Question:
In fine-tuning experiments, why does ViTime surpass both foundation and supervised models when trained on only 10% of data?
A. Because its training pipeline relies on entropy alignment bias, which allows faster adaptation by bridging synthetic and real-world distributions.
B. Because TimesFM and GPT4TS require complete datasets for comparable accuracy, whereas ViTime’s RealTS pretraining already covers trend and periodic behaviors unseen in fine-tuning data.
C. Because ViTime benefits from a channel-independent strategy that predicts each variable separately, avoiding interference across multivariate dimensions.
D. Because PatchTST demonstrates data leakage issues, while ViTime avoids them by explicitly discarding rescaled test samples during training.

Answer: ABC

In this paper, we hypothesize that a 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 series data. Real-world datasets, 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, we propose a novel time series generation algorithm, RealTS. RealTS systematically generates a large volume of synthetic 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.

ViTime: Foundation Model for Time Series Forecasting Powered by Vision
Intelligence
Luoxiao Yang 1 2 Yun Wang 3 4 Xinqi Fan 5 Israel Cohen 3 Jingdong Chen 4 Yue Zhao 1 Zijun Zhang 2
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 knowledge from classical
statistics to modern deep learning. Yet, all of
them were developed based on one fundamen-
tal concept, the numerical data fitting. Thus, the
models developed have been long known for be-
ing problem-specific and lacking application gen-
eralizability. A TSF foundation model serving
TSF tasks across different applications can re-
verse such an impression. The central question
is then how to develop such a TSF foundation
model. This paper offers a pioneering study in de-
veloping a TSF foundation model and proposes a
vision intelligence-powered framework, ViTime,
for the first time. In ViTime, a method synthesiz-
ing authentic time series periodic and trend pat-
terns is developed to enrich sample pattern diver-
sity. A deep architecture operating TSF in image
metric space is designed to achieve significantly
enhanced TSF generalizability. Extensive experi-
ments demonstrate ViTime’s SOTA performance
across multiple settings. In zero-shot scenarios,
ViTime outperforms TimesFM by 9-15%. With
just 10% fine-tuning data, ViTime surpasses both
foundation models and fully-supervised bench-
marks trained on complete datasets, with this
performance gap widening further at 100% fine-
tuning. Additionally, ViTime exhibits exceptional
robustness, handling missing data without impu-
tation and outperforming TimesFM by 20-30%
under various data perturbations.
1Department of Automation, Xi’an University of Technology,
China 2Department of Data Science, City University of Hong
Kong, Hong Kong SAR 3Electrical and Computer Engineering,
Technion, Technion, Israel 4School of Marine Science and Technol-
ogy, Northwest Polytechnical University, China 5The Manchester
Metropolitan University, UK. Correspondence to: Zijun Zhang
<zijzhang@cityu.edu.hk>.
1. Introduction
Time series forecasting (TSF) is a classic but challenging
topic that has been vigorously discussed in various appli-
cation fields, including power and energy (Sharadga et al.,
2020), environmental studies (Jacox et al., 2022), transporta-
tion studies (Lei et al., 2022), weather forecasting (Yang
et al., 2021), stock market analysis (Lin et al., 2011), pub-
lic healthcare (Liu et al., 2024). Although new heights of
accuracies were repeatedly refreshed by new studies (Zhou
et al., 2021; Wu et al., 2021; Nie et al., 2022; Zeng et al.,
2023; Patro & Agneeswaran, 2024; Wu et al., 2022), most
reported methods predominantly relied on a numerical fit-
ting based modeling paradigm so that models were often
dataset- or problem-specific and lack of application gener-
alizability. The need to repeatedly train models for various
TSF tasks has been the critical barrier of widely applying
learning-based TSF methods in practice, especially ones
with sophisticated mechanisms. Developing a TSF foun-
dation 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 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 & Mergenthaler-Canseco, 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 compe-
tent large language models, such as GPT-4 or Claude 3.5
(Zhou et al., 2023), are usually employed. Meanwhile, in
fine-tuning LLM-based TSF foundation models for han-
dling various downstream tasks demanding higher preci-
sion, the computational complexity becomes prohibitively
expensive, resulting in a large, redundant, less precise, and
price-unfriendly paradigm for the TSF foundation model
(Tan et al., 2024).
The numerical fitting-based models are trained by being
directly fit into numerical time series data, which manifests
that the primary information concerned by these models is
1
arXiv:2407.07311v3  [cs.LG]  8 Feb 2025
ViTime: Foundation Model for Time Series Forecasting Powered by Vision Intelligence
Figure 1: The pipeline comparison of the proposed ViTime
with traditional numerical TSF model. It is observable that
the proposed ViTime adopts a paradigm shift in TSF by pro-
cessing time series in the binary image space, which aligns
with human cognitive processes in time series analysis.
the numerical correlation along the temporal dimension, e.g.,
TimesFM (Das et al., 2024), ForecastPFN (Dooley et al.,
2024), etc. In contrast, human cognition tends to conjecture
trends through remembering correlations between visual
representations rather than processing numerical values di-
rectly. Studies have shown that the human brain processes
visual information more efficiently than numerical data. Pet-
tersson (Pettersson, 1993) discovered that the human brain
is more adept at processing visual information than numer-
ical data. Similarly, Dondis (Dondis, 1974) demonstrated
that the visual cortex rapidly identifies patterns, shapes, and
colors, making processing images and videos quicker than
texts and numbers. These findings lead to a hypothetical
question: On the path toward the TSF foundation model,
might employing vision intelligence for time series modeling
be more effective than conventional numerical methods?
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 spec-
trum of time series problems?
To tackle these challenges, this paper develops a novel vi-
sion intelligence-based TSF foundation model, a Visual
Time Foundation Model (ViTime), aiming to pioneer a new
computational paradigm of building the TSF foundation
model from the perspective of vision intelligence. Regard-
ing the computational principle innovation aspect, ViTime
operates by transforming numerical time series into binary
images, converting numerical temporal correlations into
pixel spatial patterns, and solving TSF tasks in binary image
space, as shown in Figure 1. To offer a large volume of
sufficiently diverse samples for training ViTime, an innova-
tive time-series-data generation method, Real-Time Series
(RealTS), is proposed. RealTS categorizes foundational
knowledge of time series analysis into ”trend” and ”peri-
odicity” and synthesizes training data during the training
of ViTime, ensuring it captures essential time series char-
acteristics. 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 perturbations,
The main contributions of this work are listed as follows:
• A Novel Vision Intelligence-Based Foundation
Model for TSF. We introduce ViTime, a pioneering
TSF foundation model with detailed theoretical anal-
ysis that leverages vision intelligence by processing
time series in the binary image space. This enables
the model to exploit pixel spatial correlation instead of
numerical temporal correlation, aligning with human
visual cognitive processes.
• RealTS: Advanced Data Generation and Augmen-
tation for TSF Foundation Modeling. To address the
training-data sample diversity challenge in 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 a wide range of time
series patterns.
• Superior Performance. Comparisons with SOTA TSF
models demonstrate ViTime’s superior performance in
zero-shot generalization, fine-tuning study, and robust-
ness against data perturbations.
2. Related work
2.1. Problem-specific model for TSF
The problem-specific TSF methods adopt a fully supervised
learning paradigm, where specific models are trained on par-
ticular datasets. Early discussions on problem-specific TSF
modeling were mainly conducted on classical statistical and
machine learning models, such as autoregressive (AR) mod-
els and AR variants (Vu, 2007), Splines and their extensions
(Lewis & Stevens, 1991), linear regressors (Montgomery
et al., 2015), support vector regressor (Montgomery et al.,
2015), neural network based regressor (Montgomery et al.,
2015), etc. In 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 former-based models
(Zhou et al., 2021; Wu et al., 2021; Nie et al., 2022), Dlinear
(Zeng et al., 2023), etc.
2.2. Foundation model for TSF
Recently, the advent of LLMs like GPT and Llama series,
which aim to use a single, generalizable approach to predict
2
ViTime: Foundation Model for Time Series Forecasting Powered by Vision Intelligence
Figure 2: The overall architecture of proposed ViTime. It consists of four key modules: (1) The RealTS synthesis module
generates diverse training samples sL by combining fundamental time series patterns of trend and periodicity. (2) The
mapping function transforms numerical time series sL into binary images vL while preserving temporal relationships
through pixel coordinates. (3) The ViTime model performs visual pattern learning and prediction v′
L on the binary image
representations vL. (4) The inverse mapping function converts the predicted binary images v′
L back to numerical time series
format s′
L. This architecture enables zero-shot generalization, allowing the model to adapt to various real-world time series
tasks without fine-tuning despite being trained only on synthetic data.
across various datasets have sparked interest in developing
foundation models for TSF.
Research on foundation models for TSF is still in its early
stages, and current approaches can be categorized into LLM-
based models, real-world dataset-based models, and syn-
thesized dataset-based models. The LLM-based models
leverage the inference capabilities of LLMs for zero-shot
TSF tasks, including TimeGPT-1 (Garza & Mergenthaler-
Canseco, 2023) and TIME-LLM (Jin et al., 2023). However,
their accuracy heavily depends on the underlying capabili-
ties of LLM, resulting in large, redundant, and less precise
models (Tan et al., 2024). The real-world dataset-based mod-
els aim to train a foundation model on extensive real-world
data to achieve zero-shot TSF, exemplified by Google’s
TimesFM (Das et al., 2024). Despite being one of the lead-
ing TSF foundation models, its reliance on real-world nu-
merical time series data limits its robustness and coverage of
various unseen time series patterns. The synthesized dataset-
based models employ synthetic data to train a foundation
model, e.g., ForecastPFN (Dooley et al., 2024). However,
the uncontrollable nature of generated data patterns in real
space often leads to lower accuracy for foundation models
trained with synthetic data than those trained with real-world
data.
3. Method
3.1. Overall architecture
The overall framework of ViTime, schematically illustrated
in Fig. 2, comprises four key modules: the RealTS syn-
thesis 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 a vast and diverse set of training sam-
ples by categorizing foundational knowledge of time series
analysis into ”trend” and ”periodicity” patterns, which en-
sures ViTime captures essential time series characteristics
across a wide range of scenarios. The core innovation of
ViTime lies in its computational principle of mapping nu-
merical time series into binary images. This approach allows
ViTime to remember temporal pattern correlations through
ordered pixel coordinates while maintaining the ability to
convert results back to numerical format. The visual model-
ing process of ViTime learns to extract relevant features and
patterns from the time series visual representation, utilizing
the historical distributions of the generated binary images
to predict future trends. Finally, the inverse mapping func-
tion is employed to convert the predicted image back into
numerical time series data for further analysis.
In the following sections, we will introduce each component
of ViTime in detail: RealTS, mapping & inverse mapping
function, and ViTime Model.
3.2. Real time series synthesis
In this paper, we hypothesize that a robust foundation
model for TSF should integrate two essential types of time
series fluctuation knowledge, the periodic and trend pat-
terns, which encompass the inherent patterns and directional
changes in time series data. Real-world datasets, however,
often lack representation of the full spectrum of these pe-
riodic and trend-based fluctuations, limiting the ability of
the model to generalize across different scenarios and effec-
tively learn underlying dynamics.
3
ViTime: Foundation Model for Time Series Forecasting Powered by Vision Intelligence
To address this challenge, we propose a novel time series
generation algorithm, RealTS. RealTS systematically gener-
ates a large volume of synthetic time series data that exhibit
diverse periodic and trend characteristics. The proposed
RealTS can facilitate more comprehensive training of foun-
dation models, exposing them to various patterns and im-
proving their ability to generalize to unseen real-world data.
The RealTS algorithm probabilistically selects between gen-
erating periodic or trend-based time series. Given the total
length L of the synthesized time series, the algorithm de-
termines the data prior hypothesis between periodic φp and
trend-based φt patterns with probability (α). The distribu-
tion of generated time series P(D) is defined as follows:
sL ∼P(D) = P (sL|L)
= α
Z
P (sL|L, Bp) P (Bp|φp) P (φp) dφp
+ (1 −α)
Z
P (sL|L, Bt) P (Bt|φt) P (φt) dφt
(1)
where sL is the synthesized time series with length L; P(φ)
represents the prior probability of hypothesis φ; P(B|φ)
is the likelihood of observing the data behavior B under
hypothesis φ. 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
Behavior (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 Appendix A.
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 series
metric space, as described in Definition 3.1.
Definition 3.1 (Binary image-based time series metric
space). The binary image-based time series metric space
is defined as a group (V, d), where V is a set of elements
defined in Equation (2):
V =

