FracTS: Hierarchical and Autoregressive Time Series Generation

Jiayu Li; Umair Afzal; Zilong Zhao; Milad Abdollahzadeh; Uzair Javaid; Biplab Sikdar

cs.LGOct 18, 2026

FracTS: Hierarchical and Autoregressive Time Series Generation

Authors: Jiayu Li, Umair Afzal, Zilong Zhao, Milad Abdollahzadeh, Uzair Javaid, Biplab Sikdar

Organizations: University of Illinois Urbana-Champaign · Betterdata AI · National University of Singapore

Abstract

Time series are ubiquitous across different domains, and the growing use of synthetic data for privacy-preserving sharing, data augmentation, and stress testing makes time series generation increasingly important. Early generative approaches are predominantly autoregressive, which can suffer from low sample diversity, error accumulation, and difficulty modeling long-range dependencies. In this paper, we introduce FracTS, a hierarchical, coarse-to-fine autoregressive model for multivariate time series generation. Inspired by fractal generative modeling developed for images, FracTS adapts the paradigm from 2D to 1D sequences by learning a multi-scale representation that captures global trajectory structure at coarse scales and local dynamics at fine scales while following the autoregressive nature of time series. In contrast to unstructured images, time series are structured, allowing easy automated or designated temporal feature processing optimized for the model, and global (static) feature extraction describing the instances that generate the corresponding time series in the collection. We incorporate them in the generation process to further improve the generation quality. The design naturally handles multivariate dependencies and improves long-horizon temporal coherence without requiring fixed-length inputs. Experiments on five datasets (two conditional, three unconditional) conducted against seven baselines show that FracTS consistently outperforms baselines across marginal distribution and predictive ML utility, and correlation preservation in the unconditional setting. In particular, the predictive ML utility is averagely improved by 100.0% and 75.0-81.0% for unconditional and conditional time series datasets compared to the best baseline (relative error computed on dataset-averaged raw values). Code is available at: https://github.com/BetterdataLabs/fracts.

Figures & tables

Figure 1: Summary comparison of FracTS (ours) and the baselines on dataset-averaged scores of Tables 3 and 4 , rescaled per axis so that larger values indicate better performance. Clockwise from top: predictive ML utility at window

W{=}1000

, marginal distribution of raw values, of means and of lengths, and correlation preservation on raw values; conditional adds ground-truth MAE at the bottom (definitions in Section 5.1 ).

Limited to	TimeGAN	WaveStitch	COSCI-GAN	TimeVQVAE	AEC-GAN	KoVAE	DiffusionTS
i) fixed length	✗		✗	✗		✗	✗
ii) single variate				✗
iii) no static			✗	✗ †	✗	✗	✗ †
iv) limited length	✗	✗	✗	✗		✗	✗
v) mixed type			✗	✗	✗	✗	✗

Table 1: Functionality and flexibility of representative time series generative models, compared to FracTS. ✗ indicates that a model is limited to the corresponding setting; for limited length , ✗ means the model contains no specific design for long time series generation. Blank entries indicate that the setting is supported. † marks models whose static conditioning is restricted to class labels.

Figure 2: Fractal decomposition and model architecture for FracTS (with $K=4$ fractal levels). At each level except for the last one, a long time series is decomposed into patches of shorter ones. The time series dataset can have variable lengths, where the gray spans indicate paddings. Completely empty patches are skipped for training. The last level $i=3$ is not a fractal decomposition of different time steps, but a series of different variables in one time step.

Figure 3: Length construction for variable-length time series data via length indicator predictors (purple trapezoids). At each fractal level, the length indicator is one network (drawn separately for separate calls). The length indicators are equivalent to is-[EOS] binary signals. Details at fractal levels 1-2 are omitted as they are the same as level 0, and the last level, unrelated to lengths, is hidden. The total length of the example in the figure is $L_{n}=(7-1)L^{[1]}+(5-1)L^{[2]}+9L^{[3]}=649$ .

