Slimming the Fat-Tail: Morphing-Flow for Adaptive Time Series Modeling
Authors: Tianyu Liu
Kai Sun
Fuchun Sun
Yu Luo
Yuanlong Zhang
Venue: ICML 2025 poster (2025)
Keywords: time series forecasting
long-term prediction
model convergence
fat-tailed distributions
Primary area: general_machine_learning->sequential_network_and_time_series_modeling
TL;DR: Morphing-Flow (MoF) stabilizes fat-tailed time series while preserving extremes. Using flow-based reshaping and test-time adaptation, it achieves superior accuracy across eight datasets.

Abstract: Temporal sequences, even after stationarization, often exhibit leptokurtic distributions with fat tails and persistent distribution shifts. These properties destabilize feature dynamics, amplify model variance, and hinder model convergence in time series forecasting. To address this, we propose Morphing-Flow (MoF), a framework that combines a spline-based transform layer (Flow) and a test-time-trained method (Morph), which adaptively normalizes non-stationary, fat-tailed distributions while preserving critical extreme features. MoF ensures that inputs remain within a network’s effective activation space—a structured, normal-like distribution—even under distributional drift. Experiments across eight datasets show that MoF achieves state-of-the-art performance: With a simple linear backbone architecture, it matches the performance of state-of-the-art models on datasets such as Electricity and ETTh2. When paired with a patch-based Mamba architecture, MoF outperforms its closest competitor by 6.3% on average and reduces forecasting errors in fat-tailed datasets such as Exchange by 21.7%. Moreover, MoF acts as a plug-and-play module, boosting performance in existing models without architectural changes.

Slimming the Fat-Tail: Morphing-Flow for Adaptive Time Series Modeling
Tianyu Liu
Kai Sun
Fuchun Sun
Yu Luo
Yuanlong Zhang
Abstract
Kurtosis Distribution
(Aross Dacasets Channe
Temporal sequences, even after stationarization, often exhibit leptokurtic distributions with fat tails and persistent distribution shifts. These properties destabilize feature dynamics, amplify model variance, and hinder model convergence in time series forecasting, To address this, we propose Morphing-Flow (MoF), a framework that combines a ghine-based transform layer (Flow) and test-time-trained method (Morph), which adaptively normalizes non-stationary, fat-tailed distributions while preserving critical extreme fea- tures. MoF ensures that inputs remain within network's effective activation space- struc- tured, normal-like distribution-even under distri- butional drift. Experiments across eight datasets show that MoF achieves
TRAFFIC
EXCHANGE
WEATHER
ELECTRICITY
ETTH
ETTh2
ETTm1
ETTm2
NIL
Data Type
Scationary
(Durs)
Leptokurtic (Far-sxt
Narmal
Parykursic (Brons-peak)
Model Performance (Average ftelacive MSE)
MoF- Mamba (Durs)
MoF-Linear
perfor- mance: With simple linear backbone architec- ture, it matches the performance of sate-offtheeart models datasets such Electricity and ETTh2. When paired with a patch- based Mamba architec- ture, MoF outperforms its closest competitor by 6.3% on average and reduces forecasting errors in fat-tailed datasets such as Exchange by 21.7% %. Moreover, MoF acts as plug-and -nd--nddlaaaa -lay module, boosting performance in existing models without architectural changes.
(2023)
TimeMachine (2024)
LiNo (2024)
GLinear
(2025)
Transformer
0222:
PatchTST
CATS (2074)
Autoformer
Transformerr
46
Informer
Parameter Count
Figure 1. Kurtosis Dynamics and Model Efficiency (Top) Sta- tionarization amplifies mill-heavnees,, while MoF restores near- Gaussian kurtosis and stabilizes variance across channels. (Bottom) MoF balances simplicity and effectiveness: lightweight variants remain competitive with the baseline, and the Mamba-based back- bone achieves 6.3% performance improvement.
1. Introduction
Time series forecasting faces inherent challenges from non- stationary, fat-tailed distributions, phenomena prevalent in critical domains such as energy grids and financial markets, where frequent extreme events (e.g., demand surges, market
shocks) deviate sharply from Gaussian assumptions. Such sequences often exhibit power-law dynamics, with fat-tailed fluctuations disproportionately influencing system behavior and destabilizing the feature AnaributionsWWilkinssss et al., 2015). While stationarization techniques (e.g., differenc- ing, detrending) mitigate non-stationarity, our analysis of nine benchmark datasets reveals critical oversight: sta- tionarization often amplifies leptokurtic behavior (Fig 1), exacerbating tail risk and inflating prediction variance. Em- pirical studies further indicate that fat-tailed, leptokurtic noise (Simsekli et al., 2019; Nguyen et al., 2019) violates the assumption of bounded variance, complicating optimiza-
"qqual contribution 'Department Computer Science and Technology, Institute for Artificial Intelligence, BNRist Center, Ts- inghua University. Beijing, China SScoool of Biomedical Engineer- ing. Tsinghua University 'Haawei Noah's 's Ark Lab 'Department of Automation, Tsinghua University. Correspondence to: Fuchun Sun <fesun @ twigguu.siggua..duuu edu.en>.
Proceedings of the nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Slimming the Fat-Tail: Morphing-Flow for Adaptive Time Series Modeling
tion (Zhang al., 2020; Wang al., 2021). This presents fundamental dilemma: aggressive constraints risk erasing critical extremal values (essential for predicting disasters or market shifts), while retaining raw distributions introduces unstable gradients and divergent training dynamics.
Gaussian Distribution, Korou 0.00
Density plot.
Existing approaches have difficulty addressing this ten- sion. dio-diifferettiabee operations (e.g.. clipping), static parametrizations, and Gaussian-centric frameworks (c.g., Box-Cox transformations) often fail to adequately ad- dress dynamic distribution shifts and fat-tailed phenomena (Chavez- -errz---emuuin& & Davison, 2012). While gradient clip- ping (Gorbunov et al., 2020; Yang et al., 2022) and decom- position architectures (RB, 1990; Hamilton, 2020) partially mitigate non-stationarity, their robustness to pervasive lep- tokurtic noise remains insufficient. This underscores the need for architectures that explicitly adapt to fat-tailed dis- tributions while balancing theoretical rigor with practical resilience.
Caudyy Didtibtton 1.46
Gaussiian Q-Q
Distributins
Optimization Process(Dutput v. Iterations)
Figure 2. Impact of Fat-Tailed Distributions Convergence. Top left: A comparison of the Probability Density Functions (PDFs) between the Gaussian (Blue, K 0) and clipped Cauchy (Red, KEsces R 1.5) distributions. Bottom left: A Q-Q plot highlights the heavier tails the Cauchy distribution. Right: Training convergence results. The Gaussian distribution rapidly converges to the correct value (indicated by the horizontal dashed line), while the sequence sampled from the Cauchy distribution converges to an incorrect value before diverging and exhibiting oscillations.
We propose Morphing-Fool Flow (MoF), a framework designed to dynamically map input distributions to a well-structured, normal-like like space within neural activation regions, preserv- ing extremal features and stabilizing training. MoF inte- grates two components:
the first moment (the mean) of non-stationary time series. Consider a time series {x, J where:
= + Co. Me kft-11
Flow: A monotonic spline transformation that learns channel-specific mappings via piecewise polynomial functions, thereby reducing kurtosis while ensuring invertibility and differentiability.
with A >> 1, constant growth factor, μ4 representing the trend, and 6, denoting noise drawn from zero-mean dis- tribution with variance o². We train linear model fo to predict 1n+1 from the historical observations 21...., Zn.
Morph: A lightweight test-time adaptation layer that iteratively adjusts Flow's parameters counteract instance-level distribution shifts and enhance robust- ness temporal drift.
int1 = 1e(01........
To compare the effects of heavy tails, we sample e, from two distributions: (1) a Gaussian (N0 o²) and (2) a clipped Cauchy distribution (bounded within [-6, 6] to satisfy the finite-moment requirements of stochastic gradient descent), whose excess kurtosis
The fully invertible architecture of the MoF enables end- to-end training without auxiliary losses, while its spline- based design adaptively balances local trends and global extremes. With a linear backbone, it matches state-of-the-an art performance despite its simplicity. By mitigating fat-tailed behavior in the output distribution, MoF implicitly stabilizes gradient dynamics, property that becomes increasingly beneficial in deeper architectures. When integrated with pasch-based Mamba backbone, MoF outperforms the closest competitor by 6.3% on average, achieving 21.7% gains on datasets with pronounced heavy-tailed behavior (e.g., Exchange--Rate) The MOF operates as a glug-and-play module, requiring no architectural modifications to existing models.
E E - -4]
K Kexcess
3 R 1.5.
0
In addition, we employ progressively diminishing learning rate to ensure training stability in the presence of these fat-tailed distributions.
Visualization of the training process is shown in Figure 2. The fat-tailed nature of Cauchy distributions (evident the Q-Q plots and PDFs of Figure 2) has a significant impact on the training dynamics. Although Gaussian noise does not hinder stable convergence, clipped Cauchy noise with mod- erate excess kurtosis (Kesees 1.46) causes oscillations, slow convergence, or divergence. Extreme deviations in the tails of the Cauchy distribution initially bias models to- ward incorrect solutions, thereby exacerbating optimization
2. Impact of Fat-Tails on Convergence
(How) Do fat-tail impede model convergence? To investi- gate the impact of fat-tailed distributions on neural network training, we design a synthetic forecasting task: predicting
2

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Slimming the Fat-Tail: Morphing-Flow for Adaptive Time Series Modeling
instability. Additional results reveal increased sensitivity to initialization and learning rates compared to Gaussian regimes, consistent with observations and theoretical analy- sis (Appendix B.2).
main's limit, and features outside this range are retained. To ensure computational efficiency and provide explicit closed-form solution for the first-order derivatives in both the forward and reverse directions, we construct a vector- ized piecewise- spline transformation. Let C denote the number of channels and B the number of bins per chan- nel, a hyperparameter that determines the discretization of the space. For each channel c {l..... ................. the learnable parameters (we,) ) t=1 and { which define the bin widths and heights respectively, are collectively denoted as
To learn high-variance features without causing instability, we propose an adaptive transformation module. It trans- forms fat-tailed distributions (e.g.. clipped Cauchy) into stable, normally distributed activations, while also reducing gradient noise. By preserving maining-friendee distributions in transformed spaces, models can leverage fat-tailed pat- terns while avoiding the high- variancr noise associated with such features.
W flow
The model computes normalized bin widths Wo, and heights He.; by applying a softplus activation function, en- suring that their respective sums each equal 27. The knot positions Te, and ye, are obtained by taking the cumulative sums of We, and He,s- The slope of each bin is precom- puted as:
3. Methodology
In this section, we introduce the proposed MoF framework, illustrated in Fig. 3. We first define the problem and then detail the Flow layer for fat-tailed feature distributions and the Morph layer to address distribution shifts.
9o,i+1 yee
Sc,
Fe,i+1 Fe,i
We address multi- variate time series forecasting: given X E RCXT where C' denotes the number of channels and T the number of time steps, the goal is to predict the next F steps Y E RCF The aim is to develop a predictor such that Y closely ap- proximates the ground-trum truth observations Y E RCXF
Problem Formulation and Notations
The forward transformation maps an input z E [-T,T] to an output y E [-T, T] using the following linear relation- ship:
y = Ye, + Se,s (x (x-Iaac Ie,s).
where the appropriate bin is determined by identifying the interval containing x.
Overall Framework The MoF framework combines two synergistic components: the Flow layer F(x | Woow that mitigates fat-tailed feature distributions via learnable reshap- ing, and the Morph module, a lightweight test-time adapta- tion mechanism that addresses temporal distribution shifts through Xmod = Morph(F(()) The Flow layer preserves the fidelity of training distributions through structured trans- formations, while the Morph module dynamically adapts to evolving patterns during deployment, collectively ensuring robustness to distribution shifts while maintaining opera- tional stability.
The inverse transformation involves first locating the corre- sponding bin for y and applying the inverse linear relation- ship through the relation:
1 -. yose). 80,5
One such transformation is defined for each input chan- nel. This formulation ensures efficient, differentiable, and vectorized forward and inverse transformations, making it suitable for applications requiring continuous and invertible mappings.
Since both the input and output spaces exhibit fat-tailed effects, the MoF model is designed to be reversible. To achieve this, we introduce both MoF and MoF-1 before and after the backbone, enabling the backbone to operate in a well-defined environment resilient to end-to-end distribution shifts.
3.2. Morph: Test-Time Trained Adaptive Spline Modification
Real-world time series often exhibit distribution shift,, where the data distribution varies both across training batches and between the training and testing datasets. Despite the representational flexibility of the Flow transformation F (æ: noo)) ). fixed W flw may fail to adapt the shifting distributions between training and testing sets.
3.1. Flow: Spline-Based Reversible Transform Layer
The residual component Ties often exhibits complex rela- tionships and elevated variance, characterized by a fat-tailed distribution that can impede model convergence.
To address this issue, we propose strictly monotonic and invertible transformation, denoted as F(x: Wnow) to map features within the bounded domain [-1, I to a new do- main with the same bounds. Here, T represents the do-
To address this issue, we propose Morph module that modifies W at test time based on the Plow-transformr
sequence F(x Wnow), enabling the module to handle distribution shifts effectively. We first apply the vectorized
3

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Slimming the Fat-Tail: Morphing-Flow for Adaptive Time Series Modeling
Flow Module
Morph Module
Training Sequence
Layer
Flow Transformed
Test-Time TimeTTime Medfiiction
Sequence
Temporal Sequence
RA N aam
murh
F(x Wiww)
Distribution
Label view
Test view
Consistent Distribution
f(0,,'')
Random Mask f(0..x)
Mesokurtic
Distribution
f(fe,
M
Distribution Shift
Testing Sequence
Morph Module
MoF-ed Sequence
Test-time trained layer
Wolld
Test time loss 4( ((Wixix)) [M è (00,x') x10,,
Projection
Morphed Transformation
F(x Wnow @ Xuod).
FP [xx Wilww @ Xmod)
Figure 3. MoF Framework. Gaussian Q-Q plots (ETTh2) demonstrate heavy-tailed training/test (red) distributions. Flow transforms these into normalized latent space (green), reducing tail effects while preserving critical features. Morph corrects instance/train-test shifts through adaptation, ensuring consistent distributions resistant to non-stationarity.
piecewise-linear spline (Flow Layer) as follows:
where is learnable step size enabling adaptive learning during test-time updates.
x' F
W now (We, he
(4)
Importantly, only the matrix W Wist is updated during this process, while all other parameters (0,, Os, 0.) remain fixed from the training phase.
where Wa,i and her denote the original bin widths and heights, respectively.
Up-Projection and Morphing. We introduce scaling matrix Xnod E Bxx using the updated weight matrix Wusttt41 and an Up-Projection layer, which is formally defined as:
Test-Time Trained Temporal Layer. Inspired by (Sun al., 2024), introduce test-time-trained temporal layer designed to adapt the parameters of W flww during inference.
We begin by constructing a self-supervised task to optimize W Yest during test time. Specifically, we define low-rank projections and binary masks as follows:
x, f(0,, W Wist.r+1
X = UpProj (x,),
(9)
where UpProj()) denotes learned linear mapping that maps the fow-dimensional embedding into B 2 matrix. This transformation enables adaptation of the feature space to a higher- sesher-dimensional space.
x = MM @ 5(0g. WT
test
(5)
X3 = x()..x'),
where /(0, x') represents a learned linear transformation parameterized by 0, and M ~ Bernoullipp) is a binary mask.
Subsequently, we apply element wise multiplication of now with Xmod to adaptively morph the Flow parameters and obtain the MoF-ed feature:
self-supervised loss function E(W ((Wies; x') is computed over batches during test time stabilize gradient flow:
x'
F
W flow
Xmod
(10)
W Test: x') = M ④ 5(0,,x') x') f(0.. x1)1
(6)
3.3. Model Backbones
During inference, iteratively update the test-time weight matrix W Yest after each forward pass multiple passes:
A prediction backbone is constructed using the MoF out- put, with its predictions fed into MoF-1, The reversible design of MoF ensures that the backbone operates within
W 1es,tt1 W Wiettt- - 1 W {( (W (Wiiet,t; x');
(7)

