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Explaining Time Series Forecasting with Horizon-Resolved Attribution

Seunghan Lee, Jun Seo, Jaehoon Lee, Junhyeok Kang, Sangjun Han, Sungdong Yoo, Minjae Kim, Tae Yoon Lim, Dongwan Kang, Hwanil Choi, Soonyoung Lee, Wonbin Ahn

arXiv:2609.12639v1cs.LGcs.AI

TL;DR

Existing TS explanation methods collapse a whole forecast into one vector, although different forecast steps may depend on different past values. HRX assigns each step its own map, and experiments report that the improvement comes from the horizon axis rather than a particular estimator.

  • Problem

    Existing TS explanation methods return one vector for an entire forecast, implicitly assuming every future value depends on the same past values.

    Source: §1 · p. 1

  • Method

    HRX forms a matrix with one explanation row per forecast step and provides a step-wise evaluation protocol and a rank criterion for judging whether the horizon axis is useful.

    Source: §1 · pp. 1–2

  • Results

    Across 56K runs, 26 backbones, and 15 benchmarks, the reported improvement comes from the horizon axis rather than a particular estimator.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The step-wise dependence is low-dimensional: leading directions of the explanation matrix recover the full gain, while a single direction recovers none.

    Source: §1 · p. 2

    Neither explanation-guided input selection nor horizon-specific pruning produced a gain distinguishable from zero, so HRX remains a measurement tool for now.

    Source: L LIMITATIONS IN DETAIL · p. 29

Abstract

from arXiv · show

Recent advances in explaining time series (TS) models have produced methods that identify which past values a prediction depends on. However, most existing methods return a single importance vector, assuming that every predicted step depends on the same past values. In this paper, we show that this assumption does not hold, as different forecast steps depend on different past values. Motivated by this observation, we propose Horizon-Resolved eXplanation (HRX), which adds a horizon axis to the explanation, so that every forecast step receives its own importance map. HRX is a simple yet effective plug-in framework with three components: 1) an estimator that reads these maps out of any differentiable forecaster without modifying the TS backbone, 2) an evaluation protocol that validates the horizon axis by measuring how much a single forecast step changes when the inputs an importance map ranks highest are removed, and 3) a rank criterion that predicts in advance whether the axis is worth resolving on a given TS. We further show that this step-wise dependence is low-dimensional, as the explanations of all steps are built from a few shared maps whose number does not grow with the forecast length. Extensive experiments across various backbones and datasets show that the improvement comes from the horizon axis and holds for estimators of previous explanation methods. Code is available at https://github.com/seunghan96/HRX.

1 INTRODUCTION

Existing TS explanation methods use one vector for an entire forecast, implicitly treating all future values as depending on the same past values. HRX gives each forecast step its own explanation map and adds an evaluation protocol and rank criterion for assessing the horizon axis.

1 INTRODUCTION

  • HRX replaces the shared explanation vector with a matrix containing one importance map per forecast step, without changing the forecaster backbone.It is a plug-in framework; its contribution is the horizon axis rather than a particular estimator.Source: §1 · pp. 1–2, Table 1 · p. 2
  • Any existing explanation-method estimator can fill a row of the matrix; input gradients are the default because they are the simplest.The framework applies to differentiable forecasters without modifying the TS backbone.Source: §1 · pp. 1–2
  • The rank of the explanation matrix predicts in advance whether resolving the horizon axis is worthwhile for a given time series.The matrix rank also indicates the low-dimensional structure of step-wise explanations.Source: §1 · p. 2
  • Across 56K runs, 26 backbones, and 15 benchmarks, the reported improvement comes from the horizon axis rather than a particular estimator.The axis also holds with prior explanation methods' metrics and against known ground truth.Source: §1 · p. 2
  • The evaluation protocol measures how much an individual forecast step changes when inputs ranked highest by its own explanation row are removed.Prior TS explanation metrics score one label per input and therefore cannot score a single forecast step.Source: §1 · p. 2

2 RELATED WORKS

The related work spans TS forecasting models and two families of explanation methods: gradient-based and perturbation-based. The paper positions HRX as adding forecast-step resolution to explanations that otherwise return one vector for the whole forecast.