Dataset	$\bm{N}$	$\bm{M_{c}}$	$\bm{\overline{L_{n}}}$	$\bm{L}$	$\bm{M}$	Type
Weather	44	0	1798	2171	17	unconditional
Sensor	6939	0	51	139	17	unconditional
Stocks	1000	0	1278	1511	6	unconditional
Walmart	45	2	9368	10474	3	conditional
Simulated	100	4	15000	15000	15	conditional

Table 2: Summary of the datasets used and their core statistics. Notations follow Section 3 ;

\overline{L_{n}}

is the average sequence length.

Model	raw	mean	quantile	skewness	autocorr.	length	raw	agg.	$W{=}10$	$W{=}100$	$W{=}1000$
	MD $\downarrow$						CP $\downarrow$		PM $\downarrow$
AEC-GAN	0.524	0.683	0.697	0.617	0.609	–	0.288	0.315	$4\,052$	$4\,171$	$8\,233$
KoVAE	0.675	0.708	0.673	0.784	0.701	0.956	0.261	0.489	$7\,461$	$7\,680$	$15\,156$
DiffusionTS	0.695	0.932	0.915	0.686	0.775	0.956	0.466	0.542	$2\,524$	$2\,598$	$5\,129$
TimeVQVAE	0.634	0.726	0.700	0.496	0.732	0.956	0.183	0.443	$6\,822$	$7\,022$	$13\,861$
COSCI-GAN	0.557	0.799	0.748	0.839	0.858	0.956	0.462	0.473	$2\,392$	$2\,462$	$4\,858$

Table 3: Unconditional generation performance averaged over the three datasets (lower is better; best in bold ). MD and CP are averaged across column pairs, and MAE uses standard-normalized values; PM is the

|\text{TSTR}-\text{TRTR}|

MAE gap at prediction window

W

. The bottom row (

\Delta

best baseline) reports, for each column, the relative error of FracTS against the best baseline, computed on these dataset-averaged raw values. Per-dataset results with standard deviations over three runs are reported in Appendix H .

Figure 5: t-SNE of real compared to generated time series data of the Weather dataset.

	MD $\downarrow$						CP $\downarrow$		PM $\downarrow$			GT $\downarrow$
Model	raw	mean	quantile	skewness	autocorr.	length	raw	agg.+cond.	$W{=}10$	$W{=}100$	$W{=}1000$	MSE	MAE
TabularARGN	0.154	0.731	0.710	0.748	0.994	0.278	0.069	0.345	0.056	0.084	0.050	1.759	0.952
WaveStitch	0.506	0.916	0.890	0.842	0.996	0.445	0.146	0.500	0.059	0.030	0.057	1.233	0.884
TimeAutoDiff	0.353	0.769	0.782	0.797	0.898	0.445	0.228	0.421	0.406	0.420	0.392	144.260	5.103
FracTS (Ours)	0.172	0.611	0.645	0.605	0.834	0.139	0.074	0.409	0.011	0.007	0.009	0.983	0.545
$\Delta$ best baseline	$-$ 11.36%	16.42%	9.02%	19.17%	7.13%	50.00%	$-$ 7.30%	$-$ 18.72%	79.28%	75.00%	81.00%	20.28%	38.35%

Table 4: Conditional generation performance averaged over the two datasets (lower is better; best in bold ). MD, CP, and GT are averaged across all column pairs, and MAE is computed on standard-normalized values; CP additionally accounts for the static conditions (agg.+cond.). PM is the

|\text{TSTR}-\text{TRTR}|

MAE gap at prediction window

W

; GT reports the error against the ground truth of the simulated dataset. The bottom row (

\Delta

Appendix figures & tables17 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 7: Comparison to normal distribution of the distribution of raw values and differences of consecutive timesteps in Q-Q plots of more features from real-life time series datasets.