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Slimming the Fat-Tail: Morphing-Flow for Adaptive Time Series Modeling
Algorithm 1 MoF Module Pseudocode
4.1. Experimental Settings
Input: Input sequence x of shape [C,
Datasets: We conduct experiments on eight widely used multivariate time series forecasting datasets, including Elec- tricity Transformer Temperature (ETThI, ETTh2, ETTml, and ETTm2) (Zhou et al., 2021), Electricity, Traffic, Weather (Wu et al., 2021), and Exchange Rate. To en- sure consistency, we follow the standard protocol (Liu et al., 2023) and split the datasets into training, validation, and test sets with a 6:2.2 ratio for ETT datasets and 7:1:2 ratio for the remaining datasets. Detailed dataset characteristies are provided in Appendix C.I.
Output: Transformed sequence x' of shape [C,
Apply the Flow transformation to the input sequence:
x' F(x| W flow)
Feed Xsow into the temporal morph module:
Xmod = Morph(xXao (Xtw))
I/ Umod has shape x
Transform the sequence using the modified parameters:
x' =F(x| W flow O mmod)
Return: x'
Following the standard Autoformer protocol (Wu et al., 2021), we adopt Mean Squared Er- ror (MSE) and Mean Absolute Error (MAE) as evaluation metrics. Due to observed variability in test results, each ex- periment is repeated five times, and Waineey U tests are used to evaluate statistical significance of performance differences. The historical horizon length is set to T = 336, with prediction lengths F E {96, 192, 336, 720]. All hy- perparameters are held constant across datasets to avoid the confounding effects of dataset-specific tuning, which might otherwise lead to overly optimistic results that fail to generalize. This unified setting may result in minor dis- crepancies from original baseline reports; nevertheless, as shown in Appendix D.4, our reproduced baselines remain competitive and consistent across benchmarks. Detailed hyperparameters for MoF Linear and MoF Mamba are in- cluded in Appendix D.
Evaluation Protocol:
its intended kurtosis range. Importantly, the MoF module is trained end-to-end without auxiliary losses to constrain its structure. To effectively demonstrate the performance of the MoF module, we design two simple and minimal backbones.
MoF Linear: A remarkably simple linear backbonn directly regresses historical time series to predict future values using a weighted sum operation (&= = Wx - b), as in LTSF-Linear (Zeng et al., 2023). To demonstrate our module's 's capabil- ity, we omit trend decomposition and apply instance-wise z-score normalization for stabilization, which standardizes residuals by removing instance-specific means and vari- ances.
MoF Mamba Adopting the patching methodology of PatchTST (Nie et al., 2023), we segment the inputs into P-length patches with a stride of S, reducing the number of tokens to [L/S] after padding. The Mamba backbone (Gu & Dao, 2023) processes these patches without positional embeddings using state-space dynamics, fully replacing traditional Transformers. For long-term stability, we de- compose the inputs using moving-average kernels, linearly predict the trend component and processing residuals with Mamba. Predictions are generated through flattening and linear head, and combined with the trend predictions.
We compare MoF against nine represen- tative models, spanning both recent state-of-the-arr methods and established baselines. Recent models include GLin- ear (Rizvi et al., 2025), LiNo (Yu et al., 2024a). TimeMa- chine (Ahamed & Cheng, 2024), CATS (Kim et al., 2024), iTransformer (Liu et al., 2024a), and PatchTST (Nie et al., 2023). Widely acknowledged baselines include DLinear (Zeng et al., 2023), Informer (Zhou et al., 2021), and Auto- former (Wu et al., 2021). These methods span Transformer- based, Linear- and Mamba-based architectures, en- abling comprehensive evaluation.
Baseline Models:
4. Experiments
4.2. Experimental Results
We evaluate the proposed model through comprehensive experiments: Section 4.1 benchmarks MoF Linear and MoF Mamba against nine state-of-the-art (SOTA) methods across eight datasets; Section 4.2 reports performance improve- ments, highlighting robustness against fat-tailed noise and non-stationarity: Section 4.3 explores module contributions via ablation studies, isolating the effects of Flow's 's normal- ization and Morph's adaptation; and Section 4.4 investi- gates hyperparameter sensitivity, efficiency trade-offs, and generalization performance when integrated with diverse backbones.
Mos-based Model Improves Over State-of-the-Art. The experimental results are summarized in Table 1, with de- tailed results for different prediction horizons and statis- tical significance tests Whitney U tests) provided in Appendix F. The MoF-based model consistently outper- forms all competitors across all 8 datasets. On average, surpasses the next best competitor by 6.3% and achieves substantial improvements over widely used models such as iTransformer and PStchTST, with gains of 14.9% and 16.9%, respectively, Furthermore, our MoF-Mamba model outperforms recent models, including TimeMachine and
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Slimming the Fat-Tail: Morphing-Flow for Adaptive Time Series Modeling
Table Performance comparison of our proposed model (MoF) with Mamba and I Linear backbones against haseline methods for multivariate long-tern term forecasting, Results are averaged across 4 horizons and represent the mean MSE from 5 independent runs using different random seeds. The best results are highlighted bold, while the second best results are underlined. Model names ending with "former" are omitted for brevity. "TimeM." " denotes Miieeaahinne Detailed results, including extended comparisons with methods like IN-Flow and FITS, are provided in Appendix and Appendix
MEDIAN
MoF (MAMBA)
MOF (LINEAR)
GLINEAR (2025)
LINO (2024B)
TIMEM. (2024)
CATS (2024)
ITRANS.
PATCHIST
DLINEAR (2023)
AUTO, (2021)
IN. (2021)
TRANS,
DATASET
Kazeess
(2023)
(2023)
(2017)
ETTHI
5.80
0,501
0.515
0.522
0.528
0.529
0.822
0.538
U.584
0.524
0.771
1.243
0.828
ETTH2
10.9
0.224
0.222
0.240
0.248
0.241
0.263
0.261
0.263
0.228
0.374
0.458
0.431
ETTMI
17.7
0,392
0.401
0.395
0.407
0.403
0.414
0.442
0.416
0.452
0,619
0,745
0.655
ETTM2
19.6
0.165
0.183
0.166
0.166
0.165
0.172
0.182
0.169
0.170
0.199
0.381
0.293
ELECTRICITY
3.42
0.163
0.164
0.167
0.175
0.166
0.221
0.165
0.164
0.167
0.221
0.354
0.291
EXCHANGE
74.2
0,305
0.402
0.394
0.403
0.432
0.397
0.462
U.544
0.390
1.198
1.676
1.224
TRAFTIC
15.6
0.416
0.430
0.428
0,458
0.417
0.937
0.451
0.416
0.435
0.648
0.811
0.691
WEATHER
30.0
0.221
0.222
0.226
0.238
0.227
0.225
0.241
0.234
0.245
0.361
0.529
0,505
1ST. COUNT
43/64
0
1
0
5
0
0
0
2ND.
AVERAGE MSE
0,298
0.317
0.317
0.328
0.323
0.431
0.343
0.349
0.326
0.549
0.775
0.615
DIFFRRENCE
0.0%
-6.4%
-6.3%
-9.9%
-8.1%
-44.6%
-14.9%
-16.9%
-9.4%
-84.0% %
-160%
-106%
LiNo, by 8.99%... Notably, on the dataset with the high- est kurtosis (Exchange), our model outperforms the closest competitor by a margin of 21.7%.
MoF-Mamba MoF-Linear
DLinear
Additionally, we incorporate IN-Flow and FITS in tended analysis (Appendix K) provide broader perspec- tive normalization-cennti approaches. While IN-Flow also builds on flow-based transformations, it focuses on using entangled flows to model non-stationarity- a funda- mentally different objective from our focus on stabilizing fat-tailed gradients. Despite this distinction, we include IN- Flow in the comparison due its similar building blocks. We observe that IN-Flow performs reasonably on smaller datasets but struggles with high- -ich-iienniionall inputs. FITS, on the other hand, prioritizes parameter efficiency through adaptive feature reweighting, though it is sensitive to hyper- parameter settings and exhibits instability under our unified evaluation protocol.
GLinear
Relative MSE
CATS
LiNo
2.0
TimeMachine
iTransformer
PatchTST
96
336
720
Prediction Length
Figure 4. Relative Mean Squared Error (MSE) across varying time horizons, averaged over eight datasets. The relative MSE is calcu- lated as the average ratio of the MSE relative to the best- cest-performing model across all prediction lengths and configurations. 4.3. Modular Analysis
MoF- Model Excels in Long-Range Prediction. As shown in Fig. 4, the MoF-based model achieves superior long-term forecasting performance, significantly outper- forming competitors by 16.5% in relative MSE (as defined by the ratio to the best-performing baseline across all hori- zons for each dataset). This improvement stems from MoF's ability to mitigate fat-tailed distributions and stabilize spo- radic, high-variance signals: the Flow module aligns residu- als a wear-Gaussian latent space (evidenced by kurtosis reduction in Fig. 1), reducing outlier-driven variance while preserving extremal patterns, and the Morph module dynam- ically adapts to distributional shifts during inference, ensur- ing stable convergence under non-stationarity. By jointly addressing fat-tailed noise and temporal instability, MoF en- hances the backbone's 's capacity to model long-range depen- dencies-critical for minimizing error compounding over extended horizons.
What the Spline (Flow) Learned. Figure illustrates MoF's distribution transformation across three phases: (1) non-stationary raw inputs with fat tails, (2) stationarized fea- tures that retain excess kurtosis (K 29), and (3) out- puts approaching tematically reduces channel-win -innel--wise excess kurtosis by 18.5% 93.3% (Appendix F). Its reversible architecture ensures optinization-friee -fienndy distributions at both model inputs (his- torical data) and outputs (future predictions), thereby provid- ing the backbone with stable and well-structured feature space.
(Kexcess < 1). MoF sys-
Gradient-level stability. Figure illustrates the evolution of gradient variance during training on the ETTm/ dataset, comparing instance- -iineeee--ieeeeee 2-score, RevIN, and our proposed MoF. MoF consistently demonstrates lower and more stable
6

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Slimming the Fai- Morphing-Flow for Adaptive Time Series Modeling
Original Data
Stationary Data
MoF Transformation Curve Soline Knoss, Auls Skowed for Corre))
MoF Transformed Data
Gaussion Guantie Guantie Pico
KGoussian Quordic Guantle Roo
Kexcess 8.73
Kexcess 29.66
Kexcess 0.94
-2
-3
& Non- stationary
Clustering
-3
Fat-tall Distribution
Learned Transformation Spline
Consistently Distribution
Normal
-2
2
Figure 5. Transformation process visualization (ETTh2 dataset, single channel): (1) Raw non-stationary data exhibits non-Gaussian characteristics in the Q-Q plot against a normal distribution, with pronounced deviation from linearity; (2) The stationarized sequence is centered at the origin but retains fat tails (excess kurtosis Kavess = +99.66);; (3) The learned MoF transformation curve obtained via end-to-end training. with vertical axis scaling adjusted for spline visualization; (4) The final transformed distribution approximates normality with excess kurtosis Kencess +0.94 after Flow processing.
Ie-7
temporal patterns while enabling stable gradient propaga- tion. Notably, this channel-aligned stabilization essential for modern channel-independent (CI) architectures, where reduced distributional disparities enhance the effectiveness of shared parameters across individual time series.
5
PatchTST + Z-Score: 12e-1111-077... a= 4.
PatchTST + ReviN: p=1.33e-0,,
PatchTST MoF: p-5.38e-08. a-3.0ee-8333
4
3
Ablation Study of the Test-Time Training Module: While the experiments above demonstrate the effectiveness of the proposed modules, further analysis reveals that the impact of the TTT module context-dependens In summary, is more beneficial in the short term but comes with minor drawbacks in the long term. As shown in Figure 7, con- ducted an ablation study of the TTT module across various datasets on an expanded range of prediction lengths. The re- sults indicate that TTT significantly improves performance in shorter-term prediction windows (up to a 5% error reduc- tion). However, for longer-term predictions, the benefits of TTT are less pronounced, as these tasks are inherently more challenging due to higher noise levels and reduced sensitivity to distributional shifts.
2
1
0
0
2000
4000
6000
8000
10000 12000
Figure 6. Gradient variance across training iterations ETTml under different normalization strategies is shown. MoF exhibits lower variance and greater stability. Statistics (mean μr, standard deviation 8) of grad-var are computed over the last 2000 iterations.
gradient variance throughout training. The summary statis- ties (mean and standard deviation) are computed over the final 2000 iterations to characterize steady-state behavior.
Superiority of MoF over other normalization methods
Although our method (MoF) primarily focuses on address- ing skewness and kurtosis in data distributions, it can struc- turally and functionally complement existing approaches targeting non-stationarity. We compared MoF with FAN (Ye et al., 2024), RevIN (Kim et al., 2021), SAN (Liu et al., 2024b), and DishTS (Fan et al., 2023) using DLinear the backbone, with the input horizon fixed at 336. Evaluations were conducted on 8 datasets across 4 prediction lengths. To demonstrate the plug-and aong-and--pay capability of MoF, all mod- ules were tested using default parameters without requiring dataset-specific tuning.
This trend is more pronounced in deeper models (e.g., PatchTST), particularly heavy-tailed datasets such Exchange. In addition to variance, track gradient norm, skewness, and (excess) kurtosis-wwere MoF again shows improved numerical conditioning and smoother training dy- namics. These gradient-level henefits correspond to the ob- served gains in accuracy under fat-tailed noise. Full statistics and additional visualizations are available Appendix M.
Cross-Channn Channel Consistency Furthermore, MoF achieves 97.27% reduction in inter-channel kurtosis variance (Ap- pendix F). This variance reduction directly stabilizes train- ing by aligning the magnitudes of gradients across channels, a step critical for preventing divergent learning trajectories in multi-series forecasting. The spline-lcarned, channel- consistent normalizing flow retains the expressiveness of
The experimental results are shown in Fig. 8, with detailed quantitative results provided in Appendix I. The results demonstrate that even when using basic instance-wise z- score normalization, MoF outperforms other models de-

---

Slimming the Fat-Tail: Morphing-Floon for Adaptive Time Series Modeling
0.50
mT
MSE
Prediction Length
Dataset
0.20
MSE v. Prediction Length
Electricity Dataset
Table 2. Hyperparameter Sensitivity Analysis of MoF
0.45
0.18
0.40
0.15
ETTHI
ETTMI
WEATHER
MSE
MAE
MSE
MAE
MSE
MAE
0.35
0.12
BIN
TAIL = 6.0
0.30
0.10
336
720
336
720
6
0.4688
0.4806
0.3802
0.4165
0.1918
0.2416
12
0.4678
0.4797
0.3780
0.4162
0.1952
0.2441
0.50
ETTml Dataset
MAE v. Prediction Length
0.28
MAE Prediction Length
Electricity Dataset
24
0,4685
0.4795
0.3778
0,4158
0.1932
0.2441
48
0,4689
0,4798
0.3777
0.4136
0.1944
0.2448
0.45
0.26
0.24
DIFF
0.23%
0.67%
1.73%
0.22%
0.71%
1.32%
0.22
TAIL
BIN COUNT 24
/wo
0.35
0.20
McF 7w
TTT
3.0
0.4688
0.4798
0.3764
0.4112
0.1927
0.2463
48
72
96
192
336
720
960
12
24
48
72
96
192
336 720
6.0
0,4672
0,4793
0.3770
0.4144
0.1927
0.2459
8.0
0.4673
0.4794
0.3787
0.4164
0.1939
0.2435
Figure Ablation study the Test- Tese-Time Training (TTT) module across different datasets. TTT significantly improves short-term performance (up to 5% error reduction) while showing less pro- nounced benefits for longer-term prodictions, where distributional shifts are relatively less impactful.
12.0
0.4622
0.4768
0.3792
0.4182
0.1961
0.2435
18.0
0.4672
0.4801
0.3903
0.4232
0.1917
0.2419
24.0
0.4656
0.4789
0.3855
0.4204
0.1902
0.2402
DIFF
1.40%
3.54%
3.02%
0.69%
2.83%
2.48%
All Datasets Averaged
MoF(Ours)
1.10
Relative MSE Prediction Length
FAN
of 3.54%. These findings confirm MoF's robustness and plug-and-play flexibility across different datasets.
SAN
DishTS
1.08
DLinear
Model Efficiency MoF achieves parameter-efficien perfor- mance, as shown Fig. Linear- MoF attains 8.5% higher accuracy than iTransformer with 21 x fewer parameters. Against TimeMachine, Linear-Mof improves performance by 1.7% using 5.28 fewer parameters, while Mamba-MoF outperforms it by 8.1 % at identical parameter scales (Ap- pendix G).
ReviN
1.06
1.04
This efficiency originates from MoF's adaptive feature trans- formation, which confines activations to stable operational ranges while mitigating training instability caused by fat- tailed, high-variance inputs. The design eliminates redun- dant parameters typically required handle distributional extremes.
96
192
336
720
Figure 8. Relative MAE comparison of MoF and other normal- ization methods across all 8 datasets and 4 prediction horizons. Relative MSE is calculated w.r.t the best-performing model for each prediction length.
Complexity Analysis The computational complexity of the proposed MoF framework consists two components: the Flow layer and the Morph module. The Flow layer operates with complexity OCT B), where C' is the number of input channels, T the sequence length, and B is the number of spline bins per channel. The Morph module introduces additional overhead of O(N CT d + d B + C'T B), where d is the internal projection dimension and N is the number of gradient updates performed at test time. The three terms respectively account for test-time adaptation, projection to Flow parameters, and second application of the Flow transformation. Since d CT, B is moderate. and Niter small (typically 1-5), the overall asymptotic overhead introduced by Morph remains bounded. Empiri- cally, Morph contributes only 17.92% of MoF's inference time on average, with less than 5% increase in total runtime over Flow-only baselines. A detailed complexity analysis and wall-clock profiling across six datasets are provided in Appendix I..
signed for non-stationary distributions, highlighting the sub- stantial impact of fat-tailed distributions.
4.4. Model Analysis
Hyperparameter Robustness of MoF Module The MoF module introduces two hyperparameters: Tail-size (I") and Bin Count (B). Our experiments demonstrate that T and B require no dataset-specific tuning and have limited impact on performance and convergence. Previous experiments fixed T 6.0 and B = 24. Increasing B allows finer feature representation but linearly increases parameter size, while a larger T enhances resilience to high-variance noise at the cost of reduced parameter density in the core feature space.
Table 2 shows the sensitivity analysis: fixing T = 6.0 and varying results in a fluctuation of up to 1.73%, whereas fixing B 24 and varying T yields a maximum difference
8