2 RELATED WORKS

  • Related work covers TS forecasting models, gradient-based explanation, and perturbation-based explanation.The paper attaches its framework to forecasting models, which it treats as black boxes without internal changes.Source: §2 · p. 2
  • Gradient-based explanations read importance from output derivatives and are described as cheap, while perturbation-based methods change inputs and are described as expensive.The cited examples include Simonyan et al. (2013) and Sundararajan et al. (2017) for gradients, and Crabbé & van der Schaar (2021) and Enguehard (2023) for perturbations.Source: §2 · p. 2
  • Existing methods return one explanation vector for the whole forecast, whereas HRX is the only compared method that also resolves the forecast-step axis.The comparison covers which axes an explanation resolves.Source: §2 · p. 2
  • Figure 2 depicts the HRX explanation matrix with one row aligned to each forecast step.Each colored row explains the correspondingly colored step in the forecast.Source: Figure 2 · p. 3

3 METHODOLOGY

The methodology gives each forecast step its own lookback-importance row, then estimates those rows, tests their step-specific value, and measures how many distinct directions they contain.

3.1 EXPLANATION MATRIX

  • E[h,t] assigns each lookback input an importance for forecast step h; aggregating rows recovers the shared vector, so the matrix refines rather than replaces it.Source: §3.1 · p. 3

3.2 ESTIMATOR

  • For differentiable forecasters, each row can be the gradient of one output step; any single-output estimator can replace it, and batched backward calls compute multiple rows efficiently.The batched implementation keeps the matrix computation within a small multiple of a forward pass.Source: §3.2 · pp. 3–4

3.3 EVALUATION PROTOCOL

  • Deletion scores each forecast step separately by comparing its own-row, shared-vector, and other-step-row AUCs; the shuffle margin isolates correct row-to-step matching.Removed positions are resampled within the same window, and error differences are divided by forecast variance.Source: Figure 3 · p. 4, §3.3 · pp. 4–5, Table 2 · p. 4

3.4 RANK OF THE EXPLANATION MATRIX

  • Effective rank uses singular-value participation to count substantial directions; rank-r decomposes horizon rows into shared lookback maps with horizon-specific weights, indicating whether resolving the axis is worthwhile.Unlike the deletion protocol, rank uses the matrix itself to assess in advance whether the axis is worth resolving.Source: §3.4 · p. 5

4 EXPERIMENTS

The experiments test whether horizon-resolved explanations recover step-specific dependencies on synthetic data and improve explanations across real-world settings and estimators. Results favor resolving the horizon axis, while the evaluation spans 26 backbones and 15 benchmarks.

4 EXPERIMENTS

  • The evaluation covers 26 backbones and 15 benchmarks, sweeping lookback length, forecast length, and model depth with three random seeds per configuration.The benchmarks span general and financial domains.Source: §4 · p. 5

4.1 SANITY CHECK ON SYNTHETIC DATASET

  • On six synthetic generators, the explanation matrix beats the shared vector on all four ground-truth metrics and three backbones, and reproduces the planted dependence rather than vertical stripes.Each result cell averages 54 runs; prior methods repeat one shared vector across forecast steps.Source: §4.1 · p. 6, Table 3 · p. 6, Figure 4 · p. 6

4.2 MAIN EXPERIMENTS

  • Across real-world runs, gmargin is consistently positive across backbones and benchmarks, and all four prior-work metrics are positive across 1,126 runs.Figure 5 reports gmargin; Table 4 reports the comparison under prior-work metrics.Source: Figure 5 · p. 7, §4.2 · p. 7, Table 4 · p. 7

4.3 APPLICATION TO OTHER EXPLANATION METHODS

  • On synthetic data, resolving the axis wins in almost every cell across six generators and 26 backbones, indicating gains robust across estimators.Each cell averages 26 backbones and six generators.Source: §4.3 · p. 8, Table 5 · p. 8
  • On real-world data, adding the horizon axis yields significantly positive gains for every tested estimator across 26 backbones, 15 benchmarks, four window shapes, and three seeds.Without the horizon axis, the three reported gain quantities are exactly zero.Source: §4.3 · p. 8, Table 6 · p. 8

5 ANALYSIS

The analyses examine when horizon-resolved explanations add value, how their maps vary, and whether the pattern differs across input axes. The reported gain is positive across domain groups, while the number of distinct maps needed is limited.

