ORBIT：通过下一频带预测实现自回归时间序列生成

# Autoregressive Time Series Generation via Next Frequency-Band Prediction
Source: [https://arxiv.org/html/2610.27000](https://arxiv.org/html/2610.27000)
DOI:[XXXXXXX\.XXXXXXX](https://doi.org/XXXXXXX.XXXXXXX)Conference:Make sure to enter the correct conference title from your rights confirmation email; ;ISBN:978\-1\-4503\-XXXX\-X/2018/06CCS:Computing methodologies Artificial intelligenceChunyi Hou,Xiangfei QiuAffiliation:School of Data Science and Engineering, East China Normal University,Shanghai,China,Hanyin ChengAffiliation:School of Data Science and Engineering, East China Normal University,Shanghai,China,Chenchen SunAffiliation:Chang’an University,Xi’an,China,Yisheng AnAffiliation:Chang’an University,Xi’an,China,Yang ShuAffiliation:School of Data Science and Engineering, East China Normal University,Shanghai,China,Jilin HuAffiliation:School of Data Science and Engineering, East China Normal University,Shanghai,ChinaandBin YangAffiliation:School of Data Science and Engineering, East China Normal University,Shanghai,China

© none

###### Abstract\.

Deep generative models provide an effective modern solution to alleviating the problem of data scarcity by synthesizing realistic time series\. However, despite the considerable progress achieved by existing deep time series generators, two key challenges remain: \(1\) learning hierarchical representations for heterogeneous temporal patterns; and \(2\) coordinating the generation process of heterogeneous temporal dynamics\. To address these key bottlenecks, in this work, we proposeORBIT, a unified deep generative framework for time series\. Specifically, we develop two key components: the Frequency\-Hierarchical VQ\-VAE \(FH\-VQ\-VAE\) and the Frequency\-Band Autoregressive Transformer \(FBAR\)\. The former learns hierarchical representations by decomposing latent temporal representations into different frequency bands and applying residual quantization with increasing token budgets\. The latter establishes a novel low\-to\-high frequency autoregressive direction, yielding a time series generation process that first constructs global temporal structures and then incrementally incorporates fine\-grained temporal details\. These compact designs not only disentangle heterogeneous temporal patterns for more effective representation learning but also mitigate potential interference among different temporal dynamics during generation\. Extensive experiments on multiple challenging real\-world benchmarks demonstrate that ORBIT achieves state\-of\-the\-art overall performance while exhibiting superior inference efficiency compared with advanced baselines\.

###### Keywords:

Time Series Generation

## 1\.Introduction

The rapid advancement of complex intelligent systems has led to a substantial growth in the volume of time series data, which are now ubiquitous across many critical real\-world domains, including meteorology\([Tian et al\., 2026](https://arxiv.org/html/2610.27000#bib.bib2)\), transportation\([Jin et al\., 2024](https://arxiv.org/html/2610.27000#bib.bib7)\), healthcare\([Ji et al\., 2025](https://arxiv.org/html/2610.27000#bib.bib8)\), energy\([Zhu et al\., 2026](https://arxiv.org/html/2610.27000#bib.bib1)\), and AIOps\([Zhang et al\., 2025](https://arxiv.org/html/2610.27000#bib.bib9)\)\. In modern networking, mobile communications, and Web of Things \(WoT\) ecosystems, such data capture evolving traffic patterns, service demands, and sensor measurements\. This provides a strong foundation for critical downstream tasks, including informed decision\-making and anomaly detection\([Qiu et al\., 2024](https://arxiv.org/html/2610.27000#bib.bib3);[Qiu et al\., 2025a](https://arxiv.org/html/2610.27000#bib.bib4)\)\. Unfortunately, access to these valuable data sources is often constrained by external factors, such as privacy concerns and extreme events\([Zhang et al\., 2026](https://arxiv.org/html/2610.27000#bib.bib6)\)\. Deep generative models offer a promising solution to this problem by generating realistic synthetic time series\([Ang et al\., 2023](https://arxiv.org/html/2610.27000#bib.bib14)\)\.

![Refer to caption](https://arxiv.org/html/2610.27000v1/figure_1.png)Figure 1\.Different frequency bands correspond to distinct temporal dynamics, with low\-frequency components capturing smooth and slow variations, while high\-frequency components characterize rapid temporal fluctuations\.![Refer to caption](https://arxiv.org/html/2610.27000v1/figure_2.png)Figure 2\.Time Series Generation with a Standard Autoregressive Backbone\. At certain inference steps \(highlighted by thered boxes\), the model is needed to jointly generate coarse\-grained global trends and fine\-grained local fluctuations\.In general, developing a deep generative model for time series involves two main stages: \(1\) learning a tokenizer that can transform original time series to compact and expressive temporal representations; and \(2\) training a generative model to fit the distribution of these tokens in the latent space\. This*two\-stage modeling paradigm*has gradually emerged as a prevalent and mainstream approach for deep time series generation\([Chen et al\., 2024](https://arxiv.org/html/2610.27000#bib.bib16);[Feng et al\., 2026](https://arxiv.org/html/2610.27000#bib.bib17);[Lyu et al\., 2026](https://arxiv.org/html/2610.27000#bib.bib19);[Li et al\., 2026](https://arxiv.org/html/2610.27000#bib.bib11)\)\.

However, real\-world time series exhibit heterogeneous temporal patterns that are superimposed in the observation space\. Existing deep time generators typically entangle these patterns within a single representation and fail to explicitly account for their distinct temporal dynamics during autoregressive generation\. For these reasons, developing a method that can effectively disentangle heterogeneous temporal patterns and coordinate their generation is crucial but challenging\. More specifically, achieving these objectives requires overcoming the following two major obstacles:

Obstacle 1: Difficulty in Learning Hierarchical Representations within Time Series\.Due to the heterogeneity of temporal patterns, time series often contain multiple dynamic patterns evolving at different levels\([Qiu et al\., 2025b](https://arxiv.org/html/2610.27000#bib.bib12);[Ding et al\., 2026](https://arxiv.org/html/2610.27000#bib.bib13);[Hou et al\., 2025](https://arxiv.org/html/2610.27000#bib.bib38)\)\. These heterogeneous patterns are superimposed in the numerical observation space, making their underlying dynamics difficult to disentangle and model separately\. As shown in Figure[1](https://arxiv.org/html/2610.27000#S1.F1), Fourier analysis provides an intuitive means from a frequency\-domain perspective to deeply understand them: low\-frequency components mainly contain slow and smooth variations, whereas high\-frequency components characterize rapid oscillations and fine\-grained changes\. Without explicitly modeling these differences in temporal dynamics, integrating diverse patterns into a single representation may make it difficult to simultaneously capture global structures and local temporal details\. Therefore, how to explicitly distinguish and learn different temporal patterns to construct hierarchical representations that preserve both global structures and local details remains a key obstacle\.

Obstacle 2: Difficulty in Coordinating the Generation of Heterogeneous Temporal Dynamics\.Conventional \(GPT\-style\) autoregressive paradigms typically produce time series sequentially along the temporal dimension, either point by point or segment by segment\([Chen et al\., 2024](https://arxiv.org/html/2610.27000#bib.bib16)\)\. Their autoregressive direction strictly follows the left\-to\-right ordering, analogous to natural language modeling, and is primarily determined by the chronological order of observations\([Bengio et al\., 2003](https://arxiv.org/html/2610.27000#bib.bib21)\)\. Although this approach exhibits reasonable generative capability, each inference step is still required to jointly generate content containing both slowly varying structures and rapidly fluctuating changes, which potentially causes interference between these heterogeneous temporal patterns, as illustrated in Figure[2](https://arxiv.org/html/2610.27000#S1.F2)\. Accordingly, another key obstacle is how to coordinate the generation of slowly varying global structures and rapidly fluctuating temporal details while preserving their dependencies and the coherence of the resulting time series\.