---

Slimming the Fat-Tail: Morphing-Flow for Adaptive Time Series Modeling
Table 3. Comparison of model performance with and without MoF module across various datasets.
MODEL
AUTOFORMER
INFORMER +MoF
DLINEAR +MoF
Raw
+MoF
DIFF.
Raw
DIFF.
RAW
DIFF.
ETT(ALL)
0.491
0.455
-5.64%
0.707
0.508
-28.1 %
0.344
0.330
-3.96 %
ELECT.
0.220
0.200
-9.05%
0.354
0.235
-33.8%
0.167
0.164
-1.59
EXCHG.
1.198
0.690
29933%
1.676
0.694
-58.6%
0.390
0.402
3.16%
TRAFFIC
0.648
0.677
-5,55%
0.812
0.742
-8.63% %
0.435
0.430
-0.96%
WEATHER
0.361
0.313
-18.4%
0.529
0.356
-32.7 %
0.245
0.222
-9.15%
AVERAGE
0.584
0.467
-20.0% %
0.816
0.507
-37.8 %
0.316
0.310
-1.93% %
0.48 0.56 00.44
0.26
Average
Average MAE
0.30
1:32 0.40
0.20
192
335
960
192
Input Sequence Length
Input Sequence Length
MoF(Ours)
FAN
ReviN
SAN
DishTS
MoF-Linear
Autoformer
ITransformer
PatchTST
Informer
Figure 9. Input-length sensitivity ETTH2, showing monotonic trends (c.g., Linear-MoF peaks at 192, iTransformer peaks 336). This figure illustrates per-dataset behavior, not overall ranking.
Figure 10. Comparison different normalization modules on the WEATHER dataset using DLinear. Similar to Fig. 8, this figure illustrates how input length interacts with normalization on single dataset rather than providing overall ranking.
4.5. MoF Generalization Analysis
ing stability even at moderate excess kurtosis levels. The proposed Morphing- (MoF) framework addresses this through components: (1) spline-based Flow layer that normalizes extremal values while preserving critical temporal patterns, and (2) Morph module mitigating distri- butional drift via test-time adaptation. Experiments across 8 benchmarks in various settings validate MoF's effectiveness, as surpasses competitors by 6.3% on average, achiev- ing 21.7% error reduction on fat-tailed datasets such as Exchange. Broader analysis confirms that MoF mitigates distributional skewness and temporal drift, promoting bal- anced gradient dynamics while maintaining computational efficiency and hyperparameter robustness. The plug-and- play design enables seamless integration into diverse archi- tectures without structural changes, offering a lightweight deployment solution. These results suggest that adaptive handling of fat-tailed distributions is critical for reliable time series forecasting in non-stationary environments.
Rethinking Input Length Most prior works fix input lengths 96 or 336, but our experiments show input length strongly impacts performance. We evaluated the Weather and ETTh2 datasets across expanded horizons (48, 96, 192, 336, 720, 960 time steps) with fixed prediction length of 192. Full results are in Appendix H. We compared Linear- MoF with other models (Fig. 9) and plinear-based methods combined with various normalization modules (Fig. 10). Re- sults indicate the optimal imput length varies across models and datasets, likely due to the data's inherent frequency and structural bias. Notably, Linear-MoP achieves superior curacy with fewer input steps and consistently outperforms other models at their respective optimal input lengths.
Generalization of MoF Across Models This section ex- plores the integration of the MoF module with various back- bone models, including Autoformer, Informer, and DLinear. Results indicate that incorporating the MoF module reduce error rate by up 58%. suggesting its potential to enhance the performance of diverse architectures. These findings highlight the adaptability of the MoF module as promising approach for enhancing various models. Detailed results and analyses are available in Appendix
A Acknowledgements
This work was partly supported by the National Key Research and Development Program of China NN.2024YFB4711102) Meanwhile, this work was jointly funded by the Xiaomi Innovation Joint Fund of the Bei- jing Municipal Natural Science Foundation (No.L233006) and the project of Tsinghua Sniversity-Toyord Joint Re- search Center for AI Technology of Automated Vehicle (No. TTAD2025-03).
5. Conclusion
In this work, we demonstrate how fat-tailed distributions, prevalent in real-world temporal data, undermine forecast-

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Slimming the Fat-Tail: Morphing-Flow for Adaptive Time Series Modeling
Impact Statement
Ilbert, R., Odonnat, A., Feofanov, V., Virmaux, A., Paolo,
G., Palpanas, T., and Redko, I. Samformer: Unlocking the potential of transformers in time series forecasting with sharpness aware minimization and channel-wise atten- tion. In Forty-first International Conference on Machine Learning, 2024.
This paper presents work whose goal to advance the field of Machine Learning. There are many potential societal consequences of our work, none which feel must be specifically highlighted here.
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Slimming the Fat-Tail: Morphing-Flow for Adaptive Time Series Modeling
A. Related Work
Fat-Tailed Features in Machine Learning. Empirical evidence and theoretical analysis have revealed the presence of fat-tailed noise and leptokurtic features in modern machine learning systems (Simsekli al., 2019; Nguyen et al., 2019). Such distributions challenge the bounded variance assumption in stochastic optimization, significantly impacting gradient stability and learning dynamies (Zhang et al., 2020; Wang et al., 2021). To address these challenges, methods like gradient clipping (Gorbunov et al., 2020; Yang et al., 2022) and adaptive normalization techniques for SGD (Hübler et al., 2024) have been proposed. Recent studies further demonstrate the prevalence of fat-tailed patterns across diverse domains, including economic indicators (Kiss & Österholm, 2020a), hydrological data (Wang et al., 2023), and epidemic transmission (Wong & Collins, 2020).
In financial indices, data with pronounced tails are commonly observed due to clustering volatility, a phenomenon linked to the heteroskedasticity of temporal sequences (Eom et al., 2019; Kiss & Österholm, 2020b). Models like GARCH (Bollerslev, 1986) are traditionally used to capture such behavior. As shown in Fig. 1, the heterogeneity of time series data results in consistently fat-tailed distributions across a wide range of domains. However, research on understanding and mitigating the effects of such characteristics in neural time series forecasting remains limited. MoF addresses this by reducing gradient skewness and kurtosis, particularly benefiting deeper backbones where outliers more severely disrupt training. Our results demonstrate that MoF yields substantially higher improvements with PatchTST than with shallow architectures like Linear, aligning with these domain-leve insights.
Non-Stationarity in Time Series Forecasting. Time series forecasting typically faces time-variant phenomena and distribution shifts, often addressed through signal decomposition to separate stationary and non-stationary components (RB, 1990; Hamilton, 2020; Zeng et al., 2023). Ogasawara et al. (Ogasawara et al., 2010) proposed a local normalization approach, while RevIN (Kim et al., 2021) introduced reversible instance normalization to better handle distribution shifts at the sample level. Subsequent models have refined this direction, including Leddam's 's learnable convolution for trend extraction (Yu et al.. 2024c), LiNo's recursive residual decomposition (Yu et al., 2024b), and FAN's frequency-based decomposition (Ye et al., 2024). While several models predict future statistical moments to better handle heteroskedasticity, they often underperform in the presence of fat-tailed noise, especially under limited batch-level statistics.
Backbones and Generalization. Time-slice-based transformers for time series forecasting struggle with weak temporal dependencies and noisy per-timestep encoding (Zhou et al., 2021; Kitaev et al., 2020). DLinear (Zeng et al., 2023) and iTransformer (Liu et al., 2023) initiated the shift toward modeling sequences holistically in temporal space. Patch- based architectures such as Crossformer (Zhang& & Yan, 2023), TimesNet (Wu et al., 2022), and Mamba- based designs (Ahamed & Cheng, 2024; Zeng et al., 2024) further enhance modeling capacity but suffer from exposure to non-stationarity due to their reliance on localized features. Several studies have noted that existing forecasting benchmarks often fail to expose the generalization challenges of such models (Han al., 2024; Ilbert al., 2024).
Recent works like IN-Flow (Fan et al., 2025) tackle non-stationarity through entangled normalizing flows, but face scalability issues on nigh-dimensional datasets. FITS (Xu et al., 2023) offers parameter efficiency via hypemnetoor- like reweighting, though it relies on expensive hyperparameter search. ESaSTST (Zhang et al., 2024) proposes a backbone-agnostic design for multi-horizon generalization, which is orthogonal to our goal of improving normalization. Our experiments (Appendix K) show that MoF complements these design strategies and remains robust even under fixed hyperparameter settings across datasets.
B. Synthetic Fat-tail Training Experiment
B.1. Detailed Experiment Setup
Dataset synthesis The dataset consists of samples drawn from either a Gaussian or Cauchy distribution, where the mean grows at each step by a factor k. At step t, the mean of the distribution is defined as f4, = kpp-11 with f40 = 1.0. Each sample 2y is generated as:
Zt = Pt + Es, == =p t= 0.1,....,.
where e, is drawn from the respective distribution (Gaussian or Cauchy), and clipped to the range [-6, 6). The dataset is structured such that the first n samples are inputs, and the last one is the output:
12

---

Slimming the Fat-Tail: Morphing-Flow for Adaptive Time Series Modeling
y
Zn
(output)
Simple Linear Model with Clipping The model is linear regression defined as:
Wæ b
where x E R is the input, and the output ý clipped within the range [-20, 20] before being passed through the linear transformation. The parameters W and are learned during training.
Training Protocol The model is trained using SGD with a learning rate ') = 0.01 and Mean Squared Error (MSE) loss. A StepLR scheduler is used to reduce the learning rate by a factor of 0. every 30 steps. The total number of iterations is calculated based on the batch size and dataset size, with each iteration consisting of forward pass, loss computation, and backpropagation.
The loss function is:
1 B
C
(1 - y1)
where B the batch size.
peppaaaraettess For both the Gaussian and Cauchy distributions, the following hyperparameters were used:
Mean growth factor (k): 1.1
Batch size (B): 128
Learning rate (1)): 0.05
Optimizer: SGD
Loss function: Mean Squared Error (MSE)
Learning rate scheduler: StepLR with step size 30 steps and decay factor y = 0.1
Total iterations: 10,000
B.2. More Results on Synthetic Datasets
Fig 1l are results with identical settings across multiple runs but different random initializations. These experiments reveal sigmificant disparities in the convergence behavior of the models when trained on sequences from distributions with heavier tails.
B.3. Stability Condition Analysis
common way to express the stability condition of SGD with a (possibly non-convex) loss L is through:
172 o²
E [10++1-00 0-]]] < (1-2 Amin (H)) 1160 - 0-11 + Amin (1))
where 8" is stationary point, H is the Hessian (or equivalent notion of curvature), and o² is upper bound on the variance of the gradient noise. From empirical standpoint:
Variance effect: 2 is large due to fat-tailed (though bounded) noise, the second term on the right-ham side grows, leading poor convergence or large stationary error.
13

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Slimming the Fat-Tail: Morphing-Flown for Adaptive Time Series Modeling
Gaussian Distibution, Rucur - 0.00
Density plot
Gaussion Distr bution, Kurcuu 0.00
Density piet
Gatasian Q-Q Piot
Cauchy Distribttio,, Rucua
1.47
Coussian 0-0 Pict
Caurtry Dieitburionn 1.53
Distributions
Optimization Process(Output Iterations)
Distributions
Optimization Wropess v. (terations)
(b) Converges to a seemingly correct value, but still above the target value.
(a) Converges to an incorrect value above the target value.
Causiann Distrbbtion, Kecar 0.00
Density piet
Denaity piot
Gaussinn O- Plot
Cauchy Distibution Kecur 1.46
Gaussien Q-0 Pict
Cauchy Distribution, Recer 1.55
Distributions
Optimization Prupuss Iterations)
Distributions
Optimization Process(Output Iterations)
(c) Converges an incorrect value below target value.
(d) Fails converge entirely.
Figure I1. Training results synthetic datasets with sequences from Gaussian and Cauchy distributions. Models trained Gaussian sequences exhibit consistent convergence, while those trained on Cauchy sequences demonstrate significantly varied behaviors, ranging from incorrect convergence to complete failure. This highlights sensitivity of models to fat-tailed distributions.
Starting point: A poor initialization On can exacerbate the challenge, especially when high high-variance updates cause parameters deviate before eventually converging to 8*, In real-world scenarios, we observed that replacing the DLinear model's "Average Initialization" with random initialization significantly affects model performance.
Other factors: Learning rate 7), local curvature Ami ( (H), and problem structure (e.g., convex vs. non-convex) all empirically influence the extent to which fat-tailed noise hinders convergence.
14

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Slimming the Fat-Tail: Morphing-Flow for Adaptive Time Series Modeling
C. Dataset Overview
C.1. Dataset Introduction
Dataset
Granularity
Dataset Split
Prediction Length
Channels
Domain
ETTh1
Hourly
(8545, 2881, 2881)
(96, 192, 336, 720]
7
Electricity
ETTh2
Hourly
(8545, 2881, 2881)
(96, 192, 336, 720)
7
Electricity
ETTml
15min
(34465, 11521, 11521)
(96, 192, 336, 720)
7
Electricity
ETTm2
15min
(34465, 11521, 11521)
(96, 192, 336, 720)
7
Electricity
Exchange
Daily
(5120, 665, 1422)
(96, 192, 336, 720
8
Economy
Weather
10min
(36792, 5271, 10540)
(96, 192, 336, 720)
21
Weather
Electricity
Hourly
(18317, 2633, 5261)
(96, 192, 336, 720)
321
Electricity
Traffic
Hourly
(12185, 1757, 3509)
(96, 192, 336, 720)
862
Transportation
NIL
Weekly
1980,193,193)
(24,36,48,60)
7
Healthcare
ETT (Electricity Transformer Temperature)': This dataset includes four subsets: two hourly-level datasets (ETTh) and two 15-minute-level datasets (ETTm). Each subset consists of seven features related to the oil temperature and load of electricity transformers, recorded from July 2016 to July 2018.
Traflic?: This dataset contains hourly road occupancy data collected from sensors on San Francisco freeways during 2015-2016.
Electricity?: This dataset contains hourly electricity consumption data from 321 clients, recorded between 2012 and
2014.
Exchange-Rate" This dataset includes daily exchange rate data for 8 countries, spanning the years 1990 to 2016.
Weather This dataset records 21 weather indicators, such as air temperature and humidity, at 10-minute intervals throughout 2020 in Germany.
The ILI dataset tracks the ratio of patients diagnosed with influenza like illness to the total number of patients, based on weekly data from the Centers for Disease Control.
ILI (Influenza-Like Illness):
C.2. Kurtosis Calculation
Kurtosis is statistical measure used to describe the shape of probability distribution, particularly its tails. It quantifies the extent to which the tails of a distribution differ from those of normal distribution. The excess kurtosis Rexeess is calculated as:
E[(X - -4)=]
Kexoess
-3
04
where: - X is a random variable, - f is the mean the distribution, - or is the standard deviation, and - -E]] denotes the expected value.
kurtosis value of kiecees 0 indicates normal distribution, with heavier tails (leptokurtic) for positive values and lighter tails (platykurtic) for negative values.
Since each channel or dimension in a time series dataset represents a different feature with potentially different statistical properties, the distribution of each dimension may vary significantly. Therefore, kurtosis is computed separately for each
'https: httpppss//ww.ccccooo/////eeee//a
15

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Slimming the Fat-Tail: Morphing-Flow for Adaptive Time Series Modeling
channel accurately capture the distinct tail behaviors across features. This channel-wise analysis allows the presence of fat-tailed distributions other characteristics specifie to individual time-series features.
C.3. Kurtosis Analysis
Kartous of Saw Data (Ouciens, Removed)
Burtoes flerenced Date (Oullers Bemoved)
Karos
natreal Bloai
intmna
weather
neteral Blosdi
EnT
weatfer
Dagaser
Dotaser
Figure 12. Kurtosis distribution across different channels. The red line represents a kurtosis of (K [eecesss = 0); indicating a normal distribution.
We computed the kurtosis for each dimension across the datasets described earlier. The kurtosis distributions for different channels are illustrated in Fig. 12. Notably, the red line the plot represents the theoretical kurtosis value of 3, which corresponds to a normal distribution (i.c., kexcess = 0). Deviations from this baseline indicate the degree of fat-tailedness present in the data.
The left plot presents the kurtosis distribution of the raw data, which are significant non-stationarity, Since the distribution appears dispersed across the feature space, the fat-tailed nature is not immediately evident. However, when we applied differentiation between adjacent time steps, the kurtosis values increased sharply, with most dimensions exhibiting pronounced fat-tailed distribution. This observation underscores the pervasive fat-tailed nature of time-series data across different domains.
Although, the common practice of normalization using instance-wise z-scores tends to balance the feature space more evenly (in comparison direct differeneing neighboring time steps). The kurtosis effect remains observable, reflecting the intrinsic properties of time-series data that persist across various datasets.
16

