resulting columns.The design has 48 columns for SP500 and 24 for each EIA dataset, with no fitted coefficient deleted after estimation.Source: A. Dense Short-role design · p. 10
  • The horizon coefficients are estimated jointly by solving block normal equations, with least-squares solutions used when λ = μ = 0.Otherwise, the block system is solved directly.Source: A. Dense Short-role design · p. 10

B. Chronological evaluation units

  • Chronological evaluation uses expanding SP500 folds and fixed 60%/20%/20% EIA blocks, with inner-origin selection followed by deployment refitting.The inner-origin rule uses the first 80% of admissible origins for fitting and the remainder for selection.Source: B. Chronological evaluation units · p. 10, Table V · p. 10

C. Paired uncertainty

  • Paired uncertainty uses length-five moving-block bootstrap resampling and a lag-four Newey–West statistic, fixed to match maximum horizon overlap.The bootstrap uses 5000 draws and reports the 2.5th–97.5th percentile interval.Source: C. Paired uncertainty · p. 10

A. Conditional predictive meaning

  • After nuisance effects are partialled out, the candidate-edge regression is equivalent to regressing residualized response on residualized candidate design.With generalized inverses, the fitted candidate contribution is identical under the same minimum-norm convention.Source: A. Conditional predictive meaning · p. 11

B. Common-trend preservation

The structural results establish preservation of source common-trend directions, distinguish matrix-valued responses from scalar-response nesting, and show coordinate consistency of the deployed operator.

B. Common-trend preservation

  • Injecting source inputs in the specified null space makes every target block vanish, and linearity extends preservation to sums of such inputs.Source: B. Common-trend preservation · p. 11

C. Scalar-response nesting

  • A general q_f × q_f response need not be proportional to C_f^-1, so the scalar-response inclusion is generally strict when q_f > 1.Source: C. Scalar-response nesting · p. 11

D. Coordinate consistency

S3. EXTENDED SYNTHETIC RESULTS

The supplement reports synthetic support and forecast diagnostics, horizon-specific paired evidence on real benchmarks, and limits on interpreting recorded computation times. It also documents baseline and reproducibility details.

S3. EXTENDED SYNTHETIC RESULTS

S4. FULL REAL-DATA RESULTS

  • The competitive-range horizon plots cover SP500 OHLC, State Electricity, and Gasoline, with Gasoline focused on Proposed, G-VARMA, and PG-VAR.Complete five-method curves are reported in the main-paper Fig. 3.Source: Table X · p. 12, Figure S1 · p. 13

S5. HORIZON-SPECIFIC PAIRED EVIDENCE

  • All five SP500 horizon-specific intervals are positive; Natural Gas and Electricity pooled intervals are positive, while Gasoline is not statistically distinguishable from G-VARMA.The declining skill with horizon is consistent with narrowing rNMSE separation in Fig. 3(a).Source: Table XI · p. 13, S5. HORIZON-SPECIFIC PAIRED EVIDENCE · p. 13

S6. RECORDED COMPUTATIONAL EVIDENCE

  • The recorded timing scopes are not end-to-end comparable because Proposed excludes the fitted G-VARMA backbone while baseline times aggregate multiple seed/fold fits.The evidence supports only the narrow claim that the incremental role head is computationally negligible once backbone forecasts and designs are available.Source: S6. RECORDED COMPUTATIONAL EVIDENCE · p. 13, Table XII · p. 13

S7. BASELINE AND REPRODUCIBILITY NOTES

  • Baseline implementations and stochastic reporting use specified constructions and the same three saved seeds on every EIA dataset.The retained publication bundle includes processed arrays, supports, indices, targets, predictions, coefficients, and figure source data for reproducibility.Source: S7. BASELINE AND REPRODUCIBILITY NOTES · p. 13

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