Horizon Activation Mapping for Neural Networks in Time Series Forecasting

Krupakar Hans; V A Kandappan

REVIEW 5 major objections 6 minor 16 references

Horizon Activation Mapping for Neural Networks in Time Series Forecasting

T0 review · 5 major / 6 minor · reviewed 2026-08-03 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A grad-CAM-style tool maps which forecast timesteps a network learns to emphasize, across model families.

desk verdict A novel masking-based visualization for time-series forecasting, but the normalization and baseline assumptions undermine the central claims. read the letter →

arxiv 2601.02094 v4 pith:REZR4ZY5 submitted 2026-01-05 cs.LG math.FA

Krupakar Hans , V A Kandappan This is my paper

classification cs.LGmath.FA MSC 68T0762M10

keywords HorizonActivationMappinggradientnormstimeseriesforecastinginterpretabilitygrad-CAMmodelselectiondeeplearningoptimizationlandscape

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper introduces Horizon Activation Mapping (HAM), a visualization technique that uses gradient norm averages over causal and anti-causal subseries of the forecast horizon to interpret what a neural network's training dynamics reveal about the horizon. The authors claim HAM is layer-agnostic, working across MLP, attention, SSM, and diffusion models, and can guide granular model selection, validation set choice, and cross-family comparison. They demonstrate HAM on the ETTm2 dataset with several state-of-the-art forecasting architectures, showing horizon-size trends, dropout effects, batch-size patterns, and attribution of certain curve shapes to architectural properties like NHITS's approximation theorem and SpaceTime's exponential autoregressive behavior. If valid, HAM would give practitioners a common interpretability lens for forecasting models that currently lack a unified diagnostic tool.

What carries the argument

The core object is the pair of HAM curves: for a horizon of length H, causal subseries [0→h] and anti-causal subseries [h→H] are masked in the loss, and the average gradient norm over the whole training set is computed for every h. A line of proportionality connecting (0,0) to (H, G) serves as a baseline for uniform activity, where G is the maximum norm average across subseries. Derived quantities include the gradient equivariant point (where causal and anti-causal norms are equal), signed area curves measuring deviation from proportionality, interpolated area plots for comparing different horizon sizes, and difference plots d(t) = [g(0→t) − g(t→H)]/max|·|.

What would settle it

Train a model with known per-timestep importance, such as a linear model with preset weights or a transformer with determinable attention. Compute HAM curves for that model and check whether the gradient equivariant point and the deviation from the proportionality line correlate with the known importance profile. If the curves are identical across random and trained models, or ignore the known importance structure, the interpretive claims of HAM would be falsified.

Watch

Extended reading notes

Core claim

HAM computes, for each subseries of the forecast horizon, the average gradient norm of the loss when that subseries is masked before backpropagation. Causal mode masks timesteps after a cut point; anti-causal mode masks timesteps before it. Plotting these norm averages against the cut point yields two curves whose shape, intersection (the 'gradient equivariant point'), and deviation from a line of proportionality indicate the model's relative emphasis on shorter versus longer horizon subseries. The paper's central claim is that these curves are not just mathematical artifacts but reflect meaningful learning behavior: they change reproducibly with batch size, dropout, early stopping, data spl

Load-bearing premise

The gradient norm of the masked loss, computed by scaling per-timestep contributions by the full horizon length, faithfully measures how much the model 'activates' for each subseries—rather than merely reflecting the increasing size of the cumulative sum.

Editorial extensions

If this is right

HAM provides a single visualization that works across fundamentally different model families (MLP, attention, state-space, diffusion), making cross-architecture comparison of gradient behavior possible without modifying the models.
Batch-size comparisons in HAM suggest that converged models exhibit a polynomial saturation in gradient norm averages as batch size increases, which could be used to estimate the effect of batch size from a few runs.
Dropout is shown to amplify gradient magnitudes, especially for longer horizon subseries, aligning with improved multivariate performance—offering a diagnostic for regularizer effects.
HAM can inform validation set selection by comparing relative variability across timesteps between candidate validation splits.
Horizon-size HAM patterns reveal family-specific inductive biases, such as NHITS's linear-combination assumption and SpaceTime's exponential weighting of longer horizons, which could guide architecture selection.
The study of early stopping shows that post-stopping epochs reduce gradient magnitudes monotonically, suggesting HAM could be used to detect when a model has left a local minimum.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the causal and anti-causal curves are cumulative sums over subseries, they are mathematically forced to be monotonically increasing and decreasing, respectively; the 'line of proportionality' comparison therefore needs a null model (e.g., random weights or a purely linear model) to calibrate whether observed deviations exceed what any model would produce.
A natural validation experiment is to compare HAM's gradient-norm curves against known per-timestep importance signals—for instance, attention weights in a transformer or learned weights in a linear model—to test whether HAM tracks ground-truth emphasis or merely reflects gradient scaling.
The paper's suggestion of layer-wise HAM is underdeveloped but could be extended to study learning dynamics within a single network, potentially revealing which layers specialize on short versus long horizon subseries.
For diffusion models, the consistently larger-than-proportional norms for small subseries hint that noise-injected training creates a characteristic gradient signature; testing whether this signature predicts denoising performance could make HAM a useful probe for generative forecasting models.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: HAM is a genuine new idea—masking the forecast horizon causally and anti-causally and plotting the norm of the resulting gradients—but the central claim that it enables granular model selection, validation-set choices, and cross-family comparisons is not supported by the evidence. The plotted quantity is not an average activation; it's the norm of a cumulative sum divided by the full horizon length, so the causal and anti-causal curves are mechanically increasing and decreasing. The 'line of proportionality' used as a uniform baseline assumes per-timestep gradient vectors are parallel, which is false for real networks. Combined with the lack of any ground-truth validation or error bars, the qualitative patterns reported in the paper are plausibly artifacts of the construction.