v ∈Rc×h×L
 vi,j,k ∈{0, 1},
i ∈[c], j ∈[h], k ∈[L],
h
X
j=1
vi,j,k = 1

(2)
where d : V × V →R is a distance function based on the
Earth Mover’s Distance (EMD), as defined in Equation (3):
d (v1, v2)
=
Z c
i=1
Z t
k=1
inf
γ∈Q(vi,1:h,k
1
,vi,1:h,k
2
)
Ex,y∼γ ∥x −y∥1 dkdi
(3)
where c represents the number of variates, L is the length of
the time series, and h is the resolution of V .
To enable the transition from numerical time-series val-
ues to the binary image-based metric space, we introduce
mapping and inverse mapping functions as follows. Let
S =

s ∈Rc×L | si,k ∈R

represent the numerical value
space of time series. The Time-Series-to-Image mapping
function f : S →V and the Image-to-Time-Series inverse
mapping function f −1 : V →S can be defined as follows:
vi,1:h,k = f (si,k) = ⟨f1 (si,k) , f2 (si,k) , . . . fh (si,k)⟩
fj (si,k) =











1,
if si,k ≥MS, j = h
1,
if si,k ≤MS, j = 1
1,
if j =
j
si,k+MS
2MS
h
k
0,
otherwise.
,
j ∈[h]
(4)
The Image-to-Time-Series inverse mapping function f −1 :
V →S can be defined as follows:
si,k = f −1 (vi,1:h,k) =
h
X
j=1

(j −0.5)2MS
h
−MS

vi,j,k
(5)
where MS > 0 denotes the maximum scale of V. Before
mapping, zero-score normalization is typically applied to
the numerical time series si,k to standardize the scale.
Given that the numerical data synthesized by RealTS are
one-channel time series, i.e., sL ∈Rl ∈R1×L, thus the
corresponding vL ∈R1×h×L is obtained via
vL = f (sL) .
(6)
3.3.1. SYSTEM ERROR ANALYSIS
The system error (SE) emerges from the bidirectional map-
ping between discrete space V and continuous space S,
which inherently impacts prediction fidelity. A rigorous
analysis of 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 3.3.
Assumption 3.2. After applying zero-score normalization,
the continuous space follows a standard normal distribution:
S ∼N(0, I)
4
ViTime: Foundation Model for Time Series Forecasting Powered by Vision Intelligence
Theorem 3.3 (System Error Upper Bound). Given a
tensor bs ∈S
⊂Rc×t, the system error defined as

f −1 (f (bs)) −bs

1 satisfies the following bound:
SE := E

f −1 (f (bs)) −bs

1 ≤g(h, MS)
= ct
"
MS
 1
h (Φ(MS) −Φ(−MS)) −2 + 2Φ(MS)

+
r
2
π e−MS2
2
#
(7)
where Φ denotes the cumulative distribution function of
N(0, I).
Denote MS
 1
h(Φ(MS) −Φ(−MS)) −2 + 2Φ(MS)

+
q
2
πe
−MS2
2
in Equation (7) as the upper bound of SE, whose
convergence is guaranteed by Proposition 3.4.
Proposition 3.4 (Asymptotic Convergence with h). For any
ε > 0, there exists δ > 0 such that when h →+∞and
MS ≥δ, the SE upper bound converges to zero:
lim
h→+∞
MS
 1
h(Φ(MS) −Φ(−MS)) −2 + 2Φ(MS)

+
r
2
π e−MS2
2
 = 0
(8)
The Proposition 3.4 reveals that when we fix MS and in-
crease the spatial resolution h, the upper bound |g(h, MS)|
of SE will reduce accordingly. On the other hand, when
h increases, the tensor sizes in V will increase exponen-
tially, 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 of SE decreases with an increase in h, it is
generally preferable to choose the largest possible value of
h based on available computational resources, resulting in a
fixed value of h for a particular computational budget.
Additional theoretical analysis of the proposed binary image-
based time series metric space and proofs are offered in
Appendix C and Appendix D.
3.4. The proposed ViTime model
Figure 3 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 Tok-
enizer and outputs embedded latent representations. Next,
the decoder network is developed to decode latent represen-
tations and produce initial prediction results. To improve
Figure 3: ViTime network. It consists of three key modules:
(1) Visual Time Tokenizer that embeds binary time series
images into latent representations, (2) Decoder that gener-
ates initial predictions from these latent representations, and
(3) Refining Module that improves final prediction quality
by refining the initial results.
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 be-
tween 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 is also adopted.
Refining Module. The transformer architecture in the De-
coder can result in discontinuities at the patch junctions,
which may affect the accuracy of the inverse mapping pro-
cess. To address this issue, the Refining Module building
with CNNs is employed. Initially, tokens decoded by De-
coder are unpatched and fed into a CNN-based backbone.
Next, the ASPP (Chen et al., 2015) module expands the
model receptive field. Finally, the output is upsampled to
5
ViTime: Foundation Model for Time Series Forecasting Powered by Vision Intelligence
the binary pixel metric space, generating the final image
prediction result.
The modeling process of ViTime is as follows:
v′
L = ViTime (vL ⊙ML)
(9)
where ML denotes temporal masks.
Loss function. The loss function employed in this study is
defined as follows:
L = d
v′
L, vL

+ αKLD
v′
L, vL

(10)
where d denotes the distance function defined in Equa-
tion (3), KLD denotes Kullback–Leibler divergence, and α
is the hyperparameter balance quantity between d and KLD.
The combined EMD and KLD loss addresses structural and
probabilistic alignment in the binary image space. EMD
minimizes spatial discrepancies in V, counteracting SE from
discretization, while KLD refines distributional consistency
to mitigate quantization artifacts. This dual approach bal-
ances geometric fidelity (via EMD) and statistical accuracy
(via KLD), crucial given the resolution-computation trade-
off governed by h.
4. Computational experiments
4.1. Experimental Configuration
Datasets and evaluation metrics
Seven popular publicly accessible datasets: Electricity, Traf-
fic, Weather, ETTh1, ETTh2, ETTm1, and ETTm2 (Wu
et al., 2021) are employed in computational experiments to
validate the effectiveness of the proposed ViTime.
Existing numerical fitting-based TSF foundation models,
e.g., TimesFM, are typically pretrained on comprehensive
real-world datasets. While the specific nomenclature of the
testing set may not be explicitly listed in the training data,
there is a possibility that the real-world dataset encompasses
similar data sources, potentially leading to issues of test set
leakage. To address this concern and ensure a more rigorous
and equitable experimental comparison, we propose two
novel metrics for zero-shot evaluation, the Rescale-Mean
Absolute Error (ReMAE) and Rescale-Mean Squared Error
(ReMSE). The fundamental principle underlying ReMAE
and ReMSE involves rescaling the test dataset across various
time resolutions, as illustrated in Equation (11). The time
series interpolation (TSI) method is employed to rescale the
original test time series of length T to βT:
SβT = TSI (ST , rescaling factor = β) .
(11)
The formulas for ReMAE and ReMSE are
ReMSE =
P
β∈U MSE

S
′
βT , SβT

len(U)
(12)
Figure 4: Radar plots comparing the average MAE of
ViTime and TimesFM across different rescale factors. The
radial axis represents MAE, with lower values (larger ra-
dius) indicating better performance. Each axis corresponds
to a specific rescale factor.
ReMAE =
P
β∈U MAE