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Slimming the Fat-Tail: Morphing-Flow for Adaptive Time Series Modeling
D. Experiment Details
D.1. Model Structure Details
For preprocessing. we use a stationary method based on instance-wise z-score normalization. This process centers each instance by subtracting its mean and dividing by its standard deviation, ensuring that each input is normalized independently.
Stationary Method:
Decomposition Method: A standard moving average (MOA) kemel is used to decompose the trend and seasonal (residual) components of the data. The kernel size for the moving average is controlled by the kernel_size hyperparameter, which defaults to 25.
D.2. Model Hyperparameters
This section provides a detailed overview of the hyperparameters used in various components of the model. Default values are specified for each parameter where applicable.
MoF Hyperparameters: For MoF modules the number of bins for the spline (spline_na) num bainbinns)) is set to 24, the tail length of the spline (splirts (elll_eteaill is set to 6.0, and the size of the b-matrix for Test Time Training (TT) is set to 92 for all the dataset. Additionally, the masking ratio p for self-supervised Test Time Training (TTT) is set to 0.5.
Except for experiments involving long-range settings (e.g., prediction lengths of 336 and 720), where the dimension count is high and the distribution across channels is similar, due to computation power limitation, same set of MoF parameters is shared across channels. In all other experiments, each channel utilizes dedicated MoF parameters.
Patch Mamba Backbone Settings: The Patch Mamba backbone is controlled by the following hyperparameters. The embedding dimension, denoted as d_model, set to 64. Dropout is applied to the fully connected layers with a rate 0.2 src_dropout), and to the attention heads with a rate of 0.0 (head dropout). The padding method for patches is to (head_dropoutt
end, meaning that padding is applied the sequence's end. Each patch has length of 16 (patch_len) and the stride for patch extraction is set to 8 (stride).
Linear Backbone Settings: The output sequence is projected to the prediction length using linear layer. The jeniiinninnn in layer projects the input sequence length (seq_len) to the prediction length (pred_len). The weights of the projection layer are initialized uniformly, with each element set to seg lon ensuring that all input elements are given equal importance at the start. This weight initialization method consistent with previous approaches in the literature, such as those described in (Zeng et al., 2023). Although we found initialization can have an positive impact on model performance, we have analyzed previously, this aspect not a novel contribution of this work.
D.3. Evaluation Protocol:
Following the protocol established by Autoformer (Wu et al., 2021), we use Mean Squared Error (MSE) and Mean Absolute Error (MAE) as the primary evaluation metrics. To account for variability in the results, each experiment is repeated five times. Since we cannot confirm the normality of the results, significance of performance differences. Specifically, we conduct Mann-Whitron U tests, which do not assume normal distribution, to determine whether observed differences in performance are statistically significant.
poramerrettiic approach is employed to assess the statistical
For the main experiment, the historical horizon length is set to T = 336, with prediction lengths F E {96, 192, 336, 720), except for the National- dataset, where T = 60 and F E (24,36; 48, 60). For other baseline models, the default set of hyperparameters and evaluation protocol are used across different datasets, ensuring consistent performance comparison.
D.4. Evaluation Protocol and Hyperparameter Settings
Unified Hyperparameter Configuration. Unlike prior works that tune hyperparameters per dataset, our experiments adopt single, unified configuration across all datasets and methods-including our MoF variants and baseline models. This design choice eliminates confounding effects from dataset-specific tuning and ensures that performance differences more directly reflect model design. While this stricter setup may introduce small deviations from original reported numbers, it reduces the risk of overfitting to test sets and improves cross-dataset generalization.
Comparison with Prior Works. Previous models such as iTransformer and PatchTST employ different hyperparameter
17

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Slimming the Fat-Tail: Morphing-Flow for Adaptive Time Series Modeling
choices for different datasets, detailed in Table These inconsistencies hinder fair comparison and obscure whether performance gains stem from model design tuning. For example, the DLinear result reported in the i'Transformer paper uses short lookback length of 96, resulting in shallow input (e.g., 96x 192 linear layer), which significantly underfits and degrades performance. In contrast, our re- re-implementatinn use unified setup with lookback length fixed at 336 across all datasets and models, eliminating sataset-specific tuning. As shown in Table 4, this fairer and more challenging protocol improves generalization and yields stronger baselines, outperforming comparable reimplemented baselines by 1.69-4.35% average MSE.
Table Comparison different series forecasting methods (DLinear, iTrnnsformer, PatchTST) reported prior works versus our re-implementations under unified hyperparameter settings (rightmost group).
Model
iT
PT
DL
PT
DL
iT
PT
DL
From paper:
iTransffrmer
Patch TST
Ours
lookback length
96
96
96
336
512
336
Patch Len/Stride
12/
16/8 to 24/2
16/8
model
128-512
16-128
512
d_ff
128-2048
128-256
512
learning_race
1c-3 to 5e-5
2.5c-4
to le- 1c-2 to le-4
IE-04
4
batchsize
16-32
24-128
8-32
4
ETTh1
0.454
0.469
0.456
0.417
0.42
0.538
0.584
0.524
ETTh2
0.383
0.387
0.559
0.331
0.43
0.261
0.263
0.228
ETTml
0.407
0.387
0.403
0.352
0.36
0.442
0.416
0.452
ETTm2
0.288
0,281
0,350
0.258
0,27
0,182
0.169
0.170
ECL
0.178
0.205
0.212
0.162
0.17
0.165
0.164
0.167
Exchange
0.360
0.367
0.354
0.462
0.544
0,390
Traffic
0.428
0.481
0.625
0.396
0.43
0.451
0,416
0.435
Weather
0.258
0,259
0,265
0.230
0,25
0,241
0,234
0,245
Average 0.345
0.353
0.403
0.307
0.33
0.343
0.349
0.326
(2.72%)
(4.35%)
(1.69%)
Abbreviations: /T=iTransfomer,, PT=PatchTST, DI=DLincar,, "=not available
Baseline Alignment. To validate experimental setup, reproduce results for low-capacity models like DLinear and compare them with original and recent benchmark papers (Table 5). Our results align well across all datasets.
Table 5. DLincar MSE comparison across papers.
Dataset
Ours
DLinear (AAAI'2)
LIFT (ICLR'24)
SAN (NeurIPS'23)
Weather
0.245
0,246
0.246
0.245
Electricity
0.167
0.166
0.166
0.166
Traffic 0.435
0.434
0.434
0.435
Stronger Baselines under Fair Settings. Even under this more stringent unified protocol, our ce-imleementaionss strong baselines achieve better results than originally reported-e.g e.g., I'Transformer implementation improves over the original by 1.69% to 19.1% in average MSE. This highlights the influence of consistent experimental conditions.
Table 6. Average MSE performance under unified settings.
Model
Ours (MSE)
i'Transfrrmer Paper PatchTST Paper
iTransformer
0.343 (-2.72%)
0.345
PatchTST
0.349 (-4.35 %)
0.353
0.307
DLinear
0.326 (-19.1 1)
0.403
0.330
MoF Gains Remain Robust.
Despite the stronger baselines, our proposed MoF module esscill-sppecially when used with the
18

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Slimming the Fat-Tail: Morphing-Flow for Adaptive Time Series Modeling
Mamba bannsone--coniitennty improves both iTransformer and PatchTST by 14.9% and 16.9% in average MSE, respectively.
19

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Slimming the Fat-Tail: Morphing-Flow for Adaptive Time Series Modeling
E. Detailed Comprehensive Experiment Results
E.1. MAE and MSE performances
We compare MoF with Mamba and Linear backbones against 10 sfate-of-the-ar models or widely acknowledged methods. Each experiment is repeated five times, and the best average results are shown in black, while the second-best results are highlighted in blue and underlined.
Duc to space limitations, the experiment results are presented across two tables.
Table 7. Model Performance Table
Model
MoF-Mamba
MoF-Linear
GLinear MAE
LiNo
TimeMachine
CATS
iTransformer
Dataset
MAE
MSE
MAE
MSE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
96
0.441
0.406
0.439
0.416
0.446
0.419
0.446
0.411
0.445
0.417
0.584
0.664
0.465
0.439
192
0.480
0.458
0.479
0.467
0.485
0.472
0.486
0.465
0.485
0.472
0.618
0.722
0.505
0.491
336
0.522
0.513
0.514
0,519
0.521
0.526
0.531
0.531
0.527
0.534
0.669
0.834
0.540
0.547
720
0.595
0.627
0.603
0.658
0.612
0.673
0.638
0.707
0.625
0.695
0.772
1.066 0.618 0.675
Avg.
0.510
0.501
0.509
0.515
0.516
0.522 0.525 0.528 0.520
0.529
0.661
0.822 0.532 0.538
96
0.286
0.173
0.279
0.166
0.288
0.175 0.290 0.175 0.290 0.177
0.315
0.198 0.297 0.189
ETTh2
192
0.314
0.207
0.310
0.204
0.320
0.213 0.324 0.217 0.322 0.216
0.344
0.235 0.333 0.233
336
0.338
0.236
0.337
0.238
0.350
0.250
0.359 0.262 0.348 0,249
0.372
0.273 0.359 0,270
720
0.394
0.281
0.371
0.282
0.400
0.322
0.414
0.340 0.399 0.324
0.420
0.347
0.410 0.353
Avg.
0.333
0.224
0.324
0.222
0.340
0.240
0.347
0.248 0.340
0.241
0.363
0.263
0.350 0.261
96
0.376
0.324
0.379
0.328
0.368
0.315
0.380
0.327
0.370
0.321
0.393
0.342
0.398 0.361
ET1ml
192
0.402
0.368
0.412
0.377
0.401
0.368
0.413
0.378
0.407 0.376
0.425
0.392
0.432 0.416
336
0.433
0.407
0.441
0.419
0.433
0.415
0.448
0.428
0.438
0.420
0.450
0.430
0.462 0.462
720
0.484
0.468
0.486
0.479
0.479
0.481
0.494
0.495
0.489
0.495
0.495
0.493
0.507 0.529
Avg.
0.424
0.392
0.429
0.401
0.420
0.395
0.434
0.407
0.426
0.403
0,441
0.414
0.450 0,442
96
0.226
0.113
0.228
0.118
0.229
0.115
0.228
0.114
0.226
0.114
0.235
0.118
0.240 0.127
BTTm2
192
0.256
0.146
0.263
0.165
0.259
0.148
0.260
0.147
0.258
0.148
0.266
0.149
0.271 0.161
336
0.282
0.175
0.292
0.200
0.286
0.179
0.287 0.180
0.284
0.180
0.298
0.186 0.302 0,200
720
0.326
0.227
0.329
0.248
0.319
0.220 0.322 0.224
0.319
0.220
0.335
0.233 0.335 0.242
Avg.
0.273
0.165
0.278
0.183
0.273
0.166 0.274
0.166
0.272
0.165
0.283
0.172 0.287 0.182
96
0.228
0.132
0.233
0.137
0.232
0.136 0.243 0.143
0.229
0.134
0.280
0.184 0.228 0.132
Electricity
192
0.242
0.149
0.246
0.151
0.247
0.153 0.259 0.161
0.244
0.152
0.297
0.209 0.249 0.153
336
0.259
0.165
0.262
0.167
0.266
0.171 0.276 0.178 0.262 0.169
0.312
0.225 0.263 0.167
720
0.294
0.205
0.293
0.201
0.298
0,209 0.308 0.217 0.295 0.207
0.342
0.264 0.301 0.209
Avg.
0.256
0.163
0.259
0.164
0.261
0.167 0.272 0.175 0.258 0.166
0.308
0.221 0.260 0.165
96
0.214
0.088
0.211
0.092
0.214
0.092 0.214 0.092 0.216 0.094
0.212
0.093 0.242 0.113
Exchange
192
0.315
0.186
0.310
0.194
0.316
0.196 0.317 0.198 0.325
0.208
0.309
0.192 0.356 0.240
336
0.426
0.325
0.434
0.363
0.436
0.362 0.446 0.375 0.447
0.383
0,431
0.354
0.470 0.409
720
0.621
0.621
0.729
0.958
0.717
0.924
0.721 0.949
0.734
1.042
0.737
0.948
0.773 1.084
Avg.
0.394
0.305
0.421
0.402
0.421
0.394
0.424
0.403
0.430
0.432
0.422
0.397
0.460 0.462
96
0.271
0.391
0.279
0.407
0.284
0.400
0.312
0.425
0.273
0.389
0.457
0.691
0.286
0.419
192
0.277
0.407
0.284
0.420
0.293
0.418
0.326 0.450
0.282
0.408
0.485
0.750
0.293 0.439
Traffic
336
0.284
0.420
0.291
0.432
0.301
0.432
0.332 0.464 0.291
0.422
0.661
1.072
0.302
0.457
720
0.300
0.447
0.313
0.463
0.318
0.461
0.345
0.491
0.309
0.451
0.735
1.235
0.322
0.490
Avg.
0.283
0.416
0.292
0.430
0.299
0.428
0.329
0.458
0.289
0.417
0.585
0.937
0.301
0.451
96
0.196
0.145
0.202
0.149
0.200
0.148
0.199
0.155
0.199
0.148
0.201
0.147
0.214
0.163
Weather
192
0.237
0.186
0.245
0.193
0.241
0.191
0.246
0.207
0.241
0.193
0.242
0.191
0.255
0.209
336
0.278
0.238
0.282
0.239
0.281
0.243
0.287 0.258
0.281
0.243
0.282
0.243
0.292
0.260
720
0.332
0.317
0.333
0.308
0.334
0.321
0.340
0.334
0.335
0.326
0.334
0.318
0.342
0.333
Avg.
0.261
0.221
0.266
0.222
0.264
0.226
0.268
0.238
0.264
0.227
0.265
0.225
0.276
0.241
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Slimming the Fat-Tail: Morphing-Flow for Adaptive Time Series Modeling
Table 8. Model Performance Table (cont.)
Model
MoF-Mamba
MoF-Linear
PatchTST
DLinear
Autoformer MAE
Informer MAE
Transformer
Dataset
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MSE
MSE
MAE
MSE
96
0.441
0.406
0.439
0.416
0.476
0.459
0.444
0.436
0.596
0.649
0.834
1.226
0.625
0.699
192
0.480
0.458
0.479
0.467
0.524
0.529
0.483
0.486
0.635
0.735
0.867
1.328
0.671
0.815
336
0.522
0.513
0.514
0.519
0.565
0.596
0.518
0.531
0.690
0.843
0.786
1.121
0.686
0.829
720
0.595
0.627
0.603
0.658
0.653
0.751
0.594
0.645
0.705
0.857
0.864
1.295
0.739
0.970
Avg.
0.510
0.501
0.509
0,515
0.554
0.584
0.510
0.524
0.656
0.771
0.838
1.243
0.680
0.828
96
0.286
0.173
0.279
0.166
0.302
0.189
0.285
0.176
0.367
0.248
0.440
0.362
0.421
0.323
B11h2
192
0.314
0.207
0.310
0.204
0.340
0.235
0.316
0.213
0.407
0.302
0.484
0.439
0.446
0.381
336
0.338
0.236
0.337
0.238
0.372
0.277
0.342
0.236
0.442
0.349
0.478
0.434
0.474
0.399
720
0.394
0.281
0.371
0.282
0.424
0.349
0.386
0.287
0.566
0.598
0.578
0.599
0.607
0.620
Avg.
0.333
0.224
0.324
0.222
0.359
0.263
0.332
0.228
0,446
0.374
0.495
0,458
0,487 0.431
96
0.376
0.324
0.379
0.328
0.385
0.336
0.386
0.355
0.559
0.617
0.558
0.621
0.508
0.523
ETTml
192
0.402
0.368
0.412
0.377
0.425
0.394
0.420
0.421
0.556
0.583
0.609
0.703
0,550 0.585
336
0.433
0.407
0.441
0.419
0.457
0.434
0.456
0.485
0.569
0.609
0.658
0.783
0.614
0.698
720
0.484
0.468
0.486
0.479
0.502
0.502
0.502
0.547
0.606
0.668
0.706
0.872
0.668
0.813
Avg.
0.424
0.392
0.429
0.401
0.442
0.416
0.441
0.452
0.573
0.619
0.633 0.745
0.585
0.655
96
0.226
0.113
0.228
0.118
0.230
0.117
0.234
0.120
0.283
0.163
0.308
0.208
0.288
0.165
ETTm2
192
0.256
0.146
0.263
0.165
0.259
0.147
0.263
0.148
0.298
0.181
0.367
0.292
0.343
0.242
336
0.282
0.175
0.292
0.200
0.290
0.183
0.292
0.181
0.315
0.205
0.404
0.350
0.403
0.308
720
0.326
0.227
0.329
0.248
0.328
0.230
0.333
0.232
0.347
0.247
0.550
0.672
0.481
0.455
Avg.
0.273
0.165
0.278
0.183
0.277
0.169
0.280
0.170
0.311
0.199
0.407
0.381
0.379
0.293
96
0.228
0.132
0.233
0.137
0.226
0.132
0.238
0.140
0.316
0.203
0.418
0.344
0.373
0.283
Electricity
192
0.242
0.149
0.246
0.151
0.242
0.149
0.251
0.154
0.329
0.213
0.433
0.364
0.390
0.304
336
0.259
0.165
0.262
0.167
0.260
0.166
0.268
0.169
0.334
0.219
0.433
0.367
0.379
0.292
720
0.294
0.205
0.293
0.201
0.295
0.207
0.301
0.204
0.352
0.247
0.415
0.342
0.374
0.287
Avg.
0.256
0.163
0.259
0.164
0.256
0.164
0.265
0.167
0.333
0.221
0.425
0.354
0.379
0.291
96
0.214
0.088
0.211
0.092
0.260
0.198
0.209
0.084
0.629
0.660
0.898
1.159
0.835
1.091
Exchange
192
0.315
0.186
0.310
0.194
0.372
0.321
0.307
0.171
0.900
1.274
0.973
1.356
0.912
1.268
336
0.426
0.325
0.434
0.363
0.512
0.581
0.435
0.332
0.916
1.313
1.049
1.635
0.994
1.505
720
0.621
0.621
0.729
0.958
0.749
1.074
0.743
0.971
0.947
1.547
1.315
2.553
0.812
1.030
Avg.
0.394
0.305
0.402 0.421
0.473
0.544
0.423
0.390
0.848
1.198
1.676 1.059
0.888
1.224
96
0.271
0.391
0.279
0.407
0.268
0.390
0.285
0.411
0.384
0.637
0.445
0.793
0.382
0.702
Traffic
192
0.277
0.407
0.284
0.420
0.274
0.407
0.290
0.424
0.397
0.660
0.416
0.744
0.382
0.696
336
0.284
0.420
0.291
0.432
0.283
0.419
0.298
0.437
0.377
0.628
0.488
0.894
0.369
0.686
720
0.300
0.447
0.313
0.463
0.299
0.448
0.317
0.467
0.406
0.668
0.450
0.815
0.364
0.681
Avg.
0.283
0.416
0.292
0.430
0.281
0.416
0.297
0.435
0.391
0.648
0.450
0.811
0.374
0.691
96
0.196
0.145
0.202
0.149
0.200
0.151
0.235
0.174
0.365
0.290
0.357
0.290
0.344
0.285
Weather
192
0.237
0.186
0.245
0.193
0.245
0.198
0.276
0.217
0.399
0.342
0.400
0.356
0.460
0.445
336
0.278
0.238
0.282
0.239
0.290
0.257
0.316
0.263
0.422
0.376
0.563
0.621
0.545
0.598
720
0.332
0.317
0.333
0.308
0.341
0.330
0.363
0.325
0,445
0.438
0.666
0.850
0.604
0.694
Avg.
0.261
0.221
0.266
0.222
0.269
0.234
0.297
0.245
0.408
0.361
0.496
0.529
0.488
0.505
21