In this paper, we address the above two key bottlenecks by introducing a unified deep generative framework, namedORBIT, which performs autOregRessive time series generation via next frequency\-Band predIcTion\. Unlike existing methods, ORBIT explicitly disentangles components across different frequency\-bands to construct hierarchical representations, and further designs a novel autoregressive direction that progresses from low\- to high\-frequency bands\.

Specifically,to address the first obstacle, we propose a Frequency\-Hierarchical VQ\-VAE \(FH\-VQ\-VAE\)\. FH\-VQ\-VAE decomposes latent temporal representations into distinct frequency bands and tokenizes these representations in a residual manner from low\- to high\-frequency bands with increasing token budgets, constructing hierarchical discrete latent representations\. In doing so, by progressively incorporating higher\-frequency components while refining earlier quantization errors, this design facilitates the disentangled learning of heterogeneous temporal patterns within time series\.

On the other hand,to address the second obstacle, we design a Frequency\-Band Autoregressive Transformer \(FBAR\), which*reformulates autoregressive time series generation as a next frequency\-band prediction task*\. FBAR generates frequency bands autoregressively from low to high frequencies, first establishing global temporal structures and then incrementally incorporating finer temporal details\. By conditioning each subsequent band on previously generated bands, FBAR explicitly models cross\-band dependencies, enabling the coordinated generation of heterogeneous temporal dynamics\.

Experimentally, we conduct extensive experiments to comprehensively evaluate ORBIT in both in\-domain and cross\-domain generation settings\. The results demonstrate that ORBIT consistently outperforms advanced baselines in generation performance, while offering substantial advantages in inference efficiency\.

Overall, our main contributions are summarized as follows:

- •We introduce ORBIT, a unified deep generative framework that enables high\-quality and high\-fidelity time series generation by combining hierarchical representation learning with a novel next frequency\-band prediction paradigm\.
- •We design FH\-VQ\-VAE, which combines explicit frequency\-band decomposition with hierarchical residual quantization to learn discrete representations of heterogeneous temporal patterns, capturing both global temporal structures and local fine\-grained dynamics\.
- •We propose FBAR, which reformulates autoregressive time series generation as a next frequency\-band prediction task, establishes a low\-to\-high\-frequency autoregressive direction, and models cross\-band dependencies to coordinate the generation of heterogeneous temporal dynamics\.
- •Extensive experiments are conducted on multiple challenging real\-world benchmarks, whose results demonstrate that ORBIT achieves state\-of\-the\-art overall generation performance while offering superior inference efficiency\.

## 2\.Related Works

### 2\.1\.Deep Time Series Generation

Early works of deep time series generation primarily focus on Generative Adversarial Networks \(GANs\), such as TimeGAN\([Yoon et al\., 2019](https://arxiv.org/html/2610.27000#bib.bib24)\)and GT\-GAN\([Jeon et al\., 2022](https://arxiv.org/html/2610.27000#bib.bib27)\)\. Contemporaneous studies also explore VAE\-based approaches\([Kingma and Welling, 2014](https://arxiv.org/html/2610.27000#bib.bib20)\), such as TimeVAE\([Desai et al\., 2021](https://arxiv.org/html/2610.27000#bib.bib25)\)\. Yet, GANs and VAEs inherently suffer from issues such as training instability and posterior collapse, respectively, making such neural architecture\-driven models difficult to train and therefore limiting their ability to generate high\-quality synthetic time series\. With continued research progress, two\-stage deep generative models are emerging as a more advanced and increasingly prevalent paradigm\([Zhang et al\., 2026](https://arxiv.org/html/2610.27000#bib.bib6)\)\. Representative efforts such as SDformer\([Chen et al\., 2024](https://arxiv.org/html/2610.27000#bib.bib16)\)first introduce the two\-stage deep generative modeling paradigm to time series generation community, where a VQ\-VAE\([Van Den Oord et al\., 2017](https://arxiv.org/html/2610.27000#bib.bib15)\)learns discrete representations and a GPT\-style autoregressive backbone\([Achiam et al\., 2023](https://arxiv.org/html/2610.27000#bib.bib10)\)subsequently models the distribution over the resulting discrete tokens\. Recently, several methodologies extend this paradigm to multi\-scale representation learning by exploiting the inherent multi\-resolution temporal characteristics of time series\. For instance, MSDformer\([Feng et al\., 2026](https://arxiv.org/html/2610.27000#bib.bib17)\)introduces a multi\-scale VQ\-VAE as the tokenizer to better capture temporal dynamics at different temporal resolutions\. Inspired by scale\-wise autoregressive modeling\([Tian et al\., 2024](https://arxiv.org/html/2610.27000#bib.bib5)\), TimeMAR\([Xu et al\., 2026](https://arxiv.org/html/2610.27000#bib.bib18)\)introduces a next\-scale prediction pipeline for time series generation\. In contrast to existing two\-stage deep generative models, ORBIT disentangles heterogeneous temporal patterns from a spectral perspective for decoupled representation learning, and further develops a novel autoregressive direction driven by next frequency\-band prediction\.

### 2\.2\.Autoregressive Modeling for Time Series Generation

Autoregressive models were initially widely adopted in computer vision\([Croitoru et al\., 2023](https://arxiv.org/html/2610.27000#bib.bib22);[Ma et al\., 2024](https://arxiv.org/html/2610.27000#bib.bib40);[Ma et al\., 2026](https://arxiv.org/html/2610.27000#bib.bib41)\), natural language processing\([Zhang et al\., 2023](https://arxiv.org/html/2610.27000#bib.bib23)\), and audio generation\([Yang et al\., 2023](https://arxiv.org/html/2610.27000#bib.bib29)\)\. This has also sparked considerable interest among researchers and practitioners in the field of time series generation, motivating extensive efforts to adapt them for generating high\-quality and high\-fidelity time series samples\. For example, GPT\-style unidirectional autoregressive models are adopted by methods such as SDformer\([Chen et al\., 2024](https://arxiv.org/html/2610.27000#bib.bib16)\), MSDformer\([Feng et al\., 2026](https://arxiv.org/html/2610.27000#bib.bib17)\), and Polarformer\([Lyu et al\., 2026](https://arxiv.org/html/2610.27000#bib.bib19)\)\. Along another line of research, BERT\-style bidirectional autoregressive models are also explored, like HALO\([Hou et al\., 2026](https://arxiv.org/html/2610.27000#bib.bib30)\), TimeVQVAE\([Lee et al\., 2023](https://arxiv.org/html/2610.27000#bib.bib28)\)and SDformer\-m\([Chen et al\., 2024](https://arxiv.org/html/2610.27000#bib.bib16)\)\. More notably, diffusion models can be viewed as a generalized autoregressive paradigm, where samples are generated through iterative denoising and information recovery\([Ho et al\., 2020](https://arxiv.org/html/2610.27000#bib.bib31)\)\. Diffusion\-TS\([Yuan and Qiao, 2024](https://arxiv.org/html/2610.27000#bib.bib33)\)incorporates seasonal\-trend decomposition into a Transformer\-based denoising network for interpretable time series generation\. TimeDP\([Huang et al\., 2025](https://arxiv.org/html/2610.27000#bib.bib34)\)enables multi\-domain time series generation through a conditional diffusion model guided by prototype\-based domain prompts extracted from a few target\-domain examples, supporting unseen\-domain generation without additional fine\-tuning\. Different from existing autoregressive models, ORBIT explicitly aligns hierarchical discrete representations with frequency bands and develops an innovative next frequency\-band autoregressive modeling paradigm, progressing from low\- to high\-frequency bands to coordinate the generation of heterogeneous temporal patterns\.

