ViTime: Foundation Model for Time Series Forecasting Powered by Vision Intelligence
Authors: Luoxiao Yang
Yun Wang
Xinqi Fan
Israel Cohen
jingdong chen
Zijun Zhang
Venue: Accepted by TMLR ()

Abstract: Time series forecasting (TSF) possesses great practical values in various fields, including power and energy, transportation, etc. TSF  methods have been studied based on kncowledge from classical statistics to modern deep learning. Yet, all of them were developed based on one fundamental concept, the numerical data fitting. Thus, the models developed have been long known for being problem-specific and lacking application generalizability. Practitioners expect a TSF foundation model that serves TSF tasks in different applications. The central question is then how to develop such a TSF foundation model. This paper offers one pioneering study in the TSF foundation model development method and proposes a vision intelligence-powered framework, ViTime, for the first time. ViTime fundamentally shifts TSF from numerical fitting to operations based on a binary image-based time series metric space and naturally supports both point and probabilistic forecasting. We also provide rigorous theoretical analyses of ViTime, including quantization-induced system error bounds and principled strategies for optimal parameter selection. Furthermore, we propose RealTS, an innovative synthesis algorithm generating diverse and realistic training samples, effectively enriching the training data and significantly enhancing model generalizability. Extensive experiments demonstrate ViTime's SOTA performance. In zero-shot scenarios, ViTime outperforms TimesFM by 9-15%. With just 10% fine-tuning data, ViTime surpasses both leading foundation models and fully-supervised benchmarks, a gap that widens with 100% fine-tuning. ViTime also exhibits exceptional robustness, effectively handling missing data and outperforming TimesFM by 20-30% under various data perturbations, validating the power of its visual space data operation paradigm.

Published in Transactions on Machine Learning Research (10/2025)
ViTime: Foundation Model for Time Series Forecasting Pow- ered by Vision Intelligence
Luoxiao Yang
School of Automation and Information Engineering, Xi'an University of Technology, Xi'am, China Electrical and Computer Engineering, Technion Israel Institute of Technology. Isruel"
Yun Wang
wanggyunjef@@@maiicccom com
Electrical and Computer Engineering, Technioo Israel Institate of Technology, Israel
Xinqi Fan
z-fon mon@rmu..ccmm
Department of Computing and Mathemalics, Manchester Metropolitan Universily, UK
Isracl Cohen
Scohen@@e..
Electrical and Computer Engineering, Technioo Israel Instilute of Technology. Isruel
Jingdong Chen
jingdongchen&hieee.rrg
Deportment of Information and Communicaatio Engineering, Northwest Polytechnical University, Xi 'n,, China
Zijun Zhang
zijhang@eeityn edu.
Department of Data Science, City University of Hong Kong, Hong Kong SAR
Reviewed on OpenReview: https: // openreviee..........mmmm net/ forum? id= XIns
Code and Pretrained Models: https: // g/tthub. com/ Ike IkeYang/ Vilime
Abstract
Time series forecasting (TSF) possesses great practical values in various fields, including power and energy, transportation, etc. TSF methods have been studied based knowledge from classical statistics to modern deep learning. Yet. all of them were developed based on one fundamental concept, the numerical data fitting. Thus, the models developed have long been known to be problem-specific and lacking application genoralizability. Practitionors expect a TSF loundation model that serves TSF lasks different applications. The central question is then how to develop such a TSF foundation model. This paper offers one pioneering study in the TSF foundation model development method and proposes vision intelligence-powered framework, ViTime, for the first time. ViTime fundamentally shifts TSF from numerical fitting to operations based on binary image-based time series metric space and naturally supports both point and probabilistic forecasting. We also provide rigorous theoretical analyses of ViTime, including quantization-induue system error bounds and principled strategies for optimal parameter selection. Furthermore, we propose RealTS, an innovative synthesis algorithm generating diverse and rcalistic training samples, effectively enriching the training data and significantly enhancing model generalizability. Extensive experiments demonstrrte ViTime's state-of-the-art performance. In zero-shot scenarios, ViTime outperforms TimesFM by 9-15%. With just 10% fine-tuning data, ViTime surpasses both leading foundation models and fully-supervised benchmarks, gap that widens with 100% fine-tuning. ViTime also exhibits exceptional robustness, effectively handling missing data and outperforming TimesFM by 20-30% under various data perturbations, validating the power of its visual space data operation paradigm.
"Work performed while XAU'T and Technion. "Coresppondig author

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Published in Transactions on Machine Learning Research (10/2025)
1 Introduction
Time series forecasting (TSF) is classic but challenging topic that has been vigorously discussed in various application fields, including power and energy
(Sharadga et al. 2020), environmental studies (Jacox et al,, weather forecasting (Yang et al. 2021), stock market analysis
2022), transportation studies (Lei et al.) 2022).
(Ln et al. 2011), public healthcare (Liu et al.
2024). Although new heights of accuracies were repeatedly refreshed by new studics (Zhou et al. 2021 Wu ct al. 2021 Nic et al. 2022 Zeng et al.) 2023 Patro &
Agneeswaran 2024 Wu el al. 2022), most reported methods predominantly relied on a numerical fitting
based modeling paradigm so that models were often dataset- or problem-specifie and lack of application generalizability. The need to repeatedly train models for various TSF tasks has been the critical barrier of promoting applications of learming- baaming-based TSF methods in practice, especially ones with sophisticated mechanisms. Developing a TSF foundation model capable of serving diverse TSF tasks across different. applications is thus of great practical value. The central question then becomes: how can we develop such a TSF foundation model?
Studying the TSF foundation model is still in its early stages, and existing efforts observed in literature are mainly devoted to exploring Large Language Model (LLM)-based and numerical fitting- based models. The LLM-based model leverages the inference capabilities of LLMs for zero-shot TSF tasks, including TimeGPT-1
Garza & Srageenthale-CCansee 2023) and TIME-LLM (Jin et al, 2023). However, the prediction accuracy of LLM-based models heavily depends on the underlying capabilities of LLM, and to achieve optimal performance, the competent large language models, such as GPT-4 or Claude 3.5 (Zhou et al. 2023a), ),
are usually employed. Meanwhile, in fine-tuning LLM-based TSF foundation models for handling various downstream tasks demanding higher precision, the computational complexity becomes prohibitively expensive, resulting in a large, redundant, less precise, and cost-ineffective paradigm for the TSF foundation model (Tan
et al. 2024).
The dominant paradigm in Time Series Forecasting (TSF) is currently centered numerical fitting- based models, such as TimesFM (Das et al. 2024) and ForecastPFN (Dooley et al. 2024). These models operate by directly learning the numerical correlations along the temporal dimension of the data. However, this purely numerical-driven approach diverges from human cognitive processes. Studies in cogmitive science indicate that for tasks like trend conjecture and forecasting, humans preferentially process and remember correlations between visual representations rather than directly handling abstract numerical sequences (Pettersson) 1993, Dondis 1974). For instance, Dondis (Dondis 1974) noted that the visual cortex is adept at rapidly identifying patterns, shapes, and colors, making the processing of visual information more efficient than that of text or numbers.
Inspired by this cognitive paradigm, the research community has begun to explore the potential of applying vision intelligence to TSF. Early attempts involved transforming time series inlo images, either through direct plotting for image-to-image forecasting (Sood et al., (021) or via multi-view visual encodings like Gramian Angular Fields (GAF), Markov Transition Fields (MTF), and Recurrence Plots (Wang & Oates)
2015: Eckmann et al.) 1987p. More recently, researchers have repurposed powerful vision backbones, such
Vision Transformers and Masked Autoencoders, for the TSF domain (Du et al. 2024 Chen el al.] 2024
While these works have shown the promising potential, they remain largely heuristic. They often lack a rigorous theoretical framework for visual quantization and metrics, and the exploration of foundation-modee paradigms within this context remains limited. These observations culminate in a fundamental question: On the path toward a TSF foundation model, could leveraging vision intelligence as a core modeling paradigm, alongside conventional rumericcl methods, offer a more promising avenue?
In addition, training data of TSF tasks typically consist of large-scale real-world datasets (Das et al. 2024 raising a critical question: Can real-world datasets comprehensively capture the diverse range of universal time series patterns? Specifically, what kind of foundational capabilities should a TSF foundation model possess to address a universal spectrum of time series problems?
To tackle these challenges, this paper develops a novel vision intelligence- based TSF foundation model, a Visual Time Foundation Model (ViTime), aiming to pioneer new computational paradigm of building the TSF foundation model from the perspective of vision intelligence. Regarding the computational principle
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innovation aspect, ViTime operates by transforming numerical time series into binary images, converting numerical temporal correlations into pixel spatial patterns, and solving TSF tasks, including both point and probabilistic forecasting, in binary image space. We provide detailed theoretical analyses of quantization- induced errors and establish principled guidelines for optimal parameter settings, ensuring precise control over the trade-off between computational complexity and prediction accuracy. To offer a large volume of sufficiently diverse samples for training ViTime, an innovative time-serics-data generation method, Real-Time Series (RealTS)) is proposed. RealT' categorizes foundational knowledge of time series analysis into "trend' and "periodicity' and synthesizes training data during the training of ViTime, ensuring it captures essential time series characteristics. Experimental results demonstrate that ViTime can achieve SOTA performance across diverse scenarios, including zero-shot generalization, fine-tuning with limited data, and robustness to data perturbalions.
The main contributions of this work are listed as follows:
Novel Theoretical Framework for Vision Intelligence Powered TSF. We introduce ViTime, a pioneering TSF foundation model grounded in a novel theoretical framework that shifts from conventional numerical fitting to operations within a formally defined binary image-based time series metric space.
RealTS: Advanced Data Generation and Augmentation for TSF Foundation Modeling. To address the training-data sample diversity challenge developing a TSF foundation model, the RealTS, a sophisticated time-series data generation method that synthesizes diverse and high-quality training data, is designed to ensure ViTime can generalize to wide range of time series patterns.
Empirical Validation of Thcoretical Advantages and SOTA Porformance. The efficacy of ViTime's theoreticalll--grouddee visual intelligence paradigı is extensively validated. ViTime significantly outperforms existing foundation models and supervised benchmarks in both zero-shot point forecasting (c.g., 9-15% improvement over TimesFM) and zero-shot probabilistic forecasting, few-shot fine-tuning, and robustness against diverse data perturbations (e.g., 20-30% better than TimesFM with missing data/ pattaa/pttubaaiios)) confirming the practical benefits of our theoretical contributions.
2
Related Work
2.1
Problem-specific Model for TSF
The problem-specific TSF methods adopt fully supervised learning paradigm, where specific models are trained on particular datasets. Early discussions on problem-specific TSF modeling were mainly conducted on classical statistical and machine learning models, such as autoregressive (AR) models and AR variants (uu 2007), Splines and their extensions (Lewis & Stevens 1991), lincar regressors (Montgomery et al., 2015). support vector regressor (Montgomery et al. 2015), neural network based regressor (Montgomery et al., 2015), etc. Tn comparison, the latest, TSF studies have shed light on modern deep learning methods, such as recurrent neural network (RNN) and RNN variants (Hewamalage et al. 2021), transformer and various transformer-basee models (Zhou et al.
2021
Wu et al.) 2021 Nie et al.) 2022 Liu et al. 2023). Dlinear (Zeng)
et al. 2023 TimeMixer (Wang et al.
2024 Mamba based method Palro & Agneeswaran
2024 etc.
2.2
Foundation Model for TSF
Inspired by recent breakthroughs of pretrained foundation models in natural language processing and computer vision, the TSF community has actively explored developing Nomain-general foundation models capable of forecasting across diverse datasets and scenarios. Current TSF foundation model studies in general fall into three categories: I.LM-based, samnerical-data-aasell and the emerging vision-based approaches.
LLM-based Models: Several recent studies have directly adapted LLMs to forecasting tasks. Methods such as PromptCast (Xue & Salim| 2023), TIME-LLM (Jin et al. 2023). GPT4TS (Zhou al.) 2023b).
TimeGPT-1 Garza & Mergenthaler- Caarreenttale--Cassccoo
2023 and LLM4TS [Chang al. 2025) recast numerical
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forecasting into text-based prompting embedding alignment tasks. Despite their promising zero-shot forecasting capabilities, these models suffer from inherent. limitations, including high computational costs, inefficiency, and domain adaptation complexity arising from fundamental discrepancies between linguistic structures and numerical temporal patterns (Tan et al. 2024).
Numerriaal-daat--bseed Models: To address these limitations, another prevalent research direction exploits large-scale collections of real-world numerical time series to train foundation models. Representative methods include T (Das et al., 2024 Moirai (Woo et al.) 2024). Chronos (Ansari et al. 2024). Moment [Goswami et al., 2024). Lag-Llama (Rasul et al. 2023), GTT (Feng et al. 2024h, and TSMamba (Ma et al. 2024). Although those real-data-basse models significantly enhance zero-shot generalization, their performance heavily depends on the quality, diversity, and representativeness of available real datasets. Moreover, they typically suffer substantial performance degradation when encountering data perturbations, missing values, or unseen temporal patterns. Furthermore, the reliance on extensive real-world datasets inherently risks test set leakage, as partial segments of test data may inadvertently appear during training, undermining true generalization evaluation. Recognizing these inherent limitations of real-world numerical data, recent. work has explored alternative data sources. ForecastPFN (Dooley et al.) 2024) trains Transformer-basel models purely on synthctic numerical data generated from predefined trend and scasonality components, demonstrating limited but promising zero-shot forecasting abilities. However, due to the uncontrolled or oversimplified synthesis patterns, these synthttic-data-based methods often fail to capture the richness and complexity of real-world scenarios, thereby limiting forecasting accuracy and robustness.
Vision-based Models:
The idea of representing time series as images to leverage powerful vision models has been explored, but its application as primary forecasting paradigm is an area of growing interest. Farly methods, such as Gramian Angular Fields (GAF) [Wang & Oates 2015 Markov Transition Fields (MTF), and Recurrence Plots (Eckmann et al. 1987p. p. transformed time series into images primarily for classification tasks, demonstrating the potential of image representations to reveal patterns not obvious in the 1D domain. Recently, this concept has been extended to forccasting. One intuitive approach is direct plotting, c.g., VisualAE (Sood et al., 2021) pioneered this by treating TSF as an image-to-inage regression lask, where a line plot is converted by a convolutional autoencoder. Other works focus on adapting vision architectures, e.g., Swin4TS (Du ct al. 2024 reshaped the time scries into 2D patches to apply the Swin Transformer. To enrich the imput, models like LDM4TS (Ruan et al., 2025 employed multi-view strategy, converting time series into multiple images using techniques like segmentation and GAFs. Recently, VisionTS (Chen) et al. 2024) proposed repurposing pretrained vision models (specifically, masked autoencoders trained on ImageNet) for TSF by reformulating forecasting as an image reconstruction problem. Nevertheless, directly reusing models pretrained on natural images introduces a significant domain mismatch that the visual features learned from natural images may not optimally represent temporal structures inherent in numerical time series. Furthermore, all these existing vision-based methods still fundamentally rely on numerical-space analyses and empirical mappings (e.g., standard plotting libraries or heuristic reshaping), lacking a rigorous theoretical framework explicitly tailored for visual representation and quantization of numerical sequences.
Our Contribution in Context: In contrast to aforementioned paradigms, our proposed ViTime framework introduces two fundamental shifts in the TSF foundation model design:
Firstly, recognizing intrinsic limitations of numerical-spacc-basee forccasting, such as limited generalization across scales and sensitivity to data perturbations, ViTime explicitly advocates modeling time series directly in principled visual representation space. Where prior visual methods use heuristic transformations, ViTime pioneering in rigorously defining dedicated visual metric space for numerical time series, providing theoretical analysis of quantization-indued errors, and offering principled guidance for optimal parameter selection. We also proved that this rigorous visual modeling framework can significantly enhance the signal-to-noise ratio (SNR) of time series and improve forecasting accuracy and interpretability.
Secondly, given the inherent challenges of relying on real-world numerical datasets (limited diversity, data leakage risks), we propose, RealTS, a controlled data synthesis strategy focusing on fundamental time series components (trend, pcriodicity) to generate structurally sound training data. The RealTS substantially mitigates data leakage risks and enriches training data diversity, enabling ViTime to generalize robustly across diverse real-world scenarios. As demonstrated by extensive experiments, ViTime sets new SOTA
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zero-shot and limited-data forecasting benchmarks, significantly outperforning existing foundation models across diverse evaluation settings.
3 Method
(a)
Real World Numerical Time Series
(b)
Directly Feed
Mapping as Image
Visual Time Tokenizer
Numerical Time
Series 0,25, 0.75,.... 0,20
W
Vision Intelligence Powered TSF Model
Decoder
Traditional Numerical TSF Model
Inverse Mapping
Numerical Prediction 0.75, 0,20, 0.30
Numerical Prediction 0.75, 0.20..... 0.30
Refining Module
MMMMVV
(c)
ViTime Visual Time
Periodic Pattern
Data generator
Mapping Function f
Numerical
Inversre Mapping Function f-1
Tendency
Array $L ....
Pattern
Numerical Array
Figure 1: ViTime architecture overview. (a) Pipeline comparison between ViTime and traditional numerical TSF models, showing ViTime's paradigm shift to binary image space processing.
(b) ViTime network with three modules: Visual Time Tokenizer, Decoder, and Refining Module. (c) Complete architecture: RealTS synthesis for diverse training samples, mapping function for numerical-to- to-tbbrrryyyyooo conversion, ViTime model for visual pattern learning, and inverse mapping for prediction output, enabling zero-shot gencralization across real-world lime series lasks.
3.1 Overall Architecture
The overall framework of ViTime, schematically illustrated in Fig. 1 (c), comprises four key modules: the RealTS synthesis module, the mapping function, the proposed ViTime model, and the inverse mapping function. To address the dataset challenge of training a robust TSF foundation model, RealTS synthesizes vast and diverse set of training samples by categorizing foundational knowledge of time series analysis into 'trend' and 'periodicity' patterns, which ensures ViTime captures essential time series characteristics across wide range of scenarios. The core innovation of ViTime lies in its computational principle of mapping numerical time series into binary images. This approach allows ViTime to remember temporal pailern correlations through ordered pixel coordinates while maintaining the ability to convert results back to numerical format. The visual modeling process ViTime learns to extract relevant features and patterns from the time series visual represontation, utilizing the historical distributions of the generated binary images predict future trends. Finally, the inverse mapping function is employed to convert the predicted image back into numerical time series data for further analysis.

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Published in Transactions on Machine Learning Research (10/2025)
In the following sections, will introduce each component of ViTime in detail: RealTS, mapping & inverse mapping function, and ViTime Model.
3.2 Real- Series Synthesis
In this paper, we hypothesize that robust foundation model for TSF should integrate two essential types of time series fluctuation knowledge, the periodic and trend patterns, which encompass the inherent patterns and directional changes in time scries data. Real-world datascts, however, often lack representation of the full spectrum of these periodic and trend-based fluctuations limiting the ability of the model to generalize across different scenarios and effectively learn underlying dynamics.
To address this challenge, propose novel time series generation algorithm, RealTS. RealTS systematically generates large volume of syuthetic time series data that exhibit diverse periodic and trend characteristics. The proposed RealTS can facilitate more comprehensive training of foundation models, exposing them to various patterns and improving their ability to generalize to unseen real-world data.
The RealTS algorithm probabilistically selects between generating periodic or trend-based time series. Given the total length of the synthesized time scries, the algorithm determines the data prior hypothesis between periodic Sp and trend-based şt. pabterns with probability (a). The distribution of generated time series P(D) is defined follows:
ST. P(D) = LL)
=a / Bo
P ((pp) dep + (1 (1 / P B)) P ((p)) di:
where St, the synthesized time series with length L; P(y) represents the prior probability of hypothesis yo; P(BBp) is the likelihood of observing the data behavior B under hypothesis sp. Data behavior B is introduced to further detail the generation behavior within different data modes. RealTS employs two data behavior modes for periodic hypothesis and three for trend hypothesis as follows:
Periodic Hypothesis: Inverse Fast Fourier Transform Bchavior (IFFTB) and Periodic Wave
Behavior (PWB).
Trend Hypothesis: Random Walk Behavior (RWB), Logistic Growth Behavior (LGB) and Trend Wave Data Behavior (TWDB)
Detailed formulas for each behavior mode and illustrative examples are provided in Supplementary Section
3.3 Binary Image- based Time Series Metric Space
In ViTime, time series are fed and operated with a binary image form, leveraging a binary image-based time serics motric space, as described Definition 3.1
Definition 3.1 (Binary based time series metric space). The binary mmage-based time series metric space is defined as group (V.d), where V is a set of elements defined in Equation
V = {
Rexhxx Ui.j. E (0,1), [c].j [M k E [Z]. Σ UajAA m.jj=11 =1}
(2)
where d: V R is a distance function based on the Earth Mover's Distance (EMD), as defined in Equation
(0), 02) v2) 00) 1 J=1 7 [ y J.b.kyy En,yyry 1|0 - wll, dkdi
inf
(3)
where represents the number of variates, T. is the length of the time series, and h is the resolution of V.