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Slimming the Fat-Tail: Morphing-Flow for Adaptive Time Series Modeling
E.2. Mann-Whitney U Test
Since each experiment is repeated five times and we cannot assume the normality of the result distributions, we conduct a one-tailed Mann- Hann-Whitneyy U test for each model against the best-performnng model in that configuration across 10 models. The p-values of the tests are presented in Table 9, where p-values greater than or equal to 0.95 are highlighted in bold, and p-values greater than 0.05 are highlighted in blue.
Interestingly, when tested on these models, the distribution of p-values across different datasets reveals that the ETTm datasets has the poorest discriminatory ability, followed by the Exchange dataset, while the ETTh scries, Traffic, and Weather datasets demonstrate better discriminatory ability.
Due to space limitations, the experiment results presented across two tables.
Table 9. Mann- Maan-Whitmcy U test
Model
MoF-Mamba
MoF-Linear
GLinear
LiNo
TimeMachine
CATS
iTransformer
Dataset
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
96
0.238
1.000
1.000
0.005
0.003
0.001
0.005 0.008 0.003 0.001
0.005
0.001 0.008
0.002
ETThl
192
0.393
1.000
1.000
0.018
0.029
0.008 0.029
0.016 0.012 0.002
0.018
0.004
0.018
0.004
336
0.200
1.000
1.000
0.114
0.029
0.014 0.018
0.008 0.012 0.005
0.018
0.008
0.018
0.008
720
0.393
1.000
0.018
0.050
0.008
0.029 0.004 0.018 0.002 0.012
0.004
0.018 0.004
0.018
96
0.029
0.029
1.000
1.000
0.003
0.003 0.003
0.003 0.003 0.003
0.005
0.005 0.005
0.005
ETTh2
192
0.029
0.029
1.000
1.000
0.029
0.029 0.029
0.029 0.012 0.012
0.018
0.018 0.012
0.012
336
0.350
1.000
1.000
0.200
0.029
0.029 0.018 0.018 0.012 0.012
0.018
0.018 0.012 0.012
720
0.050
1.000
1.000
0.500
0.029
0.029 0.018 0.018 0.012 0.012
0.018
0.018 0.018 0.018
96
0.002
0.002
0.005
0.005
1.000
1.000
0.001
0.001
0.011 0.001
0.002
0.002 0.001 0.001
ETTml
192
0.171
1.000
0.029
0.029
1.000
0.243
0.014
0.014
0.005 0.005
0.008
0.008 0.005 0.005
336
1.000
1.000
0.050
0.050
0.314
0.029
0.018
0.018
0.012 0.012
0.018
0.018 0.012 0.012
720
0.114
1.000
0.114
0.050
1.000
0.029
0.008
0.018
0.019 0.012
0.014
0.029 0.005 0.012
96
1.000
1.000
0.243
0.029
0.143
0.143
0.129
0.305
0.842
0.324
0.008
0.008 0.005 0.005
BTTm2
192
1.000
1.000
0.018
0.018
0.032
0.056
0.032
0.365
0.214
0.089
0.004 0.016 0.002 0.002
336
1.000
1.000
0.100
0.100
0.067
0.067
0.048
0.048
0.143
0.036
0.048 0.048 0.036 0.036
720
0.067
0.600
0.029
0.029
1.000
1.000
0.032
0.056
0.381
0.381
0.014 0.014 0.005 0.005
Exchange Electricity
96
0.083
0.083
0.012
0.012
0.005
0.005
0.005
0.005
0.001
0.001
0.002 0.002 0.004 0.242
192
0.571
0.571
0.012
0.012
0.005
0.005
0.036
0.036
0.001
0.001
0.002
0.002 0.001 0.001
336
1.000
1.000
0.100
0.100
0.067
0.067 0.048
0.048
0.036
0.036
0.048
0.048 0.036 0.036
720
0.600
0.100
1.000
1.000
0.050
0,050
0,018 0,018
0.018
0.018
0.029
0.029 0.012 0.012
96
0.038
0.020
0.152
0.001
0.011
0.000
0.021
0.000
0.005 0.000
0.155
0.001 0.000 0.000
192
0.030
0.011
0.261
0.002
0,011
0.000
0.030
0.000
0.002 0.000
0.168
0.000
0.000 0.000
336
1.000
1.000
0.125
0.018
0.095
0.008
0.028
0.004
0.015 0.002
0.274
0.004
0.002 0.002
720
1.000
1.000
0.029 0.029
0.014
0.014
0.008
0.008
0.005
0.005
0.014
0.014 0.005 0.005
96
0.001
0.002
0.003
0.005
0.001
0.002
0.001
0.002
0.001
1.000
0.001
0.002 0.000 0.000
Traffic
192
0.005
1.000
0.012
0.029
0.005
0.014
0.012
0.029
0.001
0.129
0.002
0.008 0.001 0.003
336
0.048
0.286
0.018
0.018
0.008
0.008
0.006 0.006
0,002
0.002
0.004
0.004 0.008 0.008
720
0.267
1.000
0.029
0.100
0.014
0.067
0.008 0.048
0.008
0.048
0.014
0.067 0.029 0.100
96
1.000
1.000
0.029
0.029
0.008
0.008
0.012 0.012 0.008
0.008
0.012
0.012 0.018 0.018
Weather
192
1.000
1.000
0.050
0.050
0.029
0.029
0.029 0.029 0.012 0.012
0.018
0.018 0.018 0.018
336
1.000
1.000
0.100
0.600
0.067
0.067
0.048
0.048
0.036 0.036
0.048
0.048 0.036 0.036
720
1.000
0.100
0.400
1.000
0.600
0.029
0.048
0.018
0.071 0.012
0.067
0.029 0.036 0.012
22

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Slimming the Fat-Tail: Morphing-Flow for Adaptive Time Series Modeling
Table 10. Mann-Whitney test (cont.)
Model
MoF-Mamba
MoF-Linear
PStchTST
DLinear
Autoformer
Informer
Transformer
Dataset
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
96
0.238
1.000
1.000
0.005
0.001
0.000
0.008
0.002
0.008
0.002
0.005
0.001
0.005 0.001
E11h1
192
0.393
1.000
1.000
0.018
0.002
0.000
0.018
0.004
0.018
0.004
0.012
0.002
0.012 0.002
336
0.200
1.000
1.000
0.114
0.002
0.001
0.036
0.008
0.018
0.008
0.012
0.005
0.012 0.005
720
0.393
1.000
0.018
0.050
0.000
0.002
1.000
0.018
0.004
0.018
0.002
0.012
0.002 0.012
96
0.029
0.029
1.000
1.000
0.001
0.001
0.008
0.008
0.014
0.014
0.008
0.008
0.008 0.008
BTTh2
192
0.029
0.029
1.000
1.000
0.003
0.003
0.018
0.018
0.018
0.018
0.018
0.018
0.012 0.012
336
0.350
1.000
1.000
0.200
0.003
0.003
0.018
0.286
0.018
0.018
0.018
0.018
0.012 0.012
720
0.050
1.000
1.000
0.500
0.003
0.003
0.018
0.018
0.018
0.018
0.018
0.018
0.012 0.012
96
0.002
0.002
0.005
0.005
0.000
0.000
0.002
0.002
0.002
0.002
0.001
0.001
0.001 0.001
BTTm2 ETTml
192
0.171
1.000
0.029
0.029
0.002
0.002
0.008
0.008
0.008
0.008
0.005
0.005
0.005 0.005
336
1.000
1.000
0.050
0.050
0.006
0.006
0.018
0.018
0.029
0.029
0.012
0.012
0.018
0.018
720
0.114
1.000
0.114
0.050
0.002
0.006
0.008
0.018
0.014
0.029
0.005
0.012
0.008
0.018
96
1.000
1.000
0.243
0.029
0.129
0.019
0.008
0.008
0.014
0.014
0.008
0.008
0.008
0.008
192
1.000
1.000
0.018
0.018
0.026
0.214
0.004
0.016
0.008
0.008
0.004
0.004
0.008
0.008
336
1.000
1.000
0.100 0.100
0.036
0.036
0.048
0.048
0.067
0.067
0.048
0.048
0.067 0.067
720
0.067
0.600
0.029 0.029
0.010
0.005
0.008
0.008
0.029
0.029
0.008
0.008
0.014 0.014
Electricity
96
0.083
0.083
0.012
0.012
1.000
1.000
0.001
0.001
0.036
0.036
0.002
0.002
0.005
0.005
192
0.571
0.571
0.012
0.012
1.000
1.000
0.001
0.001
0.036
0.036
0.002
0.002
0.005 0.005
336
1.000
1.000
0.100
0.000
0.679
0.679
0.048
0.048
0.167
0.167
0.048
0.048
0.067 0.067
720
0.600
0.100
1.000 1.000
0.886
0.200
0.029
0.029
0.100
0.100
0.018
0.018
0.029 0.029
Exchange
96
0.038
0.020
0.152
0.001
0.000
0,000
1.000
1.000
0.001
0.001
0.000
0.000
0.000
0.000
192
0.030
0.011
0.261
0.002
0.000
0.000
1.000
1.000
0.000
0.000
0.000
0.000
0.000 0.000
336
1.000
1.000
0.125
0.018
0.002
0.002
0.255
0.543
0.008
0.008
0.004
0.004
0.004 0.004
720
1.000
1.000
0.029 0.029
0.005
0.005
0.000
0.000
0.014
0.014
0.008
0.008
0.008 0.008
96
0.001
0.002
0.003
0.005
1.000
0.026
0.001
0.001
0.008
0.012
0.001
0.002
0.001 0.002
Traffic
192
0.005
1.000
0.012
0.029
1.000
0.129
0.001
0.005
0.036
0.067
0.002
0.008
0.002 0.008
336
0.048
0.286
0.018
0.018
1.000
1.000
0.002
0.002
0.018
0.018
0.004
0.004
0.004 0.004
720
0.267
1.000
0.029
0.100
1.000
0.267
0.008
0.048
0,067
0.167
0.008
0.048
0,008
0.048
96
1.000
1.000
0.029
0.029
0.003
0.003
0.002
0.002
0,018
0.018
0.012
0.012
0.012 0.012
Weather
192
1.000
1.000
0.050
0.050
0.003
0.003
0.003
0.003
0.018
0.018
0.012
0.012
0.012 0.012
336
1.000
1.000
0.100
0.600
0.013
0.013
0.013
0.013
0.048
0.048
0.036
0.036
0.036
0.036
720
1.000
0.100
0.400
1.000
0.013
0.003
0.028
0.008
0.048
0.018
0.036
0.012
0.036 0.012
23

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Slimming the Fat-Tail: Morphing-Floon for Adaptive Time Series Modeling
F. MoF Visualization
Quantile Quantile-Quantiee Flot (Original vs. Gaussian)
Quantile
Stationary (Mean-Removed)
MoF Transfomation Curve (Spline Knets, Axis Skewed Clarity)
Quantile Quantlic Flot (Trans Gaussian)
Kesaur
Yesaui
Fecusr
Charmel
-2
-2
Kesaur
Keuaur
Fecaur 5.900
Channel 1
-2
Keeosi -0.43
Kemus 2.51
Keenu 0.29
Charnel2
-2
-2
-2
Figure Visualization transformation process on three seperate dimension of ETThl dataset. After stationarization, the average Kercess is 2.34. The MoF transformation reduces the average Kencess 0.49, bringing the data distribution 79. 1% closer to normality.
Quantile-Quantie Plot (Original Gaussian)
Stationary (Mean Removed)
MoF Transfomation Curve (Spline Knots, Ald Skewed for Clarity)
Quantile-Quantil Plot (Transtormed vs. Gaussian)
Kesaur 2.62
Kesau: 1.09
Kecau: 0.18
Charmel 0
2
Keuar:
Channel 1
-2
Kesuri 9.86
Keeuri 4.50
Kecus: 1.61
Channel
11
2
Figure 14. Visualization of the transformation process on three seperate dimension of the ETTml dataset. After stationarization, the average Kexcess 2.50. The MoF transformation reduces the average absolute K Kencess 0.67, bringing the data distribution 73.1% closer to normality.
24

---

Slimming the Fat-Tail: Morphing-Flow for Adaptive Time Series Modeling
Quantile-Quante Quanbile Plot (Original Coussian)
Stationary (Mcan-Romoved)
Mof Transfomation Curve (Spline Knots, Asds Skowed Clarity)
Quantile-Qunttle Quantlle Piot (Transformed Gaussian)
Keervi
Keeeol
Chantrel
-2
-2
22
2
Kesuui 9.61
Kexus 12.50
Fecesi 1.42
Channel
Kenu
Charmel
Figure Visualization transformation process on three seperate dimension of ETTh2 dataset. After stationarization, the average Kezeuss is 22.63. The MoF transformation reduces the average absolute Kezonsa to 1.50, bringing the data distribution 93.3% closer to normality.
Quantile-Quantle Plot (Original Gaussian)
Stationary (Mean Removed)
MoF Transfomation Curve (Spline Knots, Alds Skewed for Clarity)
Quantile-Quanti Piot (Transtormed vs. Gaussian)
Kesaur 9:16
Kesar 3:57
Kecau: 3.05
Charmel
Kenre
Kearsl
Kracul
Channel 1
20.02
Keuus 147.28
2.50
Channel 2
Figure 16. Visualization of the transformation process three seperate dimension of the ETTm2 dataset. After stationarization, the average Kezeess 54.12. The MoF transformation reduces the average absolute Kezcesa to 1.27, bringing the data distribution 97.6% closer to normality.
25

---

Slimming the Fat-Tail: Morphing-Flow for Adaptive Time Series Modeling
Quantile-Quanie Quannile Plot (Original Coussian)
Stationary (Mcan-Romoved)
Mot Transfomation Curve (Spline Knots, Asds Skowed Clarity)
Quantile- Quantlle Plot (Transformed Gaussian)
Keevvl
Charnel O
Kesari
Keuast
Channel 1
-2
-2
Keaui 5.47
Feans
Axami
Channel
Figure Visualization the transformation process on separate dimension Electricity dataset. After stationarization, the average Kereess is 3.91. The MoF transformation reduces the average absolute K Kercess to 1.18, bringing the data distribution 69.7% closer to normality.
Quantile-Quantie Plot (Original Gaussian)
Stationary (Mean Removed)
MoF Transfomation Curve [Spline Knots, Alds Skewed for Clarity)
Quantile-Quanti Piot (Transtormed vs. Gaussian)
Kusuu: 8:53
Kacoer 3.7
Kamuri 0.70
Charnel 0
0.0
-0.5
-2
-2
-1.0
Keuuci
Channel 1
-2
Nacaat 2.54
Kacou 350
Neuuci -0.14
Channel 2
-2
Figure 18. Visualization of the transformation process on three separate dimension of the Weather dataset. After stationarization, the average Kencess 2.49. The MoF transformation reduces the average absolute K Kencess 0.48, bringing the data distribution 82.3% closer to normality.
26