5 ANALYSIS

  • Runs beating both naive baselines have larger gown in all three groups, but every group remains positive, so forecasting skill affects gain size rather than its presence.The groups split general and financial data by whether the model beats zero-prediction and last-value baselines.Source: §5 · p. 8, Table 7 · p. 8
  • Explanation matrices show a shifting band of important lookback positions as the forecast step advances, with three backbones differing in their number of parallel bands.The visualization covers Solar and AAPL returns, and three backbones on one benchmark.Source: §5 · pp. 8–9, Figure 6 · p. 9
  • The time axis shows a clear gain, but the channel axis shows almost none and has effective rank near one, indicating channel rows copy one vector.The reported pattern suggests forecast steps change which times the model reads, but not which variables.Source: §5 · p. 9, Table 8 · p. 9
  • A single direction recovers none of the gain, while a handful of further directions recover most; runs with effective rank near one are those where the shared vector loses almost nothing.The rank measure is used to predict the gain from resolving the horizon axis.Source: §5 · p. 9, Table 9 · p. 9, Table 10 · p. 9
  • Additional analyses report positive gains for almost every zero-shot foundation-model run; gains rise with forecast and lookback length, while mask methods cost two orders of magnitude more than gradients.The reported depth effect is in the opposite direction and substantially smaller.Source: §5 · p. 9

6 CONCLUSION

The conclusion presents HRX as a measurement tool for step-specific dependencies, not a means that improved the tested forecasts. The accompanying statements describe AI assistance, data scope, reproducibility materials, experimental configurations, datasets, and backbones.

6 CONCLUSION

  • HRX recovers different lookback dependencies across forecast steps without modifying the backbone, but no series statistic predicts effect size and tested downstream uses did not improve forecasts.The authors characterize HRX as a measurement tool for now.Source: §6 · p. 9

AI USE STATEMENT

  • A generative AI tool proofread the manuscript and provided minor coding assistance; the authors say the research framing, protocol design, and results analysis were their own.The authors state they reviewed accepted edits and code and take responsibility for the final work.Source: AI USE STATEMENT · p. 10

ETHICS STATEMENT

  • The datasets contain aggregate measurements or public stock prices, not personal or proprietary information, and no human subjects or personally identifiable data were used.The authors caution that explanation maps invite unsupported causal readings; their protocol measures model use rather than what drives the underlying process.Source: ETHICS STATEMENT · p. 10

REPRODUCIBILITY STATEMENT

  • The paper says reported numbers are generated from stored per-run records by scripts and that supplementary source code includes estimators, evaluation, data preparation, and figure and table scripts.Appendices document experiment settings and checks, including seed variation and pretrained-forecaster verification.Source: REPRODUCIBILITY STATEMENT · p. 10

A EXPERIMENTAL DETAILS

  • Experiments sweep lookback lengths, forecast lengths, and depths; configurations use three seeds, and pretrained-forecaster grids are restricted by model input and test-window requirements.Five benchmarks are sampled at even intervals rather than exhaustively when they contain hundreds of series; architectures require at least ten completed runs to be reported.Source: A EXPERIMENTAL DETAILS · p. 15

B DATASETS

  • The general-domain suite uses established public datasets, while the financial benchmark contains daily prices from 2018–2025 for 100 large US companies, represented as log prices and log returns.Trading days are used as the time grid without interpolation onto calendar days.Source: B DATASETS · p. 15

C BACKBONES

  • The evaluation spans 26 backbones across linear, convolutional, recurrent, and Transformer families, including three deliberately minimal architectures.The appendix describes individual model designs, including linear maps, temporal convolutions, recurrent models, and attention-based Transformers.Source: C BACKBONES · p. 16

D SEED REPRODUCIBILITY

Across configurations, seed-to-seed variation is smaller than variation across settings. The appendices describe the batched estimator implementation, forecasting adaptations of prior estimators, and deletion-based metrics.