What is actually good: the causal/anti-causal masking is a clean, simple extension of grad-CAM to a regression output, and it's applied to a genuinely broad set of architectures on ETTm2, with code released. The paper also is internally honest in places—it calls the batch-size trend a 'surprising polynomial curve... motivating future work' and notes that the area-based comparisons can be tedious. So the framing is exploratory rather than deceptive.

The soft spots are load-bearing. Equation (1) divides by H, not by the subseries size, so the gradient norm 'average' is a scaled partial sum. The line of proportionality is drawn from the observed maximum G, making deviations from it self-referential. The paper never tests whether the curves correlate with actual model behavior—e.g., which timesteps the model relies on, or whether the curves change when you deliberately alter the data so that certain timesteps matter more. Every conclusion is drawn from visual inspection of single runs without statistical support. The attributions to NHITS's approximation theorem and SpaceTime's exponential autoregressive activities are hand-wavy; they are not derived or tested.

Who gets value: someone thinking about gradient-based explainability for sequence models might borrow the masking construction as a starting point, but the paper as it stands is a proposal, not a validated method. A serious editor should send it to peer review, because the idea is novel and the flaws are addressable in revision—but the authors would need to fix the normalization, add a proper uniform baseline, and validate against known ground truths before the claims can be taken seriously.

Referee Report

5 major / 6 minor

Summary. The paper introduces Horizon Activation Mapping (HAM), a grad-CAM-inspired visualization technique for time series forecasting models. HAM computes causal and anti-causal masked-loss gradient norms for horizon subseries and plots them with a 'line of proportionality' and derived area/difference plots. The authors apply HAM to several model families (NHITS, CycleNet, N-Linear, FEDformer, Pyraformer, SpaceTime, Multi-Resolution DDPM) trained on ETTm2, and use visual patterns to draw conclusions about model selection, validation-set choice, early stopping, batch size, architectural variations, and cross-family comparisons. The code is released.

Significance. If HAM were a valid layer-agnostic interpretability tool, it could be a useful complement to error metrics for comparing diverse forecasting architectures. The paper's strength is its breadth: it applies a single visualization recipe across MLP, attention, SSM, and diffusion models and releases code, which would facilitate reproducibility. However, the central quantity in Eq. (1) is a scaled cumulative-sum norm, not a per-subseries average of activation magnitudes, and the 'proportionality' baseline is anchored by the observed maximum. These issues make the reported monotonic trends and model-family differences potentially mathematical artifacts rather than model-behavior signals. The paper provides no ground-truth validation linking HAM curves to actual model emphasis. The significance of the claimed utility is therefore not established.

major comments (5)