## 3\.Preliminaries

### 3\.1\.Definitions

#### Definition 1 Multivariate Time Series

For the given multivariate time series𝐗∈ℝC×O\\mathbf\{X\}\\in\\mathbb\{R\}^\{C\\times O\}, whereCCrepresents the number of channels, andOOis the number of observed time steps\. ORBIT adopts a channel\-independent strategy\([Nie et al\., 2023](https://arxiv.org/html/2610.27000#bib.bib26)\)for robust modeling\. To this end, for simplicity, we omit the channel dimension in the following formulation\.

#### Definition 2 VQ\-VAE

Let the codebook be𝐄=\{𝐞k\}1K\\mathbf\{E\}=\\\{\\mathbf\{e\}\_\{k\}\\\}\_\{1\}^\{K\}, where each𝐞k∈ℝd\\mathbf\{e\}\_\{k\}\\in\\mathbb\{R\}^\{d\}is a discrete vector \(code\)\. Given an input sample𝐗\\mathbf\{X\}, the encoderfθ​\(⋅\)f\_\{\\theta\}\(\\cdot\)produces a continuous latent temporal representation𝐳=fθ​\(𝐗\)\\mathbf\{z\}=f\_\{\\theta\}\(\\mathbf\{X\}\)\. Vector quantization maps𝐳\\mathbf\{z\}to its nearest codebook vector in𝐄\\mathbf\{E\}through a nearest\-neighbor lookup:

\(1\)𝐳q=𝐞k∗,k∗=Lookup​\(𝐳,𝐄\),\\mathbf\{z\}\_\{q\}=\\mathbf\{e\}\_\{k^\{\*\}\},\\qquad\{k^\{\*\}\}=\\text\{Lookup\}\(\\mathbf\{z\};\\mathbf\{E\}\),whereLookup⁡\(⋅,⋅\)\\operatorname\{Lookup\}\(\\cdot;\\cdot\)denotes the function that retrieves the codebook vector in𝐄\\mathbf\{E\}closest to𝐳\\mathbf\{z\}\.

The quantized vector𝐳q\\mathbf\{z\}\_\{q\}is subsequently passed through the decodergϕ​\(⋅\)g\_\{\\phi\}\(\\cdot\)to reconstruct the raw input:

\(2\)𝒳^=gϕ​\(𝐳q\)\.\\hat\{\\mathcal\{X\}\}=g\_\{\\phi\}\(\\mathbf\{z\}\_\{q\}\)\.In practice, the encoder outputs a sequence of latent temporal representations that are independently quantized at each position, yielding a sequence of discrete tokens\. An autoregressive model can then be trained over these discrete tokens to model their joint probability distribution, thereby enabling sequential generation\.

### 3\.2\.Problem Statement

Given a training set𝒟=\{X\(i\)\}i=1N\\mathcal\{D\}=\\\{X^\{\(i\)\}\\\}\_\{i=1\}^\{N\}consisting ofNNtime series samples, time series generation aims to learn a parameterized distributionpθ​\(X\)p\_\{\\theta\}\(X\)that approximates the underlying data distributionpdata​\(X\)p\_\{\\mathrm\{data\}\}\(X\)\. The model parametersθ\\thetaare estimated from the entire training set𝒟\\mathcal\{D\}by optimizing a suitable training objective, such as a likelihood\- or reconstruction\-based objective\. Once trained, the model can generate synthetic sequencesX^∼pθ​\(⋅\)\\hat\{X\}\\sim p\_\{\\theta\}\(\\cdot\)whose statistical properties and temporal dynamics resemble those of the real input data\.

## 4\.Methodologies

### 4\.1\.Overall Framework

![Refer to caption](https://arxiv.org/html/2610.27000v1/figure_3.png)Figure 3\.![Refer to caption](https://arxiv.org/html/2610.27000v1/icon.png)The overview of our proposed ORBIT\.To achieve high\-quality time series generation, we introduce ORBIT, a unified deep generative framework whose overall architecture is illustrated in Figure[3](https://arxiv.org/html/2610.27000#S4.F3)\. Specifically,*in the first stage*, we train a Frequency\-Hierarchical VQ\-VAE \(FH\-VQ\-VAE\)\. TheEncodercompresses the input time series into continuous latent temporal representations\. We then applyFrequency\-Decomposed Residual Quantizerto construct hierarchical discrete tokens from low\- to high\-frequency bands, with each level incorporating a current frequency band while refining the quantization errors left by preceding levels\. The resulting quantized representations are mapped to a common temporal resolution and aggregated before being fed into theDecoderto reconstruct the original time series\.

*In the second stage*, we freeze the FH\-VQ\-VAE and train a Frequency\-Band Autoregressive Transformer \(FBAR\) following the proposed next frequency\-band prediction paradigm\. TheHistory\-conditioned Input Constructionmodule reconstructs and accumulates preceding frequency bands from their codebook indices, then applies interpolation, projection, and position and frequency\-band embeddings to construct the Transformer inputs\. During training, these inputs are constructed from ground\-truth indices, and FBAR is optimized using cross\-entropy under aFrequency\-band Causal Attentionto predict the indices of each subsequent frequency band\. During inference, FBAR starts from a conditional or unconditional and generates frequency bands sequentially from low to high, while sampling all indices within each band in parallel\. Each generated band is converted into a refined reconstruction contribution through the frozen quantizer and added to the cumulative latent representation, which is used to construct the next frequency\-band input\. After all bands are generated, the final accumulated representation is fed into the frozen decoder of the FH\-VQ\-VAE to synthesize the time series\.

### 4\.2\.Stage 1: Frequency\-Hierarchical VQ\-VAE

Encoder\.For the encoder of FH\-VQ\-VAE, following conventional practice, we progressively compress the input time series into low\-dimensional latent representations𝐡∈ℝM×d\\mathbf\{h\}\\in\\mathbb\{R\}^\{M\\times d\}, whereMMdenotes the number of latent representations, andssis the compression ratio\. Formally, it can be expressed as follows:

\(3\)𝐡=Encoder​\(𝐗\)\.\\mathbf\{h\}=\\text\{Encoder\}\(\\mathbf\{X\}\)\.Architecturally, the encoder employs 1D convolutional layers with residual blocks, while the decoder symmetrically mirrors the encoder architecture\.

Frequency\-Decomposed Residual Quantizer\.After obtaining the encoded latent representations, we leverage frequency\-decomposed residual quantizer to obtain hierarchical discrete representations\. This component constitutes the cornerstone of both disentangled representation learning for heterogeneous temporal patterns and the frequency\-band autoregressive generation direction\.