S
′
βT , SβT

len(U)
U = [0.5, 0.66, 1, 1.5, 2] .
(13)
Model setup
The ViTime model is developed using data sequences syn-
thesized by RealTS. 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 channel-independent strategy
(Nie et al., 2022) is applied, predicting each variable sepa-
rately 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 win-
dow T = 512, and maximum prediction length l = 720.
Gaussian blur with a kernel size of (31, 31) is applied
to the binary image to facilitate faster convergence. For
a fair comparison, all considered models employ a look-
back length of 512 to forecast future sequences of lengths
96, 192, 336, 720. More details on training are available in
the Appendix B.
4.2. Comparison of ViTime to SOTA TSF Benchmarks
Under Zero-shot Setting
To assess the zero-shot performance of ViTime, we con-
sider two leading SOTA TSF models: TimesFM (Das et al.,
2024), introduced by Google Research in April 2024 and ac-
6
ViTime: Foundation Model for Time Series Forecasting Powered by Vision Intelligence
Table 1: Comparisons of zero-shot forecasting results. The best zero-shot results are bolded, the second best are underlined
and the result better than supervised PatchTST are marked with purple.
Method
ViTime-1072
ViTime
PatchTST-ZS
TimesFM
PatchTST
Training data
RealTS
RealTS
RealTS
Real world
Real world
Supervision
Zero-shot
Zero-shot
Zero-shot
Zero-shot
Supervised
Metric
ReMSE
ReMAE
ReMSE
ReMAE
ReMSE
ReMAE
ReMSE
ReMAE
ReMSE
ReMAE
Electricity
0.246
0.320
0.263
0.340
1.414
0.920
0.367
0.404
0.153
0.247
Traffic
0.677
0.365
0.760
0.408
2.053
1.002
0.875
0.518
0.353
0.256
Weather
0.262
0.276
0.263
0.279
0.910
0.584
0.284
0.307
0.222
0.263
ETTh1
0.464
0.446
0.490
0.453
1.477
0.903
0.513
0.476
0.447
0.449
ETTh2
0.330
0.373
0.338
0.380
1.096
0.775
0.355
0.391
0.365
0.406
ETTm1
0.464
0.420
0.490
0.430
1.295
0.797
0.670
0.503
0.359
0.386
ETTm2
0.293
0.348
0.297
0.339
0.805
0.612
0.336
0.359
0.257
0.320
Table 2: Comparisons of Fine-tuning forecasting results. FT is short for fine-tuning. The best results are bolded, and the
second best is underlined.
Method
Data proportion
ETTh1
ETTh2
ETTm1
ETTm2
Electricity
Traffic
Weather
TimesFM (FT)
10%
0.426
0.410
0.388
0.334
GPT4TS (FT)
10%
0.525
0.421
0.441
0.335
Not Reported
TIME-LLM (FT)
10%
0.522
0.394
0.427
0.323
ViTime (FT)
10%
0.424
0.372
0.378
0.316
0.252
0.254
0.252
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
0.393
0.396
0.327
0.274
0.291
0.281
TIMESNET
100%
0.450
0.427
0.406
0.332
0.295
0.336
0.267
ViTTime (FT)
100%
0.408
0.349
0.367
0.300
0.247
0.250
0.251
knowledged as one of the most powerful TSF model as well
as PatchTST (Nie et al., 2022), a well-established model
in supervised and transfer learning for TSF over the past
two years. All models employ a lookback length of 512 to
ensure a fair comparison. For this comparison, we consider
five variants: (1) ViTime - our proposed TSF foundation
model, trained on generative data from RealTS and adopting
a zero-shot paradigm; (2) ViTime-1072 - we allow a longer
lookback length, 1072, to test the peak accuracy of ViTime.
(3) PatchTST-ZS - trained on the same RealTS-generated
data as ViTime, using a numerical fitting paradigm to cre-
ate a zero-shot version of PatchTST. (4) Fully supervised
PatchTST - trained on training datasets to serve as a bench-
mark for maximum achievable accuracy through supervi-
sion. (5) TimesFM - powerful TSF foundation model pre-
trained on extensive real-world datasets.
Table 1 reports the average zero-shot results. The proposed
ViTime can achieve comparable accuracy to the fully super-
vised PatchTST on unseen forecasting data, which suggests
the effectiveness of the proposed ViTime framework. More-
over, it is observable that the performance of ViTime signif-
icantly surpasses PatchTST-ZS. Such a result demonstrates
that modeling TSF tasks from a vision intelligence perspec-
tive can significantly enhance model robustness and accu-
racy. Finally, ViTime consistently outperforms TimesFM
across various datasets, demonstrating an average improve-
ment of 9% in forecasting accuracy. This advantage be-
comes even more pronounced in long sequence forecasting
scenarios, where ViTime achieves an impressive 15% accu-
racy improvement.
Figure 4 illustrates the performance of ViTime and
TimesFM across different rescaling factors. It is clear that
TimesFM achieves high accuracy at rescaling factor = 1,
but the accuracy significantly decreases for other factors,
indicating potential data leakage issues in the training set.
In contrast, ViTime consistently delivers robust prediction
results across all rescaling factors.
4.3. Comparison of ViTime to SOTA TSF Benchmarks
Under Fine-tuning Settings
While zero-shot results demonstrate the predictive capability
of ViTime on unseen data, some high-precision TSF tasks
might require further fine-tuning studies to enhance pre-
diction accuracy. Thus, this section focuses on fine-tuning
studies across various specialized datasets.
To comprehensively evaluate the fine-tuning performance of
ViTime, we compare ViTime with other foundation models
and SOTA supervised TSF models. Foundational models
including TimesFM (Das et al., 2024), GPT4TS (Zhou et al.,
7
ViTime: Foundation Model for Time Series Forecasting Powered by Vision Intelligence
Figure 5: Performance with different fine-tuning data pro-
portion.
2023), and TIME-LLM (Jin et al., 2023) are fine-tuned
using 10% of the training data. Recent SOTA-supervised
TSF models such as SiMBA (Patro & Agneeswaran, 2024),
TIMESNET (Wu et al., 2022), and PatchTST (Nie et al.,
2022) use 100% of the training data, as reported in their
respective papers. We also fine-tune ViTime using from 10%
to 100% of the training data to provide a comprehensive
comparison.
Results of the fine-tuning study are provided in Table 2.
ViTime fine-tuned with 10% of the training data can outper-
form other foundational models and the latest supervised
models updated on 100% of the training data. Further-
more, as shown in Fig. 5, when the fine-tuning data propor-
tion approaches 100%, the prediction accuracy of ViTime
gradually increases and significantly surpasses all existing
models, which suggests that ViTime excels in both low-
data-availability environments (10% fine-tuning) and full-
data-availability scenarios (100% fine-tuning), consistently
outperforming both other foundation models and specialized
supervised models.
4.4. Comparison of ViTime to SOTA TSF Benchmarks
in Robust Inference
To further evaluate the generalizability of ViTime, this
section conducts a comparative analysis against TimesFM
across four data quality scenarios under zero-shot setting
(Figure 6): TSF with original time series, noise-augmented
time series, harmonic-augmented time series, and time se-
ries of missing data.
In TSF, considering the original time series, ViTime demon-
strates superior performance in capturing the underlying
periodicities and patterns. In TSF with noise-augmented
time series, both models can extract meaningful patterns,
but ViTime maintains consistent performance across longer
sequences, whereas TimesFM exhibits drift in periodic align-
ments. For harmonic-augmented time series, ViTime excels
Figure 6:
Performance comparison of ViTime versus
TimesFM on TSF tasks under various data perturbations: a.
Original time series. b. Time series with noises injected.
c. Time series with harmonic added. d. Time series with
missing data.
by accurately capturing both fundamental and harmonic
wave patterns, while TimesFM struggles to disentangle these
complex periodic structures.
The most striking difference emerges in the TSF scenario
with missing data. As TimesFM is a numerical data fitting-
based model, it requires additional imputation techniques.
At the same time, ViTime can seamlessly handle these gaps
by treating them as zero-valued pixels in its visual repre-
sentation, showcasing the robust inference capabilities of a
vision intelligence-based paradigm.
The complete numerical results and detailed ablation study
are presented in Appendix E. We also present our discussion
section in Appendix F.
5. Conclusions
This work developed a vision intelligence-powered compu-
tational paradigm, ViTime, for developing the TSF founda-
tion model compared with numerical data fitting principles
prevalently considered in literature. ViTime was inspired
by human visual cognitive processes understanding and an-
alyzing time series. By introducing a paradigm of operating
numerical data in image space and the unique deep network
based computing pipeline, ViTime elevated the SOTA per-
formance on zero-shot/fine-tuning TSF without relying on
prior data samples, demonstrating the great potential for
reshaping the computational mechanism in TSF founda-
tion model development. Moreover, data often suffer from
diverse contamination and variability in reality. ViTime
enabled robust performance under various real-world data
perturbations and alterations.
8
ViTime: Foundation Model for Time Series Forecasting Powered by Vision Intelligence
Impact Statement
This paper presents work whose goal is to advance the field
of Machine Learning. There are many potential societal
consequences of our work, none which we feel must be
specifically highlighted here.
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ViTime: Foundation Model for Time Series Forecasting Powered by Vision Intelligence
A. Details of RealTS
We present RealTS, a versatile framework for synthesizing realistic time series data. RealTS employs multiple data behavior
modes under two main hypotheses: periodic (φp) and trend (φt). This section details the various behavior modes, their
configurations, and provides visual examples.
A.1. Periodic Hypothesis Behaviors
Under the periodic hypothesis φp, 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 (14) to simulate the underlying behavior of real-world periodic time series:
P (sL|L, Bp) |Bp=IFFT =
ZZ ∞
−∞
N
Am; µAm, σ2
Am