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To enable the transition from numerical time-series values to the binary image-based metric space, we introduce mapping and inverse mapping functions as follows. Let S = {s E R 8i,6 R] represent.
the numerical value space of time series. The Time-SSries-to--lee to-Image mapping function f S V and the Image-to-Times Time-Serics inverse mapping function f V + S can be defined as follows:
Vi,.bbkk = f (8L,)) = (fi (Bi,k), (2((8 ..... Sn (8Lx))
if 8i,k 2 MS.j
1, if si S -MS
(4)
fg (sik))
j E [h]
ifj= #
2MB
0, otherwise.
The Imago-to-Time-Sere Time-Series inverse mapping function f-1 v + S can be defined as follows:
2MS
8i.k f-1 (Vi,1.h,k) Σ i 0.5)
MS
Viji
(5)
where MS denotes the maximum scale V. Before mapping, zero-score normalization is typically applied the numerical time series 8i,k to standardize the scale.
Given that the numerical data synthesized by RealTS are one-channel time series, i.e, Sr. E R2xL, thus the corresponding VL R x is obtained via
VL = f(
3.3.1
System Error Analysis
The system error (SE) emerges from the bidirectional mapping belween discrete space V and continuous space S, which inherently impacts prediction fidelity. A rigorous analysis SE is essential for ensuring reliable and robust predictions in image space V. We begin our theoretical analysis of SE with Assumption 3.2 and Theorem 3.
Assumption 3.2. After applying zero-score normalization, the continuous space follows a standard normal distribution:
S (0,1)
Theorem 3.3 (System Error Upper Bound). Given tensor & S C Rex the system error defined || (f) - 3 satisfies the following bound:
SE E (f( - 31,, (9h
cI. MS (N (B(MS) क (-MMS) (8(M))
2 V e
ar82
(7)
π
where T denotes the cumulative distribution function of NO,II).
+v V us² in Equation (7 as the upper bound of SE,
Denote MS (MMS)) - ((-MS) - 2 + 24(MS))
whose convergence is guaranteed by Proposition 3.4
Proposition 3.4 (Asymptotic Convergence with h). For any & 0, there erists h + +00 and MS 8, the SE upper bound converges to zero:
0 such that when
1 ((MS) - w(-MS)) - 2 (MS) + -
Mg2
lim 400
MS
=0
(8)