We first use the Fast Fourier Transform \(FFT\) operator to decompose the continuous latent representations𝐡\\mathbf\{h\}, obtaining its corresponding frequency\-domain representations𝐡f\\mathbf\{h\}\_\{f\}:

\(4\)𝐡f=FFT​\(𝐡\)\.\\mathbf\{h\}\_\{f\}=\\text\{FFT\}\(\\mathbf\{h\}\)\.Subsequently, we introduce a frequency\-band mask to partition𝐡f\\mathbf\{h\}\_\{f\}into multiple non\-overlapping frequency bands\. This operation disentangles heterogeneous temporal patterns by separating them in the spectral domain\. Formally, this process can be expressed as follows:

\(5\)𝐡f\(b\)=𝐌b⊙𝐡f,b=1,…,B,\\mathbf\{h\}\_\{f\}^\{\(b\)\}=\\mathbf\{M\}\_\{b\}\\odot\\mathbf\{h\}\_\{f\},\\quad b=1,\\ldots,B,where𝐌b\\mathbf\{M\}\_\{b\}denotes the frequency\-band mask corresponding to thebb\-th frequency band\. We then apply the Inverse Fast Fourier Transform \(IFFT\) to each frequency\-band representation, transforming them back to the time domain for subsequent residual quantization:

\(6\)𝐡\(b\)=IFFT\(𝐡f\(b\)\),b=1,…,B\.\\mathbf\{h\}^\{\(b\)\}=\\text\{IFFT\}\\\!\\left\(\\mathbf\{h\}\_\{f\}^\{\(b\)\}\\right\),\\qquad b=1,\\ldots,B\.
After obtaining\{𝐡\(b\)\}b=1B\\\{\\mathbf\{h\}^\{\(b\)\}\\\}\_\{b=1\}^\{B\}, we apply residual quantization to progressively construct hierarchical discrete representations:

\(7\)𝐫\(b\)=∑j=1b𝐡\(j\)−𝐡^\(b−1\),𝐡^\(0\)=𝟎,\\mathbf\{r\}^\{\(b\)\}=\\sum\_\{j=1\}^\{b\}\\mathbf\{h\}^\{\(j\)\}\-\\hat\{\\mathbf\{h\}\}^\{\(b\-1\)\},\\qquad\\hat\{\\mathbf\{h\}\}^\{\(0\)\}=\\mathbf\{0\},where𝐫\(b\)∈ℝM×d\\mathbf\{r\}^\{\(b\)\}\\in\\mathbb\{R\}^\{M\\times d\}denotes the quantization residual of thebb\-th frequency\-band level\. Considering that low\-frequency components reflect slowly varying structures while high\-frequency components contain rapid temporal variations, we adopt frequency\-dependent temporal quantization resolutions, allocating fewer tokens to lower\-frequency bands and more tokens to higher\-frequency bands:

\(8\)𝐭\(b\)=Lookup​\(Interpnb​\(𝐫\(b\)\),𝐄\),\\mathbf\{t\}^\{\(b\)\}=\\text\{Lookup\}\(\\text\{Interp\}\_\{n\_\{b\}\}\(\\mathbf\{r\}^\{\(b\)\}\);\\mathbf\{E\}\),where𝐭\(b\)∈\{1,…,K\}nb\\mathbf\{t\}^\{\(b\)\}\\in\\\{1,\\ldots,K\\\}^\{n\_\{b\}\}denotes the discrete codebook index sequence at thebb\-th frequency\-band level,nbn\_\{b\}is the number of tokens allocated to this stage, andInterpnb​\(⋅\)\\text\{Interp\}\_\{n\_\{b\}\}\(\\cdot\)resizes the residual representation tonbn\_\{b\}temporal positions\.𝐄∈ℝK×d\\mathbf\{E\}\\in\\mathbb\{R\}^\{K\\times d\}denotes the shared learnable codebook, andLookup​\(⋅,𝐄\)\\text\{Lookup\}\(\\cdot;\\mathbf\{E\}\)assigns each interpolated feature vector to the index of its closest codebook entry\. Through codebook lookup and resolution restoration, we obtain the continuous reconstruction contribution at thebb\-th stage,𝐠\(b\)∈ℝM×d\\mathbf\{g\}^\{\(b\)\}\\in\\mathbb\{R\}^\{M\\times d\}, and update the cumulative reconstruction as

\(9\)𝐡^\(b\)=𝐡^\(b−1\)\+𝐠\(b\)\.\\widehat\{\\mathbf\{h\}\}^\{\(b\)\}=\\widehat\{\\mathbf\{h\}\}^\{\(b\-1\)\}\+\\mathbf\{g\}^\{\(b\)\}\.
By allocating band\-specific token budgets, the above procedure constructs hierarchical representations tailored to heterogeneous temporal patterns\. Moreover, inspired by\([Tian et al\., 2024](https://arxiv.org/html/2610.27000#bib.bib5)\), we introduce lightweight refinement modules at each quantization stage to adjust the interpolated codeword vectors before accumulating them for latent reconstruction\. For clarity, we provide a detailed pseudo\-algorithm in Algorithm[1](https://arxiv.org/html/2610.27000#alg1)\.

Decoder\.Through the decoder, we reconstruct the final latent representation𝐡^\(B\)\\widehat\{\\mathbf\{h\}\}^\{\(B\)\}back into the time series space:

\(10\)𝐗^=Decoder​\(𝐡^\(B\)\),𝐗^∈ℝL\.\\widehat\{\\mathbf\{X\}\}=\\text\{Decoder\}\\\!\\left\(\\widehat\{\\mathbf\{h\}\}^\{\(B\)\}\\right\),\\quad\\widehat\{\\mathbf\{X\}\}\\in\\mathbb\{R\}^\{L\}\.The first stage is trained end\-to\-end by minimizing the reconstruction loss between the input time series and its reconstruction, while the discrete codebooks are jointly optimized using the straight\-through estimator\([Van Den Oord et al\., 2017](https://arxiv.org/html/2610.27000#bib.bib15)\)\. Consequently, the overall learning objective is defined as follows:

\(11\)ℒStage​\-​1=\\displaystyle\\mathcal\{L\}\_\{\\mathrm\{Stage\\text\{\-\}1\}\}=1L​‖𝐗−𝐗^‖22⏟ℒrec\\displaystyle\\underbrace\{\\frac\{1\}\{L\}\\left\\\|\\mathbf\{X\}\-\\widehat\{\\mathbf\{X\}\}\\right\\\|\_\{2\}^\{2\}\}\_\{\\mathcal\{L\}\_\{\\mathrm\{rec\}\}\}\+λcb​1B​M​d​∑b=1B‖sg⁡\[∑j=1b𝐡\(j\)\]−𝐡^\(b\)‖F2⏟ℒcb\\displaystyle\+\\lambda\_\{\\mathrm\{cb\}\}\\underbrace\{\\frac\{1\}\{BMd\}\\sum\_\{b=1\}^\{B\}\\left\\\|\\operatorname\{sg\}\\\!\\left\[\\sum\_\{j=1\}^\{b\}\\mathbf\{h\}^\{\(j\)\}\\right\]\-\\widehat\{\\mathbf\{h\}\}^\{\(b\)\}\\right\\\|\_\{F\}^\{2\}\}\_\{\\mathcal\{L\}\_\{\\mathrm\{cb\}\}\}\+λcom​1M​d​∑b=1B‖∑j=1b𝐡\(j\)−sg⁡\[𝐡^\(b\)\]‖F2⏟ℒcom,\\displaystyle\+\\lambda\_\{\\mathrm\{com\}\}\\underbrace\{\\frac\{1\}\{Md\}\\sum\_\{b=1\}^\{B\}\\left\\\|\\sum\_\{j=1\}^\{b\}\\mathbf\{h\}^\{\(j\)\}\-\\operatorname\{sg\}\\\!\\left\[\\widehat\{\\mathbf\{h\}\}^\{\(b\)\}\\right\]\\right\\\|\_\{F\}^\{2\}\}\_\{\\mathcal\{L\}\_\{\\mathrm\{com\}\}\},wheresg⁡\[⋅\]\\operatorname\{sg\}\[\\cdot\]denotes the stop\-gradient operator, and∥⋅∥F\\\|\\cdot\\\|\_\{F\}denotes the Frobenius norm\.∑j=1b𝐡\(j\)\\sum\_\{j=1\}^\{b\}\\mathbf\{h\}^\{\(j\)\}is the cumulative latent target up to thebb\-th frequency band, while𝐡^\(b\)\\widehat\{\\mathbf\{h\}\}^\{\(b\)\}is its corresponding cumulative reconstruction obtained through codebook lookup, interpolation, refinement, and accumulation\.λcb\\lambda\_\{\\mathrm\{cb\}\}andλcom\\lambda\_\{\\mathrm\{com\}\}control the contributions of the codebook and commitment losses, respectively\.