· N
ϕ; µP, σ2
P

× δ (sL −IFFT (Am, ϕ, L)) dϕdAm
(14)
where two empirical distributions of Fourier transform amplitudes and phases, N(Am; µAm, σ2
Am) and N(ϕ; µP , σ2
P ), are
maintained, and δ 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.
Figure 7: Empirical distribution I employed in IFFTB.
The empirical distributions utilized in Am and ϕ are illustrated in Figure 7-Figure 8. During experiments, we randomly
select one of two empirical distributions for generating Am and ϕ. Figure 9 shows examples of time series generated using
IFFTB.
11
ViTime: Foundation Model for Time Series Forecasting Powered by Vision Intelligence
Figure 8: Empirical distribution II employed in IFFTB.
Figure 9: Examples of time series generated using IFFTB.
12
ViTime: Foundation Model for Time Series Forecasting Powered by Vision Intelligence
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, fperiodic, with different frequencies and amplitudes:
P (sL|L, Bp) |Bp=PWB =
ZZ ∞
−∞
N

sL;
kPWB
X
i=1
Aifperiodic (ωit) , σ2
ϵ
!
× P (A) P (ω) dωdA
(15)
where P(A) and P(ω) denote predefined prior distributions of amplitudes and frequency; kP W B denotes the number of
mixed periodic functions.
For PWB, we define the prior distributions for amplitude and frequency as:
A ∼U(0.5, 5)
(16)
ln(ω) ∼U(ln(11), ln(2L))
(17)
The parameter kP W B is modeled as:
P(kP W B = k) = 1
8, for k = 1, 2, . . . , 8
(18)
Figure 10 shows examples of time series generated using PWB.
Figure 10: Examples of time series generated using PWB.
A.2. Trend Data Hypothesis Behaviors
Under the trend data hypothesis φt, 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:
P (si|si−1, L, Bp) |Bp=RWB = N
0, σ2
(19)
Figure 11 shows examples of time series generated using RWB.
A.2.2. LOGISTIC GROWTH BEHAVIOR (LGB)
The LGB models data with a logistic growth function, capturing the S-shaped growth pattern:
13
ViTime: Foundation Model for Time Series Forecasting Powered by Vision Intelligence
Figure 11: Examples of time series generated using RWB.
P (sL|L, Bp) |Bp=LGB
=
ZZ ∞
−∞
N

sL;
K
1 + e−r(L−L0) , σ2
ϵ

P(K)P(r)dKdr
(20)
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:
ln(K) ∼U(ln(1), ln(10))
(21)
ln(r) ∼U(ln(0.001), ln(0.1))
(22)
Figure 12 shows examples of time series generated using LGB.
Figure 12: Examples of time series generated using LGB.
A.2.3. TREND WAVE DATA BEHAVIOR (TWDB)
TWDB combines linear trends with periodic fluctuations:
P (sL|L, Bp) |Bp=TWDB =
ZZ ∞
−∞
N

sL; aL + b +
kTWDB
X
i=1
Aifperiodic (ωit) , σ2
ϵ
!
× P(a)P(b)P (A) P (ω) dadbdAdω
(23)
where P(a), P(b), P (A) and P (ω) are predefined prior distributions of hyperparameters.
In the TWDB, we define the probability densities for linear function random variables P(a) and P(b), as well as for the
superimposed periodic wave components P(A) and P(ω). The settings for P(A), P(ω), and kT W DB are consistent with
those used in the PWB module. The probability densities for P(a) and P(b) are detailed below:
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ViTime: Foundation Model for Time Series Forecasting Powered by Vision Intelligence
a ∼U(−1, 1)
(24)
b ∼U(−10, 10)
(25)
Figure 13 shows examples of time series generated using TWDB.
Figure 13: Examples of time series generated using TWDB.
A.3. Data Augmentation Techniques
To enhance the diversity and robustness of synthetic data, we employ various data augmentation techniques, including:
• Multiple period replication: Repeats the generated periodic data over multiple cycles to capture long-term periodic
patterns.
• Data flipping: Reverses the time series to create new patterns while preserving underlying characteristics.
• Convolution smoothing and detrending: Removes underlying trends from the data to isolate periodic components,
making it easier for the model to learn these patterns.
• Data perturbation: Introduces sudden changes or anomalies into the data, simulating real-world disturbances and
improving the model’s ability to handle unexpected variations.
More details of RealTS are offered in the code part of the Supplementary Material.
B. Training configuration
B.1. ViTime model structure
Table 3: Details of model architecture
Module
Embed dim
Depth
Patch size
Num heads
Visual Time Tokenizer
768
9
(4,32)
12
Decoder
384
4
\
12
The detailed network configuration of the proposed ViTime is reported in Table 3.
B.2. Training details
We incorporate the following technique details during training.
Data normalization. To ensure ViTime can effectively capture patterns involving sudden changes, an in-sequence data
normalization based on L2 normalization is implemented. By normalizing each sequence within the data sequence, the
model can pay more attention to abrupt variations. The normalization process is defined as follows:
15
ViTime: Foundation Model for Time Series Forecasting Powered by Vision Intelligence
Table 4: Numerically Solved Optimal MS∗
Optimal MS∗
Resolution h
k = 1
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
SL = SL −mean (∥S1:T∥2)
std (S1:T)
(26)
Temporal resolution enhancement. To improve the temporal resolution in the binary image metric space, the input
sequence is first linearly interpolated to twice its original length (2L), so that, the granularity of the temporal dimension
is increase. Moreover, due to the inherent sparsity of the binary image space, most patches exhibit low-level information
density. To improve the information density, Gaussian blurring is applied to the input binary images before being fed into
ViTime, thereby reducing sparsity and enhancing information density.
C. Theoretical analysis and Proof
C.1. Theoretical analysis of MS
Proposition C.1 investigates how the upper bound of SE varies with a MS given a fixed value of h, which provides a
theoretical guidance to choose the best MS under different computational budgets (h).
Proposition C.1 (Optimal MS Selection). For fixed h, there exists a unique optimal threshold MS∗minimizing the SE
upper bound, characterized by:
1
h (Φ(MS∗) −Φ(−MS∗)) −2 + 2Φ(MS∗) + MS∗
h
r
2
π e−MS∗2
2
= 0
(27)
C.1.1. GENERALIZATION TO SCALED VARIANCE CASE
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 ∼N(0, kI), where k > 1
captures the variance scaling.
Proposition C.2 (Optimal Threshold under Variance Scaling). Under the assumption S ∼N(0, kI) with k > 1, the optimal
threshold MS∗that minimizes the SE upper bound is characterized by the following condition:
1
h

Φ
MS∗
√
k

−Φ

−MS∗
√
k
 !
−2 + 2Φ
MS∗
√
k

+ MS∗
h
r
2
πk e−(MS∗)2
2k
= 0
(28)
Here, Φ(·) is the CDF of the standard normal distribution, h is the spatial resolution, and k is the variance scaling factor. This
result generalizes Proposition C.1 to scenarios where the latent space exhibits larger variability. In practice, it is challenging
to find an analytic solution for Equation (32). Thus, the numerical method is employed to obtain solutions of Equation (32)
in this work and the corresponding results are reported in Table 4.
16
ViTime: Foundation Model for Time Series Forecasting Powered by Vision Intelligence
C.2. Proofs
C.2.1. PROOF OF THEOREM 3.3
Theorem 3.3 (System Error Upper Bound). Given a tensor bs ∈S ⊂Rc×t, the system error defined as

f −1 (f (bs)) −bs

1
satisfies the following expectation bound:
SE := E

f −1 (f (bs)) −bs

1 ≤g(h, MS) = ct
"
MS
 1
h (Φ(MS) −Φ(−MS)) −2 + 2Φ(MS)

+
r
2
π e−MS2
2
#
(29)
where Φ denotes the cumulative density function of N(0, I).
Proof. Step 1: Error Decomposition
The total error decomposes into quantization error (|s| ≤MS) and truncation error (|s| > MS):
E∥f −1(f(s)) −s∥1 = ct


Z MS
−MS
f −1(f(s)) −s
 P(s)ds
|
{z
}
Quantization
+ 2
Z ∞
MS
(s −MS)P(s)ds
|
{z
}
Truncation


Step 2: Quantization Error Bound
When |s| ≤MS, the continuous space is quantized into h bins of width ∆= 2MS/h. The maximum per-element error is
∆/2 = MS/h:
Z MS
−MS
MS
h P(s)ds = MS
h [Φ(MS) −Φ(−MS)]
Step 3: Truncation Error Calculation
For s > MS, the reconstruction error is s −MS. Using Gaussian integrals:
Z ∞
MS
sP(s)ds =
1
√
2π e−MS2/2,
Z ∞
MS
P(s)ds = 1 −Φ(MS)
Thus:
2
Z ∞
MS
(s −MS)P(s)ds = 2

e−MS2/2
√
2π
−MS[1 −Φ(MS)]
!
Step 4: Combine Results
Summing both components:
SE ≤ct
(
MS
h [Φ(MS) −Φ(−MS)] +
r
2
π e−MS2/2 −2MS[1 −Φ(MS)]
)
Rearranging terms yields Equation (29).
C.2.2. PROOF OF PROPOSITION 3.4
Proposition 3.4 (Asymptotic Convergence with h). For any ε > 0, there exists δ > 0 such that when h →+∞and
MS ≥δ, the SE upper bound converges to zero:
lim
h→+∞
MS
 1
h(Φ(MS) −Φ(−MS)) −2 + 2Φ(MS)

+
r
2
π e−MS2
2
 = 0
(30)
17
ViTime: Foundation Model for Time Series Forecasting Powered by Vision Intelligence
Proof. As h →∞, the 1/h term vanishes. Analyze the residual terms for MS →∞:
Lemma C.3 (Mill’s Ratio). For MS ≫0,
1 −Φ(MS) ∼e−MS2/2
MS
√
2π