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The Proposition 3.4 reveals that when we fix MS and increase the spatial resolution h, the upper bound 1g(h, MS) of SE. will reduce accordingly. On the other hand, when h increases, the tensor sizes in V will increase exponentially, leading to higher computational costs. As such, the selection of h must strike a balance between the accuracy of the estimation and the computational feasibility. Since the upper bound SE decreases with an increase h is generally preferable to choose the largest possible value of based on available computational resources, resulting a fixed value of for a particular computational budget.
3.3.2 Theoretical Analysis of Optimal MS
MS determines the upper and lower limits of numerical truncation in the binary image- image-based time series metric space. Thus, is necessary to conduct a detailed thcoretical analysis of the selection of MS. Proposition 3.5 investigates how the upper bound of SE varies with MS given fixed value of h, which provides theoretical guidance to choose the best MS under different computational budgets (h).
Proposition 3.5 (Optimal MS Selection). For fired h, there erists unique optimal threshold MS mondmizing the SE upper bound, characterized by:
1 ((MS)) - 2 (MM + M h
2 π
The fidelity of predictions in binary image space V heavily depends on the bidirectional mapping between discrete space V and continuous latent space S. A key challenge arises from the SE, which quantifies the discrepancy between the original continuous representation and its reconstructed version after discretization. While Assumption 3.2 assumes S ~ N(0, I), real-world scenarios often exhibit larger variance in the latent. space due to factors such as dataset shifts or model miscalibration. This motivates our analysis of SE under the generalized assumption S ~ kT), where k >>> captures the variance scaling.
Proposition 3.6 (Optimal Threshold under Variance Scaling). Under the assumption S ~ NO, kI) with k > 1, the optimal threshold MS that minimizes the SE upper bound is characterized by the following condition:
MS Vk ))
MS vT
MS
MS
2 7k
(MS')2 e
h
-
-2 24
VT
+
h
24
0
(10)
Here, (()) is the cumulative distribution function (CDF) of the standard normal distribution, h is the spatial resolution, and k is the variance scaling factor. This result generalizes Proposition 3.5 to scenarios where the latent space exhibits larger variability. In practice, it is challenging to find an analytic solution for Equation (10). Thus, the numerical method is employed to obtain solutions of Equation in this work and the corresponding results are reported in Table [
As shown in Table the numerically computed optimal MS increases monotonically with both the spatial resolution h and the variance scaling factor k. This trend indicates that higher-resolution settings and larger latent variance require larger MS to satisfy Equation
110
3.4 Theoretical Advantages of Visual Representation for Time Series Forecasting
Representing time scries data visually, as explored by ViTime, is not merely an acsthetic or hcuristic choice; it is fundamentally advant from a signal-processing standpoint. Specifically, transforming numerical signals into structured, image-like representations can significantly boost the effective signal-to- toise ratio (SNR), thereby enhancing forecasting robustness. To formally capture and quantify this advantage, we first establish conditions under which visual representation surpasses conventional numerical representation in terms of SNR. Subsequently, we explore image-based processing techniques to further amplify these benefits.
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Table 1: Numerically Solved Optimal MS*
Optimal MS
Resolution h
k
1.5
k
2
32
2.1
2.62
3.03
64
2.38
2.95
3.41
128
2.64
3.26
3.76
256
2.88
3.53
4.08
512
3.09
3.79
4.38
3.4.1 Visual Representation and SNR Enhancement.
Consider a noisy simusoidal time series defined by:
Sk Aisnnwokk +) 77k k -=
where the signal amplitude A > 0, angular frequency wo 2n/ Perrood; phase o, and Gaussian noise terms m N(0, o²) fully specify the system. Transforming this numerical scries into a binary 'stripe' image v E {0, 13 via quantization yields notable theoretical advantages. The binary representation is defined by:
8k + MS 1)
VjAk =
(11)
with quantization step = A/h and total quantization range = 2MS. By comparing the SNR in numerical and visual domains, we obtain the following foundational result:
Theorem 3.7 (Stripe SNR Boost). Under mild assumptions that (i) the sinusoid amplitude spans at least one quantization bin (6 S AS A 5) and (ti) noise small relative quantization resolution (a < 8/4), the visual representation yields on SNR at the fundamental frequency no [L/PPerioa] satisfying:
82
0²
SNRvis 2 exp
mmmm:
(12)
802
1
where the numerical SNR is SNRaum = 220(2
Theorem 3.7 provides clear quantitative conditions for visual superiority. Specifically, visual representation surpasses numerical representation (SNRvis > SNRmum) whenever:
L 442 σ² exp
82
(13)
8
Practically, this condition is typically met for moderate sequence lengths when the quantization step is comparable to slightly larger than the noise standard deviation (e.g., 8 R 20). Under these realistic scenarios, the exponential term strongly favors visual representation, making it advantageous even at manageable L.
3.4.2 SNR Enhancement via Image Processing.
Although the theoretical advantage above is compelling, practical scenarios often involve considerable noise and subtle periodic signals. Furthermore, the binary quantization can introduce high- freefrreyyncy artifacts that obscure signal patterns. To mitigate such undesirable effects and leverage the structured nature of visual representations, we propose employing imogesprocsssing operations, notably Gaussian blurring, to enhance signal fidelity further.
Applying a Gaussian blur along the image's quantization axis (the row or "valuc" dimension) effectively smooths quantization noise while preserving meaningful temporal structures. This simple convolutional operation yields significant amplification of the visual-domain SNR, formalized follows:
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Theorem 3.8 (Gaussian Blur SNR. Boost). Under the conditions of Theorem
3.7
consider applying
one-dimensional Gaussian convolution kernel along the quanlization dimension Tows) of the binary stripe. image o:
where ==Σ exp (- 20
(-207
9j exp
20
to obtain the blurred image = 2* v. Denote the kernel's nuclear eneryy by S = Ej 9f (0, 1), and define the visually blurred SNR at the fundamental frequency no LL/ SNR Then, the following lower bounds hold:
L
8²
SN
exp
(14)
45
1802
Lo2
82
SNR
2
exp
80
SNRuums
(15)
where the numerical-domain SNR defined as SNRaum
A² 42/((22).
Consequently, the blurred visual representation amplifies the numerical-domain SNR least by a factor of:
SNR
blur
Lo²
8²
2A25 exp
(16)
SNRum
S
This result explicitly quantifies the advantage provided by Gaussian blurring in the visual representation. Notably, this amplification advantage scales linearly with the time series length L and exponentially with the squared ratio of quantization step to noise standard deviation Morcover, a smaller kernel muclear energy S, corresponding lo stronger blurring, yields greater amplification of the visual-domain SNR relative to its numerical counterpart.
In practical implementations, the choice of Gaussian kernel parameters directly influences the muclear energy S, and thus the SNR amplification factor. Typical examples include:
11 x 11 kernel (a, 2): S 0.15, providing substantial SNR amplification.
21 21 kernel (o, 4): S = 0.08, approximately doubling the amplification compared to the previous case.
31 31 kernel (a, 6): S 0.05, further significantly enhancing the amplification factor.
In summary, even moderate Gaussian blurring substantially enhances the effective visual-domain SNR, enabling siguificantly improved signal disceruibiiity and forecasting accuracy compared to traditional numerical-domain methods.
Generalization to Complex Time Series. While theoretical analysis explicitly addresses single sinusoidal component, its implications readily extend to realistic time series composed of multiple periodic components. Via linearity principles inherent in Fourier decomposition, observed visual-domain SNR advantages apply component-wise, amplifying structured periodic signals relative to unstructured and independent noise effects. Thus, real-world time series exhibiting intricate periodic behaviors benefit significantly from visual transformations and subsequent image-processing enhancements.
The rigorous theoretical results presented here establish robust mathematical foundation for employing visual intelligence in time series analysis. Beyond aligning with human cognitive patterns, visual representations structurally amplify signal fidelity through inherent quantization and subsequent image processing techniques, such as Gaussian smoothing. Consequently, visual-domain methods provide a principled, theoretically justified route toward achieving more robust, reliable, and accurate time series forecasting, especially under challenging noise conditions.
Detailed proofs and supplementary details of the theorems presented in this section are provided Supple- mentary Section
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3.5 The Proposed ViTime Model
Figure 1 (b) presents the architecture of the ViTime network, which comprises three network modules: the Visual Time Tokenizer, the Decoder, and the Refining Module. The time series binary image is first fed into the Visual Time Tokenizer and outputs embedded latent representations. Next, the Decoder is developed to decode latent representations and produce initial prediction results in the image-value axis. To improve the generative quality of patch junctions, a Refining Module is designed to generate the final smooth prediction results.
Visual Time Tokenizer. The primary role of the Visual Time Tokenizer is to segment masked binary images into multiple patches and map these patches into the feature space. By leveraging the ViT (Dosovitskiy et al. 2020) architecture, the module captures spatial relationships between patches, thereby transforming temporal dependencies of the time series into spatial dependencies within the image space.
Decoder. The Decoder translates the tokenized patches back into the binary pixel metric space, providing an initial prediction where the VIT architecture also adopted. In practice, the Decoder's prediction head applies softmax along the height. dimension j to produce probability lensor p P (defined later in Section 3.6), which reduces to a hot vector along the height dimension V if the mass collapses to single bin.
Refining Module. The transformer architecture in the Decoder can result in discontinuities at the patch junctions, which may affect the accuracy of the inverse mapping process. To address this issue, the Refining Module built with CNNs is employed. Initially, tokens decoded by Decoder are unpatched and fod into a CNN-based backbone. Next, the ASPP (Chen el. al. 2015) module expands the model receptive field. Finally, the output is upsampled to the binary pixel metric space, generating the final image prediction result. The Refiner preserves the probabilistic semantics by operating on the logits (before softmax) probability maps to maintain consistency along the j-axis.
Modeling process and masking.
The modeling process of ViTime is summarized as
v ViTime(vr. O ML),
(17)
where @ is the element-wise wise product and ML is a temporal mask that zeros out the time steps to be forecast. Concretely, we use M2 E {0, 1 xxxxI (broadcast along e and h) with
1, 1 E observed (context) time indices,
(ML)1,1
k E forecast horizon (to be predicted).
Thus, for all i [c].j e [h], k e [L],
MI,1A = 1, ViJ.k.
(VL O ML) =
0,
MI,1A 0,
i.e., the mask sets the to-be-predicted time positions to all zeros across the j-aris, removing any target information at those steps. Note that this masking operates on the input only. Although the masked columns are no longer one-hot (and hence leave V at those k), the network outputs valid distributions E P over for every (i, k) (see Section 3.6). During training, we sample masked spans to simulate forecasting: at inference, we set Mj. to zero precisely on the forecast horizon.
Loss function. The loss function employed in this study is defined as follows:
L d (vi, vL) + a KLD (Vi.
(18)
where denotes the distance function defined in Equation 3 KLD denotes Kullback Leibler divergence, and a is the hyperparameter balancing the two terms. The combined EMD and KLDD loss addresses structural and probabilistic alignment along the j-axis. EMD minimizes spatial discrepancies in V/P, counteracting dissretizationninduce shift, while KLD refines distributional consistency to mitigate quantization artifacts.
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This dual objective balances geometric fidelity (via EMD/ EMD/WasD /assersttin-n along j) and statistical accuracy (via KTDD), which is crucial under the resolution-oopptationn trade-off governed by h. In practice, to prevent. information leakage and trivial identity mapping, both d(---) and KLD are accumulated only over the masked time indices {k: M1,1 = 0}, while the unmasked indices serve as conditioning context.
From model outputs to forecasts. The Decoder Grooder/Refiner Refiner produce a probability tensor p E P over the height at each ki, For point forecasts, we apply the inverse mapping expectation (see Equation (5) by replacing Vjjjk with Pi.j. to obtain Mi,k: this coincides with the estimator in Equation (5). For proba- histogram distribution over bins and compute downstream summaries as detailed in Section 3.6
bilistic forecasts, we retain (quantiles/
intervals) (quantilesinteervlss)
3.6 Point and Probabilistic Forecasting in ViTime
Building on the binary image-based time series metric space in Section 3.3
ViTime treats forecasting
producing distribution along the height (value) axis for each variate- pair. This subsection details how ViTimm yields probabilistic forecasts and how point forecasts are recovered as expectations under the same formulation.
From one-hot images to probability tensors. We relax the one-hot constraint V. E {0,1} to a probability-sinplex oulput Pij.e E [0, 1] with E Pij.e = 1. Define
P _ p E [0, 1exxxL
Σ Pa.tke 1, vi ld, [
(19)
so that V C P and a one-hot tensor v is degenerate case of p E P. In practice, the prediction head of ViTime applies a softmax over the j-dimension to produce E P.
Let the bin width be A 2MS/h, edges by = -M,, by = MS + jA for j = l...., h, bins B, = (bb-11b)). and centers Cj = (j 0.5) MS.
Probabilistic forecast (distributional output). For each ki, ViTime interprets the h-way probability vector Pi,l,h,k as histogram (mixtuur--ffumifrmmss) predictive distribution on [-MS, MS] with
(s) A
Pij.k 1(s E
Bjh
F.((s) 2
Σ Pi + Pij.k bj-1
s Bj:
(20)
where (s) = for s < - MS and (s) for > MS. This continuous relaxation preserves the geometry of the j-axis used by the EMD metric Equation and naturally supports uncertainty quantification.
Point forecast as expectation. Under Equation 20 the predictive mean recovers the inverse mapping in Equation 5 by replacing Wij.k with Pi.a.::
2MS 0.5)
Σ
oj Pi
Σ
(j
MS Pi.j.
(21)
Thus, when only a point forecast is required, ViTime outputs (i which coincides with Equation 5 The predictive variance for uncertainty summaries is
2
Var
Σ
PijiA (ce 22,x)) 1=1
(22)
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Quantiles and prediction intervals. For E (0, 1), let
Define cumulative weights Cs,(()) _ E Pi,m with Ci,( = 0.
J, - min (j E [M] C.x() Qi,k(T) -1 T -,,
(23)
Pi
Then central (1 a:) prediction interval is (Qix(a/2), Qi,k(1 - aa/2]]
3.7 Evaluation Metrics
Existing numerical fitting-based TSF foundation models, e.g., TimesFM, are typically pretrained on com- prehensive real-world datasets. While the specifie nomenclature of the testing set may not be explicitly listed in the training data, there is possibility that the real-world dataset encompasses similar data sources, potentially leading to issues of test set leakage. To address this concern and ensure more rigorous and equitable experimental comparison, we propose novel metrics for zero-shot evaluation. For point forecasts, we introduce the Rescale-Mean Absolute Error (ReMAE) and Rescale- Mean Squared Error (ReMSE). To evaluate probabilistie forecasts, we extend this approach with the Discale-Contimund Ranked Probability Score (ReCRPS).
The fundamental principle underlying these metrics involves rescaling the test dataset across various time resolutions, as illustrated in Equation (24). The time series interpolation (TST) method is employed to rescale the original test time series of length T to BT:
SØr = TSI (Sr, rescaling factor = B).
(24)
For point forecasts, the formulas for ReMAE and ReMSE are based on the standard Mean Absolute Error (MAE) and Mean Squared Error (MSE):
ReMSE = Eseu MSE (S SKT
(25)
len(U)
ReMAE = ΣSEU MAE (S SKg len(U)
(26)
For probabilistic forecasts, the Continuous Ranked Probability Score (CRPS) is considered, which generalizes the MAE by comparing the entire predictive distribution with the ground truth. The CRPS is defined as:
CRPS(F, = f (F(x) - 1(x20)) dx
(27)
where F is the predicted cunulative distribution function (CDF) and y is the observed value, with 1 being the Heaviside step function. Following the same rescaling principle, we define ReCRPS
ReCRPS EeEu CRPS (Frr, Sgr)
(28)
len(U)
In Equation 24 Eqq-Etatt
Eer represents the point prediction, Far is the predicted distribution for the rescaled series, Sør is the rescaled ground truth, and U is the set of scaling factors:
28
U [0.5, 0.66, 1,
(29)
The proposed ReMSE, ReMAE, and ReCRPS metrics address a critical challenge in evaluating time series foundation models: mitigating test set leakage caused by overlapping data. distributions between training and
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testing phases. By rescaling the test set across multiple resolutions (3 U) via time series interpolation (TSI, Equation (24)) these metrics introduce synthetic scale variations that disrupt exact temporal patterns, thereby reducing the risk of evaluating models on memorized or overfitted data. This approach ensures a lcakago-resistant evaluation framework, as models must generalize to unseen scales rather than relying on spurious correlations learned from the training set.
A key implication of this work is the necessity of scale-agnostic evaluation in time series forecasting. Traditional single-scale metrics like MSE, MAE, and CRPS risk conflating memorization with true generalization, particularly when training data encompasses diverse real-world sources. By averaging errors across B, our rescaled metrics incentivize models to capture invariant temporal structurcs as periodicity, trends, and noise resilience that persist across resolutions. This applies to both the accuracy of point forecasts and the calibration predicted uncertainty. This aligus with recent theoretical insights in self-supervised learning, where augmentation-innun
invariance improves out-of-distribution robustness (Yao et al.) 2022). It is worth noting that in the fine-t ane-tuning study, i.c., Section 4.4 in order to ensure the consistency of the distribution between the test data and the fine-tuning data, we still adopt the traditional MSE/MAE evaluation metrics.
4
Computational Experiments
4.1
Experimental Configuration
Datasets
Seven popular publicly accessible datasets: Electricity, Traffic, Weather, ETTh1, ETTh2, ETTm1, and ETTm2 (Wu et al. 2021) are employed in computational experiments validate the effectiveness of the proposed ViTime.
Model setup
The ViTime model is developed using data sequences synthesized by RealTSS, During each training epoch, 20,000 sequences are randomly generated. After training, zero-shot testing and fine-tuning are implemented accordingly. For multivariate time series, a neaneel-ndeeenddent strategy (Nie et al., 2022) applied, prodicting cach variable separately before combining them to form the final multivariate forecast.
The default parameters for the ViTime model are set as follows: h = 128, MS = 3.5, maximum lookback window T 512, and maximum prediction length ( = 720. For fair comparison, all considered models employ a lookback length of 512 to forecast future sequences of lengths 96, 192,336, 720. Additionally, we adopt the Adamn optimizer {Kingna 2014) with a learning rate of 2x10-4 during the training process. More details on training are available in Supplementary Section
To further enhance temporal resolution and information density practically, input sequences are initially interpolated to twice their original length (2L) and the prediction results are interpolated back the original length. This interpolation increases temporal gramularity, facilitating more precise pattern extraction. Furthermore, Gaussian blurring with kernel size of 31 applied to the binary images before processing by ViTime significantly reduces sparsity and increases local information density, thereby reinforcing the theoretical advantages outlined in Section 3.4.1
4.2 Comparison of ViTime to SOTA TSF Benchmarks Under Zero-shot Point Forecasting Setting
For zero-shot performance comparison, we consider four variants: (1) ViTime our proposed TSF foundation model, trained generative data from RealTS and adopting a zero-shot paradigm; (2) ViTime-TFP variant of ViTime, which is trained the same public available dataset TimesFM. (See Supplementary Section B.3 for more information.) (3) PatchTST-ZS trained on the same RealTS- gegerateerd data as ViTime, using a numerical fitting paradigm to create a zero-shot version of PaIchTST. (4) Moirai (Woo et al. 2024), Moment (Goswami et al.) 2024), VisionTS (Chen et al.) 2024) and TimesFM (Das et al. 2024) powerful TSF foundation model pre-trained on extensive real-world datascts. All models employ lookback length of 512 to ensure fair comparison. Details of benchmark model configurations are reported in Supplementary
Section B.4
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(a) Experimental Results With Metrics of MSE and MAE
Model
ETThl
ETTh2
ETTml
ETTm2
Electricity
Trafhic
Weather
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
Numericnl Models
Moirai
0.434
0.439
0.346
0.382
0.382
0.388
0.272
0.321
0.188
0.274
1.779
0.766
0.238
0.261
Moment
0.691
0.585
0.341
0.350
0.845
0.580
0.257
0.317
0.837
0.763
1.375
0.788
0.348
0.429
VS
0.390
0.414
0.333
0.375
0.374
0.372
0.282
0.321
0.207
0.294
0.443
0.284
0.269
0. 0.292
TimosFM
0.442
0.430
0.356
0.389
0.424
0.419
0.328
0.3-17
0.151
0.245
0.369
0.245
0.229
0. 0.255
PatchTST-22
1.237
0.831
0.903
0.710
1.356
0.825
0.839
01.622
1.311
0.885
1.873
0.945
0.907
0.588
Vision-Assiston Assisted Models
ViTime--FM,
0.398
0.387
0.321
0.350
0.382
0.377
0,295
0.312
0.136
0.221
0.332
0.221
0.206
0.229
ViTine
0.545
0.449
0.284
0.344
0.409
0,398
0.302
0.341
0.196
0,250
0.730
0.386
0.286
0.280
(b) Experimental Results With Metrics of ReMSE and ReMAE
Model
ETThI
ETT12
ETTml
ETTm2
Electricity
Traflie ReMSE
Weather
ReMSE
ReMAE
ReMSE
RoMAF
RoMSE
ReMAE
ReMSE:
ReMAE
ReMSE
ReMAR
ReMAR
ReMSE
ReMAE
Numerical Models
Moina
1.144
0.722
0.754
0.167
1:118
0.819
0.455
0.397
0.859
0.676
1.116
0.891
0,706
0.114
Moment
LO89
0.498
1.138
Visiou/IS
0
1.016
0.521
0.350
0.873
0.559
0.773
0.516
0.851
(1669
1.173
0.669
0.519
0.327
TimeFM
0.396
0.744
PatchTST-22
1,777
0.903
1.097
0.775
1.295
0.798
0.805
0.613
1.414
0.921
2.054
1.002
0.911
0.584
Vision- Awsisted Models
TFM
0,481
0.451
0.314
0.354
0.519
0.455
0.276
0.325
0.301
0.350
0.718
0.460
0.237
0.261
VYTume
0.457
0.431
0.29
0.346
0.473
0.420
0.237
0.301
0.308
0.721
0.400
0.203
0.228
Table 2: Overall Experimental Results Comparison
Average MAE for Different Rescale Factors
0,66
ViTime
TimeFM
0,455
6.6%
8.65
8,5
Figure 2: Radar plots comparing the average MAE of ViTime and TimosFM across different rescale factors. The radial axis represents MAE, with lower values (larger radius) indicating better performance. Each axis corresponds to a specific rescale factor.
Table 2 summarizes the zero-shot performance of all models using traditional metrics (MSE, MAE) and our proposed scale-invariant metrics (ReMSF, ReMAE). As shown in Table 2a. our vision-assisted models demonstrate highly competitive performance. ViTime achicves the best results on the ETTh2, ETTm2, and Weather datasets, while its variant, ViTine-TFM secures the top performance Electricity and Traffic datasets. Notably, ViTime-TFM, which shares the same training data TimesFM, consistently outperforms it on most datascts, underscoring the inherent advantages of our vision-based modeling approach.
The superiority of ViTime becomes even more pronounced when evaluated with scale-invariant metrics, as shown in Table 2b ViTime demonstrates remarkable dominance by achieving the best ReMSE or ReMAE on 11 out of evaluation scttings. This highlights its robust generalization ability across different temporal resolutions zero-shot scenarios. Furthermore, ViTime significantly outperforms PatchTST-ZS across all datasets and metrics, confirming the effectiveness of visual intelligence strategies numerical fitting for