Algorithm 1Frequency\-Decomposed Residual Quantizer1:Encoded latent representation

𝐡∈ℝM×d\\mathbf\{h\}\\in\\mathbb\{R\}^\{M\\times d\}; frequency\-band masks

\{𝐌b\}b=1B\\\{\\mathbf\{M\}\_\{b\}\\\}\_\{b=1\}^\{B\}; token budgets

\{nb\}b=1B\\\{n\_\{b\}\\\}\_\{b=1\}^\{B\}; shared codebook

𝐄∈ℝK×d\\mathbf\{E\}\\in\\mathbb\{R\}^\{K\\times d\}; refinement modules

\{ϕb\}b=1B\\\{\\phi\_\{b\}\\\}\_\{b=1\}^\{B\}\.

2:Discrete codebook index sequences

\{𝐭\(b\)\}b=1B\\\{\\mathbf\{t\}^\{\(b\)\}\\\}\_\{b=1\}^\{B\}and reconstructed latent representation

𝐡^\(B\)\\widehat\{\\mathbf\{h\}\}^\{\(B\)\}\.

3:

𝐡f←FFT⁡\(𝐡\)\\mathbf\{h\}\_\{f\}\\leftarrow\\operatorname\{FFT\}\(\\mathbf\{h\}\)⊳\\trianglerightAlong the temporal dimension

4:for

b=1,…,Bb=1,\\ldots,Bdo

5:

𝐡f\(b\)←𝐌b⊙𝐡f\\mathbf\{h\}\_\{f\}^\{\(b\)\}\\leftarrow\\mathbf\{M\}\_\{b\}\\odot\\mathbf\{h\}\_\{f\}
6:

𝐡\(b\)←IFFT⁡\(𝐡f\(b\)\)\\mathbf\{h\}^\{\(b\)\}\\leftarrow\\operatorname\{IFFT\}\(\\mathbf\{h\}\_\{f\}^\{\(b\)\}\)
7:endfor

8:

𝐡^\(0\)←𝟎\\widehat\{\\mathbf\{h\}\}^\{\(0\)\}\\leftarrow\\mathbf\{0\}
9:for

b=1,…,Bb=1,\\ldots,Bdo

10:

𝐫\(b\)←∑j=1b𝐡\(j\)−𝐡^\(b−1\)\\mathbf\{r\}^\{\(b\)\}\\leftarrow\\displaystyle\\sum\_\{j=1\}^\{b\}\\mathbf\{h\}^\{\(j\)\}\-\\widehat\{\\mathbf\{h\}\}^\{\(b\-1\)\}⊳\\trianglerightConstruct the stage residual

11:

𝐭\(b\)←Lookup⁡\(Interpnb⁡\(𝐫\(b\)\);𝐄\)\\mathbf\{t\}^\{\(b\)\}\\leftarrow\\operatorname\{Lookup\}\\bigl\(\\operatorname\{Interp\}\_\{n\_\{b\}\}\(\\mathbf\{r\}^\{\(b\)\}\);\\mathbf\{E\}\\bigr\)⊳\\trianglerightObtain discrete indices

12:

𝐠\(b\)←ϕb​\(InterpM⁡\(𝐄⁡\[𝐭\(b\)\]\)\)\\mathbf\{g\}^\{\(b\)\}\\leftarrow\\phi\_\{b\}\\\!\\left\(\\operatorname\{Interp\}\_\{M\}\\bigl\(\\mathbf\{E\}\[\\mathbf\{t\}^\{\(b\)\}\]\\bigr\)\\right\)⊳\\trianglerightRestore resolution and refine

13:

𝐡^\(b\)←𝐡^\(b−1\)\+𝐠\(b\)\\widehat\{\\mathbf\{h\}\}^\{\(b\)\}\\leftarrow\\widehat\{\\mathbf\{h\}\}^\{\(b\-1\)\}\+\\mathbf\{g\}^\{\(b\)\}⊳\\trianglerightAccumulate the reconstruction

14:endfor

15:return

\{𝐭\(b\)\}b=1B,𝐡^\(B\)\\\{\\mathbf\{t\}^\{\(b\)\}\\\}\_\{b=1\}^\{B\},\\widehat\{\\mathbf\{h\}\}^\{\(B\)\}

### 4\.3\.Stage 2: Frequency\-Band Autoregressive Transformer

Building on the hierarchical discrete representations learned by FH\-VQ\-VAE, we introduce a Frequency\-Band Autoregressive Transformer \(FBAR\) to model dependencies across frequency bands\. During this stage, FH\-VQ\-VAE is frozen, and FBAR is trained to predict the next frequency\-band conditioned on preceding bands\. This establishes a progressive autoregressive direction from low\-frequency structures to higher\-frequency details\.

Algorithm 2Next Frequency\-Band Autoregressive Prediction1:Trained FBAR; frozen codebook

𝐄\\mathbf\{E\}, refinement modules

ϕb​b=1B\{\\phi\_\{b\}\}\{b=1\}^\{B\}, and decoder; token budgets

nb​b=1B\{n\_\{b\}\}\{b=1\}^\{B\}; latent length

MM; class or null condition\[BOS\]; sampling temperature

τ\>0\\tau\>0\.

2:Generated time series

𝐗^\\widehat\{\\mathbf\{X\}\}\.

3:

𝐡^\(0\)←𝟎\\widehat\{\\mathbf\{h\}\}^\{\(0\)\}\\leftarrow\\mathbf\{0\}
4:

𝐙←\[,\]\\mathbf\{Z\}\\leftarrow\[,\]⊳\\trianglerightInitialize the input prefix

5:for

b=1,…,Bb=1,\\ldots,Bdo

6:if

b=1b=1then

7:

𝐳​i\(1\)←𝐬​i​\(\[BOS\]\)\+𝐞​pos,i\(1\)\+𝐞​band\(1\),i=1,…,n1\\mathbf\{z\}i^\{\(1\)\}\\leftarrow\\mathbf\{s\}i\(\\texttt\{\[BOS\]\}\)\+\\mathbf\{e\}\{\\mathrm\{pos\},i\}^\{\(1\)\}\+\\mathbf\{e\}\{\\mathrm\{band\}\}^\{\(1\)\},\\hskip 9\.24994pti=1,\\ldots,n\_\{1\}⊳\\trianglerightConstruct BOS inputs

8:else

9:

𝐮\(b\)←Interpnb\!​\(𝐡^\(b−1\)\)\\mathbf\{u\}^\{\(b\)\}\\leftarrow\\operatorname\{Interp\}\_\{n\_\{b\}\}\!\\left\(\\widehat\{\\mathbf\{h\}\}^\{\(b\-1\)\}\\right\)
10:

𝐳i\(b\)←Proj\(𝐮i\(b\)\)\+𝐞pos,i\(b\)\+𝐞band\(b\),i=1,…,nb\\mathbf\{z\}\_\{i\}^\{\(b\)\}\\leftarrow\\operatorname\{Proj\}\\\!\\left\(\\mathbf\{u\}\_\{i\}^\{\(b\)\}\\right\)\+\\mathbf\{e\}\_\{\\mathrm\{pos\},i\}^\{\(b\)\}\+\\mathbf\{e\}\_\{\\mathrm\{band\}\}^\{\(b\)\},\\hskip 9\.24994pti=1,\\ldots,n\_\{b\}⊳\\trianglerightConstruct history\-conditioned inputs

11:endif

12:

𝐙←Concat⁡\(𝐙,𝐳\(b\)\)\\mathbf\{Z\}\\leftarrow\\operatorname\{Concat\}\\\!\\left\(\\mathbf\{Z\},\\mathbf\{z\}^\{\(b\)\}\\right\)
13:

ℓb←∑j=1bnj\\ell\_\{b\}\\leftarrow\\displaystyle\\sum\_\{j=1\}^\{b\}n\_\{j\}
14:

𝐀\(b\)←𝐀1:ℓb,1:ℓb\\mathbf\{A\}^\{\(b\)\}\\leftarrow\\mathbf\{A\}\_\{1:\\ell\_\{b\},\\,1:\\ell\_\{b\}\}⊳\\trianglerightRestrict the block\-wise causal mask

15:

𝐘←Transformer⁡\(𝐙,𝐀\(b\)\)\\mathbf\{Y\}\\leftarrow\\operatorname\{Transformer\}\\\!\\left\(\\mathbf\{Z\};\\mathbf\{A\}^\{\(b\)\}\\right\)
16:

𝐘\(b\)←𝐘ℓb−nb\+1:ℓb,:\\mathbf\{Y\}^\{\(b\)\}\\leftarrow\\mathbf\{Y\}\_\{\\ell\_\{b\}\-n\_\{b\}\+1:\\ell\_\{b\},\\,:\}⊳\\trianglerightSelect current\-band hidden states

17:

𝐏\(b\)←Softmax⁡\(Head⁡\(Norm⁡\(𝐘\(b\)\)\)/τ\)\\mathbf\{P\}^\{\(b\)\}\\leftarrow\\operatorname\{Softmax\}\\\!\\left\(\\operatorname\{Head\}\\\!\\left\(\\operatorname\{Norm\}\\\!\\left\(\\mathbf\{Y\}^\{\(b\)\}\\right\)\\right\)/\\tau\\right\)
18:

t^i\(b\)∼Categorical\(𝐏i\(b\)\),i=1,…,nb\\widehat\{t\}\_\{i\}^\{\(b\)\}\\sim\\operatorname\{Categorical\}\\\!\\left\(\\mathbf\{P\}\_\{i\}^\{\(b\)\}\\right\),\\hskip 9\.24994pti=1,\\ldots,n\_\{b\}⊳\\trianglerightSample all indices in parallel \(sampling temperature is fixed at 1\.0\)

19:

𝐯\(b\)←𝐄⁡\[𝐭^\(b\)\]\\mathbf\{v\}^\{\(b\)\}\\leftarrow\\mathbf\{E\}\[\\widehat\{\\mathbf\{t\}\}^\{\(b\)\}\]⊳\\trianglerightRetrieve codeword vectors

20:

𝐯~\(b\)←InterpM⁡\(𝐯\(b\)\)\\widetilde\{\\mathbf\{v\}\}^\{\(b\)\}\\leftarrow\\operatorname\{Interp\}\_\{M\}\\\!\\left\(\\mathbf\{v\}^\{\(b\)\}\\right\)
21:

𝐠\(b\)←ϕb​\(𝐯~\(b\)\)\\mathbf\{g\}^\{\(b\)\}\\leftarrow\\phi\_\{b\}\\\!\\left\(\\widetilde\{\\mathbf\{v\}\}^\{\(b\)\}\\right\)
22:

𝐡^\(b\)←𝐡^\(b−1\)\+𝐠\(b\)\\widehat\{\\mathbf\{h\}\}^\{\(b\)\}\\leftarrow\\widehat\{\\mathbf\{h\}\}^\{\(b\-1\)\}\+\\mathbf\{g\}^\{\(b\)\}⊳\\trianglerightAccumulate the reconstruction

23:endfor

24:

𝐗^←Decoder\!\(𝐡^\(B\)\)\\widehat\{\\mathbf\{X\}\}\\leftarrow\\operatorname\{Decoder\}\!\\left\(\\widehat\{\\mathbf\{h\}\}^\{\(B\)\}\\right\)
25:return

𝐗^\\widehat\{\\mathbf\{X\}\}

Given the discrete codebook index sequences\{𝐭\(b\)\}b=1B\\\{\\mathbf\{t\}^\{\(b\)\}\\\}\_\{b=1\}^\{B\}, FBAR factorizes their joint distribution as

\(12\)pθ​\(𝐭\(1\),…,𝐭\(B\)∣\[BOS\]\)=∏b=1Bpθ​\(𝐭\(b\)∣𝐭\(<b\),\[BOS\]\),p\_\{\\theta\}\\\!\\left\(\\mathbf\{t\}^\{\(1\)\},\\ldots,\\mathbf\{t\}^\{\(B\)\}\\mid\\texttt\{\[BOS\]\}\\right\)=\\prod\_\{b=1\}^\{B\}p\_\{\\theta\}\\\!\\left\(\\mathbf\{t\}^\{\(b\)\}\\mid\\mathbf\{t\}^\{\(<b\)\},\\texttt\{\[BOS\]\}\\right\),where𝐭\(<b\)\\mathbf\{t\}^\{\(<b\)\}denotes the index sequences of all preceding frequency bands, and\[BOS\]denotes an optional class condition\. Within each band, the token distributions are conditionally factorized and predicted in parallel:

\(13\)pθ​\(𝐭\(b\)∣𝐭\(<b\),\[BOS\]\)=∏i=1nbpθ​\(ti\(b\)∣𝐭\(<b\),\[BOS\]\)\.p\_\{\\theta\}\\\!\\left\(\\mathbf\{t\}^\{\(b\)\}\\mid\\mathbf\{t\}^\{\(<b\)\},\\texttt\{\[BOS\]\}\\right\)=\\prod\_\{i=1\}^\{n\_\{b\}\}p\_\{\\theta\}\\\!\\left\(t\_\{i\}^\{\(b\)\}\\mid\\mathbf\{t\}^\{\(<b\)\},\\texttt\{\[BOS\]\}\\right\)\.
History\-conditioned Input Construction\.Rather than directly shifting the discrete index sequence\([Chen et al\., 2024](https://arxiv.org/html/2610.27000#bib.bib16);[Feng et al\., 2026](https://arxiv.org/html/2610.27000#bib.bib17);[Xu et al\., 2026](https://arxiv.org/html/2610.27000#bib.bib18)\), we construct the input of each subsequent stage from the cumulative latent reconstruction of preceding bands\. Specifically, forb≥2b\\geq 2, the history representation is resized to the token length of the current band:

\(14\)𝐮\(b\)=Interpnb⁡\(𝐡^\(b−1\)\)∈ℝnb×d\.\\mathbf\{u\}^\{\(b\)\}=\\operatorname\{Interp\}\_\{n\_\{b\}\}\\\!\\left\(\\widehat\{\\mathbf\{h\}\}^\{\(b\-1\)\}\\right\)\\in\\mathbb\{R\}^\{n\_\{b\}\\times d\}\.Here,𝐡^\(b−1\)\\widehat\{\\mathbf\{h\}\}^\{\(b\-1\)\}is obtained through codebook lookup, resolution restoration, refinement, and accumulation, following the procedure introduced in Stage 1\. The resulting history features are projected into the Transformer embedding space and augmented with position and frequency\-band embeddings:

\(15\)𝐳i\(b\)=\{𝐬i​\(\[BOS\]\)\+𝐞pos,i\(1\)\+𝐞band\(1\),b=1,Proj​\(𝐮i\(b\)\)\+𝐞pos,i\(b\)\+𝐞band\(b\),b≥2,\\mathbf\{z\}\_\{i\}^\{\(b\)\}=\\begin\{cases\}\\mathbf\{s\}\_\{i\}\(\\texttt\{\[BOS\]\}\)\+\\mathbf\{e\}\_\{\\mathrm\{pos\},i\}^\{\(1\)\}\+\\mathbf\{e\}\_\{\\mathrm\{band\}\}^\{\(1\)\},&b=1,\\\\\[2\.0pt\] \\text\{Proj\}\\\!\\left\(\\mathbf\{u\}\_\{i\}^\{\(b\)\}\\right\)\+\\mathbf\{e\}\_\{\\mathrm\{pos\},i\}^\{\(b\)\}\+\\mathbf\{e\}\_\{\\mathrm\{band\}\}^\{\(b\)\},&b\\geq 2,\\end\{cases\}whereProj​\(⋅\)\\text\{Proj\}\(\\cdot\)maps the latent features fromdddimensions to the Transformer embedding dimensionDD\.𝐬i​\(c\)\\mathbf\{s\}\_\{i\}\(c\)combines the class embedding with a learnable start\-position embedding, while𝐞pos,i\(b\)\\mathbf\{e\}\_\{\\mathrm\{pos\},i\}^\{\(b\)\}and𝐞band\(b\)\\mathbf\{e\}\_\{\\mathrm\{band\}\}^\{\(b\)\}encode position and frequency\-band identity, respectively\. For unconditional generation, the class condition is replaced with a learned null condition \(classifier\-free guidance\)\.