D SEED REPRODUCIBILITY

  • Seed variation is smaller than configuration variation for every reported quantity, indicating that variation across architectures and benchmarks reflects those settings rather than random seeds.The three seeds are folded within each configuration before aggregation to avoid unequal weighting from configurations with different run counts.Source: D SEED REPRODUCIBILITY · p. 17, Table 11 · p. 17

E.1 FILLING THE MATRIX WITH BATCHED BACKWARD CALLS

  • Replicating each input lets one backward call return distinct forecast-step rows, reducing a 720-step horizon from 720 passes to 45 at k = 16.The larger batch trades memory for fewer backward passes; using one shared input would sum gradients instead of returning separate rows.Source: §E.1 · p. 18, Figure 7 · p. 18

E.2 IMPLEMENTATION DETAILS OF THE ESTIMATOR

  • In-place normalization can break input-gradient computation, while a decoder that ignores its input yields zero gradient and deletion effects even before training.The implementation rewrites in-place operations out of place and checks that the forward output remains identical; zero-input-dependence models are validated only after training.Source: §E.2 · pp. 18–19

E.3 PORTING THE ESTIMATORS TO FORECASTING

  • Prior estimators are reimplemented on a common forecasting interface, with each method measuring a selected forecast step rather than a class score or whole forecast.Mask methods solve one optimization problem per forecast step, and rows are estimated at eight evenly spaced steps to limit optimization cost.Source: §E.3 · p. 19

F.1 DEFINITION OF THE DELETION CURVE

  • The deletion curve perturbs the positions ranked highest by an ordering vector and tracks the selected step’s deviation from its unperturbed forecast.Selected positions are resampled from elsewhere in the same window; the reported area is evaluated over nine evenly spaced deletion fractions.Source: §F.1 · p. 20

F.2 REPORTING THE DELETION CURVE

  • The reported deletion score is the gap between most-relevant-first and least-relevant-first curve areas, which preserves their disagreement before all positions are removed.The areas converge when every position has been deleted, regardless of the explanation.Source: §F.2 · p. 20

F.3 DEFINITIONS OF THE BORROWED METRICS

  • Sufficiency and Insertion provide evidence independent of the deletion-based gain because they perturb in the opposite direction.Each metric is evaluated with a horizon-specific row and the shared vector; differences are signed so positive values mean the horizon axis helped.Source: §F.3 · p. 20

G SYNTHETIC DATA AND CALIBRATION

The synthetic generators provide controlled ground truth for step-dependent lookback windows, while a fixed-window generator serves as a negative control. Calibration and known-answer checks test whether the rank measure and recovered explanations track their targets.

Constructed ground truth

  • The six generators vary how each forecast step’s lookback set moves, with shared positions and smoothly varying unit-norm weights; held-out least-squares fits explain at least 98% of target variance.Every lookback set includes the most recent position, and the constructed labels are reused across windows.Source: G SYNTHETIC DATA AND CALIBRATION · p. 21, Table 12 · p. 21

Why only constructive labels

  • Agreement with constructed labels supports measurement soundness, but disagreement can also occur when a model finds an equally valid solution the generator did not name.The paper rejects labels that merely assert relevant inputs without ensuring that the asserted inputs are uniquely necessary.Source: G SYNTHETIC DATA AND CALIBRATION · p. 21, Figure 8 · p. 21

The negative control

  • The sparsenull negative control has one fixed lookback vector across steps, so its true effective rank is one and the horizon axis has nothing to resolve.Its measured gain is not positive on any backbone, unlike the other five generators.Source: G SYNTHETIC DATA AND CALIBRATION · p. 21, Table 14 · p. 22