1 −
1
MS2 + · · ·

Substituting into the remaining terms:
−2MS

e−MS2/2
MS
√
2π
!
+
r
2
π e−MS2/2 = −
r
2
π e−MS2/2 +
r
2
π e−MS2/2 = 0
Thus ∀ε > 0, choose δ such that for MS ≥δ, the residual < ε.
C.2.3. PROOF OF PROPOSITION C.1
Proposition C.1 (Optimal MS Selection). For fixed h, there exists a unique optimal threshold MS∗minimizing the SE
upper bound, characterized by:
1
h (Φ(MS∗) −Φ(−MS∗)) −2 + 2Φ(MS∗) + MS∗
h
r
2
π e−MS∗2
2
= 0
(31)
Proof. First-Order Condition: Differentiate the SE bound χ(MS, h) w.r.t. MS:
∂χ
∂MS = 1
h[ϕ(MS) + ϕ(−MS)] + 2ϕ(MS) −2 + MS
h
r
2
π e−MS2/2 −MS2
h
r
2
π e−MS2/2
Using ϕ(MS) =
1
√
2πe−MS2/2 and ϕ(−MS) = ϕ(MS), we obtain Equation (27).
Second-Order Condition:
∂2χ
∂MS2 =
r
2
π e−MS2/2
2 + h −MS2
h

• Convex when MS <
√
h + 2:
∂2χ
∂MS2 > 0
• Concave when MS >
√
h + 2:
∂2χ
∂MS2 < 0
Boundary Behavior:
lim
MS→0+
∂χ
∂MS = −2 < 0,
lim
MS→+∞
∂χ
∂MS = +∞> 0
By Intermediate Value Theorem and convex-concave transition, there exists exactly one MS∗where ∂χ/∂MS = 0.
C.2.4. PROOF OF PROPOSITION C.2
Proposition C.2 (Optimal Threshold under Variance Scaling). Under the assumption S ∼N(0, kI) with k > 1, the optimal
threshold MS∗that minimizes the SE upper bound is characterized by the following condition:
1
h

Φ
MS∗
√
k

−Φ

−MS∗
√
k
 !
−2 + 2Φ
MS∗
√
k

+ MS∗
h
r
2
πk e−(MS∗)2
2k
= 0
(32)
Proof. :
18
ViTime: Foundation Model for Time Series Forecasting Powered by Vision Intelligence
Step 1: Scaled Probability Measures
For S ∼N(0, kI), the probability density function (PDF) and cumulative distribution function (CDF) are scaled as:
Pk(s) =
1
√
2πk
e−s2/(2k),
Φk(x) = Φ
 x
√
k

(33)
This scaling reflects the increased spread of the latent space distribution.
Step 2: Modified Quantization Error
The quantization error term, which captures the discretization loss within the threshold [−MS, MS], becomes:
Z MS
−MS
MS
h Pk(s)ds = MS
h
[Φk(MS) −Φk(−MS)]
(34)
This term increases with MS due to the wider interval.
Step 3: Adjusted Truncation Error
The truncation error, representing the loss from discarding values outside [−MS, MS], transforms as:
2
Z ∞
MS
(s −MS)Pk(s)ds = 2
"r
k
2π e−MS2/(2k) −MS(1 −Φk(MS))
#
(35)
This term decreases with MS as more probability mass is retained.
Step 4: Unified Error Bound
Combining the quantization and truncation errors yields the SE upper bound:
SE ≤ct
(
MS
h [Φk(MS) −Φk(−MS)] +
r
2k
π e−MS2/(2k) −2MS[1 −Φk(MS)]
)
(36)
This bound balances the trade-off between discretization and truncation losses.
Step 5: First-Order Optimality Condition
Differentiating (36) with respect to MS and setting the derivative to zero gives:
∂SE
∂MS = 1
h [Φk(MS) −Φk(−MS)] −2 + 2Φk(MS)
+ MS
h
r
2
πk e−MS2/(2k) = 0
(37)
Substituting Φk(x) = Φ(x/
√
k) yields the condition in Equation (32).
Step 6: Uniqueness Guarantee
The second derivative test confirms the uniqueness of MS∗:
∂2SE
∂MS2 =
r
2
πk3 e−MS2/(2k)

2 + h −MS2
k

(38)
• Convex region: MS <
p
k(h + 2) ⇒∂2SE
∂MS2 > 0
• Concave region: MS >
p
k(h + 2) ⇒∂2SE
∂MS2 < 0
Boundary behavior ensures a unique solution:
lim
MS→0+
∂SE
∂MS = −2 < 0,
lim
MS→+∞
∂SE
∂MS = +∞> 0
By the Intermediate Value Theorem and convex-concave transition, exactly one MS∗exists.
19
ViTime: Foundation Model for Time Series Forecasting Powered by Vision Intelligence
The generalized Proposition C.2 provides a principled framework for selecting the optimal threshold MS∗when the latent
space exhibits larger variance. Key insights include:
• Scaling Law: The optimal threshold scales as MS∗∝
√
k, reflecting the need to accommodate a larger MS∗when
facing the dramatically fluctuating application scenarios. We have offered further analysis in Appendix E.1 and
Appendix F.3.
• Practical Guidance: For real-world applications where k is unknown, adaptive methods can estimate k and dynamically
adjust MS∗.
• Consistency Check: When k = 1, Proposition C.2 reduces to the original Proposition C.1, validating the generalization.
D. Theoretical Advantages of Visual Intelligence for Time Series Forecasting
The integration of visual intelligence into TSF transcends a mere architectural decision, presenting a comprehensive
framework rooted in cognitive science, mathematical rigor, and empirical robustness. This section delineates the theoretical
underpinnings of ViTime, our proposed model, through three pivotal dimensions: cognitive alignment, pattern preservation,
and geometric regularization. Each dimension addresses specific advantages inherent in visual intelligence approaches to
TSF.
D.1. Cognitive Alignment
Analysts inherently interpret time series data by discerning visual patterns rather than processing raw numerical values.
This cognitive inclination leverages the brain’s proficiency in recognizing spatial relationships and temporal trends visually.
ViTime capitalizes on this natural predisposition by converting time series data into binary images, thereby enabling the
model to emulate human-like analytical reasoning. This alignment is operationalized through isometric spatiotemporal
mapping, which ensures that temporal patterns in the time series are preserved as spatial patterns in the visual representation.
Definition D.1 (Spatiotemporal Isometry). For a time series s ∈S and its visual embedding f(s) ∈{0, 1}h×T , the mapping
satisfies:
1
T
T −1
X
t=1
|st+1 −st| = λ
T −1
X
x=1
∥f(s)x −f(s)x+1∥F
where λ is a scale invariance factor that controls the relationship between temporal changes in the time series and spatial
changes in the visual representation.
This isometry ensures that slope changes in the time series manifest as proportional edge densities in the image. By aligning
the data representation with human cognitive processes, ViTime not only enhances interpretability but also leverages the
brain’s innate ability to recognize spatial relationships and temporal trends. This alignment is particularly important for
applications where human-AI collaboration is essential, such as in healthcare or finance.
D.2. Pattern Preservation
A paramount challenge in TSF is the preservation of critical temporal structures—such as trends, seasonality, and periodici-
ties—through data transformations. ViTime addresses this challenge by ensuring spectral fidelity, thereby maintaining the
essential frequency components of the original time series within its visual representation.
Theorem D.2 (Spectral Fidelity). For any time series s(t) with Fourier transform Ft(ω), its visual representation v(x, y)
satisfies:
∃C > 0 : ∥Fx,y(v) −C · Ft(s)∥∞≤ϵ(h),
where ϵ(h) →0 as the image height h →∞.
Proof. Consider a discrete time series s(t) ∈R of length T. The visual representation v(x, y) ∈{0, 1}h×w is constructed
via thresholding as follows:
20
ViTime: Foundation Model for Time Series Forecasting Powered by Vision Intelligence
1. Thresholding Process: The value range of s(t) is divided into h equal intervals. For each time step t, the pixel v(t, y)
is set to 1 if:
s(t) ∈
y −1
h
∆, y
h∆

,
and 0 otherwise, where ∆= max(s) −min(s).
2. 2D Fourier Transform of v(x, y): The 2D Fourier transform is given by:
Fx,y(v)(u, v) =
T −1
X
x=0
h−1
X
y=0
v(x, y)e−i2π( ux
T + vy
h ).
3. Relation to 1D Fourier Transform: Focusing on the v = 0 frequency component:
Fx,y(v)(u, 0) =
T −1
X
x=0
 h−1
X
y=0
v(x, y)
!
e−i2π ux
T .
The inner sum P
y v(x, y) approximates s(x) scaled by h/∆, yielding:
h−1
X
y=0
v(x, y) ≈h
∆s(x) + O
 1
h

.
Let C = h
∆, then:
Fx,y(v)(u, 0) ≈C · Ft(s)(u) + O
T
h

.
4. Error Bound Analysis: The approximation error comprises:
• Discretization Error: O(1/h) per time step.
• Boundary Effects: Diminishing as h →∞.
Consequently, the supremum norm of the error over all frequencies is:
ϵ(h) = max
u
|Fx,y(v)(u, 0) −C · Ft(s)(u)| = O
T
h

,
which tends to zero as h increases.
5. Extension to 2D Spectrum: For non-zero v-frequencies, energy concentrates at v = 0 due to threshold averaging:
∥Fx,y(v)∥∞= ∥Fx,y(v)(·, 0)∥∞,
thereby ensuring the bound holds across the entire 2D Fourier spectrum.
Theorem D.2 substantiates that ViTime’s binary image transformation preserves the critical spectral characteristics of
the original time series. This preservation ensures that both high-frequency dynamics, such as short-term fluctuations,
and low-frequency components, like long-term trends, are accurately represented and hence effectively captured by the
forecasting model.
21
ViTime: Foundation Model for Time Series Forecasting Powered by Vision Intelligence
D.3. Geometric Regularization through Visual Embedding
A key challenge in TSF is the sensitivity of models to noise and perturbations in the input data. Traditional numerical
approaches often struggle to maintain robustness under such conditions, as they operate directly on raw time series values
without explicit structural constraints. By mapping time series data to binary images, we introduce a form of geometric
regularization that smooths the underlying data manifold, making it less sensitive to small perturbations in Theorem D.3.
This is particularly important in real-world applications where data quality is often compromised by noise and outliers. The
computational results are offered in Appendix E.4.
Theorem D.3 (Geometric Regularization through Visual Embedding). Let Ms ⊂RT denote the manifold of raw time series
data, and Mv ⊂{0, 1}h×w its corresponding visual embedding. Assume the time series s(t) has a bounded amplitude
range ∆= max(s) −min(s) and is Lipschitz continuous with constant Lip(s). Then, the visual embedding process imposes
the following constraints on Mv:
1. Local Perturbation Stability: For any time series s ∈Ms and its visual embedding v ∈Mv, small perturbations in
the input are compressed into sparse changes in the output:
∀ϵs > 0,
∥s′ −s∥∞≤ϵs =⇒∥v′ −v∥0 ≤hTϵs
∆
+ O(1/h),
where ∥v′ −v∥0 is the Hamming distance between the original and perturbed visual embeddings.
2. Path Complexity Constraint: Any continuous path γ : [0, 1] →Mv in the visual embedding space has a bounded
number of pixel transitions:
Jump(γ) ≤T ·
h · Lip(s)
∆