Figure 8: Comparison to normal distribution of the distribution of raw values and differences of consecutive timesteps in Q-Q plots of simulated features from mathematical formulae.

col1: Linear trend	col2: Cosine wave with phase shift
$0.001kx+b$	$\cos(0.05\pi x+b)$
col3: Alternating constant pattern	col4: Rounded step function
$\left\lfloor\frac{x+\lfloor 25b+0.5\rfloor}{100}\right\rfloor\bmod 2$	$\lfloor-0.01w_{1}x+0.5\rfloor$
col5: Convex curve	col6: Non-convex shape (tanh function)
$\sqrt{0.01(w_{1}+1)x+10+b}$	$\tanh\left(5k\left(\frac{x}{L_{n}}-0.5\right)\right)$
col7: Linear and sinusoidal combination	col8: Cosine wave with exponential decay

Appendix

Table 5: Generating formulas of the 15 time series features of the simulated dataset. Each feature is a function of the time step

x

and the static conditions

w_{1},w_{2},b,k

Table 7: Computational cost of FracTS on the two datasets used in the efficiency analysis. Memory is measured on an NVIDIA H100 GPU with 80 GB. Training batch sizes are selected for data throughput, so the memory figures are not a controlled comparison against baselines with different batch-size configurations.

Panel of Figure 9 — Figure 9: Training loss curves across the five evaluated datasets. The x-axis represents the training steps, demonstrating that the models consistently converge over the course of the training regime.

Panel of Figure 10 — Figure 10: Validation loss curves across the five evaluated datasets. The x-axis represents the training epochs, illustrating that the models consistently converge within the 200-epoch training regime.

Panel of Figure 11 — Figure 11: Windowed t-SNE visualizations of real and generated samples for the unconditional datasets at different window sizes. ( $W=1000$ is omitted for Sensors because every Sensors series is shorter than 1000 time steps, so no such window exists.)

Panel of Figure 12 — Figure 12: Windowed t-SNE visualizations of real and generated samples for the conditional Walmart dataset at different window sizes.

Panel of Figure 13 — Figure 13: Windowed t-SNE visualizations of real and generated samples for the conditional Simulated dataset at different window sizes.

Figure 14: t-SNE of real compared to generated time series data of the Sensors dataset.

Panel of Figure 15 — Figure 15: Examples of generated versus ground-truth simulated data for six representative generating formulas (Table 5 ). Each panel overlays FracTS samples on the analytic ground truth.

	FracTS (Ours)		TabularARGN		WaveStitch		TimeAutoDiff
Pattern	MAE $\downarrow$	MSE $\downarrow$	MAE $\downarrow$	MSE $\downarrow$	MAE $\downarrow$	MSE $\downarrow$	MAE $\downarrow$	MSE $\downarrow$
1. Linear trend	$\mathbf{0.525}_{\pm 0.102}$	$\mathbf{0.590}_{\pm 0.250}$	$0.963_{\pm 0.015}$	$1.547_{\pm 0.059}$	$0.885_{\pm 0.083}$	$1.232_{\pm 0.213}$	$0.896_{\pm 0.055}$	$1.466_{\pm 0.208}$
2. Cosine wave	$\mathbf{0.292}_{\pm 0.047}$	$\mathbf{0.166}_{\pm 0.055}$	$1.144_{\pm 0.001}$	$1.993_{\pm 0.001}$	$0.914_{\pm 0.097}$	$1.312_{\pm 0.258}$	$3.298_{\pm 0.526}$	$14.337_{\pm 3.971}$
3. Alternating constant	$\mathbf{0.168}_{\pm 0.027}$	$\mathbf{0.335}_{\pm 0.055}$	$1.272_{\pm 0.125}$	$2.547_{\pm 0.249}$	$0.881_{\pm 0.055}$	$1.217_{\pm 0.145}$	$1.158_{\pm 0.102}$	$2.696_{\pm 0.484}$
4. Rounded step	$\mathbf{0.467}_{\pm 0.057}$	$\mathbf{0.428}_{\pm 0.104}$	$0.838_{\pm 0.057}$	$1.240_{\pm 0.157}$	$0.967_{\pm 0.106}$	$1.453_{\pm 0.283}$	$1.444_{\pm 0.062}$	$3.195_{\pm 0.232}$
5. Convex curve	$\mathbf{0.157}_{\pm 0.223}$	$\mathbf{0.139}_{\pm 0.197}$	$1.024_{\pm 0.008}$	$1.609_{\pm 0.034}$	$0.870_{\pm 0.057}$	$1.187_{\pm 0.135}$	$1.506_{\pm 0.029}$	$3.224_{\pm 0.075}$

