Slimming the Fat-Tail: Morphing-Flow for Adaptive Time Series Modeling

Thank you for raising this concern.

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Question:
Regarding baseline performances

Prior works, such as iTransformer and PatchTST, adopt different hyperparameters for different datasets, as summarized in the table below. This practice makes it difficult to disentangle performance gains stemming from model design versus hyperparameter tuning—potentially introducing test-set overfitting.

As noted above, baseline performance in prior works often varies due to differing experimental setups. For low-hyperparameter models like DLinear, our results closely match both the original paper and other reimplementations under comparable settings on overlapping datasets.

Our baseline experiments use a single, unified hyperparameter configuration across all methods and datasets (including ours; see Appendix C), enabling a fairer and more challenging comparison by removing dataset-specific tuning. Even under this stricter setup, our re-implementations of strong baselines (e.g., iTransformer, PatchTST, DLinear) outperform reported results—e.g., those from the iTransformer paper—by 1.69%–19.1% on average across datasets (see summary table below; full results here).

Our proposed MoF module (with Mamba backbone) still achieves substantial gains, despite these stronger baselines, improving over iTransformer by 14.9% and over PatchTST by 16.9% in average MSE.

Thank you for raising this concern. We will emphasize these clarifications in the revised paper to preempt any potential reader confusion.

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We hope this addresses your concern and if you have any further questions or concerns, we are happy to address them.

Thank you for your comments. Please kindly find our response below.

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Comment 1:
Regarding the contribution

As shown in Fig.1, while stabilizing methods normalize data, they often expose heavier tails(often exceeding those in our synthetic setups) in real datasets—outliers that disrupt gradient flow. Our Flow remaps outliers into a stable range, and Morph adapts to evolving data at test time.

In summary, with new ablations on PatchTST, our contributions are:

• Flow as the Primary Contributor mitigates outlier-driven gradient spikes and yields a 5.95% MSE reduction—exceeding the gain from any other individual component. Showing that fat-tail suppression is the principal driver of MoF’s effectiveness.

• Morph for Distribution Shifts provides a 3.11% MSE gain by adapting to distributional shifts via test-time tuning of RevIN’s affine parameters.

• Flow and Morph work synergistically: Morph benefits from the stabilized gradient flow, resulting in a 7.17% MSE reduction over PatchTST.

We revised the introduction to highlight these insights.

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Question 1:
Regarding the importance of dealing with fat-tailed distributions and performance comparison with Linear Backbone(Appendix I)

Real data (finance [1], climate [2], health [3]) often has frequent outliers that dominate loss and disrupt training dynamics. The Linear backbone used in Appendix I often underfits, hiding these effects (Flow and RevIN both show limited gains).

Added gradient analysis (link) shows MoF reduces skewness/kurtosis for smoother gradients, crucial in larger model & fat-tailed data (ETT -> Exchange).

Across datasets, MoF yields +15.0% with PatchTST vs +2.75% with Linear, indicate that more powerful backbones magnify the impact of outliers and thus benefit more from Flow-based normalization.

[1].Fat tails in leading indicators(Econ Lett 2020)

[2].Emergence of heavy tails in streamflow distributions: the role of spatial rainfall variability (Adv Water Res 2023)

[3].Evidence that coronavirus superspreading is fat-tailed(PNAS 2020)

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Question 2: Regarding the fariness of test-time training

As suggested, We updated RevIN’s affine param via Morph(TTT) and achieved +3.11% gain(RevIN ± Morph). Meanwhile, Flow (no TTT) gives +5.95%, and Flow+Morph reaches +7.17%, showing that Flow is the key driver, with Morph benefiting stable grad. Our TTT starts from a fixed $W_0$ per instance and modifies only $W$, ensuring no future leakage.

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Question 3:
Have you considered combining the backbone of PatchTST with MoF to check how MoF advance over RevIN?

Yes. PatchTST+MoF improves MSE by 7.17% over PatchTST+RevIN across 8 datasets (wins 7/8). [Results] to be added to Table 3 & Appendix J.