,
where Jump(γ) counts the number of pixel transitions along the path γ.
Proof. Part 1: Local Perturbation Stability Consider a time series s(t) and its visual embedding v(x, y). A perturbation
ϵs in the input time series can cause the value s(t) to shift by at most ϵs. In the visual embedding, this corresponds to a
vertical shift of the active pixel within a column. The maximum vertical shift is bounded by:
∆y ≤
hϵs
∆

.
Since each column contains exactly one active pixel, the total Hamming distance between the original and perturbed
embeddings is:
∥v′ −v∥0 ≤T · ∆y ≤T
hϵs
∆+ 1

.
Ignoring the constant term, we obtain:
∥v′ −v∥0 ≤hTϵs
∆
+ O(1/h).
This shows that small input perturbations are compressed into sparse output changes, with sparsity controlled by the bin
height h.
Part 2: Path Complexity Constraint Consider a continuous path γ(τ) in Mv, parameterized by τ ∈[0, 1]. The
corresponding time series path s(τ, t) is Lipschitz continuous, so the difference between adjacent time steps is bounded by
Lip(s). In the visual embedding, this translates to a vertical shift of at most:
|yt+1 −yt| ≤
h · Lip(s)
∆

.
The total number of pixel transitions along the path is therefore bounded by:
Jump(γ) ≤T ·
h · Lip(s)
∆