Appendix

Table 8: Per-column MAE/MSE against ground truth on the simulated dataset (lower is better; best in bold , second best underlined ). Column numbers correspond to the generating formulas in Table 5 .

Table 9: Windowed discriminative evaluation, measured as

\mathrm{Accuracy}-0.5

(lower is better, with

0

indicating ideal indistinguishability). (a) compares FracTS against the strongest baseline per dataset; (b) isolates the effect of the feature subset used for evaluation on Weather while keeping the generator fixed.

		MD						CP		PM
	Model	raw	mean	quantile	skewness	autocorr.	length	raw	agg.	$W{=}10$	$W{=}100$	$W{=}1000$
Weather	AEC-GAN	$0.503_{\pm 0.000}$	$0.845_{\pm 0.000}$	$0.887_{\pm 0.000}$	$0.613_{\pm 0.000}$	$0.697_{\pm 0.000}$	–	$0.393_{\pm 0.000}$	$0.465_{\pm 0.000}$	$0.076_{\pm 0.000}$	$0.077_{\pm 0.000}$	$0.075_{\pm 0.000}$
	KoVAE	$0.692_{\pm 0.238}$	$0.861_{\pm 0.113}$	$0.877_{\pm 0.099}$	$0.781_{\pm 0.170}$	$0.775_{\pm 0.082}$	$1.000_{\pm 0.000}$	$0.169_{\pm 0.010}$	$0.438_{\pm 0.001}$	$0.286_{\pm 0.291}$	$0.284_{\pm 0.279}$	$0.280_{\pm 0.275}$
	DiffusionTS	$0.805_{\pm 0.002}$	$1.000_{\pm 0.000}$	$1.000_{\pm 0.000}$	$0.714_{\pm 0.015}$	$0.821_{\pm 0.045}$	$1.000_{\pm 0.000}$	$0.361_{\pm 0.113}$	$0.523_{\pm 0.045}$	$0.062_{\pm 0.010}$	$0.082_{\pm 0.014}$	$0.069_{\pm 0.020}$
	TimeVQVAE	$0.466_{\pm 0.049}$	$0.720_{\pm 0.069}$	$0.705_{\pm 0.068}$	$0.563_{\pm 0.022}$	$0.840_{\pm 0.012}$	$1.000_{\pm 0.000}$	$0.156_{\pm 0.013}$	$0.511_{\pm 0.028}$	$0.047_{\pm 0.008}$	$0.049_{\pm 0.007}$	$0.045_{\pm 0.006}$
	COSCI-GAN	$0.559_{\pm 0.001}$	$0.879_{\pm 0.009}$	$0.894_{\pm 0.021}$	$1.000_{\pm 0.000}$	$1.000_{\pm 0.000}$	$1.000_{\pm 0.000}$	$0.745_{\pm 0.001}$	$0.677_{\pm 0.140}$	$0.125_{\pm 0.004}$	$0.125_{\pm 0.005}$	$0.135_{\pm 0.005}$

Appendix

Table 10: Unconditional generation performance with standard deviations over three runs (lower is better; best in bold ).