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Slimming the Fat-Tail: Morphing-Flow for Adaptive Time Series Modeling
Quantile-Quamin Quannile Plot (Original Coussian)
Stationary (Mcan-Romoved)
Mot Transfomation Curve (Spline Knots, Asds Skowed Clarity)
Quantile-Qunttle Quantlle Piot (Transformed Gaussian)
Channel O
Kesas 4.27
4.00
Channel 1
Kesaui 5.08
Kesur 5.50
Kecur: 14.12
Charmel
Figure Visualization of transformation process on three separate dimension the NIL dataset. After stationarization, the average Kezeess is 5.52. The MoF transformation reduces the average absolute Kezessa to 7.58, bringing the data distribution 37.4% further to normality.Dut to limited data size, MoF wasn't able train sufficiently this dataset
Quantile Quantile Plot (Original Caussian)
Stationary (Mean-Removed))
MoF Transfomation Curve [Spline Knots, Asis Skowed Clarity)
Quantile Quantile Piot (Transformed Gaussian)
Keoaui
Kecasi
Kecuu
Chanire
-2
-2
-2
-2
2
Kesaur 0.88
Keour 4.52
Fecurl 0.37
Channel 1
Keeusl
Keersi
Krewol
Channel 2
22
-2
-2
-2
Figure Visualization the transformation process on three separate dimension of the Exchange dataset. After stationarization, the average Kereeas is 3.88. The MoF transformation reduces the average absolute Kencuss to 0.57, bringing the data distribution 85.3% closer to normality.
27

---

Slimming the Fat-Tail, Morphing-Flon for Adaptive Time Series Modeling
Quantile-Qual Quaililleeeeleeeeeeeeeee Plot (Original vs. Goussian)
Stationary Recan-)e
Mol Transformtionn Curve 1Spline Knots, Ards Skewed for Clarity)
Quantile-Quantle Quantile Piot ITranstormed vs. Gaussian)
Quantile
Kineri 16.28
Kenri 16:35
Kremsi 12003
Channdl O
-2
-2
22
2
2
Keuas 4:035
Kenai 1.88
Fenmu 5, 1.52
Channel
-1
-2
-3
-2
Kenii 3.92
Kenri 3.59
Krami 2.3
Channel
-2
-2
-2
Figure 21. Visualization of the transformation process on three separate dimension of the Traffic dataset. After stationarization, the average Kercess is 8.21. The MoF transformation reduces the average absolute K Kerceas 6.69, bringing the data distribution 18.5% closer to normality.
28

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Slimming the Fat-Tail: Morphing-Flow for Adaptive Time Series Modeling
G. Model Efficiency Comparison
This section provides detailed explanation the columns and metrics presented table below, comparing our proposed model, MoF, with two backbones (Mamba-64 and Linear), against several state-of-the-art baselines.
The number of floating-pont point operations in billions, measured using the Evcore package. Note that this value is not entirely exclusive and auxiliary computations may occur.
FLOPs:
Time (ms): The running time per 100 iterations milliseconds, reflecting the practical efficiency of the models.
Params: The total number of trainable parameters in millions, also computed with fvcore.
The average squared emoor (MSE) and mean absolute error (MAE), computed as follows: each experiment is repeated five times, and the average result recorded. For comparison, the MSE/MAE is normalized against the best-performing model in each dataset, which typically corresponds to the shortest prediction length (due to lower error).
(el.atelat MSE and MAE:
FLOPs Ratio and Param Ratio: These columns show the ratio of FLOPs and parameter counts relative MoF with the Linear backhone, serving baseline.
The percentage improvement or decline in performance compared to the baseline model, MoF with the Linear backhone.
Perf. Boost:
Time (ms)
Rel. MSE
Rel. MAE
FLOPs Ratio
Param Ratio
Perf. Boost
Model
FLOPs
Params
MoF(Mamba664)
1,02E+09
1173
1.46E+06
1.563
1.248
10.37
8.74
6.48%
MoF(Linear)
9.86E+07
712
1.66E+05
1.733
1,273
1.00
1.00
0.00%
Autoformer
9.22E+09
1784
6.49E+06
3.403
1.860
93.46
39.02
-75.06%
Transformer
1.10E+10
962
6.29E+06
3.819
2.019
111.25
37.80
-94.20%
Informer
8.62E+09
1269
7.08E+06
4.862
2.263
87.38
42.53
-37.005%
DLinear
1.26E+08
163
3.40F+05
1.752
1.310
1.27
2.04
-1.86%
TTransformer
2.17E+09
497
3,50E+06
1.893
1.337
22.01
21.04
-7.45%
PatchTST
7.99E+10
488
1.04E+07
1.980
1.331
810.17
62.57
-10.14%
GLinear
1.26E+08
185
2.27E+05
1.719
1.279
1.27
1.36
0.24%
CATS
2.38E+10
360
1.53E+06
2.059
1.501
241.57
9.21
-18.45%
LiNo
2.60E+09
486
4.29E+06
1.775
1.315
26.34
25.79
-2.80%
TimeMachine
2.71E+08
823
8.79E+05
1.777
1.281
2.75
5.28
-1.74%
Performance Analysis. The results demonstrate that our proposed model, MoF, achieves competitive or superior perfor- mance compared to state-of-the-arn models across various metrics. Notably, MoF consistently delivers strong results in Avg. MSE and Avg. MAE*, despite having significantly smaller parameter counts. However, it is important to note that the Mamba-64 backbone introduces binning process that negatively impacts running time, reflected in the Time (ms) column.
H. Comprehensive Comparison of MoF with Expanded Input Horizon
29

---

Slimming the Fat-Tail: Morphing-Flow for Adaptive Time Series Modeling
0.30
0.40
Aeerage MAE
MAE
0358
033
0.36
0.32
0.34
152
356
720
560
40
1.92
236
720
500
192
330
720
900
26
102
336
720
900
Inpur Seng uence Length
Input Sequennce Length
Nof
FAN
Reuli
SAN
DinnS
Autofomer
Informar
-
i'ranaoomer
NoP Linear
(a) Different normalization method compared on weather datasct with prediction length fixed at 720.
(b) Different backbone method compared on weather dataset with prediction length fixed at 192.
0.32
Average KAE
Averpp MAE
0.30
0.22
0:25
0.24
152
Naf
FAN
ReefN
SAN
Dints
Auteforme:
Informe
-
l'Handammer
PrteNTST
NoT-Linwe
(c) Different normalization method compared weather
(d) Different backbone method compared on weather dataset with prediction length fixed 720.
dataset with prediction length fixed 192.
5.20
0.336
Average MAEE
0,224
0.328
0.220
6312
152
720
560
48
335
720
95
392
316
720
900
40
20
132
330
720
980
Inpur
Sequennce Length
Input Sequence Langtn
TAN4
Rev
DieNTS
Aunafomer
Informer
-
[Esnntommer
PrteNTST
NoF-1 Iner
(e) Different normalization method compared on ETTh2 dataset with prediction length fixed at 192.
(I) Different backbone method compared on ETTh2 dataset with prediction length fixed at 720.
0.00
Average MekE
0.30
0.39
Averge MAE
0.48
0.38
0.28
0.37
150
720
510
335
720
132
950
56
192
460
Miput Sequenccs Length
Input Sequenoc Langth
NoF
FAN
Revli
SAN
DENTTS
Autofomer
Informe
-
iTaondomerr
PoocoTST
NDF Linar
Different normalization method compared on ETTh2 dataset with prediction length fixed at 720.
(g)
(h) Different backbone method compared on ETTh2 dataset with prediction length fixed at 192.
Figure Comparison of different normalization and backbone methods on ETTh2 weather datasets prediction lengths fixed 192 and 720.
30

---

Slimming the Fat-Tail: Morphing-Flow for Adaptive Time Series Modeling
I. Comprehensive Comparison of MoF and Other Normalization Methods
MoF applies simple instance-wise z-score normalization to address non-stationary distributions. All methods are tested with the same parameters across datasets and settings, without dataset-specific tuning.
Model
MoF
RevIN
FAN
DishTS
SAN
DLinear
Dataset
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAB
96
0.415
0.438
0.418
0.439
0.414
0.445
0.421
0.444
0.426
0.448
0.435
0.444
ETTh1
192
0.466
0.478
0.468
0.476
0.467
0.483
0.465
0.478
0.477
0.484
0.486
0.483
336
0.518
0.514
0.526
0.516
0.515
0.521
0.526
0.523
0.539
0.524
0.531
0.518
720
0.657
0.602
0.667
0.604
0.634
0.598
0.656
0.608
0.676
0.608
0.644
0,594
Avg.
0.514
0,508
0.520
0.509
0.508
0.512
0.517
0.513
0.529
0.516
0.524
0.510
96
0.165
0.278
0.165
0.277
0.173
0.288
0.165
0.279
0.172
0.285
0.176
0.285
ETTh2
192
0.203
0.310
0.206
0.313
0.211
0.318
0.195
0.304
0.207
0.312
0.212
0.315
336
0.238
0.337
0.240
0.341
0.237
0.341
0.222
0.327
0.232
0.335
0.236
0.342
720
0.281
0.371
0.317
0.394
0.279
0.382
0.261
0.356
0.288
0.386
0.286
0.386
Avg.
0.222
0.324
0.232
0.331
0.225
0.332
0.211
0.316
0.225
0.329
0.228
0.332
96
0.328
0.379
0.333
0.373
0.319
0.370
0.325
0.368
0.328
0.384
0.355
0.385
ET1m]
192
0.376
0.412
0.385
0.404
0.368
0.404
0.377
0.400
0.373
0.412
0.420
0.419
336
0.418
0.441
0.435
0.436
0.416
0.438
0.422
0.432
0.418
0.447
0.484
0.455
720
0.478
0.485
0.497
0.479
0.467
0.479
0.478
0.473
0.474
0.483
0.547
0.501
Avg.
0.400
0.429
0.412
0.423
0.393
0.423
0.401
0.418
0.398
0.432
0.452
0.440
96
0.118
0.228
0.113
0.226
0.117
0.230
0.112
0.225
0.114
0.224
0.120
0.234
ETTm2
192
0.164
0.263
0.145
0,256
0.148
0,259
0.140
0.251
0.148
0.256
0.148
0,262
336
0.200
0.292
0.171
0.277
0.181
0.287
0.169
0.277
0.175
0.282
0.180
0.292
720
0.248
0.329
0,219
0.315
0.219
0.321
0.210
0.309
0.215
0.317
0.232
0.333
Avg.
0.182
0.278
0.162
0.268
0.166
0.274
0.158
0.265
0.163
0.270
0.170
0,280
Electricity
96
0.137
0.232
0.140
0.236
0.139
0.238
0.138
0.235
0.134
0.231
0.140
0.237
192
0.151
0.245
0.154
0.248
0.156
0.255
0.154
0.250
0.150
0.246
0.153
0.250
336
0.166
0.261
0.171
0.265
0.174
0.274
0.170
0.268
0.166
0.265
0.169
0,268
720
0.201
0.293
0.210
0.297
0.212
0.310
0.207
0.303
0.201
0.298
0.203
0.301
Avg.
0.164
0.258
0.169
0.261
0.170
0.269
0.167
0.264
0.163
0.260
0.166
0.264
Exchange
96
0.092
0.211
0.090
0,210
0.136
0.274
0.126
0.278
0.123
0.249
0.084
0.209
192
0.194
0.310
0.188
0.307
0.252
0.374
0.268
0.389
0.193
0.323
0.171
0.306
336
0.363
0.433
0.346
0.425
0.594
0.589
0.445
0.545
0.353
0.440
0.332
0.435
720
0.958
0.728
0.915
0.715
1.306
0.847
1.904
1.084
1.201
0.775
0.970
0.742
Avg.
0.401
0.421
0.385
0.414
0.572
0.521
0.686
0.574
0.467
0.447
0.389
0.423
96
0.407
0.279
0.411
0.281
0.411
0.290
0.423
0.295
0.417
0.283
0.411
0.284
Traffic
192
0.420
0.283
0.424
0.286
0.428
0.300
0.437
0.302
0.435
0.290
0.423
0.290
336
0.431
0.291
0.437
0.293
0.446
0.312
0.451
0.310
0.451
0.298
0.436
0.297
720
0.462
0.313
0.465
0.309
0.481
0.333
0.482
0.329
0.481
0.314
0.466
0.317
Avg.
0.430
0.291
0.434
0.292
0.441
0.309
0.448
0.309
0.446
0.296
0.434
0.297
96
0.148
0.202
0.173
0.222
0.153
0.210
0.166
0.240
0.146
0.208
0.174
0.234
Weather
192
0.193
0.244
0.216
0.258
0.197
0.257
0.207
0.278
0.192
0.254
0.216
0.276
336
0.239
0.281
0.263
0.292
0.246
0.294
0.258
0.323
0.241
0.292
0.263
0.315
720
0.308
0.333
0.331
0.339
0.314
0.344
0.330
0.382
0.318
0.348
0.324
0.363
Avg.
0.222
0.265
0.246
0.278
0.228
0.276
0.240
0.305
0.224
0.275
0.244
0.297
(ALL)
0.317
0.347
0.320
0.348
0.338
0.365
0.371
0.354
0.327
0.354
0.356
0.326
(%)
0.00
0.00
-0.94
-0.12
-6.55
-5.15
-6.93
-11.48
-3.15
-1.86
-2.50
-2.81
31

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Slimming the Fat-Tail: Morphing-Flow for Adaptive Time Series Modeling
J. Module generalization test
Model
Autoformer
Informer
DLinear
w/o MoF
w/ MoF
w/o MoF
w/ MoF
w/o MoF
w/ MoF
Dataset
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
96
0.596
0.649
0.568
0.631
0.834
1.226
0.606
0.716
0.444
0.436
0.439
0.416
ETThi
192
0.635
0.735
0.576
0.632
0.867
1.328
0.648
0.782
0.483
0.486
0.479
0.467
336
0.690
0.843
0.623
0.734
0.786
1.121
0.663
0.797
0.518
0.531
0.514
0.519
720
0.705
0.857
0.715
0.875
0.864
1.295
0.763
1.096
0.594
0.645
0.603
0.658
Avg.
0.656
0.771
0.620
0.718
0.838
1.243
0.670
0.848
0.510
0.524
0.509
0,515
96
0.367
0.248
0.357
0.255
0.440
0.362
0.372
0.291
0.285
0.176
0.279
0.166
ETTh2
192
0.407
0.302
0.380
0.275
0.484
0.439
0.400
0.319
0.316
0.213
0.310
0.204
336
0.442
0.349
0.389
0.302
0.478
0.434
0.434
0.382
0.342
0.236
0.337
0.238
720
0.566
0.598
0.410
0.322
0.578
0.599
0.417
0.352
0.386
0.287
0.371
0.282
Avg.
0.446
0.374
0.384
0.289
0.495
0.458
0.406
0.336
0.332
0.228
0.324
0.222
96
0.559
0.617
0.533
0.570
0.558
0.621
0.493
0.558
0.386
0.355
0.379
0.328
ETTml
192
0.556
0.583
0.557
0.636
0.609
0.703
0.521
0.608
0.420
0,421
0.412
0.377
336
0.569
0.609
0.550
0.595
0.658
0.783
0.558
0.631
0,456
0.485
0.441
0.419
720
0.606
0.668
0.576
0.631
0.706
0.872
0.577
0.687
0.502
0.547
0.486
0.479
Avg.
0.573
0.619
0.554
0.608
0.633
0.745
0.537
0.621
0.441
0,452
0.429
0.401
96
0.283
0.163
0.285
0.167
0.308
0.208
0.264
0.153
0.234
0.120
0.228
0.118
ETTm2
192
0.298
0.181
0.304
0.196
0.367
0.292
0.310
0.208
0.263
0.148
0.263
0.165
336
0.315
0.205
0.323
0.219
0.404
0.350
0.336
0.247
0.292
0.181
0.292
0.200
720
0.347
0.247
0.346
0.248
0.550
0.672
0.377
0.306
0.333
0.232
0.329
0.248
Avg.
0.311
0.199
0.315
0.207
0.407
0.381
0.322
0.228
0.280
0.170
0.278
0.183
Electricity
96
0.316
0.203
0.293
0.192
0.418
0.344
0.311
0.211
0.238
0.140
0.233
0.137
192
0.329
0.213
0.296
0.192
0.433
0.364
0.323
0.226
0.251
0.154
0.246
0.151
336
0.334
0.219
0.302
0.196
0.433
0.367
0.329
0.231
0.268
0.169
0.262
0.167
720
0.352
0.247
0.319
0.219
0.415
0.342
0.358
0.271
0.301
0.204
0.293
0.201
Avg.
0.333
0.221
0.302
0.200
0.425
0.354
0.330
0.235
0.265
0.167
0.259
0.164
Exchange
96
0.629
0.660
0.476
0.424
0.898
1.159
0.403
0.307
0.209
0.084
0.211
0.092
192
0.900
1.274
0.494
0.456
0.973
1.356
0.510
0.523
0.307
0.171
0.310
0.194
336
0.916
1.313
0.599
0.667
1.049
1.635
0.642
0.733
0.435
0.332
0.434
0.363
720
0.947
1.547
0.828
1.213
1.315
2.553
0.789
1.212
0.743
0.971
0.729
0.958
Avg.
0.848
1.198
0.599
0.690
1.059
1.676
0.586
0.694
0,423
0.390
0.421
0,402
24
1.338
3.716
1.377
3.646
1.518
4.964
1.166
3.173
1.322
3.554
1.072
2.623
NIL
36
1.235
3.273
1.373
3.701
1.495
4.816
1.220
3.584
1.336
3.680
1.053
2.567
48
1.212
3.244
1.322
3.597
1.522
4.875
1.190
3.349
1,356
3.808
1.076
2.660
60
1.200
3.290
1.167
3.003
1.610
5.397
1.252
3.714
1.413
4.206
1.099
2.780
Avg.
1.246
3.381
1.310
3.487
1.536
5.013
1.207
3.455
1.357
3.812
1.075
2.658
96
0.384
0.637
0.367
0.671
0.445
0.793
0.402
0.723
0.285
0.411
0.279
0.407
Traffic
192
0.397
0.660
0.379
0.692
0.416
0.744
0.382
0.691
0.290
0.424
0.284
0.420
336
0.377
0.628
0.364
0.672
0.488
0.894
0.413
0.747
0.298
0.437
0.291
0.432
720
0,406
0.668
0.369
0.674
0.450
0.815
0.441
0.804
0.317
0.467
0.313
0.463
Avg.
0.391
0.648
0.370
0.677
0.450
0.811
0.410
0.741
0.297
0.435
0.292
0.430
96
0.365
0.290
0.290
0.246
0.357
0.290
0.273
0.224
0.235
0.174
0.202
0.149
Weather
192
0.399
0.342
0.331
0.305
0.400
0.356
0.327
0.346
0.276
0.217
0.245
0.193
336
0.422
0.376
0.343
0.321
0.563
0.621
0.360
0.435
0.316
0.263
0.282
0.239
720
0.445
0.438
0.368
0.380
0.666
0.850
0.385
0.420
0.363
0.325
0.333
0.308
Avg.
0.408
0.361
0.333
0.313
0.496
0.529
0.336
0.356
0.297
0.245
0.266
0.222
32