.
This demonstrates that the complexity of paths in Mv is linearly controlled by the bin height h.
22
ViTime: Foundation Model for Time Series Forecasting Powered by Vision Intelligence
Table 5: Empirical Forecasting Performance under Different MS Values
MS
2.38
2.64
2.88
3.09
3.50
5.00
6.00
ReMSE
0.4423
0.4404
0.4400
0.4348
0.4178
0.4780
0.4724
ReMAE
0.3818
0.3812
0.3811
0.3788
0.3759
0.3990
0.3959
Theorem D.3 establishes that the visual embedding process introduces geometric regularization by compressing input
perturbations and constraining the complexity of paths in the embedding space. This has two key implications:
• Robustness to Noise: The local perturbation stability property ensures that small input perturbations result in sparse
changes in the visual representation. This makes the model less sensitive to noise, as minor fluctuations in the input are
effectively filtered out. By adjusting h, practitioners can control the trade-off between robustness and model complexity.
Larger values of h enhance robustness (see in Figure 14) but may increase computational cost.
• Controlled Complexity: The path complexity constraint limits the number of pixel transitions in the visual embedding,
preventing the model from overfitting to complex patterns in the data. This regularization effect is controlled by the bin
height h, which can be tuned to balance robustness and expressiveness.
The geometric regularization theorem provides a rigorous foundation for the robustness and stability of ViTime. By
transforming time series data into binary images, ViTime introduces structural constraints that mitigate the effects of noise
and prevent overfitting. This theoretical insight not only enhances the interpretability of the model but also guides practical
implementation, ensuring reliable performance in real-world applications.
D.4. Summary
ViTime’s adoption of visual intelligence for time series forecasting is a theoretically grounded advancement that effectively
addresses the limitations of conventional numerical approaches. By aligning with human cognitive processes, ensuring the
preservation of essential temporal patterns, and embedding intrinsic robustness against noise, ViTime establishes a robust
and interpretable framework for accurate and reliable forecasting. This triad of theoretical foundations not only enhances
forecasting performance but also promotes greater trust and usability in practical applications.
E. Additional results of computational experiments
E.1. Ablation study
E.1.1. ABLATION OF MS
Proposition C.2 establishes the theoretical relationship between the optimal MS threshold and the variance scaling factor
k in the latent space. For stationary data (S ∼N(0, I), i.e., k = 1), Proposition C.2 reveals that with h = 128, the
optimal MS should be 2.64. However, real-world time series often exhibit non-stationary characteristics. Our analysis of the
target variable’s variance after input-based standardization (see KL divergence results in Section 4.2) demonstrates that the
effective k value for the prediction horizon falls within [1.5, 2] across all benchmark datasets.
Table 4 provides numerically solved optimal MS∗values under different k and h configurations. For h = 128 (our
experimental setting) and k ∈[1.5, 2], the theoretical optimal MS ranges between 3.26-3.76. This motivates our selection of
MS = 3.5 as a balanced configuration within this interval.
To validate this choice, Table 5 presents the average relative MSE and MAE across six benchmark datasets. The results
demonstrate that MS = 3.5 achieves the minimum forecasting error, reducing ReMSE by 4.1% and ReMAE by 1.8% com-
pared to the stationary optimal MS = 2.64. This strong alignment between theoretical predictions (Table 4) and empirical
performance (Table 5) confirms that our MS selection strategy effectively minimizes system error while accommodating
real-world data characteristics.
23
ViTime: Foundation Model for Time Series Forecasting Powered by Vision Intelligence
Table 6: Ablation study of loss function components on prediction performance.
Metric
Loss Configuration
EMD Only
JSD+EMD (Ours)
JSD Only
Average ReMAE
0.3941
0.3759
0.3956
Average ReMSE
0.4586
0.4178
0.4637
E.1.2. ABLATION OF LOSS FUNCTION
In this section, we conducted ablation studies on the loss function components using the ViTime architecture. Table 6 com-
pares model performance under three configurations: (1) EMD alone, (2) our proposed loss function in Equation (10),where
α = 0.2 to balance the quantity level, and (3) JSD alone. The results demonstrate that our dual-objective loss achieves
optimal performance on both ReMSE and ReMAE.
E.1.3. ABLATION OF OTHER CONFIGURATION
(a) Forecasting performance varying with
h
(b) Forecasting performance varying with
the length of lookback window
(c) Forecasting performance across differ-
ent model sizes
Figure 14: Ablation studies with zero-shot forecasting.
Note: The model size of ViTime used in computational experiments is 93M parameters version.
In this section, we perform several ablation studies to gain deeper insights into ViTime model configuration. The results are
reported in Figure 14. Figure 14 a depicts the influence of varying spatial resolutions (h) on model accuracy. Although
increasing h slightly improves the prediction results, the associated computational cost increases exponentially. Thus, setting
h to 128 is more economical and efficient. Figure 14 b illustrates the effect of different lookback window lengths (T)
on prediction accuracy. It is evident that a longer lookback window length significantly enhances the model’s prediction
accuracy. Figure 14 c 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 with TimesFM, which is over 200M parameters, further demonstrating the efficiency and effectiveness of ViTime.
E.2. Zero-shot study
The full results of the zero-shot study are reported in Table 7.We also illustrate zero-shot TSF examples with prediction
length equals 720 of the proposed ViTime versus TimesFM in Figure 18 - Figure 23. It is observable that ViTime consistently
demonstrates superior zero-shot prediction performance compared to TimesFM across a range of rescale factors.
E.3. Fine-tuning study
Complete results of the fine-tuning study are reported in Table 8.
E.4. Robust inference study
To thoroughly evaluate the robustness of ViTime against TimesFM under various data quality scenarios, we conducted
extensive experiments with different types of data perturbations under zero-shot setting. Table 9 presents the comparative
24
ViTime: Foundation Model for Time Series Forecasting Powered by Vision Intelligence
Table 7: Full computational results of zero-shot study. The best zero-shot results are bolded, the second best are underlined
and the result better than supervised PatchTST are marked with purple.
Method
ViTime-1072
ViTime
PatchTST-ZS
TimesFM
PatchTST
Training data
RealTS
RealTS
RealTS
Real world data
Real world data
Supervision
Zero-shot
Zero-shot
Zero-shot
Zero-shot
Supervised
Metric
ReMSE
ReMAE
ReMSE
ReMAE
ReMSE
ReMAE
ReMSE
ReMAE
ReMSE
ReMAE
Electricity
96
0.216
0.297
0.219
0.308
1.336
0.896
0.269
0.338
0.121
0.216
192
0.229
0.308
0.234
0.320
1.425
0.924
0.322
0.375
0.139
0.234
336
0.244
0.320
0.258
0.338
1.443
0.929
0.381
0.414
0.156
0.251
720
0.296
0.355
0.344
0.395
1.451
0.932
0.496
0.488
0.194
0.286
Average
-
0.246
0.320
0.263
0.340
1.414
0.920
0.367
0.404
0.153
0.247
Traffic
96
0.665
0.357
0.709
0.394
1.964
0.975
0.771
0.459
0.327
0.242
192
0.658
0.354
0.723
0.391
2.073
1.008
0.818
0.495
0.342
0.249
336
0.660
0.357
0.747
0.401
2.071
1.011
0.878
0.524
0.359
0.259
720
0.725
0.391
0.844
0.447
2.105
1.012
1.033
0.595
0.387
0.274
Average
-
0.677
0.365
0.760
0.408
2.053
1.002
0.875
0.518
0.353
0.256
Weather
96
0.179
0.210
0.172
0.209
0.836
0.561
0.171
0.215
0.146
0.197
192
0.233
0.258
0.227
0.258
0.922
0.588
0.236
0.273
0.195
0.244
336
0.285
0.295
0.287
0.300
0.938
0.592
0.303
0.327
0.241
0.281
720
0.352
0.341
0.368
0.352
0.945
0.594
0.427
0.411
0.307
0.328
Average
-
0.262
0.276
0.263
0.279
0.910
0.584
0.284
0.307
0.222
0.263
ETTh1
96
0.425
0.416
0.432
0.414
1.398
0.877
0.409
0.413
0.365
0.396
192
0.438
0.428
0.462
0.433
1.490
0.908
0.486
0.458
0.414
0.431
336
0.464
0.447
0.496
0.457
1.507
0.912
0.535
0.488
0.457
0.461
720
0.530
0.493
0.574
0.509
1.511
0.914
0.623
0.545
0.550
0.506
Average
-
0.464
0.446
0.490
0.453
1.477
0.903
0.513
0.476
0.447
0.449
ETTh2
96
0.252
0.315
0.258
0.320
1.032
0.752
0.265
0.324
0.290
0.349
192
0.285
0.346
0.301
0.355
1.115
0.782
0.316
0.362
0.347
0.390
336
0.351
0.389
0.346
0.390
1.115
0.781
0.365
0.401
0.372
0.414
720
0.431
0.443
0.447
0.458
1.122
0.783
0.472
0.476
0.451
0.469
Average
-
0.330
0.373
0.338
0.380
1.096
0.775
0.355
0.391
0.365
0.406
ETTm1
96
0.428
0.394
0.416
0.392
1.215
0.773
0.576
0.454
0.296
0.347
192
0.451
0.411
0.454
0.413
1.309
0.801
0.608
0.479
0.338
0.374
336
0.467
0.424
0.503
0.437
1.329
0.808
0.705
0.517
0.381
0.398
720
0.508
0.449
0.588
0.479
1.327
0.807
0.792
0.563
0.421
0.423
Average
-
0.464
0.420
0.490
0.430
1.295
0.797
0.670
0.503
0.359
0.386
ETTm2
96
0.211
0.293
0.203
0.277
0.749
0.593
0.214
0.284
0.174
0.262
192
0.263
0.330
0.265
0.319
0.823
0.618
0.295
0.330
0.227
0.301
336
0.309
0.360
0.322
0.356
0.817
0.617
0.368
0.377
0.279
0.336
720
0.388
0.409
0.400
0.406
0.829
0.621
0.465
0.443
0.348
0.382
Average
-
0.293
0.348
0.297
0.339
0.805
0.612
0.336
0.359
0.257
0.320
25
ViTime: Foundation Model for Time Series Forecasting Powered by Vision Intelligence
Table 8: MAE of different methods in fine-tuning study.
Method
VITIME (FT) TimesFM (FT) GPT4TS (FT) TIME-LLM (FT) PatchTST SiMBA TIMESNET PatchTST
Data proportion
10%
100%
10%
10%
10%
10%
100%
100%
100%
ETTh1
96
0.397
0.383
0.398
0.485
0.460
0.485
0.395
0.402
0.400
192 0.414
0.401
0.424
0.524
0.483
0.524
0.424
0.429
0.429
336 0.427
0.410
0.436
0.550
0.540
0.550
0.443
0.469
0.440
720 0.460
0.438
0.445
0.610
0.604
0.610
0.469
0.500
0.468
Average
-
0.424
0.408
0.426
0.542
0.522
0.542
0.433
0.450
0.434
ETTh2
96
0.324
0.302
0.356
0.389
0.326
0.389
0.339
0.374
0.337
192 0.354
0.331
0.400
0.414
0.373
0.414
0.390
0.414
0.382
336 0.377
0.352
0.428
0.441
0.429
0.441
0.406
0.452
0.384
720 0.431
0.411
0.457
0.480
0.449
0.480
0.431
0.468
0.422
Average
-
0.372
0.349
0.410
0.431
0.394
0.431
0.393
0.427
0.381
ETTm1
96
0.341
0.333
0.345
0.419
0.388
0.419
0.360
0.375
0.346
192 0.364
0.353
0.374
0.434
0.416
0.434
0.382
0.387
0.370
336 0.386
0.373
0.397
0.454
0.426
0.454
0.405
0.411
0.392
720 0.420
0.410
0.436
0.556
0.476
0.556
0.437
0.450
0.420
Average
-
0.378
0.367
0.388
0.466
0.427
0.466
0.396
0.406
0.382
ETTm2
96
0.260
0.237
0.263
0.274
0.261
0.274
0.263
0.267
0.256
192 0.293
0.278
0.309
0.317
0.314
0.317
0.306
0.309
0.296
336 0.325
0.313
0.349
0.353
0.327
0.353
0.343
0.351
0.329
720 0.382
0.371
0.415
0.427
0.390
0.427
0.399
0.403
0.385
Average
-
0.316
0.300
0.334
0.343
0.323
0.343
0.327
0.332
0.317
Electricity
96
0.231
0.226
Not Reported
0.222
0.235
0.253
0.272
0.222
192 0.239
0.235
0.240
0.250
0.262
0.289
0.240
336 0.256
0.249
0.259
0.270
0.277
0.300
0.259
720 0.289
0.279
0.290
0.315
0.305
0.320
0.290
Average
-
0.252
0.247
0.253
0.268
0.274
0.295
0.253
Traffic
96
0.247
0.237
Not Reported
0.249
0.268
0.268
0.321
0.249
192 0.247
0.239
0.256
0.274
0.317
0.336
0.256
336 0.251
0.246
0.264
0.282
0.284
0.336
0.264
720 0.271
0.279
0.286
0.319
0.297
0.350
0.286
Average
-
0.254
0.250
0.264
0.286
0.291
0.336
0.264
Weather
96
0.188
0.186
Not Reported
0.198
0.221
0.219
0.220
0.198
192 0.230
0.228
0.241
0.261
0.260
0.261
0.241
336 0.271
0.270
0.282
0.300
0.297
0.306
0.282
720 0.317
0.322
0.334
0.351
0.349
0.359
0.334
Average
-
0.252
0.251
0.264
0.283
0.281
0.267
0.264
26
ViTime: Foundation Model for Time Series Forecasting Powered by Vision Intelligence
Figure 15: Average MAE of ViTime across different DM rates.
results under Gaussian noise (GN) with standard deviations of 0.1 and 0.3, as well as data missing (DM) with a probability of
0.3. As shown in Table 9, ViTime consistently outperforms TimesFM across all data quality scenarios. ViTime’s resilience
to GN is particularly noteworthy, where it maintains superior performance even under higher noise levels (std=0.3). This
robust performance can be attributed to the model’s visual representation learning mechanism, which effectively filters out
noise through its vision feature extraction process.
Furthermore, we conducted an in-depth analysis of ViTime’s performance under varying degrees of missing data, ranging