[Eq. (1), §2] The masked loss in Eq. (1) divides the cumulative loss by the full horizon length H, not by the length of the masked subseries. Consequently, the causal-mode value at subseries 0→h is ||Σ_{t≤h} ∇θ l_t|| / H, i.e., a scaled norm of a partial sum. The 'gradient norm average' is thus not an average over the subseries; it is a cumulative quantity. Under this definition, a linear proportionality curve is only expected if per-timestep gradient vectors are parallel and equal in norm. In realistic networks, gradient vectors from different timesteps can cancel or be orthogonal, causing the cumulative norm to grow sublinearly even under perfectly uniform per-timestep activity. The monotone increases/decreases of the causal and anti-causal curves, their 'equivariant points,' and their proximity to the proportionality line could therefore reflect the geometry of vector summation rather than the mode
[Abstract, §1, §3–§4] The central claim—that HAM enables 'granular model selection, validation set choices and comparisons across different neural network model families'—is not validated against any ground truth. There are no ablation studies, no synthetic tasks with known horizon importance, no comparison with alternative attribution methods, and no quantitative measure of whether HAM curves correlate with model behavior. All conclusions are drawn from visual inspection of single training runs without error bars or statistical tests (e.g., Figs. 2–9). For instance, the batch-size claim in §3.2 is based on three configurations and no variance estimate. Without a validation experiment, the paper's empirical conclusions are unfalsifiable and the utility claim is unsupported.
[§2, 'line of proportionality', Eq. (2)] The line of proportionality is defined using G = max over all subseries of the gradient norm average. This makes the baseline self-referential: every curve intersects or touches the line at its own maximum by construction. The 'areas' computed in Eq. (2) are deviations from an anchor that depends on the observed data, so 'closer to the line' statements (e.g., §3.3, §4.1) are not calibrated across models or runs. A meaningful uniform-activity baseline should be independent of the observed maximum, for example a line based on the expected cumulative norm under a stated assumption about per-step gradients. As written, the proportionality line cannot support claims that a model's activity is 'more uniform' or 'more concentrated' than another's.
[Abstract vs. §3.2] The abstract states that batch-size differences 'seem to indicate potential for existence of an exponential approximation across them per epoch relative to each other,' while §3.2 and Figure 4 describe 'a polynomially saturating trend' and 'polynomial approximation across batch sizes.' These are contradictory. Moreover, the claim is based on only three batch sizes (500, 2000, 4000) with no repeats, and the interpolated plots use a single model each. The text should commit to one functional form and provide evidence (e.g., residual analysis or multiple seeds) before making this claim.
[§3.6 (CycleNet cycle queue)] In the architectural-change study, the authors plot 'CycleNet’s cycle queue’s norms ... instead of all the layers’ to prevent too much averaging.' This changes the quantity being visualized from the full-model gradient norm defined in Eq. (1) to a submodule's gradient norm. The paper does not specify how the partial gradient is computed (e.g., whether the loss is masked with respect to all parameters or only the queue parameters), nor does it discuss whether HAM's interpretation remains valid for a submodule. This ambiguity makes the corresponding conclusions about cycle length effects difficult to interpret.

minor comments (6)

[§2, Eq. (1)] Notation is inconsistent: the loss L is written as a scalar average over h, but the masked loss then appears as a vector or sum. The precise meaning of the '·' product and the index ranges should be clarified. Also, the text says masks are defined over h∈[1,H], but for anti-causal mode the empty/zero cases are not handled explicitly.
[§3.2, Fig. 4] Figure 4 caption says 'polynomially saturating trend' while the text calls it 'polynomial curve'; this inconsistency should be resolved.
[§3.3, Fig. 5] The four early-stopping runs are presented as a time series (epochs 20,30,40,50) but no stopping criterion is described. Please specify how 'after early stopping' is defined and what the epochs correspond to.
[§3.5, Fig. 7] The text says 'High UseFul Load' but the ETTm2 dataset's variable is usually 'HUFL'; please correct the typo.
[§4.1, Table 3] Area values are mentioned in the text ('anti-causal areas converge to 1') but the table reports only MSE/MAE and plots. It would help to report the numerical area values or point to a supplementary table.
[General] There are several typos and grammatical issues (e.g., 'alongwith', 'Differences plots over 4 models' in Fig. 5, 'Interpolated Norms w.r.t. Lines of Proportionality' in Fig. 8). A careful proofread is needed.

Circularity Check

1 steps flagged · score 2.0 of 10

HAM's construction is independent, but the 'line of proportionality' benchmark is defined from the observed maximum G, so uniform-distribution claims are self-referential; central model-selection claim is not circular but is under-validated.

self definitional [Section 2, definition of line of proportionality after Eq. (1); used in Sections 3.2–3.5 and 4.1]
"A line of proportionality connecting y=0 and y=G is used in both causal and anti-causal modes corresponding to a uniform distribution of gradient norm averages by the subseries, where G= max ∀i(µθ(∇θL(fθ(x(g)), y(g), i))) represents the maximum gradient norm average across all the subseries."

G is the maximum of the very causal/anti-causal gradient-norm-average curves that the line is used to interpret. The line through (0,0) and (H,G) is a chord determined by the observed curve's own maximum, not an independent uniform-activation baseline. Statements like 'saturating closer to lines of proportionality' (Sec. 3.2) describe the cumulative partial-sum norm's shape relative to its own scale. Also, a linear cumulative norm equals uniform per-timestep gradients only if per-timestep gradient vectors are collinear; Eq. (1) plots the norm of a sum, so that correspondence is an extra assumption, not implied by the definition.

full rationale

No significant circularity in the central construction. HAM is a new visualization computed from trained-model gradients via Eq. (1); the causal/anti-causal curves are not fitted to any target, and no parameter is renamed as a prediction. There are no load-bearing self-citations: the cited NHITS, SpaceTime and MRL works are external, and the paper does not invoke a self-authored uniqueness theorem. The main interpretive claims (proximity to the line of proportionality, gradient equivariant points, area curves) are descriptive statistics of the same cumulative-norm curves; the paper treats these shapes as evidence about model behavior without ground-truth validation, which is a validation/correctness weakness rather than a circular derivation. The one mildly circular component is the line of proportionality: because G is the observed maximum of the curves being interpreted, 'uniform distribution' is not an independent baseline. This affects the strength of some qualitative readings but does not force the central model-selection claim, so it is scored as a minor self-referential element (2/10).