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zero-shot forecasting. The strong performance of ViTime ReMSE and ReMAE, compared to the still-strong but less consistent performance of ViTime-TFM, suggests that the synthetic training data from RealTS is crucial for enhancing zero-shot generalization across varying temporal scales.
To further assess robustness, Figure 2 presents the performance across different rescaling factors. TimesFM exhibits optimal accuracy only at the original scale (3 = 1), suffering significant degradation when evaluated at other scales. Such behavior indicates sensitivity scale-specific patterns and suggests potential data leakage from the original resolution. In contrast, ViTime maintains consistently robust forecasting performance across all rescaling factors, evidenced by stable ReMSE and ReMAE metrics. This illustrates ViTime's ability to learn intrinsic temporal relationships independent of specific time resolutions, further reinforcing the robustness and generalization benefits of vision-based modeling trained on RealTS data.
Large-scale benchmark note. A large-scale zero-shot evaluation the community GIFT-EVAL bench- mark [Aksu et al. 2024 is provided in Supplementary Section D.2
4.3 Comparison of ViTime to SOTA TSF Benchmarks Under Zero-shot Probabilistic Forecasting Settings
Table 3: Comparison of probabilistic forecasting performance. Within each scenario, the best results (lower is better) for each dataset are bolded.
Method
ETTh1
ETTh2
ETTml
ETTm2
Electricity
Traffic
Weather
CRPS
Moirai
0.506
0.274
0.538
0.309
0.522
0.626
0.506
Lag-Llama
0.441
0.401
0.435
0.432
0.470
0.548
0.394
ViT'mee
0.356
0.319
0.344
0.286
0.267
0.327
0.244
ReCRPS
Moirai
0.537
0.382
0.631
0.329
0,609
0.684
0.620
Lag-Llana
0.478
0.442
0.463
0.420
0.514
0.629
0.377
VITime
0.358
0.318
0.346
0.283
0.266
0.324
0.241
For zero-shot probabilistic forecasting, we evaluate three variants: (1) ViTime our dision-assiste TSF foundation model trained on generative data. from RealTS and deployed in zero-shot paradigm; (2) Moirai 2024) strong TSF foundation model pretrained
large-scale real data; and (3) Lag- Llama Rasul et al. 2023 an LLM-based probabilistic forecaster. All models adopt. a lookback length of 512 to ensure a fair comparison. We report both CRPS and our rescaling invariant metric, ReCRPS.
Woo ct al.
As summarized in Table 3 ViTime delivers stato-of-tho-ar zero-shot probabilistic performance. Under the standard CRPS metric, ViTime achieves the best results on out datasels, with particularly substantial gains on large multivariate benchmarks. For instance, compared to the strongest baseline, ViTime achieves relative CRPS reductions of approximatcly 43% on Electricity, 40% on Traffic, and 38% Weather. On the ETTh2 dataset, while Moirai attains slightly lower CRPS, this narrow advantage is reversed when scale effects are controlled for. Using the ReCRPS metric, ViTime achieves the best performance on all 7 datasets, demonstrating superior distributional calibration across scales. These results indicate that ViTime not only produces sharper forecasts but also maintains this accuracy and calibration robustness across diverse temporal resolutions, establishing it as a highly reliable zero-shot probabilistic forecaster.
Overall, ViTime emerges as a robust, accurate, and reliable zero-shot time series forecasting model. The effectiveness of ViTime stems from two key innovations: vision-assisted modeling and synthetic training data generated by RealTS, Together, these features enable ViTime to generalize effectively across heterogeneous datascts and varying temporal scales. The strengths of ViTime are evident in both point and probabilistic forecasting singsng. For point forecasts, ViTime delivers strong performance across diverse applications. For probabilistic forecasts, ViTime demonstrates exceptional qualities it maintains scale-robust calibration and consistently achieves lower CRPS and ReCRPS scores across different datasots and resolutions. These comprehensive results establish ViTime as dependable zero-shot forecaster that excels in both point istimation and distributional prediction tasks.
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4.4 Comparison of ViTime to SOTA TSF Benchmarks Under Fine-tuning Settings
Table 4: Comparisons of Fine-tuning forecasting results with MAE. FT is short for fine-tuning: The best MAE results are bolded, and the second best are underlined. Standard deviations shown in the second row for ViTime.
Method
Data proportion
ETThl
ETTh2
ETTml
ETTm2
Electricity
Traffic
Weather
TMmesFM (FT)
10%
0.426
0.410
0.388
0.334
GPTTTS (FT)
10%
0.542
0.431
0.466
0.343
Not Reported
TIME-LLM (FT)
10%
0.522
0.394
0.426
0.323
0.422
0.370
0.376
0.312
0.250
0.251
0.252
ViTime (FT)
10%
(+000034)
(+0.007)
(=00003)
(=0.008)
(=0.008)
(#00008)
(=0.006)
PatchTST
10%
0,542
0,431
0,466
0.343
0.268
0.286
0.283
PatchTST
100%
0.434
0.381
0.382
0.317
0.253
0.264
0.264
SiMBA
100%
0,433
0.392
0.396
0.328
0.274
0.291
0.281
TIMESNET
100%
0.450
0.427
0.406
0.333
0.295
0.336
0.286
Transformer
100%
0.407
0.410
0.332
0.270
0.282
0.278
TimeMixer
100%
0.423
0.384
0.376
0.316
0.246
0.263
0.262
0.406
0.344
0.366
0.297
0.245
0.248
0.249
ViTime (FT)
100%
(4:0.039)
(4:0.004)
(1.0.003)
(100.017)
(4:0.004)
(4:0.005)
(#00004)
Original series
Augmented series
VITime TImesFM
Average MAE Proportion
Original series
5.0
Performance Impe Improveen ovementt
4.1%
Aterage MAE
0.320
0,320
3.1%
Real World
Noise
Pecformance Impooveceett over 1005
Time Series
0.315
515
Harmonic
0.115
0.312
0.311
0.310
www.mmmm
Data
Missing
2.1%
Data
Figure 4: Porformance comparison of ViTime versus TimesFM on TSF tasks under various data pertur- bations: Original time series. b. Time series with noises injected. c. Time series with harmonic added. d. Time series with missing data.
Figure 3: Performance with different fine-tuning data proportion.
While zero-shot results demonstrate the predictive capability of ViTime unseen data, some high-precision TSF tasks might require further fine-tuning studics to enhance prediction accuracy. Thus, this section focuses fine-tuning studies across various specialized datasels.
To comprehensively evaluate the fine-tuning performance of ViT'ime, we compare ViT'me with other foundation models and SOTA supervised TSF models. Foundational models including TimesFM (Das et al.] 2024). GPT4TS (Zhou et al. 2023a), and TIMF-LLM (Jin et al., (023) are fine-tuned using 10% of the training data. Recent SOTA-supervised TSF models such as SiMBA (Patro & Agnceswaran 2024), TIMESNET (Wu al., 2022), iTransformer [Lu al.] 2023), TimeMixer (Wang et al. 2024) and PatchTST (Nie et. al., 2022) use 100% of the training data, reported in their respective papers. We also fine-tune ViTime using between 10% to 100% of the training data to provide comprehensive comparison.
Results of the fine-tuning study are provided in Table ViTime fine-tuned with 10% of the training data can outperform other foundational models and the latest supervised models updated on 100% of the training data. Furthermore, as shown Figure 3 when the finc-tuning data proportion approaches 100%, the prodiction accuracy of ViTime gradually increases and significantly surpasses all existing models, which suggests that ViTime excels in both low-dataa-vailability environments (10% fine-tuning) and full-data-availabblle scenarios
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(100% fine-tuning), consistently outperforming both other foundation models and specialized supervised models.
4.5 Robust Inference and Generalizability Analysis
Table 5: Comparison of average ReMAE forecasting results.
Method
ETThl
ETTh2
ETTml
ETTm2
Electricity
Traffic
Weather
GN standord deviations
0.1
TimesIM
0.40110.002
0.39310.001
0.10220.012
0.35211.0001
0.40310.004
0.51210. 016
0.280.10.00
VITime
0.449=0.002
0.360=10.001
0.429=0.0001
0.329=0.0002
0.340=#00006
0.402=10.014
0.279=0.007
GN standard deviations 0.3
TimesFM
0.47310.007
0.39011.000
0.48510.010
0.344
0.41411.0006
0.51810.011
0.28811.0005
VITime
0.455=0.000
0.363+0.000
0.434=100002
0.333#0.003
0.355=0.000
0.426=0.0016
0.258#0.009
GN standard deviations 0.5
TimeFFM
0.479+0.006
0.392+0.003
0.488+0.009
0.345+0.00
0.433++0.00
0.529+0.012
0.295+0.005
VITime
0.466=10.003
0.370=10.002
0.445=10.003
0.337=0.003
0.371#0.007
0.461#0.011
0.294+100013
GN standard deviations= 0.7
TimesFM
0.483+00000
0.394+00.003
0.492+00.006
0.349+0.005
0.450+0.0005
0.543+0.015
0.302+0.0005
ViTime
0.484400.004
0.394:10.003
0.443:10.003
0.346:10.005
0.377:10.006
0.510:10.021
0..01
GN standard deviations-
I.0
TimesFM
0.487
0.399=00003
0.500++0.00%
0.359+0.000
0.475++0.000
0.567+0.000
0.312++0.00
ViTime
0.007:10.006
0.408100.005
0.401:10.004
0.358:10.000
0.415:10.010
0.546:10.022
0.305
DM P 0.3
ViTime
0.453+0.002
0.378
0.432+00.001
0.337=0.003
0.343+0.006
0.417=00001
0.281=0.013
Overall Performance: Average Across All Datavels Datasets
(u) Average ReMAE Across
(b) Averuge ReMAE Chunge Rate
TimesPM (Arg)
TinesFM (Ang Rate)
Mean Absolute Errot (Average)
TimesEM (+3e dauseet)
Vitimo (Ang Ratc)
5.50-
ViTime (Arg)
ViTime (1ln acress desseeto
MMEEOON (Averape)
4.30
0.43
0.3
0.3
10
Gaussian Naise Level
Gaussian Noise Level
Figure 5: Robustness analysis under increasing Gaussian noise levels.
rigorously assess the robustness and generalizability of ViTime, we conducted comprehensive zero-shot experiments comparing its performance against TimesFM under various data perturbation scenarios, including original time series, Gaussian noise (GN), harmonic augmentation, and missing data (DM). These scenarios represent challenges often encountered in practical forecasting tasks, evaluating the capabilities of models in maintaining predictive accuracy amidst compromised data quality.
In our analysis of robustness against noise, we consider two complementary dimensions:
Absolute Robustness, defined as the ability of model to maintain a superior absolute performance (i.e., lower prediction error) across varying levels of noise.
Relative Robustness, which refers to the rate of performance degradation as the noise intensity increases. A model with higher relative robustness exhibits smaller increase in error for given increase in noise.
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648
Average ReMAR
1.4
DM level
Figure Average ReMAE of ViTime across different DM rates.
Our extended experimental results as depicted in Figure and detailed in Table 5 provide a nuanced view of the behavior of models. In terms of absolute robustness, Figure (a) clearly shows that the ReMAE of ViTime is consistently and significantly lower than that of TimesFM M across all tested Gaussian noise levels, from standard deviation of 0.1 to 1.0. This demonstrates that ViTime reliably delivers more accurate predictions in both low- and high-noise environments. We argue that this sustained performance advantage is critical aspect of robustness for real-world applications, where the primary goal is achieve the highest possible accuracy under given conditions.
Regarding relative robustness, the analysis of the performance degradation rate shown in Figure (b), confirms the observation that the performance of ViTime is more sensitive to increasing noise. The slope of ViTime's ReMAE curve is generally steeper than that of TimesFM, indicaling a lower relative robustness. We posit that this is an expected phenomenon; as noise approaches an infinite magnitude, the predictive power any model should diminish, and their error rates will converge towards a high value determined by the inherent scale of data.
In summary, while ViTime is more sensitive increases in noise (lower relative robustness), its foundational performance is so strong that its absolute prediction accuracy remains superior to TimesFM across the entire spectrum of tested noise levels. For instance, as shown in Table 5 even at high noise level of 0.7, ViT'mme outperforms or matches TimesFM on all datasets. In practice, an end-user is more concerned with which model provides a more reliable result (lower RcMAE) in a given noisy environment, rather than which model's performance curve is flatter. Therefore, ViTime's exceptions absolute robustness makes more dependable and effective choice for forecasting in the presence of noise.
The most distinctive performance disparity emerges under the scenario of data missing (DM). As shown for the DM P=0.3 case in Table 5 ViTime decisively outperforms the baseline across all tested datasets. TimesFM, being reliant on numerical fitting, would necessitate explicit imputation strategies to handle such data gaps. Conversely, ViTime robustly accommodates missing values by interpreting them as zero- valo-valued pixels within its visual representations. Consequently, ViTime leverages spatial dependencies among available data points effectively, maintaining high prediction accuracy even amidst substantial data sparsity.
To further validate the robustness of ViTime varying degrees of missing data, we systematically evaluate its forecasting accuracy across data missing ratios ranging from 10% to 90% (Figure 6). Results reveal that ViTime sustains remarkable forecasting performance with minimal degradation until data missingness surpasses 50%, underscoring its exceptional resilience to incomplete data scenarios.
Collectively, these extensive evaluations substantiate the superior robustness of ViTime and generalizability compared traditional numerical fitting-basod methods. Its inherent capability to effectively mitigate perturbations through visual representation learning positions it as highly promising approach for real-world forecasting applications, where consistent data quality cannot always be guaranteed.
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Table 6: Empirical Forecasting Performance under Different MS Values
MS
2.38
2.64
2.88
3.09
3.50
5.00
6.00
ReMSE ReMAF
0.4423
0.4404
0.4400
0.4348
0.4178
0.4780
0.4724
0.3818
0.3812
0.3811
0.3788
0.3759
0.3990
0.3959
4.6 Ablation Study
4.6.1 Ablation of MS
establishes the thooretical relationship between the optimal MS threshold and the variance scaling factor & in the latent space. For stationary data (8 ((0,, I), i.e., k = 1); Proposition 3.6 reveals that with h 128, the optimal MS should be 2.64. However, real-world time series often exhibit non-stationary characteristics. Our pre-analysis of the target variable's variance after input-based standardization (see Supplementary Section B.2 demonstrates that the effective k value for the prediction horizon falls within [1.5.2 2] across all benchmark datasets.
Proposition 3.6
Table provides numerically solved optimal MS* values under different & and h configurations. For h = 128 (our experimental setting) and &o E [1.5, 2]. the theoretical optimal MS ranges between 3.26-3.76. This motivates selection of MS 3.5 balanced configuration within this interval.
To validate this choice, Table 6 presents the average relative ReMSE and ReMAE across six benchmark
datasets under zero-shot setting. The results demonstrate that MS = 3.5 achieves the minimum forecasting
error, reducing ReMSE by 4.1% and ReMAF by 1.8% compared to the stationary optimal MS = 2.64. This
strong alignment between theoretical predictions (Table [ and empirical performance (Table 6) confirms that our MS selection stralegy effectively minimizes system error while accommodating real-world data characteristics.
4.6.2 Ablation of Loss Function
Table 7: Ablation study of loss function components prediction performance.
Loss Configuration
Metric
EMD Only
JSD+EMD (Ours) JSD Only
Average ReMAE
0.3941
0.3759
0.3956
Average ReMSE
0.4586
0.4178
0.4637
In this section, we conducted ablation studies on the loss function components of ViTime under zero-shot. setting. Table 7 compares model performance under three configurations: (1) EMD alone, (2) our proposed loss function in Equation demonstrate that our obje-viee loss achieves optimal performance on both ReMSE and ReMAF.
a = 0.2 to balance the quantity level, and (3) JSD alone. The results
4.6.3 Ablation of Other Configuration
In this section, perform several ablation studics to gain deeper insights into ViTime model configuration. The results are reported in Figure] 7 Figure 7 a depicts the influence of varying spatial resolutions (h) on model accuracy. Although incrcasing h slightly improves the prediction results, the associated computational cost increases exponentially. Thus, setting h to 128 is more economical and efficient. Figure 7 b illustrates the effect of different lookback window lengths (T) on prediction accuracy. is evident that a longer lookback window length significantly enhances the model's prediction accuracy. Figure 7 reports the prediction accuracy across different model sizes. The data shows that models with more parameters tend to perform better. Moreover, the proposed ViTime achieves superior performance with only 93M parameters compared
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0.6
0.4
ReMAB
0.40
ReMAF
ReMAE
0.5
0.35
0.2
0.4
0.30
64
128
256
250
500
750
1000
24
93
Input Length
Parameters(M)
(a) Forecasting performance varying
(b) Forecasting performance varying with the length lookback window
(c)
Forecasting performance across different model sizes
with
Figure 7: Ablation studies with zero-shot forecasting. Note: The model size of ViTime used in computational experiments is 93M parameterss version.
(a) Grad-CAM heatmap showing attention key trend changes.
wwmmmm
(b) Attention maps at different prediction positions demon- strating temporal dependencies.
Figure 8: Visualization of ViTime's attention mechanism. Despite not using autoregressive paradigm, ViTime exhibits sequential processing patterns through its multi-layer self-attention modules.
with TimesFM, which is over 200M parameters, further demonstrating the efficiency and effectiveness of ViTime.
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4.7 Interpretation of ViTime
Figure illustrates the attention mechanism of ViTime through Grad-CAM (Sevaraauu et al 2017 heatmaps and position-specifie attention maps. The Grad-CAM results demonstrate that ViT'ime focuses strongly periods of fundamental trend changes. Further analysis through attention maps at different prediction positions reveals an interesting ing pattern: despite not adopting an auloregressive paradigm, ViTime's multi-layer self-attention modules process information in a temporal sequence. The imput data and the predicted results from previous time steps determine the spatiotemporal distribution of prodictions at cach time step. This aligns with human cognitive patterns, where information is processed from the recent the distant past while maintaining awareness of known information.
Prediction
Prediction
35-
3-
Input Sequence
Input Sequence
30
3-
1000
2-
10-
1000
(a) MS=3.5
(b) MS=3.5
Prediction
Prediction
Imput Sequence
Input Sequence
30
100
3000
1200
1000
1256
(c) MS-7
(d) MS-7
Figure 9: Resolution analysis for explosive growth patterns: (a-b) With MS=3.5, ViTime incorrectly predicts peak decline due to spatial constraints. (c-d) Doubling MS to enables accurate growth trend capture.
5
Discussion
While ViTime demonstrates state-of-the-art performance in accuracy and robustness, two key challenges warrant further investigation.
5.1
Resolution Constraints & Adaptive Enhancement.
The mapping function's truncation imposes resolution limits, particularly evident explosive growth patterns (Figure b-b). A key limitation of ViTime arises from its assumption of ~ N(0 I), which fails to capture
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the high-variance nature of explosive growth data that typically follows 8 ~ ( &I) with k > 1. As shown in Proposition 3.6 the optimal threshold MS* scales vh, implying that fixed thresholds (e.g,, MS = 3.5 for k = 1.5) become suboptimal for high-variance scenarios, introducing significant system errors and degrading prodiction accuracy.
Our empirical analysis reveals that doubling the MS parameter from 3.5 7 significantly improves prediction fidelity for explosive growth patterns (Figure Ir-d). However, excessively large MS values increase system error, demonstrated in Theorem 3.3 leading to computational inefficiency. This trade-off suggests two complementary research directions:
Elastic Resolution Enhancement: Techniques to dynamically adjust spatial resolution h based on data variance, ensuring sufficient gramularity for high-variance regions without unnocessary computational overhead.
Adaptive MS Estimation: Algorithms to estimate the variance scaling factor i and compute the optimal M in real-time, balancing prediction fidelity with spectral efficiency.
These enhancements would enable ViTime to handle explosive growth patterns more effectively while maintaining computational tractability.
5.2 Future Directions for Synthetic Data Generation in ViTime
RealTS, our synthetic data generation algorithm, plays a crucial role in performance of ViTime by creating diverse and realistic training samples that significantly enhance model gencralizability. The algorithm enriches the training data through sophisticated pattern synthesis, enabling ViTime to achieve superior zero-shot and few-shot learning capabilities demonstrated in our experiments.
While RealTS has proven effective in the current framework, several enhancements could further improve ViTime's predictive quality: 1) More advanced pattern injection mechanisms to capture complex real-world dynamics such as non-stationary processes and regime-switching scenarios. 2) Development of quantitative metrics for assessing simulation fidelity across different temporal regimes. 3) Extension to multivariate time series generation. Although thcoretically feasible by incorporating additional channels in RealTS, this extension presents practical challenges in computational demand and in generating synthetic daía that preserves realistic inter-variable correlations. These represent important directions for strengthening ViTime's data generation capabilities.
6 Conclusions
This work developed a vision intelligence-powered computational paradigm, ViTime, for developing the TSF foundation model, as compared with the numerical data fitting principles prevalently considered in literature. ViTime was inspired by human visual cognitive processes in understanding and analyzing time series. By introducing a paradigm of operating numerical data in image space and unique deep network based computing pipeline, ViTime is capable of handling both point and probabilistic forecasting, elevating the SOTA performance on zero-shot mapsshot/finee /fine-tuning TSF without relying on prior data samples. This demonstrates the great potential for reshaping the computational mechanism in TSF foundation model development. Moreover, as data often suffer from diverse contamination and variability in reality, ViTime's visual approach enables robust performance under various real-world data perturbations and alterations, showcasing its superior resilience.
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China under grant 62402384, in part by the Hong Kong RGC General Research Fund Project under grant 11213124, in part by the Hong Kong RGC Collaborative Research Fund Project under grant C1049-24GF, in part by the Shenzhen- Hong
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Kong-Macau Science and Technology Category C Project under grant SGDX22020053111205555 in part by the Hong Kong ITC Innovation and Technology Fund Project under grant ITS /TS/034/22MS, and in part by InnoHK initiative, The Government of the HKSAR, and Laboratory for Al-Powered Financial Technologies.
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A Details of RealTS
We present RealTSS, a versatile framework for synthesizing realistic time series data. RealTS employs multiple data behavior modes under two main hypotheses: periodic (pp) and trend (sx)). This section details the various behavior modes, their configurations, and provides visual examples.
A.1
Periodic Hypothesis Behaviors
Under the periodic hypothesis 4p, we employ two distinct data behavior modes:
A.1.1 Inverse Fast Fourier Transform Behavior (IFFTB)
To ensure the synthesized data adequately reflects the variation paradigms of real-world time series, we utilize IFFT as expressed in Equation (30 to simulate the underlying behavior of real-world periodic time series:
P LL Bp) 1B,-IFFT = // N (AmilAm )am)) (p, x 6 - IFFT (Am o, L)) dódAm (30)
where two empirical distributions of Fourier transform amplitudes and phases, N(Am: MAm : aamm) and ((;: up, of),, are maintained, and 5 denotes the Dirac delta function. By sampling from empirical distributions, we can obtain the amplitude and Phase vector, which is then inversely transformed back to the time domain via IFFT.
Amplitude Spectrum with Error Shading
Magnitide
2000
Magnitude
1000
-500
Frequency
Phase Spectrum with Error Shading
Phase (radians)
Phase
Frequency
Figure 10: Empirical distribution employed in IFFTB.
The empirical distributions utilized in Am and & are illustrated in Figure 10 Figure 11 During experiments, we randomly select one of two empirical distributions for generating Am and d. Figure 12 shows examples of time series generated using IFFTB.