Checking the matrix against a known answer

  • On the controlled synthetic generator, the recovered explanation reproduces the planted causal-window staircase, providing a known-answer check unavailable on real data.The accompanying real-data example applies the same procedure to Electricity with PatchTST.Source: G SYNTHETIC DATA AND CALIBRATION · p. 23, Figure 10 · p. 23

Instrument calibration

  • Across 349 runs, measured effective rank tracks planted rank on every backbone, though the measured increase is exact on some and shallower on others.This calibration supports interpreting effective rank as a property of the explanation matrix rather than only of the estimator.Source: Figure 9 · p. 23, G SYNTHETIC DATA AND CALIBRATION · p. 23

H RESULTS ACROSS ARCHITECTURES AND BENCHMARKS

Across architectures and benchmarks, the horizon-axis results are mostly positive but include exceptions and backbone-dependent variation. Explanation complexity is similar across two domains, and endpoint sweeps show trends with forecast and lookback length alongside estimator costs.

H RESULTS ACROSS ARCHITECTURES AND BENCHMARKS

  • The ideal direction holds across benchmarks except PEMS04, which does not reach significance; three backbones are at or below zero, with Pyraformer’s interval entirely below zero.Tables 15 and 16 report the three protocol quantities across architectures and benchmarks; only backbones with at least 10 completed runs are listed.Source: H RESULTS ACROSS ARCHITECTURES AND… · p. 24, Table 15 · p. 24, Table 16 · p. 25

I RESULTS BY DOMAIN

  • Effective rank is similar in the general and financial domains, at 1.84 and 1.85, but which backbone exploits the axis changes by domain.The vanilla Transformer shifts from among the weakest in the general domain to among the strongest in the financial domain, so backbone rankings are not general.Source: I RESULTS BY DOMAIN · p. 25

J QUALITATIVE PANELS IN FULL

  • Across the displayed domains, explanation matrices show banding, but effective lookback varies across rows and the sharpest banding can come from a model that cannot forecast.Figure 11 includes naive-baseline verdicts; Figure 12 shows effective lookback for three Electricity backbones.Source: Figure 11 · p. 26, Figure 12 · p. 26, J QUALITATIVE PANELS IN FULL · p. 26

K.1 SWEPT AXES

  • Endpoint comparisons show gain rising with forecast and lookback length, while depth moves oppositely by an order of magnitude less; intermediate levels are not monotone.The endpoint contrasts hold backbone, dataset, seed, and the other two settings fixed; only endpoint contrasts are significant, so the pattern is a trend rather than a law.Source: §K.1 · p. 27, Table 17 · p. 27, Figure 13 · p. 27

K.2 COST OF EACH ESTIMATOR

  • Mask methods cost two orders of magnitude more than a plain gradient because the horizon axis requires a separate optimization per forecast step.The default input-gradient estimator also needs multiple backward passes to fill the matrix.Source: §K.2 · p. 28, Table 18 · p. 28

K.3 THE ESTIMATORS AS PLAIN ATTRIBUTION METHODS

Without the horizon axis, the default input gradient is competitive but not the best estimator. The paper treats the axis, rather than the estimator that fills it, as the contribution.

K.3 THE ESTIMATORS AS PLAIN ATTRIBUTION METHODS

  • The default input gradient is competitive but not best in its marginal form, underscoring that estimator rankings without the axis do not measure the axis’s contribution.The comparison uses each estimator in its published marginal form under the standard deletion gap.Source: §K.3 · p. 28, Table 19 · p. 28

L LIMITATIONS IN DETAIL

The discussion limits how HRX’s rank criterion and distance-based comparisons should be interpreted: the rank describes magnitude structure, while measured differences between forecast steps depend on their distance.