from 10% to 90% missing values. Figure 15 illustrates the model’s average ReMAE across different DM ratios. The results
demonstrate that ViTime maintains remarkable prediction accuracy when the DM ratio remains below 50%, with only
gradual performance degradation observed. This robustness can be attributed to the model’s ability to treat missing values as
zero-valued pixels in its visual representation, effectively leveraging spatial relationships in the remaining data points to
maintain prediction quality.
These comprehensive results further validate the superior generalizability and robustness of ViTime compared to traditional
numerical fitting-based approaches, particularly in scenarios with compromised data quality. The model’s ability to maintain
consistent performance under various perturbations makes it a promising solution for real-world applications where data
quality cannot always be guaranteed.
E.5. 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 a single Nvidia
3090 GPU.
The baseline TimesFM model requires substantial computational resources with 18.1GB GPU memory and 200M parameters,
while achieving an average ReMAE of 0.423. In contrast, our proposed ViTime 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 (27× 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 - 3.1GB GPU memory and 2.89s inference time per batch, though still maintaining a 5.8× 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-
constrained 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 computation-accuracy
27
ViTime: Foundation Model for Time Series Forecasting Powered by Vision Intelligence
Table 9: Performance comparison of TimesFM and ViTime across different robust inference scenarios
Dataset
PredLen
GN (0.1)
GN (0.3)
DM (0.3)
TimesFM
ViTime
TimesFM
ViTime
ViTime
TimesFM
ReMSE ReMAE ReMSE ReMAE
ReMSE ReMAE ReMSE ReMAE
ReMSE ReMAE -
-
ETTh1
96
0.410
0.414
0.431
0.416
0.427
0.423
0.444
0.436
0.433
0.414
-
-
192
0.460
0.445
0.460
0.434
0.473
0.452
0.469
0.452
0.463
0.433
-
-
336
0.513
0.478
0.493
0.458
0.525
0.485
0.501
0.474
0.496
0.456
-
-
720
0.617
0.545
0.572
0.509
0.632
0.553
0.585
0.525
0.574
0.507
-
-
Average
0.500
0.471
0.489
0.454
0.514
0.478
0.500
0.472
0.492
0.453
-
-
ETTh2
96
0.269
0.326
0.256
0.321
0.265
0.329
0.268
0.335
0.255
0.319
-
-
192
0.326
0.367
0.301
0.356
0.314
0.366
0.308
0.367
0.297
0.352
-
-
336
0.378
0.406
0.347
0.391
0.366
0.404
0.354
0.401
0.343
0.387
-
-
720
0.480
0.475
0.449
0.460
0.464
0.470
0.450
0.465
0.441
0.455
-
-
Average
0.363
0.394
0.338
0.382
0.352
0.392
0.345
0.391
0.334
0.378
-
-
ETTm1
96
0.521
0.444
0.428
0.404
0.532
0.442
0.436
0.425
0.408
0.392
-
-
192
0.572
0.476
0.468
0.425
0.575
0.469
0.467
0.440
0.451
0.413
-
-
336
0.642
0.508
0.517
0.450
0.640
0.501
0.509
0.462
0.507
0.440
-
-
720
0.719
0.551
0.594
0.490
0.688
0.538
0.587
0.501
0.595
0.484
-
-
Average
0.614
0.495
0.502
0.442
0.609
0.488
0.500
0.457
0.490
0.432
-
-
ETTm2
96
0.212
0.282
0.200
0.277
0.200
0.280
0.209
0.290
0.199
0.274
-
-
192
0.291
0.329
0.260
0.319
0.262
0.321
0.266
0.327
0.262
0.317
-
-
336
0.358
0.370
0.322
0.358
0.323
0.360
0.320
0.360
0.320
0.354
-
-
720
0.459
0.430
0.403
0.407
0.420
0.418
0.392
0.406
0.398
0.403
-
-
Average
0.330
0.353
0.296
0.340
0.301
0.345
0.297
0.344
0.295
0.337
-
-
Electricity
96
0.266
0.338
0.225
0.317
0.285
0.363
0.252
0.354
0.222
0.311
-
-
192
0.320
0.375
0.240
0.328
0.333
0.398
0.262
0.361
0.235
0.323
-
-
336
0.377
0.413
0.263
0.345
0.390
0.440
0.285
0.377
0.260
0.341
-
-
720
0.488
0.486
0.348
0.402
0.527
0.532
0.368
0.432
0.343
0.397
-
-
Average
0.363
0.403
0.269
0.348
0.384
0.433
0.292
0.381
0.265
0.343
-
-
Traffic
96
0.770
0.457
0.724
0.398
0.765
0.478
0.778
0.463
0.782
0.406
-
-
192
0.830
0.489
0.723
0.395
0.815
0.505
0.776
0.461
0.772
0.400
-
-
336
0.850
0.516
0.743
0.404
0.873
0.534
0.799
0.470
0.792
0.409
-
-
720
0.979
0.582
0.844
0.450
0.988
0.600
0.896
0.514
0.886
0.454
-
-
Average
0.857
0.511
0.759
0.412
0.860
0.529
0.812
0.477
0.808
0.417
-
-
Weather
96
0.157
0.202
0.178
0.209
0.170
0.222
0.178
0.230
0.170
0.209
-
-
192
0.214
0.253
0.229
0.257
0.228
0.272
0.232
0.273
0.231
0.261
-
-
336
0.272
0.299
0.289
0.300
0.283
0.314
0.284
0.309
0.288
0.302
-
-
720
0.378
0.369
0.372
0.353
0.380
0.376
0.361
0.355
0.366
0.353
-
-
Average
0.255
0.281
0.267
0.280
0.265
0.296
0.264
0.292
0.264
0.281
-
-
Table 10: Model Performance and Computational Resource Requirements
Model
GPU Memory
Inference Time (s/batch)
Parameters
Avg. ReMAE
(MB)
Total
Mapping & Inverse Mapping
(M)
TimesFM
18,154
0.130
-
200
0.423
ViTime w/ Refining Module
3,120
2.890
0.0068
95
0.376
ViTime w/o Refining Module
667
0.082
0.0068
74
0.381
28
ViTime: Foundation Model for Time Series Forecasting Powered by Vision Intelligence
trade-off.
E.6. Interpretation of ViTime
(a) Grad-CAM heatmap showing attention on key trend changes.
(b) Attention maps at different prediction positions demonstrating
temporal dependencies.
Figure 16: Visualization of ViTime’s attention mechanism. Despite not using an autoregressive paradigm, ViTime exhibits
sequential processing patterns through its multi-layer self-attention modules.
Figure 16 illustrates the attention mechanism of ViTime through grad-cam (Selvaraju et al., 2017) heatmaps and position-
specific attention maps. The grad-cam results demonstrate that ViTime focuses strongly on periods of fundamental trend
changes. Further analysis through attention maps at different prediction positions reveals an interesting pattern: despite
not adopting an autoregressive paradigm, ViTime’s multi-layer self-attention modules process information in a temporal
sequence. The input data and the predicted results from previous time steps determine the spatiotemporal distribution of
predictions at each time step. This aligns with human cognitive patterns, where information is processed from the recent to
the distant past while maintaining awareness of known information.
F. Discussion
F.1. Mechanistic Insights: Temporal Dynamics in ViTime
Despite eschewing autoregressive paradigms, ViTime’s multi-layer self-attention modules inherently maintain temporal
ordering through spatiotemporal dependency learning. As shown in Figure 16(b), current predictions’ attention patterns
demonstrate systematic dependence on both input data characteristics and previous prediction steps, mirroring human
cognitive patterns in time series reasoning.
29
ViTime: Foundation Model for Time Series Forecasting Powered by Vision Intelligence
F.2. Why Using ReMSE and ReMAE as metrics?
The proposed ReMSE and ReMAE metrics address a critical challenge in evaluating time series foundation models:
mitigating test set leakage caused by overlapping data distributions between training and testing phases. By rescaling the
test set across multiple resolutions (β ∈U) via time series interpolation (TSI, Equation (11)), 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 leakage-resistant evaluation framework, as models must generalize to unseen
scales rather than relying on spurious correlations learned from the training set.
The empirical results in Figure 4 highlight the efficacy of this strategy. TimesFM, while achieving low error at β = 1,
exhibits significant performance degradation at β ̸= 1, suggesting its predictions at the original scale may exploit leaked
patterns. In contrast, ViTime maintains consistent accuracy across all β values, indicating robust learning of intrinsic
temporal dynamics rather than scale-specific memorization. This stark divergence underscores how ReMSE/ReMAE expose
leakage by penalizing models that overfit to the training data’s implicit resolution.
A key implication of this work is the necessity of scale-agnostic evaluation in time series forecasting. Traditional single-
scale metrics like MSE/MAE risk conflating memorization with true generalization, particularly when training data
encompasses diverse real-world sources. By averaging errors across β, ReMSE/ReMAE incentivize models to capture
invariant temporal structures—such as periodicity, trends, and noise resilience—that persist across resolutions. This
aligns with recent theoretical insights in self-supervised learning, where augmentation-induced invariance improves out-of-
distribution robustness (Yao et al., 2022).
The proposed ReMSE and ReMAE establish multi-scale evaluation as a paradigm for leakage-free benchmarking of
time series models. The success of ViTime under ReMSE/ReMAE suggests that architectural inductive biases for scale
invariance—such as decoupled frequency-time processing (Wu et al., 2022)—are critical for developing generalizable
foundation models. This aligns with broader machine learning principles where invariance to augmentations correlates with
robustness (Yao et al., 2022). Extending this framework to multi-variate and non-stationary settings remains an impactful
direction for future research.
F.3. Limitations and Future Directions
While ViTime demonstrates state-of-the-art performance in accuracy and robustness, two key challenges warrant further
investigation:
F.3.1. RESOLUTION CONSTRAINTS & ADAPTIVE ENHANCEMENT.
The mapping function’s truncation imposes resolution limits, particularly evident in explosive growth patterns (Figure 17
a-b). A key limitation of ViTime arises from its assumption of S ∼N(0, I), which fails to capture the high-variance
nature of explosive growth data that typically follows S ∼N(0, kI) with k ≫1. As shown in Proposition C.2, the
optimal threshold MS∗scales as
√
k, 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 prediction accuracy.
Our empirical analysis reveals that doubling the MS parameter from 3.5 to 7 significantly improves prediction fidelity for
explosive growth patterns (Figure 17c-d). However, excessively large MS values increase spectral energy, as 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 granularity for high-variance regions without unnecessary computational overhead.
• Adaptive MS Estimation: Algorithms to estimate the variance scaling factor k and compute the optimal MS∗in
real-time, balancing prediction fidelity with spectral efficiency.
These enhancements would enable ViTime to handle explosive growth patterns more effectively while maintaining computa-
tional tractability.
30
ViTime: Foundation Model for Time Series Forecasting Powered by Vision Intelligence
(a) MS=3.5 (Ground Truth)
(b) MS=3.5 (Prediction)
(c) MS=7 (Ground Truth)
(d) MS=7 (Prediction)
Figure 17: 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 7 enables accurate growth trend capture. Comparison demonstrates the
critical role of dynamic resolution adjustment.
31
ViTime: Foundation Model for Time Series Forecasting Powered by Vision Intelligence
F.3.2. ENHANCED DATA GENERATION.
ViTime’s predictive quality fundamentally depends on RealTS’s synthetic data generation capabilities. The current
methodology faces challenges in simulating complex real-world temporal dynamics, particularly for non-stationary processes
and regime-switching scenarios. Future work should develop 1) Advanced pattern injection mechanisms for synthetic data
generation, and 2) Quantitative metrics for simulation fidelity assessment across different temporal regimes.
32
ViTime: Foundation Model for Time Series Forecasting Powered by Vision Intelligence
(a) Rescale factor=0.5
(b) Rescale factor=1
(c) Rescale factor=2
Figure 18: Illustrative example of Electricity dataset.
33
ViTime: Foundation Model for Time Series Forecasting Powered by Vision Intelligence
(a) Rescale factor=0.5
(b) Rescale factor=1
(c) Rescale factor=2
Figure 19: Illustrative example of Traffic dataset.
34
ViTime: Foundation Model for Time Series Forecasting Powered by Vision Intelligence
(a) Rescale factor=0.5
(b) Rescale factor=1
(c) Rescale factor=2
Figure 20: Illustrative example of ETTh1 dataset.
35
ViTime: Foundation Model for Time Series Forecasting Powered by Vision Intelligence
(a) Rescale factor=0.5
(b) Rescale factor=1
(c) Rescale factor=2
Figure 21: Illustrative example of ETTh2 dataset.
36
ViTime: Foundation Model for Time Series Forecasting Powered by Vision Intelligence
(a) Rescale factor=0.5
(b) Rescale factor=1
(c) Rescale factor=2
Figure 22: Illustrative example of ETTm1 dataset.
37
ViTime: Foundation Model for Time Series Forecasting Powered by Vision Intelligence
(a) Rescale factor=0.5
(b) Rescale factor=1
(c) Rescale factor=2
Figure 23: Illustrative example of ETTm2 dataset.
38