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Amplitude Spectrum with Error Shading
Magnitude
3000
Magnitude
1000
-1000
Frequency
Phase Spectrum with Error Shading
Phase (radians)
Phase
Frequency
Figure 11: Empirical distribution Il employed in IFFТB.
Tear
Figure 12: Examples of time series generated using IFFТВ.
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A.1.2 Periodic Wave Behavior (PWB)
This behavior generates data by superimposing multiple periodic waves, which is modeled as a sum of sin, cos, and other periodic functions, speriodte: with different frequencies and amplitudes:
/ʃ
kewB
P Ls L,
B,= B,
N
SL:
A.fprriodie (wit), of XP(A)P(w) dwdA
(31)
where P(A) and P(w) denote predefined prior distributions of amplitudes and frequency; kew B denotes the number of mixed periodic functions.
For PWB, we define the prior distributions for amplitude and frequency as:
A ~ 5).5,55)
(32)
In(w) ((In(11), (nnL)))
(33)
The parameter kPwB modeled as:
P(kewn k) k)= 1 g' for k 1, ..,
(34)
Figure
13 shows examples of time series generated using PWB.
VWV
76a
Figure 13: Examples of time series generated using PWB.
A.2
Trend Data Hypothesis Behaviors
Under the trend data hypothesis Pt, we employ three distinct data behavior modes:
A.2.1
Random Walk Behavior (RWB)
The RWB models data as a stochastic process where each value is the previous value plus a random step:
(( L, Bp) =w, N (0,
(35)
Figure 14 shows examples of time series generated using RWB.
A.2.2 Logistic Growth Behavior (LGB)
The LGB models data with logistic growth function, capturing the S-shaped growth pattern:
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Figure 14: Examples of time series generated using RWB.
P P(SL L, Bp) BB,BLGB
(36)
( LL- La) 1 P(K)P(r)dKdr
N
SL: 1
where P(K) and P(r) denote predefined prior distributions of S-shaped function hyperparameters. For LGB, we define the probability densities for Carrying Capacity K and Growth Rate r as:
In(K) (IIn(1),
(37)
In(r) ~ U(((((000)))
(38)
Figure 15 shows examples of time series generated using LGB.
Tax
Taa
Figure 15: Examples of time series generated using LGB.
A.2.3 Trend Wave Data Behavior (TWDB)
TWDB combines linear trends with periodic fluctuations:
(st] 1R, = 1//0 N st,aL+ +b+ b+ AsSperiotic (2))) x ((a)P(()P(AA) (A) (w) dadbdA
STwDB
(39)
where P(a), P(b), P (A) and P (w) are predefined prior distributions hyperparameters.
In the TWDB, we define the probability densities for linear function random variables P(a) and P(b), well for the superimposed periodic wave components P(A) and P(w). The settings for P(A), P(w), and
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kew DB consistent with those used in the PWB module. The probability densities for P(a) and P(b) are detailed below:
a (--11)
(40)
b n~ U(-10,10)
(41)
Figure 16
shows examples of time series generated using TWDB.
00 M
Figure 16: Examples of time series generated using TWDB.
A.3 Data Augmentation Techniques
We sample IFFTBI, IFFTBII, PWB, RWB, LGB, and TWDB included in RealTS according to the following probability distribution: P(IFFTBI) 0.3, P(IFFTBII) 0.3, P(PWB) 0.16, P(RWB) 0.08, P(LGB) 0.08, P(TWDB) = 0.08. To enhance the diversity and robustness of synthetic data, we employ various data augmentation techniques, such as:
Multiple period replication - repeating the generated periodic data over multiple cycles to capture long-term periodic patterns.
Data flipping reversing the time series to create new patterns while preserving underlying charac- teristics.
Convolution smoothing and detrending removing underlying trends from the data to isolate periodic components, making casier for the model to learn these patterns.
Data perturbation introducing sudden changes or anomalics into the data, simulating real-world disturbances and improving the ability the model to handle unexpected variations.
More details of RealTS offered in the code part of the Supplementary Material.
B
Training Configuration
B.1 ViTime Model Structure
The detailed network configuration of the proposed ViTime is reported in Table
B.2 Data Normalization
To ensure ViTime can effectively capture patterns involving sudden changes, an in-sequence data normalization based on L2 normalization implemented. By normalizing each sequence within the data sequence, the model pay more attention to abrupt variations. The normalization process is defined as follows:
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Table Details model architecture
Module
Embed dim
Depth
Patch size
Num heads
Visual Time Tokenizer
768
(4,32)
12
Decoder
384
12
Refining Module
Component
Configuration
Dotails
Backbone
MobileNetV2
Downsample factor: 16
ASPP
branches
Dilation rates: 1,6,12,18,GAI
Low-level features
Convix1
24-+48 channels
Feature fusion
Conv3x3x2
256 channels
Upsampling
DeconvASPP
6 branches with SE attention
Sr. Sr. mean (||SI.Tl2))
(42)
std (S
B.3 Training Set of ViTime- TFM
The complete training datasets of ViTime- TFMimee-FM are detailed in Table
Table 9: Training dataset composition for ViTime-THE TFM
Dataset
Granularity
Time series
Time points
Synthotic
Hourly
3,000,000
6,144,000,000
Electricity
Hourly
321
8,443,584
Traffic
Hourly
862
15,122,928
Weather
10 Min
42
2,213,23 232
Favorita Sales
Daily
111,840
139,179, ,538
LibCity
15 Min
6,159
34,253,622
M4 hourly
Hourly
414
353,500
M4 daily
Daily
4,227
9,964,658
M4 monthly
Monthly
48,000
10,382,411
M4 quarterly
Quarterly
24,000
2,214,108
M4 yearly
Yearly
22,739
840,644
B.4 Benchmark Model Configurations
To ensure full transparency and reproducibility, this section details the configurations for the bascline models used in our comparisons.
1. Zero-Shot Forecasting (Table 2): For the zero-shot forecasting results, used the following publicly available pre-trained models:
TimesFM: 1imesfm-1.0-2000
Moirai: noirai-1.1-R-latt
Moment: MOMENT-1-large
VisionTS. base
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PatchTST: PatchTST/64
2. Fine-Tuning (Table 4): The performance scores reported for the fine-tuning experiments are the official results cited directly from the original publications of the respective baseline models.
3. Probabilistic Forecasting (Table 3): For the probabilistic forecasting experiments, we configured the baseline models as follows:
Moirai: We used the pre-trained model moirai-1.1-R-large
Lag-Llane Llama: We utilized the publicly available implementation from the original Lag-Llama paper and provided checkpoint, lag-llama. ckpt.
C
Proofs
This section provides the detailed proofs for the theorems and propositions presented in the main text.
C.1 Proof of Theorem 3.3
(System Error Upper Bound)
Theorem C.1 (Theorem 3.3 restated). Given tensor S C ReXL, where 8 follows NO,, 1) as per
Assumption 3.2 the system error (SE) from mapping to V and back, defined || (f (S) the Jollowing expectation bound:
$ I satisfies
SE :E: (f) a11 oL MS (1 (+(MS) 2(-MS)) 2 (0(MM)
2 arg2
+ - 2
(43)
where & is the cumulative distribution function (CDF) of (0, 1), e is the namber of variates, L is the time series length, h the image height (resolution), and MS is the manimum scale.
Proof. The proof considers the error for single clement of 8 and then scales by cL. Let P(s) be the PDF of (O, 1). The expected absolute error for single element is (J()) s). This error be decomposed into two parts: quantization error for |s| < MS and truncation error for |s| > MS.
1. Quantization Error (Is) S MS):
When |s| S MS, the value is mapped to bin j = ((s MS)/(2MS/h)) The inverse mapping f (f))s) reconstructs this as the midpoint of the bin, (j - (25) - MS. The maximum error in this case is half the bin width, 8/2 = (2MS/h)/2 = MS/h.
The expected quantization crror is:
MS
EQ
(f(s P(s(ss
(44)
-MS
MS
MS
P(s)ds= h { P(s)ds
MS
MS
(45)
-MS
h
MS
(#(MS) -(MM)
(46)
h
2. Truncation Error (|s) > MS):
If MS, fs) maps to the highest bin h, and f ff((s()) MS (MS/h). The error is (MS MS/h).
I MMS, [s) maps the lowest bin 1, and (f(()) MS MMS+(M//M) The error M-/S+
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For simplicity in bounding, we consider the error magnitude as |s] MS when |s| > MS. The expected truncation error is:
00
(s P(sdds + [
Er =
(-MS - s)
(47)
MS
2
(s Ps)ds (by symmetry of P(s))
(48)
MS
2
PP(s)ds - MS
P(s)ds
(49)
MS
MS
We know JMS PP(s)ds
SMS 8 /22T
12 p.- 12
222T
And JMs P(s)ds= =1 ((MS))
So,
ET 2
1
MS 12 MS(1 ((MS))
(50)
V2TT
MSM 2M5(1- ((MS)).
(51)
3. Total Expected Error per Element: The total expected absolute error for one element is Eq + Ey:
MS (Q(S)) - ((-MS)) /. - -e------ -M5 2M - ((MS))
2
{)(s) s| S
(52)
h
= MS
1 -
- -0--55(()) - 2(1 ((MSS))
+
- e
2
Ma2
(53)
h
1 (0(M)) - (S-MS) - 2 20(5M))
M
MS
+
(54)
Multiplying by cL (number of elements) gives the bound for (BI)))(((3)) - -
C.2 Proof of Proposition 3.4 (Asymptotic Convergence with h))
Proposition C.2 (Proposition 3.4 restated). For any 0, there crists 80 0 such that when h + 400 and MS da, the per-element SE upper bound
( (MMS) D(-MS)) - 2 (
2 +
ma²
91 (h, MS) MS
E
(55)
π
converges zero.
Proof. Let g1 MS) be the per-element upper bound from Theorem 3.3
91 MS) = MS h (#(MS) 4 - 2MS(1 - ((M))) +
2 M
-
(56)
π
As + +oc, the term MS (*(MS) - ((-MS) - 0 since @(MS) ((-MS) s
The remaining terms are R(MS) -MMS(1 - -(MM)) + v 2 MS²
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We use Mill's ratio for the tail probability of standard normal distribution: for MS >
1 ((M))~
((MS)
1
-MM8//2
as MS - oc.
(57)
MS
MS 2T
So,
2MM( ((MS) 2MS
-MS 12
(58)
2T
2
MS²/2
(59)
π
Thus, as MS + 00,
R(MS)
V
2 MS
+
2
/2
(60)
n
π
= 0.
(61)
Therefore, for any e > O, we can find a do such that for MS 2 do: [R(MS) < <//2.
And for any MS 8o, we can find an Ho such that for Ho.
MS ($(MS) 2 2(-MS)) < <
(62)
This implies that 4o. 91 (h, MS) = 0. More precisely, for a fixed large enough MS, as + oo, the limit is R(MS), which can be made arbitrarily small by choosing MS large.
The statement asks for convergence h for MS 2 60-
Let MS 2 6g. Then
lim 91 MS) -MMS(1 ( + v 2 - +
(63)
This limit itself tends to 0 as MS xo. The proposition asks for the expression to be small when h + AND MS 2 5g.
Taking the limit as - oo first, we get:
/2 MS ((MS)) ( (11s))-((MMS)) ((-MS) - 20(MS) + / =
-2MS(1 +(MS)) + / /2 mn2
lim
(64)
This term goes to 0 as MS + co. So, for any s > 0, there exists on such that for MS 2 do, the term is less than E.
C.3 Proof of Proposition 3.5
(Optimal MS Selection)
Proposition C.3 (Proposition 3.5 restated). For a fixd h there exists unique optimal threshold MS* > thot minimizes the per-element SE upper bound 9i (h, MS). This MS is the solution to:
1 - -0-((-(M)))))) - (4( h 20(MS*)= 0,
MS*
(65)
h
where x(x) = 6 28(MS)))
12/2 the PDF of N(O,1) 1). (Note: The original equation had -2/TeeMS** which is
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Proof. Let (M(S) = MS (+( - 4(-MS) - 20 + / -2-"
M252
We want find MS* such that ((M5) = 0. Using & f(-x)
(x) and (-=x) (xx), we have (MS)) - 4(-MS) 20( 1.
So, 91 (MS) MS (2 1) 20(MS) (x(M))
Derivative with respect to MS:
dgi
20(MS)
(20( 1) 2 2 + MS 2 (MMS) -1 -2 2 + 2MSD(MS) h
+ 22(M1)) + +22'(MS)
(66)
dMs
h
+ 2MS6(MS) 2MS0(MS) (since d' = -M5(MS))
h
(67)
20(MS) - 2M(M)) +
(MS)
(68)
h
Setting dg1/dMS- = 0:
2 (MSM) - 2 2D(M + 2MS (MMS) =0
(69)
h
This matches the condition the proposition since (MS'*) - 2 (SS) = (2 1.
To show uniqueness and minimality, we examine the second derivative or the behavior of the first derivative.
Let f(MS) dgM/
f(0) = (0- (00-1)/ 1)/h 2 2(0.5) +0 2 -- 1/h 0.
As MS + oo, (bM5) - and 6(M5) - 0.
limms-4s (MS) = 1/h 2+ 1/h (assuming 0).
Since f(MS) continuous and goes from negative to positive, there must be at least one root MS* >
The second derivative:
dg 20(MS)
+ +2((MS) + 20(MS) + (MSØ((MS)
(70)
dMS
h
h
+ 20(MS) - 2MS 2MS) ((MS)
20(MS)
h +1
(71)
M
22(MS)
(1+
(72)
For small MS, d /d/dM > 0, indicating convexity. If 1 + 2/h-2hh-M 115h-M57 MS > 0, ie., MS² < <h+
If MS < vh+2 then 91 (MS) is convex at MS", ensuring a local minimum.
The function (MS) starts negative, becomes positive, and its derivative / is positive for MS /h 2 and can become negative for MS > +h+22 + 2. This structure ensures unique minimum for MS > 0. ]
C.4 Proof of Proposition 3.6 (Optimal Threshold under Variance Scaling)
Proposition C.4 (Proposition 3.6 restated). Under the assumplion S ~ N kI) with k > 1, the optimal threshold M that minimizes the per-element SE upper bound is characterized by:
MS
MS*
))
MS
MS
2
(MB",2 2h
h
Vk
-
+
Vk
- 2 2
Vk
+
h
Tk
= 0.
(73)
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Proof. Let k). Then s' = s/Vk 1).
The original values are scaled by Vk. The mapping function f operates on s. The bins are from -MS to MS. Bin width 6, = 2M
The SE upper bound for a single element g*(MS) -. - a/--
This is equivalent to scaling the original problem. Let = vkz where 2 ~ N(0, 1). The function operates on s. The effective range for z is MS/Vk to MS/Vk.
The quantization error part:
in [- (-MS, MS). The error is bounded by MS/h.
MS MS Pa
MS MSS 1
JMS 22
3 (2h) ds
(74)
MS h
h
Let u = 8/vk. Then ds = vkdu. Limits become - -MS/ to MSIVE.
MS
MS/VE
a² 12 dus
(75)
h
-SS/VK
22T
MS
MS
MS
(76)
Vk
vk k
The truncation error part: Ms (s)ds.
[
20
3 1/(A2) do MS
f00
0
3 1(2)) de
(77)
MS
2
JMS 2
The first integral:
1
122()2)) ds Vk 1
1
uu² 2 du
(78)
MS
v 2
MS/VK
2TT
VT 1
(MS///R)11
(79)
2TT
lo V 2m
e -MM2//22
(80)
The sccond integral: MS (1 (1-0
MS vK
)
So,
b V22
e /S2/(2))) MS 1 1 Tb
(SS)
Er, =2
(81)
2T
Vie
The per-element SE bound 91 (MS) is:
MS (MS)
[ (MS) 4x(-MS) + V
2k
MS² -S2/(22) 2MS(1 (MM))
(82)
h
where (x(x)) ((x/VK).
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This can be written
(MM) = MS (((((M)) +x(-MMS) - 2 (Ms) +
2k
c
(83)
To find the optimal MS", we differentiate 91,x (MS) with respect to MS and set to zero.
Let (x)
kv)
be the PDF of N(0, k) in terms of ø.
LE (MS)
d JMS
(MMSV
6SMS/ k) J (M()).
aMS
2k e
2M
MS
2e Ma²
-MMS
aMS
2k
Vk
V
The derivative
d is: 7MS
= ( (x(-M))) - 2 (MS)) MS (f(0L(M))) - (-0x(MS))) 20x(M)))
2M (MSON(MS)
(84)
(MS) @n(- 2+ (MS)
2M 2M
(85)
h
Sctting this to zcro gives:
1
MS
MS +
(- Vk (-<])
)) - + 24
MS
2MS
MS*
-
④
= 0
(86)
h
VK
VT
h
vk
VT:
Substituting 26(x/ (/le/////V -2 (M ((2N), we get the stated condition.
The uniqueness follows similar argument to Proposition 3.5
C.5 Proof of Theorem 3.7 (Stripe SNR Boost)
Theorem C.5 (Theorem 3.7 restated). Let the length lime series 84 A sin(wnk σ) + k= o......, L-1, 1, with amplitude A > 0, angular frequency wo 2n/ / Pperiod (P (Pperiod E N+ and i.t.d. Gaussian noise TΙ N(0, 2²) be visualised as the binary stripe image E 10, defined through Uj.A = 1(j = [ss + MS)/
where = A/h, A = 2MS.
Denote SNBaum = A2/(20?) and SNRvia [2D 10.100//////////2200000 nol1]. with no = LL/P Pperiod]
Assume (i) 8 A < A 8 and (ii) 8/4.
Then, for every L > Pperiod:
L
82
a²
SNRwis 2
exp
SNR
(87)
4
802
A²
L
8²
SNRvis
2
exp
(88)
4
8a2
Proof. Let Velean be the image from A ssnn(wak Φ) and Ucloan Unoise where Unoise is the change due to T7k-
1. Deterministic Signal Power in Visual Domain.
The 2D Discrete Fourier Transform (DFT) is
Fad(v)(m,nj
Σ
Σ Uy
(89)
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We interested in the coefficient at (m, (0,no), where no L/ L/P.ee period (assuming L is multiple of
P period for simplicity, or (V/Pporiod) otherwise).
The transform of the clean signal component at (0, no) is
4-1h-
F2D (Fclean) 10, nol
Σ E
(Velean)5.4 j.ke (2x(nos/h)
(90)
k= j=0
Following the provided analysis, [[]FaD[Pctemm]1, nol =
2. Probability of Quantization Flip.
A flip means j changes due to noise Th- This occurs if 8 Sp.clean 7k crosses quantization boundary
0g = jō - MS.
The clean value Sk,eloan falls into bin jo- flip occurs 8 falls into jo + 1, 2
The closest boundaries are juô MS and (jo 1)8 MS.
8k,eloan at least from any boundary. A flip to adjacent bin occurs if [ns]
The condition < 8/4 implies noise is small. A flip occurs if mh moves sp to another bin. This primarily happens if Sy crosses Sk,clean + 8/2 (approximately).
So, Ptip
Pr Pr(Imal> 8/2). Using Gaussian tail bound Pr([X] > S 2c 22/20/2200
(6/2)²
8²
Ptlip exp
2 2exp
(91)
20
80
3. Energy of the Noise Image Unoise.
Unoise has entries 1, 1,,0. O. If flips from bin jo to j1: (Uuoise)jo.k = -1, (F 1.
Umoisee 11:- = Eng (Uuoise) Ja,-- Each flip changes two pixels, so contributes 12 + (-1)² = to this sum.
[[[] Unoise [|::] Σn Elcontribution at k] L (2 p.ip).
4. Bound on a Single DFT Coefficient of Noise.
By Parseval's identity for 2D DFT: Em,n F2D (Unolse) [m, n2|2 || (with appropriate normalization).
Thus, for any specific (m, n), [FDD (FDD(unoor (Unoise)(mm n2|2 < || Unoise 11
Therefore, F2D (Unoise) [0, R0] < [[[ [ 2LPrip
Visual SNR. Bound.
I
SNRwis
(92)
El| (Unoise) ) j0, no]
I
(93)
2T-Paip L
(94)
2pHip
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Using Pfip 2exp(-88" (0000-80(887))) (880)))
SNRvis 2
(95)
2 2 exp((=pp (8//(82//// (8)(())))
8²
4 exp
(96)
80
This is the second part of the result.
6. Relation to Numerical SNR.
SNBaum =2²2(20²).
SNRvis
202
SNRvis
(97)
SNRaum
A2
8²
20²
2 exp
(98)
4
80
A2
This yields
L
82
a²
SNR 2
exp
SNR
Ruum
(99)
2
Aa
Thus, both inequalities in the theorem statement are proven.
C.6 Proof of Theorem 3.8
(Gaussian-Blur SNR Boost)
apply a 1D normalized Gaussian convolution 9j (1/2) exp((j"/(2of) with E, 99 1 along the row direction of to get 9 *j***
Thcorem C.6 (Thcorem 3.8
restated). Under the assumptions of Theorem 3.7
Let S = Σ, E (0, 1) be the filter's nuclear eneryy.
Define SNR = [ (Wotea)) (0, /0 (2
(Wnosse) )0,
n)] Then,
L
82
SNR
cxp
(100)
45
80
Lg²
82
SNR
exp
SNR
(101)
2
S
The visual SNR is amplified by least 1/S >1s>1 compared to the unblurred case.
Proof. Let G(m,) be the DFT of the 1D filter gj with respect to the value-axis frequency my.
F2D(W)[m, m.] (2D.)) m.]-
We are interested in the frequency (0, no); where no is the time-axis frequency index.
1. Signal Power after Blurring.
Using the frequency index (mg, n2) for (time, value/row), we examine the coefficient FeD(w) ny].
The specific coefficient in focus is FaD (Weleau) no], where no the time index.
Signal power:
(no, 0 Since Σ, 93 1, we have G(0) 1. So signal power remains 2,2
(102)
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2. Noise Power after Blurring.
The noise image is Wnisse = 9 { Vnoise
The total energy of Wnoise:
Wnoise
Σ llg (Unoise) )
(103)
For each column k ( is a vector. Convolution is along j. Sil( (noiss))
||g * (Unoise) ).
(104)
where S 119122 = Ej 97.
So,
Unoise P. Vnosse F
(105)
E| Wnosse 2 S BBIl Unoise []]
(106)
S (2Lpaip)
(107)
3. Bound on Single DFT Coefficient of Blurred Noisc.
El]F22(Wnokk (no, 0]]3] S [E[ Wnoise 11:
(108)
2T. Saip
(109)
4. SNR after Blurring.
I.2
SNR blur
(110)
BII 2D (Wnosse) [mo, 1
1,2
(111)
2LSpaip
L
(112)
2Spaip
Using Pfip 2expp(-88 (eapp-2/((882)))
SNR blur
2
(113)
2S 2 (aap--22/(8o2))
L
exp
(114)
4S
802
This means SNRblur blur 2 (1/S) SNN (unblurred). For the relation to numerical SNR:
L
82
SNR blur
exp
(115)
48
80
L
8²
20
A
exp
(116)
4S
802
A²
202
Lo²
82
exp
SNR
(117)
2A25
8
Since S 1, the factor 1/5> provides amplification of the SNR compared to the unblurred case. 7
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D Additional Results of Computational Experiments
D.1 Zero-shot Study
The full results of the zero-shot study are reported in Table 10 Table 14
We also illustrate zero-shot TSF
examples with prediction length equals 720 of the proposed ViT imee versus TimesFM in Figure 19
Figure 24
It is observable that ViTime consistently demonstrates superior zero-shot prediction performance compared to TimesFM across a range of rescale factors.
D.2 Additional Zero-Shot Results the GIFT-EVAL Benchmark
To rigorously assess cross-domain generalization under Iruly zero-shot setup, evaluate ViTime the communty-adopped GIFT-EVAL benchmark (Aksu et al. 2024), which provides a standardized protocol and strong baselines, including TimesFM (Das et al. 2024 We report Mean Absolute Scaled Error (MASE; lower is better) across short-, medium-, and long- horizon settings. TimesFM results are taken from the official GIFT-EVAL repository
As shown in Table 15 ViTime achieves substantial gains on medium- and long-horizon forecasting, improving MASE by 38.3% and 28.9%, respectively. The short-horizon result is comparable but slightly weaker than TimesFM, which motivates more careful analysis of potential training test overlaps in baseline data.
Centolld analysis. We hypothesize that short-horizon performance of TimesFM may benefit from latent overlap with the M4 dataset family, since its large-scale pre-training includes extensive Google Trends data that is plausibly correlated with M4 categories. Because M4 contributes substantially to short-horizon tasks within GIFT-EVAT, such overlap could compromise strictly zero-shot comparison. To mitigate this risk, we conduct a leakage-controlled evaluation by excluding all M4 datasets from GIFT-EVAL and recomputing motrics for both models.
The controlled results Table 16 are consistent: after excluding M4, ViTime outperforms TimesFM across short, medium, and long horizons. This substantiates ViTime's robustness under strict zero-shot conditions and clarifies the discrepancy short-horizon tasks.
Reproducibility. To facilitate transparency and reprodueibility, we release complete evaluation logs and full GIFT-EVAL results in our repository
D.3 Probabilistic Forecasting Study
The complete results of the study on zero-shot probabilistic forecasting are detailed Table 17 and Table 18 Furthermore, we present sample zero-shot TSF generated by our proposed ViTime model for a prediction horizon of 720 in Figure Figure 36 These visualizations include both 90% and 50% prediction intervals. ViTime consistently demonstrates superior performance across a range of
As shown in Figure 25 Figure
scaling factors.
D.4 Fine- tine-tnnig study
Complete results of the fine-tuning study are reported in Table 19
D.4.1 Analysis of Fine- Tuning on Limited and Noisy Data
An intriguing obscrvation from our fine-tuning experiments is that, under certain conditions, zero-shot (ZS) performance can surpass that of models fine-tuned small data subsets. For instance, the ETTh2 dataset, the 7S MAE for ViTime is 0.344, which is superior the 0.370 MAE achieved after fine-tuning 10% of phenooomenn the data. This counterintuitive phenomenon warrants deeper investigation.
http://ghhhhaceccceeeeeeeeeee/ee htgp::://gthuub..om/tteeeee
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Table 10: Computational results for Scale 0.5 (Landscape) The best results are bolded, the second best are underlined
Patch1SST-2S
MAE
0.919
0.953
0.951
0.952
0.918
0.930
0.928
0.929
0.805
0.823
0.838
0.831
0.606
0.639
0.650
0.647
0.854
0.870
0 0.860
0.867
0.597
0.623
0.620
0.629
0.822
0.846
0.848
0.851
MSE
1.425
1.519
1.513
1.521
1.359
1.407
1.408
1.409
1.251
1.311
1.355
1.335
0.719
0.818
0.846
0.840
1.575
1.544
1.595
1.529
0.871
0.955
0.954
0.988
1.131
1.195
1.201
1.229
MAE
2.941
3.050
3.450
4.109
2.875
2.987
3.353
4.023
0.398
0.382
0.393
0.433
0.355
0.345
0.349
0.367
0.355
0.303
0.330
0.342
0.301
0.329
0.367
0.459
0.399
0.331
0.374
0.362
Vision
MSE
1.673
1.675
1.810
2.055
1.650
1.642
1.772
2.027
0.668
0.622
0.609
0.622
0.547
0.531
0.536
0.564
0.687
0.686
0.661
0.691
0.337
0.508
0.589
0.695
0.533
0.498
0.549
0.538
MAE
0.102
0.475
0.352
0.880
0.402
0.400
0.513
0.617
0.407
0.303
0.871
0.771
0.180
0.197
0.373
0.342
0.423
0.451
0.490
10204
0.380
0.454
0.619
0.726
0.372
0.422
0.305
1.228
Moirai
MSE
0.310
0.450
0.601
1.575
0.375
0.338
0.518
0.825
0.407
0,370
1.642
1.182
0.080
0.093
0.407
0.303
0.757
0.732
0.737
2.459
03367
0.527
1.065
1.246
0.323
0.395
0.575
23333
MAE
4. 1.666
4.781
5.044
5.506
3.027
3.171
3. 3.635
1.503
0.606
0.597
0.582
0.565
0.563
0.598
0.726
0.824
0.798
0.805
0.799
0.804
0.332
0.344
0.350
0.461
0.795
0.805
0.819
0.820
Moment
MSE
2.331
2.349
2.438
2.601
1.560
1.615
1,83
2.196
0.884
0.869
0.539
0.786
0.505
0.531
0.645
0.738
1.149
1.195
1.164
1,663
0.538
0.552
0.567
0.702
0.951
0.963
0.980
0.982
MAE
0.361
0.383
0.383
0.580
0.344
0.377
0.434
0.548
U.403
0.427
0.454
0.493
0.317
0.357
0.415
0.483
0.383
0.402
0.429
0.505
0,253
0.312
0.360
0.432
0.279
0.308
0.347
0.432
TimesFM
MSE
0.308
0.335
0-412
0.617
0.305
0.333
0.406
0.585
0.411
0.447
0.485
0.537
0.269
0.342
0.433
0.522
0.515
0.539
0.612
0. 0.795
0.209
0.289
0. 0.365
0. 0.471
0.184
0.220
0.277
0. 600
Vil'me-TE -T
MAE
0.366
0.384
0.424
0.534
0.236
0.243
0.266
0.312
0.362
0.382
0.401
0.417
0.259
0.291
0.320
0.362
0.283
0.274
0.281
0.306
0.242
0.280
0.312
0.355
0.277
0.281
0.296
0,342
MSE
0.335
0.308
0.439
0.639
0.164
0.173
0.197
0.212
0.362
0.309
0.442
0.462
0.188
0.234
0.277
0.335
0.374
0.362
0.372
0.423
0.197
0.245
0.295
0.361
0.193
0.194
0.211
0.276
MAE
0.448
1.449
0 1.4599
0.510
0.328
0.348
0.362
1.435
0.442
0.450
0.477
0.297
0.336
3,366
0.411
0.431
40405
0. 103
0. 409
0.212
0.252
297
0:339
1.339
0.343
0.363
0.385
Vilime
MSE
0.154
0.437
0.454
0.555
1.266
0.288
0.298
0.399
0.563
0.571
0.588
0.595
0.222
1.280
0.321
0.395
0.839
0.724
7223
0.093
0.172
0. 0.208
0.272
0.338
0.207
0.264
0. 2880
0. .200
0.
192
330
720
336
720
96
192
336
720
96
192
336
720
96
192
336
720
96
192
336
720
192
336
720
Electricity
Dataset
FTTh
ETTh2
ETTml
ETTm2
Traffic
Weather
43