L LIMITATIONS IN DETAIL

  • No tested candidate predicted the effect size on real data, and the apparent stacked-period relationship weakened to -0.169 after target scale was fixed.The earlier relationship came from a generator that scaled target variance with the number of periods.Source: L LIMITATIONS IN DETAIL · p. 29

M THE RANK OF THE EXPLANATION MATRIX

  • The participation ratio of the singular values of |E| counts directions carrying magnitude, not merely nonzero directions; structure in signs alone is invisible.Order two is less sensitive to the long tail of small singular values and, by Jensen’s inequality, does not overstate the number of directions in use.Source: M THE RANK OF THE EXPLANATION MATRIX · p. 29
  • The truncation sweep pairs each rank truncation with the same corruption pattern, so Monte Carlo variation does not obscure the analytic rank-one value of zero.The distance sweep in Appendix N also uses pairing; the paper says pairing is not used elsewhere.Source: M THE RANK OF THE EXPLANATION MATRIX · p. 29
  • Normalizing by forecast variance preserves each run’s sign but can change benchmark rankings, so reported comparisons use normalized gains.The normalization focuses the comparison on whether the horizon axis carries information rather than on forecast scale.Source: M THE RANK OF THE EXPLANATION MATRIX · p. 29

N DISTANCE BETWEEN FORECAST STEPS

  • The margin is positive even between neighboring steps, then grows with distance without saturating within the horizon; the uniform-permutation result reflects distant steps.The sweep uses a fixed distance from each step, falling back to the opposite direction at horizon ends; its uniform-permutation result lies near the large-distance end.Source: N DISTANCE BETWEEN FORECAST STEPS · p. 30, Table 20 · p. 30

O PRETRAINED FORECASTERS IN DETAIL

The horizon axis yields positive gains for almost every zero-shot time-series foundation model on GIFT-Eval. The paper also situates HRX among forecasting, explanation, and evaluation approaches that do not resolve a prediction by forecast step.

O PRETRAINED FORECASTERS IN DETAIL

  • The horizon-axis gain is positive for almost every one of eight zero-shot time-series foundation models evaluated on GIFT-Eval, a benchmark held apart from pretraining corpora.The sign matches the trained-backbone setting, though the gain size differs.Source: O PRETRAINED FORECASTERS IN DETAIL · p. 31, Table 21 · p. 31
  • Timer and VisionTS do not reach significance; Timer rarely beats the naive baseline, while VisionTS was pretrained on images and has never seen a time series.The paper notes that baseline skill is especially relevant when evaluating zero-shot models.Source: O PRETRAINED FORECASTERS IN DETAIL · p. 31
  • Across six metrics, signs agree over 16,729 runs; by dataset, 32 of 66 are significantly positive and 15 significantly negative.The gain rises with forecast length, while gmargin remains near +0.118 across settings.Source: O PRETRAINED FORECASTERS IN DETAIL · p. 32, Table 23 · p. 32, Table 24 · p. 32

P.1 LINES THIS WORK SITS BESIDE

  • Forecasting architectures span transformer, linear or MLP, and convolutional families, while HRX attaches to them as a black-box framework.Gradient-based explainers use derivatives, whereas perturbation-based methods change inputs and measure output movement; the latter solve an optimization problem for each input.Source: §P.1 · p. 34

P.2 LINES ADJACENT TO THIS WORK

  • Prior TSFM work applies pretrained forecasters to unseen series, but does not report what a released model reads for a given forecast step.The cited GIFT-Eval benchmarks are kept apart from the models’ pretraining corpora.Source: §P.2 · p. 34
  • Interpretable-by-design models expose architecture-specific structure, while spectral explanations resolve frequency rather than time or forecast step.The horizon axis could also be applied to a spectral explanation; established explanation metrics, meanwhile, score a whole prediction rather than one forecast step.Source: §P.2 · pp. 34–35

Q FULL RESULTS BY BENCHMARK

The appendix unfolds pooled estimator results into benchmark-specific tables, with each table covering every backbone and estimator and reporting median normalized gain over swept settings.

Q FULL RESULTS BY BENCHMARK

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