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The method has no fitted parameters, but it relies on several unvalidated interpretative assumptions, most importantly that gradient norms of masked losses reveal 'activation' of subseries. The proportionality line and theoretical attributions are additional domain assumptions without independent evidence.

assumptions (4)

domain assumption The gradient norm of the masked loss reflects the model's emphasis on each subseries.
The paper assumes that the computed gradient norms are meaningful activations without any empirical validation that they correlate with actual model internals.
domain assumption ETTm2 dataset is representative and sufficient for cross-model comparisons.
All experiments use only ETTm2, yet the paper generalizes to claims about model families; no other dataset is used.
domain assumption The line of proportionality (straight line from (0,0) to (H,G)) represents a uniform distribution of gradient norm averages.
The paper claims deviations from this line correspond to non-uniformity, but this is an interpretive assumption not derived from any statistical principle.
domain assumption NHITS's neural approximation theorem and SpaceTime's exponential autoregressive properties are applicable to explain HAM trends.
The paper attributes observed curve shapes to these cited theoretical properties without demonstrating a rigorous link.

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Cite this review

Pith. "Pith review of Horizon Activation Mapping for Neural Networks in Time Series Forecasting." pith.science (2026). https://pith.science/paper/REZR4ZY5

@misc{pith2026260102094,
  author       = {Pith},
  title        = {Pith review of: Horizon Activation Mapping for Neural Networks in Time Series Forecasting},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/REZR4ZY5}},
  note         = {Machine review of arXiv:2601.02094}
}

read the original abstract

Neural networks for time series forecasting have relied on error metrics and architecture-specific interpretability approaches for model selection that don't apply across models of different families. To interpret forecasting models agnostic to the types of layers across state-of-the-art model families, we introduce Horizon Activation Mapping (HAM), a visual interpretability technique inspired by grad-CAM that uses gradient norm averages to study the horizon's subseries where grad-CAM studies attention maps over image data. We introduce causal and anti-causal modes to calculate gradient update norm averages across subseries at every timestep and lines of proportionality signifying uniform distributions of the norm averages. Optimization landscape studies with respect to changes in batch sizes, early stopping, train-val-test splits, architectural choices, univariate forecasting and dropouts are studied with respect to performances and subseries in HAM. Interestingly, batch size based differences in activities seem to indicate potential for existence of an exponential approximation across them per epoch relative to each other. Multivariate forecasting models including MLP-based CycleNet, N-Linear, N-HITS, self attention-based FEDformer, Pyraformer, SSM-based SpaceTime and diffusion-based Multi-Resolution DDPM over different horizon sizes trained over the ETTm2 dataset are used for HAM plots in this study. NHITS' neural approximation theorem and SpaceTime's exponential autoregressive activities have been attributed to trends in HAM plots over their training, validation and test sets. In general, HAM can be used for granular model selection, validation set choices and comparisons across different neural network model families.

Figures

Figures reproduced from arXiv: 2601.02094 by the authors.

**Figure 3.** H = 720 CycleNet models with different batch sizes show norm averages increasing over number of epochs. The numbers next to CycleNet correspond to epochs with 0 indicating the randomly initialized model’s. The plots are darker corresponding to increases in epochs. From [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** Although different in HAM magnitudes when batch sizes change over more [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

Figures from the paper (4 more)

**Figure 6.** Figure 6: HAMs across different dataset splits between randomly initialized and [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 7.** Figure 7: CycleNet’s multivariate and univariate High UseFul Load series’ HAMs highlight the difficulties involved in the spatial aspects of long horizon forecasting. polated area plot finds them closer with slight increases that don’t reflect beyond shorter horizon subseries. …

**Figure 8.** Figure 8: HAMs and Difference Plots for H = 720 N-Linear with and without its normalization and H = 720 CycleNet’s Cycle Queue of |Q| ∈ {64, 128, 256}. norm averages don’t in NHITS, and vice versa in Pyraformer, and given the non-autoregressive anti-causal increase trends in the…

**Figure 9.** Figure 9: Difference plots between gradient norm averages of first [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]

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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

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