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Question4: Regarding Figure 8 limited to ETTh2 dataset

Figure 8 shows non-monotonic input-length effects on given dataset (Linear-MoF peaks at 192, iTransformer at 336), not meant for overall ranking. More results in Appendix H.

In fact, our gains aren’t ETTh2-specific, Table 1 shows simple MoF+Linear outperforms iTransformer by 7.5% and PatchTST by 9.2% across 8 sets, trailing only on ETTm2. Stronger backbones can further amplify gains.

We clarified this intent in the updated version.

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Question 5:
Regarding Figure 9 limited to the Weather dataset

Thank you for pointing this out.

Fig.9, like Fig.8, shows how input length interacts with normalization on one dataset, not a universal ranking. We replaced it with PatchTST-based results, showing MoF’s consistency at larger scale, avoiding future confusion.

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Regarding References Not Discussed

We have added the suggested works to our related section.

IN-Flow[1] addresses nonstationarity(not fat-tail) via entangled flows, which can be complex for high-dimensional data. FITS[2] is parameter-efficient but needs hyperparameter grid search, which are rather costly to train. ElasTST[3] focuses on backbone design for varied-horizon forecasting, orthogonal to our normalization approach. Implementing [1] (no public code) worked for small data but was unstable on high-dim benchmarks. [2] was erratic with our unified hyperparams. [Full Result]

We will include these findings in our revision.

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We hope these address your concerns satisfactorily. If you have any further questions or concerns, we are happy to address them.

Thank you for your feedback!

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Suggestions 1:
Section 2: The term "excess kurtosis" is used without defining it.

Thank you for the helpful comment.

A definition of "excess kurtosis" was included in Appendix C.2: it measures the tailedness of a distribution relative to a Gaussian distribution, which has zero excess kurtosis.

We've revised the wording and added a pointer from the main text to improve accessibility.

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Questions 1:
Can the authors provide more details on the computational overhead introduced by the Morph module during inference? Specifically, how does the Morph module's test-time adaptation affect the runtime performance compared to models without this adaptation?

Thanks for the question.

The Morph module introduces test-time adaptation overhead, which scales as:

O(N_iter × C × T × d)  # test-time gradient updates  
+ O(d × B)             # up-projection  
+ O(C × T × B)         # re-applied Flow pass

With small projection dimension d in Morph, a moderate number of bins B in the Flow, and a fixed, small number of test-time iterations N_iter, the overall asymptotic complexity remains dominated by the shared Flow component, i.e., O(C × T × B).

We report average per-batch inference times (ms/iter) across six datasets:

On average, Morph accounts for ~17.92% of MoF’s inference time, with less than 5% overhead on overall model runtime.

We’ve included this in the revised version with detailed complexity breakdowns.

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Question 2:
The Flow layer uses a spline-based transformation to normalize fat-tailed distributions. Can the authors provide more interpretability analysis or case studies to explain how the spline transformation affects specific features or time series patterns?

Thank you for the suggestion.

Our Flow layer applies a monotonic and differentiable spline transformation to re-map frequent outliers into a more regular, Gaussian-like range. This normalization compresses extreme values, re-centering the effective support into the region where activations and gradients are more stable. The result is improved numerical conditioning during optimization.

We added two new visual components:

• A case study visualizing how Flow attenuates heavy-tailed spikes while preserving underlying temporal structure (Illustration).
• A gradient dynamics study demonstrating that Flow stabilizes backpropagation (reduced norm, skewness, and kurtosis), detailed in (Gradient Statistic Dynamics).

In Figure 4, subfigure (2) shows a heavy-tailed (green) distribution from ETTh2, which was transformed by Flow (subfigure (3)) into a more symmetric, near-Gaussian form (blue, subfigure (4)). These results are now further contextualized in the revised text with a dedicated paragraph discussing how Flow impacts both distribution geometry and optimization dynamics.

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We hope these address your concerns satisfactorily. If you have any further questions or concerns, we are happy to address them.