---

Published in Transactions on Machine Learning Research (10/2025)
Table 11: Computational results for Scale = 0.66 (Landscape). The best results are bolded, the second best are underlined
PatchTST -25
MAE
1.011
1.041
1.051
1.051
0.843
0.803
0.803
0.863
0.825
0.860
0.861
0.853
0.622
0.642
0.650
0.651
1.126
1.148
1.150
1.153
0.502
0.582
0.590
0.589
1.067
1.093
1.083
MSE
1.827
1.953
1.979
1.978
1.273
1.330
1.328
1.328
1.344
1.403
1.477
1.449
0.787
0.848
0.872
0.866
2.437
2.570
2.516
2.593
0.800
0.864
0.892
0.886
1.858
1.915
1.967
1.938
MAE
0.539
0.540
0.542
0.605
0.354
0.373
0.381
0.450
0.794
0.659
0.660
0.642
0.613
0.543
0.560
0.566
1.091
1.172
1.045
0. 0.852
0.291
0.324
0.345
0.366
0.985
1.046
0.951
0.804
Vision
MSE
0.827
0.820
0.788
0.840
0.572
0.583
0.575
0.656
1.210
0.978
0.978
0.959
0.989
0.835
0.862
0.871
1.495
1.591
1.474
1,273
0.460
0.505
0.543
0.577
1.195
1.273
1.108
1.065
MAE
0.529
0.393
0.770
0.742
0.362
0.357
0.449
0.516
0.697
0.700
0.873
0.753
0.222
0.238
0.395
0.347
0.796
0.834
0.943
0.863
0.363
0.452
0.643
0.680
0.802
0.843
1.036
1.180
Moirai
MSE
0.647
0.711
1.282
1.106
0.340
0.292
0.423
0.554
1.164
1.239
1,690
1.248
0.122
0.139
0.472
0.373
1.579
1.618
1,966
1.748
0.315
0.472
0.952
1.424
1.026
1.118
2.213
3.476
MAE
0.558
0.576
0.601
0.656
0.332
0.353
0.375
0.472
0.643
0.636
0.631
0.602
0.488
0.567
0.673
0.784
0.802
0.802
U. 0.804
0.810
0.374
0.374
0.369
0.351
0.774
0.780
0.797
0.829
Moment
MSE
0.778
0.799
0.825
0.878
0.508
0.527
0.549
0.672
0.936
0.924
0.919
0.871
0.142
0.513
0.608
0.708
1.195
1.180
1.177
1.214
0.537
0.544
0.560
0.564
0.928
0.934
0.952
0.994
MAE
0.516
0.555
0.588
0.645
0.342
0.394
0-431
0.509
0.441
0.461
0.495
0.351
0.303
0.362
0.409
0.474
0.672
0.737
0.807
0.865
0:233
0.296
0.341
0.126
0.599
0.674
0.752
0.335
TimesFM
MSE
0.587
0.6.5
0.711
0.798
0.277
0.344
0.392
0.508
0.490
0.502
0.562
0.669
0.258
0.384
0.436
0.531
1.133
1.279
1.432
1.642
0.182
0.258
0.317
0.451
0.602
0.746
0.903
1.072
ViTime TFM
MAE
0.389
0-405
0-123
0.492
0.247
0.268
0.295
0.328
0.352
0.374
0.398
0.425
0.261
0
0.328
0.369
0.317
0.317
0.328
0.341
0.221
0,260
0.295
0.348
0.294
0.299
0.347
MSE
0.387
0.408
0.433
0.552
0.170
0.194
0.229
0.266
0.318
0.357
0,412
0.461
0.191
0.249
0.288
0.345
0.467
0.466
0.496
0.523
0.175
0. 2.221
0.272
0.339
0.203
0.207
0.221
0.274
MAE
0, 453
0. 458
0 47
0 2.503
202
0.819
0.357
0.414
0.411
0.433
0.447
0.475
0.278
0.320
0.356
0.307
0. 433
0.42 4:426
424
0.4 1:43
0.199
0.238
0.281
0.328
0.334
0.348
0.372
ViTime
MSE
0 A992
0 485
0.606
0 0.553
25
0.303
0.372
0.475
0.522
0.547
0.596
0.199 199
0. 262
0.313
0.376
0.742
0 88837
0. 705
0.7 747
0.164
0.200
0.251
0.323
0.247
0.241
0.257
0. 2,293
96
192
333
720
946
192
336
720
56
192
336
720
96
192
336
720
96
192
336
720
96
192
336
720
96
192
336
720
Electricity
Dataset
ETTh1
BTTh2
ETTml
FTT'22
Traffii
Weather
44

---

Published in Transactions on Machine Learning Research (10/2025)
Table 12: Computational results for Scale 1.0 (Landscape). The best results are bolded, the second best are underlined
PatchTST ---
MAE
0.814
0.837
0.836
0.837
0.716
0.715
0.711
0.804
0.831
0.596
0.633
0.632
0.626
0.922
0.955
0.950
0.963
0.506
0.596
0.588
0.600
0.858
0.804
0,892
0.896
MSE
1.191
1.251
1.249
1.254
0.870
0.916
0.917
0.909
1 1.209
1.375
1.385
1.365
0.775
0.867
0.865
0.849
1.786
1.911
1.863
1 1.933
0.840
0.926
0.914
0.949
1.223
1.337
1.334
1.349
MAE
0.383
0.110
0 0.423
0.441
0.328
0.367
0.381
0.422
0.347
0.360
0.374
0.405
0.282
0.305
0.328
0.370
0.298
0.256
0.284
0.299
0.257
0.275
0.299
0.337
0.266
0.277
0.296
0.337
Vision/S
MSE
0.304
0.626
0.638
0.637
0.521
0.573
0.587
0.623
0.584
0.600
0.614
0.645
0.477
0.512
0.541
0.586
0.630
0.641
0.678
0.711
0.489
0.494
0.329
0.374
0.421
0.434
0.455
0.506
MAE
0.304
0.443
0.807
0,828
0.222
0.243
0.364
0.399
0.309
0.471
0.745
0.217
0.256
0.450
0.401
0.470
0.329
1.071
0.949
0.402
0.517
0.610
0.721
0.366
0.353
0.920
0.994
Moirai
MSE
0.373
0.151
1.420
1.299
0.120
0.132
0.300
0.332
0.380
0.564
1.887
1.300
0.113
0.162
0.518
0.480
0.975
0.598
2-496
2,055
0.351
0.557
0.830
1.287
0.336
1.599
1.601
MAE
0.410
0.420
0.434
0.472
0.345
0.359
0.381
0.439
0.349
0.359
0.416
0,260
0.280
0.309
0.387
0.282
0.286
0. 0.293
0.310
0.209
0 0.229
0.258
0.336
0.242
0.252
0.266
0.305
Moment
MSE
0.387
0.397
0.413
0.454
0.288
0.306
0.332
0.408
0.293
0.310
0.336
0.405
0.170
0.200
0.244
0.363
0.391
0.400
0.414
0.450
0.154
0.179
0,216
0.315
0.138
0.149
0.106
0.211
MAE
0.390
0.116
0.434
0..799
0.3:0
0.306
0.403
0.456
0.369
0.406
0.431
0.470
0.270
0.327
0.358
0.433
0.225
0.233
0.246
0.276
0.16G
0.207
0.275
0.371
1.209
1.228
052
0.292
TimesPM
MSE
0.379
0. 029
0. 453
0.506
0.285
0.329
0.374
0.438
0.345
0.405
0.447
0.501
0.197
0.300
0.345
0.470
0.324
0:338
0.886
0. 42428
0.121
0.162
0.217
0.386
0.109
0.128
0.157
0.20 2.209
ViTime: TFM
MAE
0.368
0.378
0.396
0.406
0.335
0.343
0.358
0.365
0.346
0.362
0.392
0.408
0. 0.296
0.304
0.320
0.327
0.206
0.213
0.229
0.286
0.221
0.225
0.233
0.237
0.209
0.215
0.227
0.233
MSE
0.377
0.392
0.400
0.424
0.288
0.308
0.324
03644
0.322
0.351
0.451
0.231
0.270
0.316
0.360
0.306
0.320
0.345
0.357
0.181
0.191
0.215
0,237
0.118
0.126
0.139
0.162
MAE
0.424
0.138
0.445
0 0.489
0.300
0.327
0.346
0.402
0.362
0.381
0.404
0.445
0,280
0.322
0.356
0.409
0.375
0.375
0.379
0.416
0.217
0.263
0.311
0.363
0.257
0.265
0.279
0.318
ViTime
MSE
0.512
0.536
0.529
0.603
0,232
0.264
0.281
0.359
0.341
0.375
0.421
0.499
0.207
0.273
0.329
0.421
0.703
0.708
0.719
0.790
0.196
0.216
0.314
0.391
0.164
0.175
0.194
0.250
96
192
336
720
96
192
336
720
96
192
336
720
96
192
336
720
96
192
336
720
96
192
336
720
96
192
336
720
Electricity
Dataset
ETTh
BTTh2
ETTml
ETTm2
Traffic
Woather
45

---

Published in Transactions on Machine Learning Research (10/2025)
Table 13: Computational results for Scale 1.5 (Landscape). The best results are bolded, the second best are underlined
PatchTSST -2
MAE
0.828
0.86-1
0.806
0.871
0-701
0.770
0.770
0.785
0.742
0.775
0.780
0.782
0.587
0.598
0.588
0.600
0.998
1.033
1.041
1.042
0.549
0.564
0.574
0.573
0.889
0.924
0.918
0.928
MSE
1.270
1.362
1.374
1.387
0.960
1.137
1.138
1.177
1.146
1.257
1,256
1.270
0.781
0.808
0.777
0.808
2.056
2.186
2.146
2.217
0.837
0.918
0.941
0.931
1.294
1.412
1.396
1.425
MAE
0.508
0.632
0.608
0.574
0.258
0.285
0.206
0:243
0.902
0.726
0.714
0.702
0.713
0.593
0.593
0.608
1.013
1.081
1.027
0.859
0.249
0.306
0.359
0.460
0.876
0.937
0.916
0.813
Vision TS
MSE
0.929
0.959
0.896
0.863
0.442
0.424
0.389
0.361
1,342
1.009
1.034
1.030
1.097
0.912
0.912
0.935
1.356
1.476
1.444
1.206
0.420
0.472
0.526
0.637
1.071
1.148
1.114
0.991
MAE
0.609
0.683
0 0.903
0.721
0.310
0.331
0.590
0.456
0.727
0.788
1.001
0.805
0.263
0.339
0.526
0.475
0.793
0.843
1.163
1.114
0.408
0.419
0.489
0.370
0.681
0.729
0.915
0.891
Moirai
MSE
0.926
1.083
2.037
1.089
0.231
0.236
0.943
0.462
1.319
1.545
2.478
1.502
0.153
0.266
0.634
0.631
1.447
1.543
2.880
2.095
0.381
0.391
0.628
0.701
0.767
0,871
1.575
1.320
MAE
0.392
0.607
0.609
0.584
0.266
0.252
0.240
0.235
0,648
0.638
0.642
0.630
0.433
0.543
0.684
0.721
0.768
0.754
0.764
0,771
0.341
0.364
0.406
0.194
0.682
0.667
0.682
0.733
Moment
MSE
0
0.909
0.913
0.867
0.302
0.306
0.345
0.336
0.944
0.930
0.941
0.917
0.396
0.493
0.620
0.652
1.111
1.060
1.103
1.144
0.494
0.517
0.368
0.668
0.796
0.799
0.816
0.883
MAE
0.416
0.146
0.485
0:534
0.310
0.339
0.374
0.434
0.487
0.496
0.552
0.604
0.260
0.299
0.359
0.409
0.523
0:557
0.580
0.678
0.216
0.273
0.328
0,106
0.306
0.333
0.358
0.438
TimesFM
MSE
0.412
0.451
0.521
0.614
0.238
0.335
0.413
0.604
D. 9.622
0.927
0.168
0.223
D.319
0. 0.302
0.941
0 9999
0.0996
1.249
0.176
0.237
0.297
0. 104
D. 2.227
9.254
1.283
0. 0.393
ViTime- TFM
MAE
0.387
0. 0.7
0. 0.421
0.428
0.256
0.275
0.294
0.325
0.329
0.350
0.394
0.229
0.258
0.291
0.328
0.285
0.281
0.288
0.294
0. 192
0.237
0.283
0.341
0.246
0.251
0.260
0.285
MSE
0.377
0.419
0.443
0.445
0.177
0.200
0.232
0.273
0.300
0.329
0.360
0.405
0.144
187
0.238
0.290
0.410
0.416
0.417
0.438 438
0.157
0.109
0.252
0.339
0.148
0.156
0.168
0.200
MAE
0.413
0.435
0.444
0.463
0.297
0.334
0.350
0.379
0.360
0.398
0.418
0.451
0.225
0.266
0.298
0.346
0.384
0.371
0.371
0.418
0.141
0.189
0.231
0.286
0.262
0.266
0.275
0.302
ViTime
MSE
0.450 450
0.45 19X
0.503
0.535
0.220
0.209
0.290
0.329
0.351
0.430
0.454
0.524
0.139
0.180
0.229
0.301
0.695
0.630
0.693
0.746
0.120
0.158
0.200
0.267
0.171
0.172
0.186
0.223
96
192
336
720
96
192
336
720
96
192
336
720
96
192
336
720
96
192
336
720
96
192
336
720
96
192
336
720
Electricity
Dataset
ETTh1
ETTh2
ETTml
ETTm2
Traffie
Weather
46

---

Published in Transactions on Machine Learning Research (10/2025)
Table 14: Computational results for Scale 2.0 (Landscape). The best results are bolded, the second best are underlined
Patch MRxchTTST-Z2 ---
MAE
0.818
0.841
0.856
0.860
0.601
0..632
0.632
0.629
0.600
0.719
0.729
0.743
0.555
0.582
0.567
0.586
0.977
1.035
1.045
1.050
0.534
0.578
0.588
0.582
0.848
0.895
0.
MSE
1.279
1.366
1.420
1.419
0.701
0.786
0.789
0.789
1.037
1.139
1.173
1.219
0.686
0. 0.775
0.730
0.784
1.966
2.157
2. 2.236
2.256
0.836
0.948
0.992
0.974
1.177
1.270
1.318
1.315
MAE
0.537
0.557
0.558
0.575
0.329
0.347
0.327
0.283
0.837
0.647
0.660
0.634
0.681
0.547
0.581
0.595
1.010
1.128
0. 8.892
0.837
0.194
0.263
0.305
0.375
0.863
0.956
0.764
0.715
Vision
MSE
0.834
0.8433
0.823
01891
0.500
0.506
0.471
0.411
1.256
0.987
1.012
L.007
1.052
0.842
0.894
0.915
1.420
1.570
1.317
1.270
0.326
0.417
0.468
0.553
1.034
1-167
0.977
0.890
MAE
0.444
0.561
0.928
1.052
0.387
0.426
0.561
0.693
0.822
0,800
0.981
1.086
0.333
0.391
0.516
0.567
0.522
0-497
1.206
1.352
0.371
0.330
0.427
0.529
0.376
0.416
0.950
1.037
Moirai
MSE
0.674
0.855
1.427
1.562
0.02
0.629
0.855
0.983
1.311
1.250
1.539
1.651
0.489
0.582
0.763
0.835
0.990
0.911
1.823
1.989
0.600
0.539
0.799
0.885
0.565
0.592
1.275
1.302
MAE
0.543
0.557
0.578
0.605
0.340
0.307
0.298
0.269
0.662
0.650
0.650
0.641
0.540
0.553
0.623
0.718
0.522
0.497
0.751
0.758
0.282
0.301
0.314
0.373
0.654
0.651
0.654
0.606
Moment
MSE
0.791
0.820
0.855
0.909
0.447
0.433
0.385
0.955
0.938
0.947
0.489
0.502
0.565
0.6449
0.990
0.911
1.071
1.00 1.092
0.413
0.439
0.463
0.540
0.798
7.7
0.798
0.805
MAE
0.384
0.115
0.414
0.508
0.295
0.336
0.303
0.432
0.560
0.005
0.655
0.700
0.268
0.304
0.346
0.417
0.492
0.547
0.561
0.656
0.205
0.276
0.331
0.419
0.299
0.334
0.361
0.447
TimesFM
MSE
0.358
0. 406
0-154
0.581
0.219
0.283
0.316
0.415
1.029
1.068
0.178
0. 0.223
0.285
0.408
0.943
0.983
0.957
1.145
0.167
0.231
0.291
0.413
0.223
0.263
0.285
0.399
ViTime- TFM
MAE
0.362
0.381
0. 0.400
0.418
0.238
0.288
0.304
0.335
0.324
0.342
0.366
0.395
0.219
0.244
0 0.276
0.318
0. 0.268
0.265
0 0.268
0.282
0.182
0. 0,230
0. 0,257
0
0.234
0.242
1.219
0. 1.275
MSE
0.352
0.388
0.431
0.458
0.187
0.227
0.253
0.298
0.306
0.395
0.137
0.167
0.212
0.273
0.412
0.412
0.407
0 1.28
0.145
0.188
0.216
0.278
0.146
0.155
0161
1.193
MAE
0.429
0.446
0.449
0.474
0:203
0.326
0.356
0.401
0.342
0.395
0.407
0.448
0.206
0.244
0.278
0.333
0.392
0.390
0.375
0.387
0.133
0.171
0.212
0.270
0.266
0.274
0.282
0,296
WiTime
MSE
0.521
0.541
0.537
0.579
0:210
b.258
0.300
0-367
0.308
0.408
0.420
0.500
0.115
0.158
0.201
0.282
0.766
0.719
0.670
0.765
0.120
0.136
0.183
0.254
0.181
0.188
0.198
0.216
96
192
336
720
192
3:3
720
96
192
336
96
192
336
720
96
192
336
720
192
336
720
96
192
336
720
Electricity
Dataset
ETTh1
ETTTh2
ETImm
FTTm2
Traffic
Weather
47

---

Published in Transactions on Machine Learning Research (10/2025)
Table 15: Zero-shot performance on the full GIFT-EVAL benchmark (MASE; lower better). TimesF results from the official repository.
Forecast Horizon
FM (MASE)
ViTime (MASE)
Relative Improvement
Long
2.3706
1.4636
38.3%
Medium
1.9621
1.3939
28.9%
Short
2.3548
2.4557
-4.3%
Table 16: Zero-shol performance on GIFT-EVAL with M4 datasets excluded (MASE; lower better).
Forecast Horizon
TimesFM (MASE)
ViTime (MASE)
Relative Improvement
Long
2.3706
1.4636
38.3%
Medium
1.9621
1.3939
28.9%
Short
2.6148
2.5687
1.8%
We hypothesize that this performance degradation is attributed to a combination of data-specific challenges and the behavior of fall-parameter fine-tuning.
Data Characteristics: The ETT datasets, particularly ETTh2, present significant challenges with their low signal-to-noise ratio. In long-horizon forecasting tasks (e.g., 720 steps), while the data shows clear seasonal cycles, it lacks other predictable patterns that models can reliably use. Most. variations outside of seasonality appear random or too noisy for effective learning.
Overfitting and Forgetting: When fine-tuing the entire model on a small (e.g., 10%) and potentially non-representative slice the data, the model is prone to overfitting to short-term noise and atypical scasonal variations presenting in that specific subset. This process can lead to a form of catastrophic forgetting, where the robust and generalized seasonal priors learned during large-scale pre-training are partially overwritten.
To validate this hypothesis, conducted extended analysis the ETTh2 dataset, comparing three different fine-tuning strategics across varying data proportions, full fine-tuning (Full FT), as well as parameter- efficient fine-tuning (PEFT) using LoRA with 10% and 1% of trainable parameters. The results are visualized in Figure 17
As depicted in Figure the performance of full fine-tuning exhibits 'U-shaped' trajectory: the MAE first increases, surpassing the ZS baseline, and then gradually decreases more data becomes available, eventually outperforming the 7S model only when fine-tuned on the full dataset. This confirms hypothesis that full fine-tuning limited data is detrimental.
In stark contrast, the LoRA-based methods significantly alleviate this issue. The 10% LoRA strategy shows a much smaller initial performance dip, while the 1% LoRA strategy consistently improves upon or matches the ZS baseline across all data ratios. This demonstrates that by updating only a small fraction of the parameters, PEFT methods are less susceptible to overfitting on noisy small-scale data and are more effective at preserving the valuable priors acquired during pre-training.
This analysis not only explains the initially counterintuitive results but also yields a crucial insight: for adapting large pre-trained models to downstream tasks with limited or noisy data, parameter-efficion fine-tuning methods like LoRA represent a more robust and reliable strategy than conventional full-model fino-tuning.
D.5 Robust Inference Study
Complete results of the robust inference study reported in Table 20 Table 22
48