Frequency\-band Causal Attention\.We concatenate the stage inputs and apply a block\-wise causal mask to organize attention along the frequency\-band axis\. Letb⁡\(i\)b\(i\)denote the frequency\-band stage associated with theii\-th input position\. The additive attention mask is defined as

\(16\)Ai​j=\{0,b⁡\(j\)≤b⁡\(i\),−∞,b⁡\(j\)\>b⁡\(i\)\.A\_\{ij\}=\\begin\{cases\}0,&b\(j\)\\leq b\(i\),\\\\ \-\\infty,&b\(j\)\>b\(i\)\.\\end\{cases\}The mask permits attention to positions within the same stage and preceding stages while blocking future stages\. Importantly, the inputs of the current stage are constructed exclusively from preceding bands, allowing interactions among history\-conditioned positions without exposing the current target indices\. The resulting hidden states are normalized and passed through a classification head to obtain a distribution over theKKcodebook entries at each position\.

Training and Inference\.During training, teacher forcing\([Bengio et al\., 2003](https://arxiv.org/html/2610.27000#bib.bib21)\)constructs each stage input using the preceding ground\-truth index sequences\. All stages are evaluated in a single forward pass under the block\-wise causal mask\. FBAR is optimized using token\-level cross\-entropy:

\(17\)ℒStage​\-​2=−1∑b=1Bnb∑b=1B∑i=1nblogpθ\(ti\(b\)∣𝐭\(<b\),\[BOS\]\)\.\\mathcal\{L\}\_\{\\mathrm\{Stage\\text\{\-\}2\}\}=\-\\frac\{1\}\{\\sum\_\{b=1\}^\{B\}n\_\{b\}\}\\sum\_\{b=1\}^\{B\}\\sum\_\{i=1\}^\{n\_\{b\}\}\\log p\_\{\\theta\}\\\!\\left\(t\_\{i\}^\{\(b\)\}\\mid\\mathbf\{t\}^\{\(<b\)\},\\texttt\{\[BOS\]\}\\right\)\.
During inference, FBAR starts from the class\-conditioned or unconditional start embeddings and proceeds sequentially from low\- to high\-frequency bands\. At each level, allnbn\_\{b\}indices are sampled in parallel from the predicted categorical distributions\. The sampled indices are converted into a reconstruction contribution using the frozen codebook and refinement modules, and added to the cumulative latent reconstruction\. This reconstruction is then resized to construct the input for the next frequency\-band\. After allBBstages, the final accumulated latent representation is fed into the frozen decoder:

\(18\)𝐗^=Decoder⁡\(𝐡^\(B\)\)\.\\widehat\{\\mathbf\{X\}\}=\\operatorname\{Decoder\}\\\!\\left\(\\widehat\{\\mathbf\{h\}\}^\{\(B\)\}\\right\)\.Thus, generation requiresBBsequential band\-level steps, with token positions within each band generated in parallel\. Benefiting from this frequency\-band\-level parallel decoding capability, ORBIT also achieves improving inference efficiency\. For clarity, we present the inference procedure in Algorithm[2](https://arxiv.org/html/2610.27000#alg2)\.

Table 1\.Maximum mean discrepancy \(MMD\) and Kullback–Leibler divergence \(K\-L\) for time series generation at sequence lengths 24, 96, and 168\. Lower values indicate better performance\. Results at individual lengths are reported as mean±\\pmstandard deviation\. Best means are highlighted in red and bold, and second\-best results are highlighted in blue and underlined\.

## 5\.Experiments

### 5\.1\.Datasets

We evaluate ORBIT on 12 datasets across four domains: energy \(Electricity, Solar, Wind\), transportation \(Traffic, Taxi, Pedestrian\), environment \(Air Quality, Temperature, Rain\), and economics \(NN5, FRED\-MD, Exchange\)\. These benchmarks are sourced from GluonTS\([Alexandrov et al\., 2020](https://arxiv.org/html/2610.27000#bib.bib35)\)and the Monash Time Series Forecasting Repository\([Godahewa et al\., 2021](https://arxiv.org/html/2610.27000#bib.bib36)\)\. Following\([Huang et al\., 2025](https://arxiv.org/html/2610.27000#bib.bib34)\), we jointly train on the combined multi\-domain dataset and evaluate each domain separately\. Dataset details are provided in Appendix[A](https://arxiv.org/html/2610.27000#A1)\.

### 5\.2\.Baselines

We compare our method against a set of cutting\-edge deep generative baselines, including TimeGAN\([Yoon et al\., 2019](https://arxiv.org/html/2610.27000#bib.bib24)\), GT\-GAN\([Jeon et al\., 2022](https://arxiv.org/html/2610.27000#bib.bib27)\), TimeVAE\([Desai et al\., 2021](https://arxiv.org/html/2610.27000#bib.bib25)\), TimeVQVAE\([Lee et al\., 2023](https://arxiv.org/html/2610.27000#bib.bib28)\), TimeVQVAE\-C \(the class\-conditional variant of TimeVQVAE\)\([Lee et al\., 2023](https://arxiv.org/html/2610.27000#bib.bib28)\), TimeDP\([Huang et al\., 2025](https://arxiv.org/html/2610.27000#bib.bib34)\), and TimeMAR\([Xu et al\., 2026](https://arxiv.org/html/2610.27000#bib.bib18)\)\. Detailed descriptions of these baselines are provided in Appendix[B](https://arxiv.org/html/2610.27000#A2)\.

### 5\.3\.Implementation Details

All experiments are conducted on a single NVIDIA RTX 4090 GPU using Python 3\.10\.2 and PyTorch 2\.9\.1\([Paszke et al\., 2019](https://arxiv.org/html/2610.27000#bib.bib37)\)\. Both stages are trained for up to1×1061\\times 10^\{6\}iterations with a batch size of 256 and an initial learning rate of1×10−41\\times 10^\{\-4\}, using a1×1031\\times 10^\{3\}\-iteration linear warm\-up followed by cosine decay\. We consider a basic generation length of 24 and further evaluate longer lengths of 96 and 168 to examine the robustness of different models in long\-sequence generation\. More implementation details are provided in Appendix[C](https://arxiv.org/html/2610.27000#A3)\.

### 5\.4\.Evaluation Metrics

To evaluate generation quality, we assess both the distributional fidelity between real and synthetic time series and the preservation of their temporal characteristics\. Consequently, we adopt two metrics: Maximum Mean Discrepancy \(MMD\), and Kullback–Leibler \(K\-L\) divergence\.MMDmeasures the distributional discrepancy between real and synthetic samples in a kernel\-induced feature space, whileK\-Ldivergence quantifies the discrepancy between their probability distributions\. For all metrics, lower values indicate better generation quality\.