---

Slimming the Fat-Tail: Morphing-Floon for Adaptive Time Series Modeling
K. Comparison with IN-Flow and FITS
Model
MoF-Linear
MoF-Mamba
INFloW-Linear
INFlow-
Patch
FITS
Len
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
96
0.441
0.406
0.439
0.416
0.440
0.417
0.451
0.421
0.682
0.909
ETThl
192
0.480
0.458
0.479
0.467
0.477
0.468
0.489
0.482
0.696
0.937
336
0.522
0.513
0.514
0.519
0.518
0.528
0.535
0.550
0.718
0.979
720
0.595
0.627
0.603
0,658
0.607
0.670
0.633
0.709
0.777
1.096
Avg.
0,510
0.501
0.509
0.515
0.510
0.521
0.527
0,540
0.718
0.980
96
0.286
0.173
0.279
0.166
0.278
0.165
0.292
0.181
0.350
0.246
ETTh2
192
0.314
0.207
0.310
0.204
0.311
0.203
0.325
0.217
0.366
0.268
336
0.338
0.236
0.337
0.238
0.343
0.243
0.358
0.263
0.385
0.295
720
0.394
0.281
0.371
0.282
0.394
0.318
0.414
0.337
0.418
0.348
Avg.
0.333
0.224
0.324
0.222
0.332
0.232
0.347
0.249
0.380
0.289
96
0.376
0.324
0.379
0.328
0.372
0.331
0.383
0.335
0.649
0.852
ETTml
192
0.402
0.368
0.412
0.377
0.405
0.385
0.419
0.383
0.660
0.871
336
0.433
0.407
0.441
0.419
0.436
0.434
0.447
0.421
0.676
0.898
720
0.484
0.468
0.486
0.479
0.480
0.497
0.494
0.487
0.696
0.936
Avg.
0.424
0.392
0.429
0.401
0.423
0.412
0.436
0.406
0.670
0.889
96
0.226
0.113
0.228
0.118
0.227
0.114
0.226
0.113
0.296
0.176
ETTm2
192
0.256
0.146
0.263
0.165
0.255
0.144
0.258
0.146
0.309
0.194
336
0.282
0.175
0.292
0.200
0.279
0.173
0.288
0.177
0.326
0.220
720
0.326
0.227
0.329
0.248
0.316
0.220
0.330
0.235
0.352
0.259
Avg.
0.273
0.165
0.278
0.183
0.269
0.163
0.276
0.168
0.321
0.212
Electricity
0.228
0.132
0.233
0.137
0.248
0.149
0.271
0.172
0.757
0.835
192
0.242
0.149
0.246
0.151
0.262
0.165
0.299
0,200
0.760
0.845
336
0.259
0.165
0.262
0.167
0.275
0.178
0.494
0.454
0.764
0.857
720
0.294
0.205
0.293
0.201
0.305
0.217
0.614
0.652
0.773
0.883
Avg.
0.256
0.163
0.259
0.164
0.273
0.177
0.419
0.370
0.763
0.855
Exchange
96
0,214
0.088
0.211
0.092
0.212
0.093
0.211
0.090
0.416
0.307
192
0.315
0.186
0.310
0.194
0.309
0.190
0.328
0.208
0.483
0,413
336
0.426
0.325
0.434
0.363
0.435
0.368
0.431
0.360
0.581
0.577
720
0.621
0.621
0.729
0.958
0.724
0.947
0.745
1.020
0.813
1.143
Avg.
0.394
0.305
0.421
0.402
0.420
0.399
0.429
0.419
0.573
0.610
24
1.077
2.508
1.072
2.623
1.128
2.808
1.077
2.597
1.570
4.865
NIL
36
1 1.055
2.439
1.053
2.567
1.109
2.774
1.068
2.570
1.568
4.913
48
1.066
2.529
1.076
2.660
1.119
2.801
1.101
2.714
1.574
5,012
60
1.079
2.615
1.099
2.780
1.123
2.843
1.116
2.729
1.603
5.216
Avg.
1.069
2.523
1.075
2.658
1.120
2.806
1.091
2.652
1.579
5.002
0.271
0.391
0.279
0.407
0.301
0.429
0.687
1,508
0.794
1.378
Traffic
192
0.277
0.407
0.284
0.420
0.308
0.448
0.308
0.445
0.798
1.393
336
0.284
0.420
0.291
0.432
0.315
0.459
0.396
0.566
0.800
1.408
720
0.300
0.447
0.313
0.463
0.332
0.489
0.597
1.780
0.804
1.428
Avg.
0.283
0.416
0.292
0.430
0.314
0.456
0.497
1.075
0.799
1.402
96
0.196
0.145
0.202
0.149
0.218
0.166
0.201
0.153
0.311
0.256
Weather
192
0.237
0.186
0.245
0.193
0.254
0.209
0.244
0.198
0.328
0.284
336
0.278
0.238
0.282
0.239
0.291
0,256
0.284
0.251
0.347
0.317
720
0.332
0.317
0.333
0.308
0.340
0.327
0.338
0.324
0.380
0.368
Avg.
0.261
0.221
0.266
0.222
0.276
0.239
0.267
0.231
0.342
0.306
Table 11. Comparison with ---low and FITS/Best in bold). IN-Flow focuses on nonstationarity (not fat-tail), showing decent performance on smaller ETT but struggling with large-channel data (Electricity, Traffic). FITS parameter-officient but relies on extensive hyperparameter tuning and is unstable under unified setting.
33

---

Slimming the Fat-Tail: Morphing-Flow for Adaptive Time Series Modeling
L. Temporal Complexity Analysis
In this section, we analyze the time (temporal) complexity the two core components our proposed MoF framework: the Flow layer (Section 3.1) and the Morph module (Section 3.2). We focus on the key (hraeerpas and how they affect runtime with respect to the sequence length T, the number of channels C, and the number of bins B used in our piecewise-lineat linear spline models.
L.I. Flow Layer Complexity
Recall that the Flow layer (ef. Section 3.1) applies piecewise-linear spline transform each input channel. Each channel C (1, (I,...,C) C) has B bins, and each bin is defined by width {w2,)) } and height (he,)), where i {1,...,B), B).
Forward pass. In the forward direction, every input element Ze,t for channel e and time step t E {l...., T is mapped to some output Ve.z- To find the appropriate bin for Te,t, one can:
1. Perform a (potentially) linear scan refix-sum-based search among the B bins.
2. Compute the output via one linear operation within the chosen bin.
direct (linear) scan select bin requires O(B) time, and this is repeated for each channel (C) and time step (T). Thus, the worst-case complexity for the forward pass across all elements is:
x
a more efficient bin-search strategy is employed (e.g., binary search), the factor B could be reduced to log B. However, we assume the simpler linear approach here, yielding the O(C B) cost.
Inverse pass. The inverse transformation proceeds analogously: one locates the bin for each output element Ye. in O(B) and then applies a linear mapping to recover Te,t. As above, this yields:
OC
time complexity for the inverse pass. Since both forward and inverse passes are required for training end-to-end (and for any invertible steps within the backbone), the overall complexity for the Flow layer remains OCT B).
L.2. Morph Module Complexity
The Morph module (ef. Section 3.2) consists of two main parts:
1. The test-time-trained temporal layer, which optimizes a low-rank weight matrix W using self-supervised loss.
2. The Up-Projection step that uses the updated W to generate the scaling factor Xmod, then re-applies the Flow transformation with modified parameters.
Test-time training. During inference, each incoming batch (or sequence) of length T' and channel dimension C' is first transformed by the Flow layer into x'. The Morph module then:
Projects x' into query and key embeddings, incurring OCCTd) costs if these linear projections map x' into internal dimension d.
Computes a self-supervised loss E(Wrs. res; x') in O(CT) time (since it involves elementwise differences under a binary mask M).
Performs one or more gradient steps to update W Wiest Each gradient step requires backpropagation through the above projections, adding another OCT d) factor per update.
If we denote by Nuer the number of gradient steps performed per test batch, the test-time training complexity scales as:
O x (CTd))
In practice, Nuw typically small (often to 5) to maintain real-time inference speed.
34

---

Slimming the Fat-Tail:
Morphing-Flow for Adaptive Time Series Modeling
Up-Projection and final Flow morph. After updating W rest: the Morph module uses Up-Projection map the low-rank embedding X B 2 matrix (Xmol); incurring O(d B) cost. Finally, the Flow layer is applied again with the morphed parameters Wftow flow Xmod- Section L.1.
This second Flow transformation has the same O(CT B) complexity discussed in
Overall cost of Morph. Hence, for a single batch of length 7' and channels C, test-time training plus the final Flow morph yields:
( x C + (d
+ OCTTB.
self-supervised updates
up- projection
Flow morph
Since B and d are typically much smaller than C (and since Niter is kept small), the overall additional time cost of Morph remains reasonable in practical settings.
L.3. Morph vs. Flow-Only Computation
Flow-Only Baseline. Recall that applying the Flow layer to a single batch of length T and C channels costs O B) in each forward (or inverse) pass, where B is the number of bins per channel. In typical forward-only inference setting, if no further transformations are applied, the runtime per batch is dominated by OCTT B).
Added Steps from Morph. By contrast, when the Morph module is enabled, each batch must go through:
1. The initial Flow pass (as in Flow-only): OCCTB)
2. Test-time-trained temporal layer:
Projection to internal dimension (query/key/valuee) computed in O(CT)) d). Self-supervised loss and gradient-based updates on W Wistt repeated Nise times for each batch. Each update backpropagates through the projections, adding another O(CT d) per iteration.
In total, this yields ONNiee C)....
3. Up-Projection from dimension d to the bin-height shape (B x 2), costing Ol B).
4. Final Flow pass with morphed parameters, which again is OCCT B).
Hence, compared to a Flow-only system, the new steps due to Morph amount to:
ON CT + Od
O
test-time updates
up- projection
re-applied Flow pass
in addition to the baseline Flow cost.
L.4. Relative Serper-Parametery Influencing Overhead
Bins per channel, B. In Flow-only, the cost scales as OCTT B). Morph re-applies the Flow transform once more, increasing the effective factor on OCTT B). For small B, the overhead from Morph is more pronounced in the self-supervised step; for large B, the additional Flow pass also becomes non-trivial.
Hidden dimension, d. The internal dimension used in the Morph module's query/key/value projections directly affects the OCCT d) cost. A modest d keeps the test-time updates relatively cheap, while large d can dominate the overall overhead, potentially exceeding the Flow cost if d > B.
Number of iterations, Niter- Each additional gradient update test time scales linearly. Even if d is small, large Nise can inflate the Morph overhead. Balancing between adaptation accuracy and runtime is key.
Empirically, one typically chooses B to be moderate (e.g., B = 8 to 16) and d to be significantly smaller than C. T, keeping the Morph overhead at a fraction the total runtime. Furthermore, a small Niw (often 1 1-5) can be sufficient to stabilize inference under mild distribution shifts, ensuring that Morph's 's overhead remains practical compared Flow-only.
35

---

Slimming the Fat-Tail: Morphing-Flow for Adaptive Time Series Modeling
L.5. Wall-Clock Runtime Comparison
We report average per-batch inference times (in milliseconds per iteration) across six datasets in Table 12. All experiments conducted using single NVIDIA Titan RTX GPU paired with Intel 19- 10900BB CPU. Each runtime is measured with input length T 96, batch size B 24, and model dimension d 92.
Table 12, Average inference time (ms/iter) for different configurations. The final column reports Morph's 's share of total MoF runtime.
Dataset (C)
w/ RevIN
w/ Flow
w/ MoF
Morph in MoF (%)
Traffic (C=862)
107.1
115.5
122.2
44.37%
Weather (C=21)
21.7
30.3
32.5
20.37%
ETThl (C=7)
22.1
32.5
32,9
3.70%
E (C=7)
23.7
32.7
33.0
3.23%
Electricity (C=321)
42.9
48.5
51.5
34.88%
Exchange (C=8)
22.2
32.2
32.3
0.99%
Average
17.92%
On average, Morph contributes only 17.92% of MoF's inference time, and incurs less than 5% overhead on total runtime compared to Flow-based baselines. A detailed computational complexity analysis is included in the revised version.
Summary.
Combining both Flow and Morph, each inference batch involves:
An initial Flow pass: OCC' B).
Morph test-time update and re- trtaasfformaiinn ON CTd+
While the precise constants depend on implementation details (e.g., bin-search optimization or parallel GPU vectorization), the asymptotic behavior governed by the linear dependence on C, T, and B, plus the modest overhead of test-time adaptation steps.
M. Gradient-statisties analysis
For each training step s we collect the gradient tensor of the model parameters and compute:
Norm(s) = 11
n
Var(s) 1
Σ
(ge.) - -9a)²
1=1
Skew(s) - (00.i - ]x
(Var(s)) 3/2
Kurt(s) EE (90.; - 96) )
3,
(Var(s))²
where n the number of gradient elements and Zs their mean. Gaussian).
The kurtosis is reported in its excess form (zero for
The first column of Fig. 23 shows only minor differences among Instance-wise 2-score, RevIN, and MoF when a shallow Linear backbone is used. In contrast, the second column (PatchTST on ETTh1) reveals visibly smaller gradient norm and variance for MoF, with skewness and kurtosis converging more rapidly. The third column (PatchTST on Exchange) highlights the strongest effect: gradients under MoF remain well- behaved throughout training, reflecting the heavier tails of the dataset and explaining the larger performance improvements.
36

---

Slimming the Fat-Tail: Morphing-Floo for Adaptive Time Series Modeling
Linear Variants
PatchTST (ETTm1)
PatchTST (Exchange)
Linerr 2-Scare p=2.300 Ma-03 2m
FriShTST Z.ccoe und.7 3333
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Rookk
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0.0125
Patch ST Score
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0.0100
Pakcr"ST
RextMl
0.006
PaRchTST Beuil
PatenTST
0.004
0.0075
0.004
0.002
0.0050
0.002
0.0025
2000
4000
12000
2000
5000
8000
10000
12000
1000
2000
3000
4000
5000
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1e-7
10-7
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0.8
1.25
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1.00
0.75
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2000
4000
12000
8000
10000 12000
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4000
0.03
[incar 2 Score 573 C6
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PaRAIST 2 2trre Me 07
PocceTST 2 Score 353 03 o- 146
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2000
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[Ihear 2 Scone p 1.22 20. 8 1:35e 63
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0.08
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0.02
0.04
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0.01
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0.0
0.00
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2000
4000
6000
10000 12000
2000
8000 10000
12000
1000
3000
4000
5000
Step
Stop
Stop
Figure 23. Gradient statistics over training. Columns: (i) Linear backbone on Exchange, (ii) PatchTST ETTh/. (ii) PatchTST on Exchange. Rows: gradient norm, variance, skewness, and kurtosis. MoF (green) yields lower norms/variances and markedly smaller higher-order espcnls--speciilly the heavy-tailed Exchange whileet-while differences modest for the shallow Linear model. Shaded areas denote one standard deviation over five runs.
37