		MD						CP		PM			GT
	Model	raw	mean	quantile	skewness	autocorr.	length	raw	agg.+cond.	$W{=}10$	$W{=}100$	$W{=}1000$	MSE	MAE
Walmart	TabularARGN	$0.113_{\pm 0.027}$	$0.812_{\pm 0.116}$	$0.775_{\pm 0.109}$	$0.741_{\pm 0.098}$	$1.000_{\pm 0.000}$	$0.556_{\pm 0.000}$	$0.080_{\pm 0.002}$	$\mathbf{0.370}_{\pm 0.047}$	$\mathbf{0.007}_{\pm 0.001}$	$0.069_{\pm 0.001}$	$\mathbf{0.005}_{\pm 0.001}$	–	–
	WaveStitch	$0.653_{\pm 0.015}$	$1.000_{\pm 0.000}$	$0.944_{\pm 0.096}$	$0.932_{\pm 0.069}$	$1.000_{\pm 0.000}$	$0.889_{\pm 0.000}$	$0.163_{\pm 0.021}$	$0.619_{\pm 0.045}$	$0.081_{\pm 0.008}$	$0.018_{\pm 0.008}$	$0.076_{\pm 0.006}$	–	–
	TimeAutoDiff	$0.247_{\pm 0.182}$	$0.778_{\pm 0.185}$	$0.802_{\pm 0.113}$	$0.667_{\pm 0.161}$	$0.852_{\pm 0.148}$	$0.889_{\pm 0.000}$	$0.237_{\pm 0.005}$	$0.435_{\pm 0.019}$	$0.142_{\pm 0.083}$	$0.181_{\pm 0.133}$	$0.153_{\pm 0.096}$	–	–
	FracTS (Ours)	$\mathbf{0.077}_{\pm 0.034}$	$\mathbf{0.583}_{\pm 0.167}$	$\mathbf{0.633}_{\pm 0.130}$	$\mathbf{0.549}_{\pm 0.043}$	$\mathbf{0.781}_{\pm 0.088}$	$\mathbf{0.278}_{\pm 0.073}$	$\mathbf{0.050}_{\pm 0.006}$	$0.503_{\pm 0.018}$	$\mathbf{0.007}_{\pm 0.002}$	$\mathbf{0.003}_{\pm 0.003}$	$\mathbf{0.005}_{\pm 0.002}$	–	–
Simulated	TabularARGN	$\mathbf{0.195}_{\pm 0.004}$	$0.650_{\pm 0.032}$	$\mathbf{0.644}_{\pm 0.030}$	$0.756_{\pm 0.021}$	$0.988_{\pm 0.003}$	$\mathbf{0.000}_{\pm 0.000}$	$\mathbf{0.057}_{\pm 0.002}$	$0.319_{\pm 0.008}$	$0.104_{\pm 0.017}$	$0.099_{\pm 0.016}$	$0.095_{\pm 0.015}$	$1.759_{\pm 0.080}$	$0.952_{\pm 0.034}$

Appendix

Table 11: Conditional generation performance with standard deviations over three runs (lower is better; best in bold ).

Weather

\downarrow

Walmart

\downarrow

Data proc.

Var.-length handling

Length source

Global cond.

l^{[i]}

Backbone

Cond. model

q_{\phi}

W{=}10

W{=}100

W{=}1000

W{=}10

W{=}100

W{=}1000

FracTS (default)

att. mask

predictor

✓

GPT-2

CTGAN

0.010

0.010

0.010

0.009

0.007

0.002

w/o differences

–

0.021

0.020

0.022

0.027

0.028

0.009

w/o bins

–

0.104

0.109

0.118

0.003

0.024

0.017

k=2

bins

–

0.018

0.024

0.034

0.003

0.006

0.008

k=200

bins

–

0.034

0.035

0.034

0.009

0.007

0.005

Appendix

Table 12: Ablation of FracTS components, reporting PM (predictive ML utility) MAE at prediction windows

W

on Weather and Walmart (lower is better). – denotes “same as the default setting” (top row, highlighted).

Perceiver compression	$W{=}10$	$W{=}100$	$W{=}1000$
Yes (default)	$0.010$	$0.010$	$0.010$
No	$0.010$	$0.011$	$0.011$

Appendix

Table 13: Ablation of the Perceiver-based condition compression module on Weather. Values report PM (predictive ML utility) MAE; lower is better.

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