---

Published in Transactions on Machine Learning Research (10/2025)
Lag- Llama
0.336
0. 0.392
0 0.501
0.537
0.441
0.314
0.333
0.450
0.507
0.401
0.366
0.412
0.459
0.495
0.435
0.345
0.403
0.484
0 0.495
0.432
0.318
0.378
0.543
0.631
0.467
0. 0.364
0.446
0.658
0.724
0.548
0.2899
0.363
0.418
0.507
0.394
Scale ScaleValue ValleValle-111 1.0
Moirai
0.288
0.325
0.754
0.659
0.506
0.164
0.189
0.386
0.357
0.274
0.295
0.356
0.855
0.648
0.538
0.159
0.211
0.455
0.410
0.309
0.284
0.271
0.800
0.731
0.521
0.390
0.262
1.045
0.808
0.626
0.316
0.417
0.596
0.696
0.506
ViTime
0.332
0.346
0.353
0.392
0.356
0.268
0.300
0.322
0.386
0.319
0.313
0.330
0.349
0.384
0.344
0.224
0.266
0.301
0.352
0.286
0.243
0.252
0.266
0.307
0.267
0.314
0.318
0.322
0.354
0.327
0.177
0.222
0.264
0.312
0.244
Table 17: CRPS Performance Comparison (Part 1)
Lag- Llama
0.456
0.486
0.499
0.592
0.508
0.374
0.444
0.513
0.545
0.469
0.435
0.475
0.502
0.511
0.481
0.361
0.431
0.477
0.516
0.446
0.495
0.507
0.556
0.681
0.560
0.618
0.595
0.670
0.679
0.641
0.312
0.399
0.431
0.501
0.411
ScaleValue 1.00000.66
Moirai
0.399
0.451
0.697
0.631
0.544
0.275
0.280
0.418
0.444
0.354
0.500
0.522
0.847
0.674
0.638
0.167
0.193
0.415
0.378
0.288
0.557
0.590
0.896
0.994
0.759
0.578
0.609
0.798
0.713
0.674
0.276
0.352
0.600
0.631
0.465
ViTime
0.321
0.342
0.366
0.431
0.365
0.259
0.302
0.341
0.426
0.3332
0.317
0.339
0.363
0.396
0.354
0.240
0.286
0.323
0.379
0.307
0.252
0.262
0.279
0.335
0.282
0.307
0.310
0.328
0.368
0.328
0.196
0.248
0.286
0.329
0.265
Lag- Llama
0.452
0.474
0.500
0.613
0.510
000
0.444
0.514
0.568
0.481
0.457
0.473
0.513
0.538
0.495
0.401
0.448
0.500
0.517
0.467
0.499
0.544
0. 0.595
0.708
0.586
0.611
0.588
0.614
0.698
0.628
0.331
0.410
0.457
0.510
0.427
SealeValue=D0 0.5
Moirai
0.295
0.348
0.455
0.640
0.434
0.301
0.322
0.423
0.566
0.403
0.303
0.299
0.855
0.687
0.536
0.135
0.153
0.380
0.330
0.249
0.285
0.534
0.424
0.931
0.543
0.319
0.340
0.385
0.928
0.493
0.278
0.359
0.630
0.721
0.497
ViTime
0.321
0.338
0.381
0.462
0.376
0.278
0.302
0.362
0.485
0.357
0.319
0.338
0.357
0.384
0.350
0.248
0.291
0.329
0.374
0.310
0.262
0.275
0.297
0.350
0.296
0.300
0.293
0.304
0.338
0.309
0.199
0.246
0.281
0.327
0.263
96
192
336
720
Avg
96
192
333
720
Avg
96
192
336
720
Avg
96
192
336
720
Avg
96
192
336
720
Avg
96
192
336
720
Avg
96
192
336
720
Avg
Electricity
Dataset
ETTh
ETTh2
ETTm1
ETTm2
Traffic
Weather
49

---

Published in Transactions on Machine Learning Research (10/2025)
Lag- Llama
0.351
0.456
0.488
0.544
0.460
0.371
0.417
0.441
0.492
0.430
0.418
0.446
0.473
0.490
0.457
0.248
0.359
0.400
0.458
0.366
0.313
0.392
0.519
0.693
0.479
0.673
0.695
0.625
0.708
0.675
0.244
0.273
0.328
0.355
0.300
SealeValue=24
Moiral
0.310
0.398
0.907
0.763
0.595
0.265
0.300
0.620
0.657
0.461
0.564
0.553
0.939
0.824
0.720
0.266
0.291
0.523
0.567
0.412
0.272
0.328
0.910
0.772
0.571
0.373
0.391
1.215
1.248
0.807
0.345
3.237
0.453
0.535
1.143
Table 18: CRPS Performance Comparison (Part 2)
ViTime
0.314
0.336
0.357
0.390
0.349
0.242
0.282
0.312
0.344
0.2995
0.303
0.323
0.349
0.391
0.342
0.196
0.232
0.270
0.318
0.254
0.227
0.237
0.248
0.294
0.252
0.319
0.322
0.314
0.353
0.327
0.142
0.186
0.223
0.276
0.207
Lag- Llama
1:438
0.453
0.500
0.508
0.475
0.364
0. 0.407
0.443
0. 19498
0.428
0.419
0.442
0.457
0.478
0.449
0.312
0.357
4.422
0.471
0.301
0.410
0.454
0.478
0.559
0.475
0.658
0.653
0.693
0.612
0.654
0.254
0.306
0.389
0.467
0.354
SealeValue=14. 1.5
Moirai
0.446
0.508
0.836
0.624
0.604
0.236
0.258
0.643
0.542
0.420
0.552
0.599
0.970
0.774
0.724
0.197
0.264
0.549
0.544
0.389
0.482
0.587
0.867
0.675
0.653
0.587
0.644
1.153
0.899
0.822
0.380
0.467
0.501
0.610
0.489
ViTime
0.316
0.335
0.352
0.375
0.345
0.245
0.272
0.294
0.330
0.285
0.305
0.327
0.348
0.390
0.343
0.198
0. 0.237
0.277
0.326
0.260
0.216
0.224
0.240
0.283
0.241
0.316
0.307
0.325
0.366
0.329
0.150
0. 200
0.246
0.301
0.224
96
336
720
Avg
96
192
333
720
Avg
96
192
336
720
Avg
96
192
336
720
Avg
96
192
336
720
Avg
96
192
336
720
Avg
.96
192
336
720
Avg
Electricity
Dataset
ETTh
ETTh2
ETTm1
ETTm2
Traffic
Weather
50

---

Published in Transactions on Machine Learning Research (10/2025)
|TimMMse Miver
100%
0.399
0.414
0.429
0.460
0.423
0.330
0.402
0.396
0.4008
0.384
0.254
0.205
0.330
0.383
0.316
0.3
0.305
0.381
0.417
0.376
0.2440
0.250
0.270
0.281
0.263
0.197
0.239
0.280
0.330
0.202
0.224
0.220
0.255
0.288
0.246
Patch i'Trassorree
100%
0.405
0.436
0.458
0.191
0.448
0.3:19
0.400
0.432
0.445
0.407
0.264
0.309
0.344
0.407
0.332
0.368
0.391
0.420
0.459
0.410
0.268
0.276
0.283
0.302
0.282
0.214
0.254
0.296
0.347
0.278
0.240
0.253
0.269
0.317
0.270
Baseline Methods
100%
0. 0.400
0. 0.4229
0.440
0. 0.468
0. 0.434
0.337
0.382
0. 0.422
0.381
0.256
0. 0.2996
0. 0.329
0 0.385
0.317
0.3/16
0.370
0. 0.392
0. 0.420
0.382
0. 0.249
0. 0.256
0. 0.264
0. 0.286
0. 0.264
0.198
0. 0.241
0. 0.282
0.334
0. 0.264
0.222
0.240
0.2 0.259
0. 0.290
0, 0.253
TIMESNET
100%
0.402
0.429
0-469
0.300
0,450
0.374
0.414
0.452
0.468
0.427
0.267
0.309
0.351
0.408
0.333
0.375
0.387
0.411
0.450
0.406
0.321
0.336
0.336
0.350
0.336
0.220
0.261
0.306
0.359
0.286
0.272
0.289
0.300
0.320
0.295
Table 19: MAE of different methods in fine- toningnng study.
SIMBA 100%
0.395
0.424
0.443
0.469
0.433
0.339
0.390
0-406
0.431
0.392
0.263
0.306
0.3-13
0.399
0.328
0.360
0.382
0.405
0.437
0.396
0 0.268
0.317
0.284
0.297
0.291
0.219
0.260
0 0.297
0.349
0.281
0.253
0.262
0.277
0.305
0.274
PatchTST
10%
0.485
0.524
0.550
0.610
0.342
0.389
0.4.14
0.441
D-480
0-431
0.274
0.317
0.353
0.427
0.343
0.419
0.434
0.454
0.556
0.166
0.268
0.274
0.282
0.319
0.286
0.221
0.261
0.300
0.351
0.283
0.235
0.250
0.270
0.315
0.268
TIME-LLM LLM
(F'T) 10%
0.400
0.483
0.540
0.604
0.522
0.326
0.373
0.429
0.449
0.304
0.261
0.314
0.327
0.390
0.323
0.388
0.416
0.426
0.476
0.426
Not Reported Not Reported Not Reported Not Reported
Not Reported
Not Reported
Not Reported
Not Reported
Not Reported
Not Reported
Not Reported
Not Reported
Not Reported
Not Reported
Not Reported
GPT4TS
(FT) 10%
0.485
0.524
0.550
0.610
0.542
0.380
0.414
0.441
0-480
0.431
0.274
0.317
0. 0.358
0. 0.427
0.343
0.419
0.434
0.454
0.556
0.466
TimesFM FM
(FT) 10%
0.398
0-424
0-436
0.415
0,426
0.356
0.400
0.428
0.457
0.410
0.263
0.309
0.344
0.415
0.334
0.345
0.374
0.397
0.436
0.388
0.057
0.050
0.040
0.007
0.002
0.003
0.004
0.004
= 0.012
0.2750 + 0.017
0.3108 + 0.024
0.3096 0.017
0.3319 = 0.004
0,3518 + 0.003
0.3702 # 0.002
0.4083 0.002
0.2336 + 0.008
0.2376 + 0.001
0.2437 1 0.002
0.2775 + 0.005
0.002
0. 0.2234 + 0.001
0.2686 + 0.003
0.3198 1t 0.001
4: 0.005
0.001
0.005
0.2777 0.003
100%
0.4057
0.3438
0.2970
0.3656
0,2481
0 0,2486
0 0.2454
VITIME (PT)
0.3814
0.3978
0.4076
0.4355
0.2970
0.3270
0.3450
0.4060
0. 0.2328
0.1827
0,222
0. 2339
0.2476
0.033
0,039
0,001
0.012
0.010
0.008
0.005
0.008
+ 0.004
0.006
0.016
0.009
0.3393 + 0.001
0.3624 + 0.002
0.002
0.007
+ 0.033
0.005
0.2472 + 0.002
0.2670 + 0.005
0.004
0.007
0.008 006
+ 0.010
0.012
0.009
0.2522 + 0.004
0.28867 + 0.007 002
10%
0.4218
0.3608
0.3120
0.3761
0.2508
0.2492
0.2504
0.3933
0.4098
0.4256
0-4578
0.3230
0.3520
0.3750
0,4290
0.2579
0,28066
0.3224
0.3779
0.3839
0.4189
0.2446
0.2111
0.1865
0.2281
0,2696
0.3126
0.2275
0.2350
96
192
336
720
Avg
96
192
336
720
Avg
96
192
336
720
Avg
96
192
336
720
Avg
96
192
336
720
Avg
96
192
336
720
Avg
96
192
836
720
Avg
Electricity
Dataset
ETT12
ETTm2
ETTml
hoarher
Trafii
11

---

P Published in Transactions on Machine Learning Research (10/2025) (
MAE
0.419 000
.00
0.458
0.003
0.509
.0002
0.455
0.307=0.000
000
0.2793:0.000 000 0.340=10.000
000
0.372::0.000
0.433310.000
0.363
0.400
0003
0.416 .000
0.410 0.0
0.479=0.0000
.002
0.434
0.276 0.004
0.315
0.00040
4.3484:0.003
0.307 0.392+:0.002
0.333
0.326 00088
0.336-10.000 .007
0.352 .006
0.406= 000
0.355
0.412
0.410=:0.018
0.41910.015
0.4621:0.014
0.426
0.223 007
0.270 011
0.306-1-..01 011
0.355 00.0
0,288
0.435
Vil'mm GN((.3)
MSE
0. 4271:0.008
0.458+:0.011
0.485
0000
0.5681:0.001
0.483
.02
0.3211:0.002
0.411
0.312
00.0 0.404$00004
0.4361:0.000
0.4821:0.006
0.557++0000
0.470
0.005
004 0.255+10.004
0000
0.375110.00
0.283
0.225
006 0.238110000
0.423 000000 0.260++0.011
0.341::0.012
0.266
0.694 00.395
0.700#:0.342
0.52210.010 0.801
0.71610.331
0.728
0.184110.012
0.238110.025
0.289-10.023
0.305
0.269
0.197
0.238
Table 20: Performance comparison of ThnesFM and ViTime: Gaussian noise levels 0.1 and 0.3
MAE
0.4191:0000
0.004
0.447
0. 0.48111.000
0.546-10.000
0.473
0000
0.3611:0.000
0.402110.000
0.470--0.001
0,390
0.43940.000
0.4661-0.008
0.497-10.015
0.536=
0.485
0.278
00.00
0.320
0.3591:0.00
0.419400000
0.344
0.348--0000
0000
0.502110.000
0.414
-467-00.01
0.495-10.000
0.587
0.518
0.21310.000
0.006
0.263 003
0.305
0.370
0.288
0.325-10
0.384110
TimesF GN(0.3)
0.
0.
MSE
0.419:10.000 .07
0.464
0.5151:0.02
0.61810.066
0.504
0.2611:00001
10001
0.3124
0.3631
0.4611-0.001
0.350
0.51940.027
0.5571:0.021
0.6194:00771
0.680=
04044
0.504
0.199::0.00
0.284+-0.000 00000
0.324+0.000
0.425:0.007
0.303
0.273+-.012
0.325+1.000
0 0.37810.008
0.49511.005
0.368
2622
0.81210.133 132
0.85410.188
0.981-00115
0.853
0.164110.00 00.
0.220 .009
0.272+00007
0.372+0.006
0.257
0. 0.766=:0.22
0
MAE
0. 0.4233:0.000 005 0.411
002
0.429
003
0.452 002
0.503 001
0.449
0.304+:..00 001
0.336 000
0.368:10.001
001
0.4341:0.00
0.360
0.3941:0000.
0.41210.001
0.4371:0.001
0.474+0.000 001
0.429
0.270
003
0.309
002
0.345+00.001
0.39310.001
0.329
0.309 007
0.319410.000
0.337
004
0.393110.004
0.340
0.389
00.01
0.385+:0.012
0.39410.012
0.438110.013
0.402
0.209-1.000
0. 0.256110.00 000
0.300-10.009
0.353
0.279
ViTime GN(0.1)
MSE
0. 0.452
0066
0.484
0.563110.00
0.480
0.238=0.000
0.27710.001
0.3181100011
0.41410.001
0.312
0.407
0.44210.002
0.489110.002
0.557+0.003
0.474
0.003
0.19410.000
0.251
0.306=:0.001
0.38010..00
0.283
0.217+0.014
0.2291100013
0.254=0.007
0.3371:0000
0.259
0.723 28.26
0.71920.318
0.741
1.835110.51
0.754
1.16110.008
0. 0.227-0.002
0.288-10.021
0.37 0..01
0.266
MAE
0.415=10.002
0.461 0.445=0.0000
.00
0.47810.003
0.54110.002
0.471
0.326:10000
0000
0. 0.36710.001
0.405-10.001
0.475+0.001
0.393
0.44410.000
0. 0.475-0.001
0.509
0.551
0.495
0.281
0.328+0.005
0. 0.369+0.003
0. 0.430100008
0.352
0.338+00.000
0006
0.376-0.000 004
0. 0.414=0.000 0.00
0.4861100001
0.403
0.457:0.021
0.4891:0.017
0.51710.013
0.980-10.159 0.58110.010
0.512
0.20210.004
0.253
0.29810.007
0.375+0.006 0..00300003
0.280
TimesPM GN(O.1)
0.
MSE
0.4111:0.003
003
0.51310.004
0.6171100008
0.500
0.208 1:0000000000 0.000
0. 0.326110.000 0.000
0. 0.480++0.000
0.363
0.519 025
1.005-
0.6411:0.046
0. 0.720=0.122 123
0.612
0.211
0. 0.209
0. 0.015
0. 0.0017
0.329
0.267+00.010
0.322100000 0.008
0.378 000 006
0.486:100000
0.363
0.757 4300
0.815400.360 0.5
0.861110.188 189
0.853
0.15610.000
0.21310.007
0.270100.007
0.254
0.377
96
192
336
720
Mean
96
192
333
720
Mean
96
192
336
720
Mcan
96
192
336
720
Mean
96
192
336
720
Mean
96
192
336
720
Mean
96
192
336
720
Mcan
Electricity
Dataset
ETThl
ETTh2
ETTml
ETTm2
Traffic
Weather
52

---

Published in Transactions on Machine Learning Research (10/2025) (
MAE
0.05
0.456
0.00
0.462
004
0.480=00.00
0.004
0.5391:0.00
0.003
0.484
0000 003
0.342::0.00
0.369 0.003
0.4021:0.000
.002
0.4611:0.000
0..002
0.394
0.4181:0.000
003
0.425+:0.002
0.003
0.4111
0.033
0.484+000000
.003
0.443
0.000
0.2921:0.00
0.326
0.006
0.358::0.000
0.004
0.407-00.002
0.346
0.357 0.006
0.350
0000
0.371
000
0.419 000
0.377
0. 0.503+ 026 026
0.498 0210
0.50210.019
0.5381:0.015
0.510
0.240110.010
0.280100.017
0.317
0.366:10.011
0.301
Vil'mm GN((..7
MSE
0. 0.471=00000
00.008
0.477=:0.000
0.506
00000
0.6061:00000
0.515
0.2751:0000
000
0.310+=0.000
000
0.3551:0.000
000
0.441
0.345
0.433
0.41510.000
0.480110000
0.550+00000
0.477
01211::1000
100000
000000
0.311 10.000
0.389-100000
0.293
0.257
0.260-10.013
0.277+:00113
0.3551:0012
0.287
03.44
0.8834006554
0.8151:0.886
486
0.82910.338
0.9181:0.361
0.849
0.188110.016
0.237110022
0.291110.024
0.37210.019
0.272
000000 0.202
Table 21: Performance comparison of ThnesFM and ViTime: Gaussian noise levels 0.5 and 0.7
0.
MAE
0.428110.000
0.009
0.456:10.009
0. 0.4900100000
0.556110.008
0.483
.003
0.3691:0 0,003
0.405-10.000 0000
0.471
0.394
0.44810.000
0.473
0.505-:0.000
000
0.542+0.000
0.492
0.285100000
0.327
0.3641=0.006
0.421
10.004
0.349
0.377
+0.044
000
0.457-00000
0.554110.000
0.450
0.49510.011
0.51910.020
0.547100012
0.613-00011
0.543
0.230110.000
0.005
0.278110.000 0.00
0.320-10.005
0.381=00000
0.302
0.3324:0
0.133:0
TimesF GN(U.T)
MSE
0.43410.044
0.82::0.041
0. 0.53110.0553
0.63810.004
0.521
0.268=00000
0000
0.31910.003
0.368-10.004
0.467+00.02
0.355
0.54610.033
0.5801:0.016
0.645=000010
0.693=0.022
0.616
0.205 10000
0.270=
0006
0.331=00000
0 4201:00006
0.308
0.207--0.010
0.347
0. 1.405=00000
0.55610.010
0.401
2.00
0.83110.277
0.4551100112 0.871
1.00100.182
0.877
0.1771:0.005
0.234 008
0.290110.005
0.386=0.009
0.272
0.297:
0.
MAE
0.432 004
0.447 003
0.468=000002
002
0.518110.003
0.466
0.315 003
0.346 002
0.378100.02
0.440100.001
0.370
0.415 003
0.429 002
0.41910.000
0.486+0..000
003
0.445
0.283-100003 20.00
0.31910.000 003
40.00
0.395
0.337
0.34440..00
0.352100007
0.368
006
0.4221:0.008
0.371
348 0.447 01.033
0.4461:0.012
0.4961100013
0.461
0.23310.000
0.27611.0116
0.310110.014
0.35610.012
0.294
ViTime GN((.5)
MSE
0.441 006
10000
0.4651:0.000
0.494
0.57611.009
0.494
0.24440.000
1.28310.002
0.326110.022
0.41620.001
0.317
0.41910.005
0.44810.003
0.486110.000
0.559100.07
0.478
0.202--00000
0.257+0.000 .00
0.308=:0.004
0.377
0.286
0.242+0.015
0.252400015
0.273+0.013
0.357
0.281
0.747
0. 0.755+:0.260
0.776.10.22 222
0.8621:0.212
0.785
1.18-10.014
0. 0.238-0.0029
0.286-10.024
0.358
0.268
MAE
0.4241000003
0.451=0.007
0. 0.4861:0000
0.553100.00
0.479
0.327100000
0.366100.003
0.4031100002
0.470+0.002
0.392
0.44210.012
0.469-0.008
0.500+0.000
0.539
.007
0.488
0.27910.004
0.321 +0.00000004
0. 0.300+0.007
0.41810.000
0.345
0. 362-000001
0.39840.000
006
0.05
0. 0.440=1.0000
0.532-10.006
0.433
0. 0.478+00006
0.50710.008
0.533
0.903-10.148 0.600-0.012
0.529
0.22210.004
0.271+0.003
0.313-00.005
0.376=+0.005
0.295
TimesPM GN(O.5)
0.
0.
MSE
0.428110.010
0.473++00020
0. 0.52510.014
0.634110.014
0.515
0.2621:0000 0.002
0.31410.000 000
0.305
0. 0.461+0.002
0.351
0.534100.033 0.035
0.003
0.6222
0. 0.695=0.025 028
0.608
0.19910.005
0.203
0. 0.324
0.41910.007
0.301
00008
0.283+0.008
0.334100011
0.300
0.5281000113
0.384
0.761 268
0. 0.825::0.13 139
0. 0.86310.095
0.860
0.1711:0.000
0.225110.006 0.0
0.282100.010
0.379+0.009
0.264
96
192
336
720
Mean
96
192
333
720
Mean
96
192
336
720
Mcan
96
192
336
720
Mean
96
192
336
720
Mean
96
192
336
720
Mean
96
192
336
720
Mean
Electricity
Dataset
ETThl
ETTh2
ETTml
ETTm2
Traffic
Weather
53