Reviews:
[{"review_id": "i5abe9jEwV", "reviewer": "Reviewer_E595", "summary": "The work introduces Morphing-Flow (MoF), a framework to address the challenges of fat-tailed distributions in time series forecasting through adaptive normalization and test-time adaptation. MoF combines a spline-based Flow layer for distribution normalization and a Morph module for dynamic adaptation, achieving state-of-the-art performance across multiple datasets. The framework is efficient, robust to hyperparameters, and can be easily integrated into various models, offering a practical solution for improving forecasting accuracy in non-stationary environments.  MoF achieves state-of-the-art performance on multiple benchmark datasets, outperforming existing models by an average of 6.3%, and operates efficiently with a simple linear backbone, achieving comparable performance to complex models while using significantly fewer parameters.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your feedback!\n\n---\n\n> **Suggestions 1:**  \n> *Section 2: The term \"excess kurtosis\" is used without defining it.*\n\nThank you for the helpful comment. \n\nA 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. \n\nWe've revised the wording and added a pointer from the main text to improve accessibility.\n\n---\n\n> **Questions 1:**  \n> *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?*\n\nThanks for the question.\n\nThe Morph module introduces test-time adaptation overhead, which scales as:\n\n```\nO(N_iter × C × T × d)  # test-time gradient updates  \n+ O(d × B)             # up-projection  \n+ O(C × T × B)         # re-applied Flow pass\n```\n\nWith 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)`.\n\nWe report average per-batch inference times (ms/iter) across six datasets:\n\n| _Runtime(ms/iter)_ | **w/ RevIN** | **w/ Flow** | **w/ MoF** | **Runtime of Mo in Mof(%)** |\n|--------------------|--------------|------------------|------------|---------------------|\n| Traffic(C=862)     | 107.1        | 115.5            | 122.2      | 44.37%              |\n| weather(C=21)      | 21.7         | 30.3             | 32.5       | 20.37%              |\n| ETTh1(C=7)         | 22.1         | 32.5             | 32.9       | 3.70%               |\n| ETTm1(C=7)         | 23.7         | 32.7             | 33         | 3.23%               |\n| electricity(C=321) | 42.9         | 48.5             | 51.5       | 34.88%              |\n| exchange(C=8)      | 22.2         | 32.2             | 32.3       | 0.99%               |\n|                    |              |                  |            | **17.92%**          |\n*Tested with T=96, B=24, d=92*\n\nOn average, Morph accounts for ~17.92% of MoF’s inference time, with less than 5% overhead on overall model runtime.\n\n\nWe’ve included this in the revised version with detailed complexity breakdowns.\n\n---\n\n> **Question 2:**  \n> *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?*\n\nThank you for the suggestion. \n\nOur 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.\n\nWe added two new visual components:  \n- A case study visualizing how Flow attenuates heavy-tailed spikes while preserving underlying temporal structure ([Illustration](https://anonymous.4open.science/r/Materials-1D2F/illus.pdf)).  \n- A gradient dynamics study demonstrating that Flow stabilizes backpropagation (reduced norm, skewness, and kurtosis), detailed in ([Gradient Statistic Dynamics](https://anonymous.4open.science/r/Materials-1D2F/gradient_stats_grid.pdf)).\n\nIn **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.\n\n---\n\nWe hope these address your concerns satisfactorily. If you have any further questions or concerns, we are happy to address them.", "claims_and_evidence": "I think most claims that are made in the paper are supported by clear evidence. For example,\n\n1. The authors demonstrate the effectiveness of MoF in reducing fat-tailed distributions through strong experiments MoF significantly reduces excess kurtosis (a measure of fat-tailed distributions) in the transformed data compared to raw or stationarized data. This is supported by visualizations and quantitative results across multiple datasets (e.g., ETTh2, ETTm1, Electricity, Weather).\n\n2. The authors demonstrate that MoF is plug and play through experiments with different model architectures (Autoformer, Informer and DLINEAR).", "methods_and_evaluation_criteria": "The proposed methods and evaluation criteria in the paper \"Slimming the Fat-Tail: Morphing-Flow for Adaptive Time Series Modeling\" are overall aligned with the problem of handling non-stationary, fat-tailed distributions in time series forecasting.\n\nFor the proposed Flow Layer (Spline-Based Transformation), it addresses a critical issue in time series forecasting—fat-tailed distributions that destabilize model training and prediction. By using a spline-based transformation to normalize these distributions, the Flow layer directly targets the problem of high kurtosis and skewness.\n\nFor the proposed Morph Module, it tackles distribution shifts between training and testing data, which is particularly important in real-world applications where data characteristics can change over time.\n\nFor the metrics, the authors use MSE and MAE, which are standard for evaluating time series forecasting models and provide a clear measure of prediction accuracy. By using both MSE and MAE, the authors capture different aspects of model performance (sensitivity to outliers and overall error magnitude).", "theoretical_claims": "No theoretical claims.", "experimental_designs_or_analyses": "Yes I checked the soundness of experimental designs, from datasets, metrics, baselines to the experiment results. I didn't find apparent issues.", "relation_to_broader_scientific_literature": "The MoF framework directly addresses the challenge of fat-tailed distributions by using a spline-based transformation (Flow layer) to normalize the data. This approach is novel in the context of neural network-based time series forecasting and provides a structured way to mitigate the impact of fat-tailed noise on model convergence and performance. MoF can integrate with various backbones, including Transformer-based and Mamba-based architectures.", "essential_references_not_discussed": "I didn't find essential references that are not discussed.", "other_strengths_and_weaknesses": "The paper is overall written with clear logic.", "other_comments_or_suggestions": "Section 2: The term \"excess kurtosis\" is used without defining it. Adding a brief definition or reference to the appendix would be helpful for readers unfamiliar with the term.", "questions_for_authors": "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?\n\n2. 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?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "DR6eyKrIGJ", "reviewer": "Reviewer_fqDp", "summary": "This paper proposes Morphing-Flow, a spline transformation module coupled with test-time adaptation, to counter fat-tailed distributions and distribution shifts.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your comments. Please kindly find our response below.\n\n---\n> **Comment 1:**  \n> *Regarding the contribution*\n\nAs 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.\n\nIn summary, with *new ablations on PatchTST*, our contributions are:\n\n- **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.\n   \n- **Morph for Distribution Shifts** provides a **3.11%** MSE gain by adapting to distributional shifts via test-time tuning of RevIN’s affine parameters.\n   \n- **Flow and Morph work synergistically**: Morph benefits from the stabilized gradient flow, resulting in a **7.17% MSE reduction** over PatchTST.\n\nWe revised the introduction to highlight these insights.\n\n|Patch TST +|Average MSE across 8 datasets|-Δ (%)|\n|-|-|-|\n|no Norm|0.370|-4.33%|\n|RevIN|0.349|0.00%|\n|Morph|0.333|3.11%|\n|Flow|0.319|5.95%|\n|Morph+Flow|0.314|**7.17%**|\n\\* [[Full Result Here]](https://anonymous.4open.science/r/Materials-1D2F/abla.pdf)\n\n---\n> **Question 1:**  \n> *Regarding the importance of dealing with fat-tailed distributions and performance comparison with Linear Backbone(Appendix I)*\n\nReal 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).  \n\nAdded gradient analysis ([link](https://anonymous.4open.science/r/Materials-1D2F/gradient_stats_grid.pdf)) shows MoF reduces skewness/kurtosis for smoother gradients, crucial in larger model & fat-tailed data (ETT -> Exchange). \n\nAcross 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.\n\n|Avg. of avg.(MSE)|+RevIN|+MoF|\n|-|-|-|\n|Linear|+1.83%|+2.75%|\n|PatchTST|+5.65%|+15.0%|\n\n[1].Fat tails in leading indicators(Econ Lett 2020)\n\n[2].Emergence of heavy tails in streamflow distributions: the role of spatial rainfall variability (Adv Water Res 2023)\n\n[3].Evidence that coronavirus superspreading is fat-tailed(PNAS 2020)\n\n---\n> **Question 2:**\n> *Regarding the fariness of test-time training*\n\nAs 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.  \n\n---\n> **Question 3:**  \n> *Have you considered **combining the backbone of PatchTST with MoF** to check how MoF advance over RevIN?*\n\nYes. PatchTST+MoF improves MSE by 7.17% over PatchTST+RevIN across 8 datasets (wins 7/8). [[Results]](https://anonymous.4open.science/r/Materials-1D2F/patch_mof.pdf) to be added to Table 3 & Appendix J. \n\n---\n> **Question4:**\n> *Regarding Figure 8 limited to ETTh2 dataset*\n\nFigure 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.\n\nIn 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.\n\nWe clarified this intent in the updated version.\n\n---\n> **Question 5:**  \n> *Regarding Figure 9 limited to the Weather dataset*\n\nThank you for pointing this out. \n\nFig.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.\n\n---\n> *Regarding References Not Discussed*  \n\nWe have added the suggested works to our related section.\n\n*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]](https://anonymous.4open.science/r/Materials-1D2F/ref_cmp.pdf)\n\nWe will include these findings in our revision.  \n\n---\nWe hope these address your concerns satisfactorily. If you have any further questions or concerns, we are happy to address them.", "claims_and_evidence": "The paper claimed that fat-tailed distributions have negative effects on model convergence. This claim is supported by some empirical experiments on synthetic data.", "methods_and_evaluation_criteria": "Empirical experiments on synthetic data seem to suggest that fat-tailed distributions could be a severe problem affecting model convergence. However, existing time-series approaches seem to work well in practical data, and this paper also introduces test-time adaptation to adjust the parameters of the normalizaiton module. These contradictions made me hard to believe what is the real contribution behind Morphing-Flow.", "theoretical_claims": "No. This paper does not include theoretical proofs.", "experimental_designs_or_analyses": "I have checked the experiments and analyses.", "relation_to_broader_scientific_literature": "The fat-tailed distribution problem may also exist in other scenarios.", "essential_references_not_discussed": "This paper lacked some discussions and comparisons with more recent papers. For example, [1] is closely related to this work because it also inroduces normalizaiton flow to address distribution shifts; [2] is an efficient and effective time-series model that operates in the frequency domain; [3] is a successor of PatchTST that is capable of delivering forecasts for any horizons using a single model. Both [3] and PatchTST leverage RevIN as the normalizaiton method and seem to work well in practice.\n\n[1] IN-Flow: Instance Normalization Flow for Non-stationary Time Series Forecasting, https://arxiv.org/abs/2401.16777\n[2] FITS: Modeling Time Series with 10k Parameters, https://arxiv.org/abs/2307.03756\n[3] ElasTST: Towards Robust Varied-Horizon Forecasting with Elastic Time-Series Transformer, https://arxiv.org/abs/2411.01842", "other_strengths_and_weaknesses": "This paper proposes an interesting angle. But current experiments can hardly convince me that fat-tailed distribution is a severe problem in time-series forecasting in real-world scenarios.", "other_comments_or_suggestions": "I would like to see some concrete and real evidences demonstrating the importance of tackling fat-tailed distributions in time-series forecasting. Current results make me feel that RevIN is already good enough while MoF only makes marginal improvements but at huge costs and with complicated designs.", "questions_for_authors": "- I have concerns on the real importance of dealing with fat-tailed distributions in time-series data. Although the synthetic experiments (figure 2) seem to suggest this could be a severe problem, concrete experiments on real-world data, comparing MoF with other normalization methods (Appendix I, page 28) does not show significant performance gains, especially when compared with the simple and effective baseline, RevIN. I would like to understand why the authors claimed this as a severe problem in time-series forecasting and designed such a complicated method to tackle it.\n\n- If the MoF module does not play a criticle role, the remaining contribution of this paper would be test-time adaptation. However, in this context, directly comparing test-time tuned MoF models with traditional time-series models only trained on the training data is not fair. For example, you can also apply test-time adaptation to the parameters of RevIN, which may also help.\n\n- Moreover, if MoF is designed as a model-agnostic module to normalize data distributions, have you tried to combine this with more forecasting models beyond DLinear and Mamba? Table 3 includes similar experiments but using some baselines that have well-known performance issues, such as Informer and Autoformer, in the liteature. It is well-known that PatchTST is equipped with RevIN by default. Have you considered combining the backbone of PatchTST with MoF to check how MoF advance over RevIN?\n\n- Figure 8 compares Linear-MoF with other baselines, but only on ETTh2 dataset. It is well known that on this dataset, the Linear model could be better, but on other complicated scenarios, PatchTST and iTransformer could still significantly improve Linear. Only showing results on one dataset could be very misleading.\n\n- Similarly, Figure 9 compares MoF with other normalization methods, but only on the Weather dataset. After checking Appendix I, I find that MoF does not always ensure the best normalizaiton across diverse datasets. Hence Figure 9 could largely mislead readers, too.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "KWgyBg1umP", "reviewer": "Reviewer_sm3N", "summary": "This paper tries to address the challenge of forecasting temporal sequences characterized by non-stationarity and leptokurtic (fat-tailed) distributions. The proposed Morphing-Flow (MoF) framework innovatively integrates a spline-based transform layer (“Flow”) with a test-time-trained adaptation method (“Morph”) to normalize these distributions while preserving essential extremal features. The numerical experiments are conducted.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for raising this concern.\n\n---\n\n> **Question:**  \n> *Regarding baseline performances*\n\nPrior 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*.\n\n\n| Hyper-param      | Ours          | iTransformer        | PatchTST             |\n|--------------------|---------------|----------------------|-----------------------|\n| lookback length    | 336           | 96                   | 336                   |\n| Patch Len/Stride   | 16/8          | 12 / unknown    | 16/8 to 24/2          |\n| d_model            | 512           | 128–512              | 16–128                |\n| d_ff               | 512           | 128–2048             | 128–256               |\n| learning_rate      | 1e-4          | 1e-3 to 5e-5         | 1e2 to 1e-4           |\n| batchsize          | 4             | 16–32                | 8–128                 |\n\nAs 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.\n\n| *Dlinear(MSE)*    | Ours  | [DLinear](https://arxiv.org/abs/2205.13504) (AAAI 2023) | [LIFT](https://arxiv.org/pdf/2401.17548)(ICLR 2024) | [SAN](https://openreview.net/forum?id=5BqDSw8r5j) (NeuraIPS 2023) |\n|-------------|-------|-----------------------------|-----------------|-------------------|\n| Weather     | 0.245 | 0.246                       | 0.246           | 0.245             |\n| Electricity | 0.167 | 0.166                       | 0.166           | 0.166             |\n| Traffic     | 0.435 | 0.434                       | 0.434           | 0.435             |\n\n\n**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](https://anonymous.4open.science/r/Materials-1D2F/param_cmp.pdf)).\n\n| **Avg. MSE**     |  by us           |    by iTrans.   |       by PatchTST     |\n|--------------------|---------------|----------------------|-----------------------|\n| iTrans             | **0.343 (-2.72%)**| 0.345                | -                     |\n| PatchTST           | **0.349 (-4.35%)**| 0.353                | 0.307                 |\n| DLinear            |**0.326 (-19.1%)**| 0.403                | 0.330                 |\n\n\n**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.\n\nThank you for raising this concern. We will emphasize these clarifications in the revised paper to preempt any potential reader confusion.\n\n---\n\nWe hope this addresses your concern and if you have any further questions or concerns, we are happy to address them.", "claims_and_evidence": "Since this draft doesn't contain a specific section to directly summarize the major contributions. The evaluations are based reviewer's personal understanding.\n\n1. The methods part\n\n-- The proposed methods are well-motivated.\n\n2. The experiments part\n\n-- After the details results in Table 1 and Appendix E. I obverse significant performance mismatch for benchmarks and their results in literature. For example, in iTransformer original paper the average MSE results are as follows\n\n| Dataset    | Itransformer | PatchTST | Dlinear | \n| -------- | ------- | --- | ---| \n| ETTh1  | 0.454   |0.469 | 0.456|\n| ETTh2 | 0.383     | 0.387| 0.559|\n| ETTm1    | 0.407   | 0.387 | 0.403 | \n| ETTm2   | 0.288   | 0.281|  0.350|\n| ECL    | 0.178   | 0.205 | 0.212 | \n|Exchange | 0.360 |  0.367 |  0.354| \n| Traffic    | 0.428   | 0.481 |  0.625|\n| Weather    | 0.258   | 0.259 | 0.265 | \n\nCan the authors elaborate more on the reason for the mismatch?", "methods_and_evaluation_criteria": "My major concern is the mismatch of the benchmarks' performance in the numerical section. Based on the current presentation, it is hard for me to make fair evaluation.", "theoretical_claims": "N/A", "experimental_designs_or_analyses": "Please see the above sections.", "relation_to_broader_scientific_literature": "N/A", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "N/A", "other_comments_or_suggestions": "N/A", "questions_for_authors": "At this stage, the aforementioned mismatch issue blocks me from providing a comprehensive evaluation. I will defer my final decision until this concern is adequately addressed.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {"ethical_review_concerns": "N/A"}}]

Author rebuttal:

Meta-review: This submission proposes Morphing-Flow, a two-component normalization framework that (i) uses a spline-based Flow layer to tame fat-tailed inputs and (ii) applies a light test-time Morph update to track distribution drift. 

Strengths
- Clearly motivated attack on the under-studied “fat-tail” failure mode; Flow alone already yields sizable gains.
- unified tuning, ablations isolating Flow vs. Morph, new PatchTST+MoF results added in the rebuttal, and detailed runtime analysis (Morph ≈ 18 % of inference cost).
- Plug-and-play design that fits multiple backbones without architectural changes.

Weaknesses / how addressed
- authors showed that earlier papers tune per-dataset, whereas their single-config evaluation is stricter; recreated baselines actually perform better than literature, yet MoF still wins.
- confusion came from an appendix using a deliberately weak Linear backbone. PatchTST+MoF gains (+7.2 % MSE, wins 7/8 datasets) clarify impact.

I vote for a weak accept.
Decision: Accept (poster)