---

Published in Transactions on Machine Learning Research (10/2025)
Table 22: Performance comparison of TimesFM and ViTime: Gaussian noise level 1.0 and data missing (DM 0.3)
MAB
0.414:0.000
0.002
0.433 003
0.45610.002
0.000
0.507
0.453
0.318
000
0,8521
100210.00
0.374
0.8923
100
1O1310.00
11001000001
$305-0.000 0.48411.000
0.432
02741100
000
11000004
0.
1000000
0.40310.000
10.00
0.337
1911111000
0.3233
007
0.34110.000
0.397
0.343
0
0.40010.000
0.409
10.012
0.44410.013
0.417
0.209 100000
0.26110.016
0.9024000014
0.353+00012
0.281
ViTime DM((33)
1.001000001 100 0
.004 0
MSE
0. 43330
000
O, 0,463:10 .006
O. 0.496.0 .005
0.574.10.000
1.00
0.491
0.2500
10000000
1000000
0. 0
1000
00 1.343
50.00 0.11100.000
1334
0.334
400
1000
.008 000007000
0,490
0000
000
0.3694:0.000 (.3300
1105000000200 0.39810
0
0.295
0.290
100 0 0.3971000000 0
018
000
00000
0.463#10.010 0.343
0.265
782-
04.4488
0.782
0.772
256
0.792.10.22
0.886:10.122
0.808
0.170 414
0.23110.022
0,288 0.21
0.366:10.022
0.264
0.
0.
20,000000
MAE
0.459
000000
0.472
00007
0.487=0.000
.003
0.530
0.003
0.487
0.35621
0.385 10.00
0.410:100000 0.370 1 0-41
0.478=0.00
0.408
007 0.474130
17740000000
0.5001000000
0.471
0:307
0.415
0.358
0.389
0.410==0.010 0000
.0010
0.415
416 0.536=00.025
025
020
0.536=00.022
0.53810.020
0.953-10.30 000 0.57410.019
0.546
0.250 0.000
0.2881000111
0.320
0.36110.009
0.305
Vilime GN(1.))
0
0.4880
007 0
0.57920.008 0
MSE
0.481 2000
0.502
013
0.522#0.006
0.004
0.599
0.526
902101100
008
0,0 0,8:0:11:000
005
0.00000003
0.364
0.491 10011110000
0.524=0.000
0.516 0
0.22
0,279
0.404=10000
0.309
10.00
0.300200021
0.326-0.024
0.417=0.020
0.336
0.846
0.3
0.869
0.0
0.881
0.19710.011
10.011
0.24411.001
1001-00018
0..011
10.014
0.273
0.299
0.4700
0.306
0.858.
0.291
0.361
0000000000000000 0.37
0.
0006 0..0
MAE
0.436-100000
0.008
0.462+:0.010
0.4933:0.011
0.55510.011
0.487
.002
0.3791100004 1.47310.004
0.399
04811:00000
0.5101:0.000
0.545
10000
0.500
100000
.000
13321-0.000
0.431
0.350
14401000000
006
0.385 0,480110.000
0.575110.00
5000
0.475
1111111:000
05711000000
900000
1.03710.140 0.633-100009
0.507
0.2411:0.003
005
0000
100008
0.302+0.011 0.390110.007
0.312
0.340=2:0000
0.3751:0.000
0.459
0.295
0.519
0.541
0.288
TimesPM GI(1L))
0.283 0.33300
0.5
0.
0.
MSEE
0.436110.050 0.005
0.483 042
0.537 096
0.636.10.22 227
0.523
0.280=00.0 002
0.328 003
0.474+0.006
0.365
0.550=1..00 053
1..89
0.635+0.035
0.695 +0.00
0.617
0.217100000 000
000
0.343+0.010
0.42110.000 .009
0.321
0.09
0000
0.311200012
0.583100012
0.429
0.814+0.128 123
0.862 .95
0.0
0.905
0.188110.00 006
0.2441100000 009
0.300
0.283
0.319+0.000
0.90710.0000
96
192
336
720
Mean
96
192
336
720
Mean
96
192
336
720
Mean
96
192
336
720
Mean
96
192
336
720
Mean
96
192
336
720
Mean
966
192
336
720
Mcan
Electricity
Dataset
ETTh1
ETTTh
ETTml
ETT m2
Traffic
Weather
54

---

Published in Transactions on Machine Learning Research (10/2025)
Performance Comparison Across Fine-tuning Methods and Training Data Proportions
Full Panmeter Fine-tuning
LoRA-128 Fine-tuning
0.350
LoRA-16 Fire-tuning
Zere-shot Baseline
0,370
Performancce Score
0.360
0.350
0.330
0.1
0.1
0.9
Training Proportion
Figure 17: Performance comparison different fine-tuning strategies on the ETTh2 dataset with varying data ratios. The zero-shot (ZS) performance is shown a dashed line for reference. Full fine-tuning (Full FT) initially degrades performance small data subsets before improving. In contrast, parameter-effiieet LoRA methods, especially with fewer parameters (1% LoRA), effectively mitigate this degradation by preserving pre-trained knowledge.
D.5.1 Analysis on Pre-Denoising Methods
To examine the elfectiveness of pre-denoising algorithms in improving numerical model accuracy under moderate noise interference, we conducted series of experiments evaluating standard denoising techniques applicd to time series data before model input. The experiments were performed on the Electricity datasct, to which we added Gaussian noise with a variance of o² 0.5 to simulate moderate noise conditions. We then applied three common pre-processing denoising methods before feeding the data into the baseline TimesFM model. The methods evaluated are:
3-Sigma Rule: A simple outlier detection and removal method.
Moving Average: A smoothing filter with a window size of 5.
Savitzky-Golay Filter: polynomial-based smoothing filter (polyorder=3).
The results shown in Figure 18 illustrate the performance improvements and trade-offs associated with these pre-processing denoising steps under moderate noise conditions.
As depicted in the figure, under moderate noise interference, the effectiveness of pre-denoising algorithms in improving model accuracy varies significantly depending on both the method and the prediction horizon. The 3-sigma method demonstrates modest. accuracy improvements for short prediction lengths (96 and 192), achicving positive ReMAE improvement rates. However, its performance degrades and even bocomes detrimental for longer horizons. This limitation arises because the method only removes extreme outliers but fails to address the underlying noise distribution that significantly impacts long-term trend forecasting accuracy.
Smoothing-basd based denoising methods like moving average and Savitaky-Golay (savgol) show contrasting results for numerical model accuracy under moderate noise. These methods exhibit poor performance for short-term predictions, showing negative improvement rates and actually roducing model accuracy. This degradation occurs because smoothing techniques inherently average out recent fluctuations and introduce temporal lag in the data. While this characteristic helps reveal long-term trends and improves accuracy for
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3sigma
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Figure 18: Performance comparison of TimesFM combined with various pre-denoising methods the Electricity dataset under moderate Gaussian noise (o2 = 0.5). The y-axis represents the ReMAE improvement rate (%) compared to the baseline TimesFM without any denoising. Our proposed method, ViTime, is included as a reference to showcase its superior performance.
extended horizons (as evidenced by the moving average's positive performance at the 720-step horizon), it corrupts the fine-grained, high-frequency patterns essential for accurate short-term forecasting.
In stark contrast, our proposed ViTime method, serving as a reference line in the chart, consistently achieves significant ReMAE improvements across all prediction lengths under moderate noise conditions. This superior performance demonstrates that ViTime's intrinsic architecture for handling time series data provides inherent robustness against moderate noise interference. Unlike traditional pre-processing denoising pipelines that often involve accuracy trade-offs between different prediction horizons, ViTime successfully denoises and extracts meaningful features while maintaining high numerical precision, thereby establishing its superior robustness in moderately noisy euvironments.
D.6 Computational Complexity Analysis
We conduct extensive experiments to analyze the computational complexity and prediction accuracy of our proposed models. All experiments are performed with batch size 4, input sequence length 512, and prediction horizon 720 on single Nvidia 3090 GPU.
Table 23: Model Performance and Computational Resource Requirements
Model
GPU Mem.
Inference Time (s/batch)
Params
Avg. ReMAE
(MB)
Total
Map. & Inv. Map.
(M)
TimesFM
18,154
0.130
200
0.423
ViTime w/ Refining
3,120
2.890
0.0068 0.0068
95
0.376
ViTime w/o Refining
667
0.082
74
0.381
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The results reported in Table 23 The baseline TimesFM model requires substantial computational resources with 18.1GB GPU memory and 200M parameters, while achieving average ReMAE of 0.423. In contrast, our proposed ViT'mm architecture demonstrates remarkable improvements in both efficiency and accuracy. The basic version without the refining module strikes an optimal balance between computational efficiency and performance- it requires only 667MB GPU memory (27x reduction), achieves faster inference at 0.082s per batch, uses 63% fewer parameters (74M), while maintaining competitive accuracy with an average ReMAE of 0.381.
For applications prioritizing prediction accuracy, the ViTime variant with refining module achieves the best performance with an average ReMAE of 0.376, representing an 11.1% improvement over TimesFM. This comes at the cost of increased computational overhead - 3.IGB GPU memory and 2.89s inference time per batch, though still maintaining a 5.8x reduction in memory compared to TimesFM. Notably, the mapping & inverse mapping between image space and numerical space in ViTime variants consume only 0.0068s, representing 8.3% and 0.24% of the total inference time for the basic and refined versions, respectively.
These results demonstrate that our proposed architecture offers flexible deployment options: the basic version for resource-constrainet scenarios requiring good accuracy and computational efficiency, and the refined version for applications where prediction accuracy is paramount. Both variants significantly outperform the baseline in terms of the compulation-accuraay trade-off.
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Figure 19: Illustrative example of Electricity dataset.
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Figure 20: Illustrative example of Traffic dataset.
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Figure 21: Illustrative example of ETThl dataset.
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Figure 22: Illsstrative example of ETTh2 dataset.
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Figure 23: Illustrative example of ETTml dataset.
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Figure 24: Mustrative example of ETTm2 dataset.
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Probabilistic Time Series Forecasting Analysis of ViTime
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Figure 25: Illustrative example of dataset electricity (Part 1).
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Probabilistic Time Series Forecasting Analysis of ViTime
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Figure 26: Illustrative example of dataset electricity (Part 2).
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Probabilistic Time Series Forecasting Analysis of ViTime
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Figure 27: Illustrative example of dataset traffic (Part 1).
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Probabilistic Time Series Forecasting Analysis of ViTime
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Figure 28; Illustrative example of dataset traffic (Part 2).
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Probabilistic Time Series Forecasting Analysis of ViTime
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Figure 29: Illustrative example of dataset ETThl (Part 1).
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Probabilistic Forecasting Analysis ViTime
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Figure 30: Muutsatitv example of dataset ETTh1 (Part 2).
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Figure 31: Illustrative example of dataset ETTh2 (Part 1).
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Probabilistic Time Series Forecasting Analysis of ViTime
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Figure 32: Mustrative example of dataset ETTh2 (Part 2).
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Probabilistic Time Series Forecasting Analysis of VITime
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Figure 33: Illustrative example of dataset ETTml (Part 1).
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Probabilistic Time Series Forecasting Analysis of ViTime
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Figure 34: Mustrative example of dataset ETTml (Part 2).
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Forecasting Analysis
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Figure 35: Illustrative example of dataset ETTm2 (Part 1).
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Probsbilistic Time Series Forecasting Analysis of ViTime
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Figure 36: Mustrrtive example of dataset ETTm2 (Part 2).
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Reviews:
[{"review_id": "v6WLv8uhRS", "reviewer": "Reviewer_9Dt3", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths:**\n1. The paper introduces an interesting idea to model time-series (TS) data as images and leverage the ViT (vision transformer) model's ability for effective zero/few-shot prediction on new tasks.\n2. The effectiveness of synthetic data is highlighted, which also conveys a message that one does not have to rely heavily on real-world time-series data to train such models effectively.\n3. The role of quantizing the TS data and the parameters involved is studied in detail.\n4. The performance (ReMSE, ReMAE) of the ViTime model is consistently better than other approaches across the datasets considered.\n\n**Weaknesses:**\n\n1. Details about the model architecture, computational resources, and optimizers used for training are not discussed. (Please let me know if I missed anything). This is important since Table 15 shows that the inference latency of ViTime is quite lower than TimesFM while having better performance with a smaller number of parameters. Do these inference numbers consider the overhead of converting TS data to images?\n2. In Table 2(b), how can ViTime outperform ViTime-TFM when the data used for ViTime-TFM is much more diverse? Details about this dataset is also missing.\n3. The authors mention that the TS -> image conversion process can act as a denoiser and lead to better performance. What happens if we can apply some standard denoising techniques on the TS data itself before passing it through, say, TimesFM?", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper presents a methodology to train a time-series foundation model by transforming numeric time-series data into visual inputs. The key contributions lie in:\n1. Transforming numeric sequences into images.\n2. Creating synthetic time-series data with diversity in trend and periodicity.\n3. Training the vision transformer model to predict the visual \"patch\" tokens and then converting it back into numeric time-series data.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "major: see weakness comments above.\n\nI am unsure about the validity of the results without knowing the model/data recipe design. It would be great if the authors could provide information about:\n1. ViTime model design: layers, encodings, depth, etc.\n2. data recipes used to train the model, i.e, the fraction of TS data created for trend/periodicity etc.\n3. experimental settings to obtain the benchmark performance of various models considered (ex: for Table 2)\n4. The details of the ViTime-TFM data recipe and training time/resources used.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "DK4S0PvjqW", "reviewer": "Reviewer_bwkY", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "__Strengths:__\n- The idea of treating timeseries as images is an interesting methodology. In particular, the use of image-based representation of timeseries provides some interesting solutions to treating missing data, enhancing signal-to-noise ratio, etc... as discussed by the authors.\n- Theoretical results on quantization error and signal-to-noise ratio enhancement are sound and provide justification for the proposed methodology.\n- The RealTS framework for generating timeseries is interesting as it is both a simple framework and able to generate timeseries with quite large diversity.\n\n__Weaknesses:__\n- Looking at some of the qualitative results in Appendix D, the fit to ground truth doesn't always seem to be great. Of course, the task of predicting irregular time series is challenging; hence, it might be more practical to consider probabilistic forecasts instead of pointwise forecasts considered in this work. This is true especially when modelling time series with random walk-like behaviour (Appendix A.2.1), where it doesn't really make sense to consider deterministic forecasts as the properties can vary vastly depending on the noise realization. \n- While RealTS is interesting and seems promising, I also wonder if only using five degrees of freedom (IFFTB, PWB, RWB, LGB, TWDB) leads to some bias in the learned model, due to the imposed structures. Perhaps we can augment RealTS with real timeseries data to learn and capture different characteristics in the various time series, beyond the five characteristics considered.\n- The framework only seems to model timeseries of a single variable; however, many timeseries in practice have multiple variables, whose correlations can be useful to improve predictions. Can the methodology be extended to these multi-output settings?\n- The authors make it sound like the use of a vision-based approach is completely novel for time-series forecasting. However, a quick search shows that there are several related works that are not cited (e.g., [1] and [2]). I would advise the authors to perform a more extensive literature search on related ideas and perhaps include a **Related works** section discussing these and how the ViTime methodology differs from them.\n\n[1] Yunfei Du, Yin Wang, Ya Cong, Weihao Jiang, Shiliang Pu, \"Long-term Time Series Forecasting with Vision Transformer\", International Conference on Learning Representations (ICLR), 2024\n\n[2] Weilin Ruan, Siru Zhong, Haomin Wen, Yuxuan Liang, \"Vision-Enhanced Time Series Forecasting via Latent Diffusion Models\", arXiv preprint arXiv:2502.14887, 2025", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "In this paper, the authors propose **ViTime**, a foundational model for time series modelling, powered by vision intelligence, in contrast to traditional approaches based on numerical fitting. This proceeds by converting the time series data into binary images and blurring by convolution; these processes are furthermore shown to be theoretically grounded (i.e., quantization errors vanish as resolution is increased, and blurring enhances signal-to-noise ratio). The model is trained on **RealTS**, a timeseries dataset of diverse time series patterns, proposed by the authors. Experiments demonstrate the superiority of ViTIme over various SOTA timeseries foundation models, in addition to various ablation studies.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "- I would recommend conducting a more thorough literature search for vision-based methodologies for timeseries modelling as it is definitely not the first one. The authors should be aware of such works and place the current work in better context.\n- Some figures can be improved with bigger labels. Most notably, Figure 5.\n- The results in the tables only diaplay a single number, however, to demonstrate further robustness, I would recommend adding errors (e.g. standard deviation) across multiple seeds/experiments.\n- Equation (17) is not very informative. Please could you be more concrete regarding the modelling process.", "broader_impact_concerns": "There are no ethical implications of the work as far as I can see.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "PagEMHH33s", "reviewer": "Reviewer_oM7Z", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths:**\n- The paper tackles a creative and promising problem formulation, which may open up new directions for time-series research.\n- The experimental setup is comprehensive, comparing against strong baselines across multiple datasets.\n- Theoretical bounds on quantization error are derived, providing practical guidance for system parameter design.\n- The paper is clearly written, with a logical presentation of theory and experiments.\n\n**Weaknesses:**\n- The zero-shot performance is often better than the fine-tuning setting, which is counterintuitive. For example, on the ETTh2 dataset, the MAE of TimesFM and ViTime under zero-shot is 0.389 and 0.344 respectively, but after 10% fine-tuning, they increase to 0.410 and 0.372. The paper does not provide any explanation for this unexpected phenomenon.\n- In the robustness section, the authors claim that ViTime is more robust than TimesFM under Gaussian noise. However, robustness is typically defined as smaller performance degradation as noise increases. The paper mainly focuses on outperforming TimesFM, but does not explicitly analyze sensitivity to noise. According to the reported results, ViTime appears to suffer larger performance drops as the Gaussian noise level increases, which seems inconsistent with the claimed robustness. It would improve the analysis if the authors could include results under higher levels of Gaussian noise and provide a more detailed comparison of the degradation trends between ViTime and TimesFM.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper proposes a novel paradigm for time-series forecasting. Instead of performing numerical fitting in traditional value space, ViTime formulates the task in a formally defined binary image time-series metric space, enabling end-to-end visual modeling. The authors demonstrate through experiments that this approach can yield stronger pretrained models, and they introduce a dedicated synthetic data generator, RealTS, to support scalable pretraining.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "I am not an expert in time-series forecasting, but from my understanding, this paper explores an interesting direction with solid writing. I would encourage the authors to address the two issues mentioned above—particularly the unexplained zero-shot vs fine-tuning behavior and the robustness discrepancy—to improve the completeness and clarity of the work.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)