Figure 4\.MMD and K\-L divergence in cross\-domain settings of Web datasets for sequence length 168\. U, F, and P denote unconditional generation, fine\-tuning, and sample prompting, respectively\. ORBIT uses CFG to achieve unconditional generation\.
### 5\.5\.Main Results

In\-Domain Generation\.As shown in Table[1](https://arxiv.org/html/2610.27000#S4.T1), ORBIT achieves state\-of\-the\-art overall generation performance across 12 datasets and 3 sequence lengths, demonstrating strong distributional fidelity under both MMD and K\-L divergence\. These improvements can be attributed to its frequency\-hierarchical representation learning and next frequency\-band autoregressive direction\. Specifically, FH\-VQ\-VAE separates frequency components and progressively refines quantization errors, facilitating the representation of global temporal structures and fine\-grained details\. FBAR further captures cross\-band dependencies through next frequency\-band prediction, enabling coordinated coarse\-to\-fine generation\. The combination of these complementary designs contributes to ORBIT’s strong generation performance across diverse time series datasets\.

TimeMAR\([Xu et al\., 2026](https://arxiv.org/html/2610.27000#bib.bib18)\)and TimeDP\([Huang et al\., 2025](https://arxiv.org/html/2610.27000#bib.bib34)\)remain competitive, yet ORBIT delivers stronger overall performance\. Although TimeMAR learns multi\-scale representations, it retains token\-level causal decoding, which may accumulate prediction errors and compromise generation fidelity\. In contrast, ORBIT generates tokens within each frequency band in parallel, reducing sequential decoding steps while modeling cross\-band correlations\. TimeDP relies on prototype\-based domain prompts, whose effectiveness depends on how accurately the inferred prototype assignments reflect target\-domain semantics\. Imperfect alignment may introduce conditioning noise and compromise generation fidelity\. ORBIT instead explicitly organizes temporal representations and their generation order by frequency, providing a structured inductive bias for modeling heterogeneous temporal patterns\. In summary, frequency\-hierarchical representation learning and band\-level parallel generation contribute to ORBIT’s improved generation quality and fidelity\.

Cross\-Domain Generation\.Benefiting from classifier\-free guidance \(CFG\), HALO supports unconditional generation and can be used to unseen domains excluded from the multi\-domain training set\. Following\([Huang et al\., 2025](https://arxiv.org/html/2610.27000#bib.bib34)\), we use 3, 10, and 100 samples as both prompts and fine\-tuning data for other models, with the MMD and K\-L divergence results for a generation length of 168 reported in Figure[4](https://arxiv.org/html/2610.27000#S5.F4)\. ORBIT demonstrates superior and robust cross\-domain generation capability, substantially outperforming other unconditional baselines and even methods specifically fine\-tuned on the target domains\. Notably, ORBIT also maintains a clear advantage over TimeDP\([Huang et al\., 2025](https://arxiv.org/html/2610.27000#bib.bib34)\), which leverages samples from unseen target domains as prompts\. This further demonstrates ORBIT’s strong cross\-domain generalization capability and highlights its potential as a scalable foundation for large\-scale time series generative models\.

### 5\.6\.Model Analysis

Ablation Studies\.To assess each component, we evaluate three ablations:*Variant 1*uses a single quantization level;*Variant 2*quantizes frequency bands independently without residual error compensation; and*Variant 3*allocates equal token budgets across bands\. As shown in Table[2](https://arxiv.org/html/2610.27000#S5.T2), ORBIT outperforms all variants across both sequence lengths and metrics\. The degradation of Variants 1 and 2 supports the importance of hierarchical representations and progressive error correction\. Variant 3 incurs the largest performance drop, suggesting that frequency\-dependent token allocation is essential for balancing the representation of global structures and fine\-grained details\. Overall, these results validate the effectiveness of ORBIT’s frequency\-hierarchical quantization design\.

Table 2\.Ablation study results\. We report MMD and K\-L averaged across datasets for long\-sequence time series generation task\. Improvement denotes the relative reduction achieved by the full model over each variant, computed from the average scores across lengths 96 and 168\.Inference Efficiency Analysis\.To investigate the inference efficiency advantages of ORBIT, we conduct a comparative efficiency analysis\. Specifically, we benchmark ORBIT against TimeMAR\([Xu et al\., 2026](https://arxiv.org/html/2610.27000#bib.bib18)\), a representative state\-of\-the\-art baseline\. As shown in Figure[5](https://arxiv.org/html/2610.27000#S5.F5), ORBIT consistently achieves superior efficiency across different sequence lengths, with lower generation time, peak allocated memory, and peak reserved memory\. This advantage is primarily attributed to ORBIT’s ability to perform parallel decoding at the frequency\-band level\. The efficiency advantages of ORBIT revealed by the efficiency evaluation strongly underscore the necessity of further exploring more accurate and efficient parallel decoding strategies\.

Figure 5\.Comparison of generation time, peak allocated memory, and peak reserved memory between ORBIT and TimeMAR\([Xu et al\., 2026](https://arxiv.org/html/2610.27000#bib.bib18)\)for generating 5,000 samples on the Electricity dataset with a fixed batch size of 1,000\.Visualization\.To qualitatively assess generation quality, we use t\-SNE\([Van der Maaten and Hinton, 2008](https://arxiv.org/html/2610.27000#bib.bib39)\)to project real and synthetic time series from representative datasets into a two\-dimensional space, visualizing their distributional overlap\. As shown in Figure 6, ORBIT exhibits greater overlap between real and synthetic samples in the t\-SNE space across representative datasets, suggesting closer alignment of their underlying representations\.

Figure 6\.t\-SNE visualization of real and synthetic time series samples on the Electricity and Wind benchmarks with a sequence length of 168\. Additional qualitative visualizations are provided in Appendix[D](https://arxiv.org/html/2610.27000#A4)\.

## 6\.Conclusions

In this paper, we propose ORBIT, a unified deep generative framework, which enables high\-quality and high\-fidelity time series generation\. Its Frequency\-Hierarchical VQ\-VAE combines frequency\-band decomposition with residual quantization to better learn both global temporal structures and fine\-grained temporal details\. Its Frequency\-Band Autoregressive Transformer models cross\-band correlations through next frequency\-band prediction, enabling low\-to\-high\-frequency generation with parallel decoding within each band\. Experiments across 12 datasets demonstrate state\-of\-the\-art overall generation performance and improved inference efficiency, while ablation studies support the effectiveness of the core components\. These findings highlight the potential of frequency\-hierarchical representations and effective autoregressive modeling for high\-quality and efficient time series generation\.

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## Appendix ADataset Description

We provide detailed descriptions of the 12 datasets used for model training and in\-domain generation evaluation\.

Electricity\. This dataset contains hourly electricity consumption measurements, expressed in kilowatts \(kW\), from 321 clients between 2012 and 2014\. It is sourced from the UCI Machine Learning Repository\.

Solar\. This dataset contains 137 time series of hourly solar power production in Alabama, USA, during 2006\.

Wind\. This dataset comprises a single long time series of wind power production, measured in megawatts \(MW\) at 4\-second intervals starting on August 1, 2019\. It is sourced from the Australian Energy Market Operator \(AEMO\) online platform\.

Traffic\. This dataset contains 440 daily records spanning 15 months\. It records freeway lane occupancy rates in the San Francisco Bay Area, with values ranging from 0 to 1\.

Taxi\. This dataset contains spatiotemporal records of New York City taxi rides across 1,214 locations, collected at 30\-minute intervals during January 2015 and January 2016\.

Pedestrian\. This dataset contains hourly pedestrian counts from 66 sensors across Melbourne, Australia, starting in May 2009\. Although the source dataset is continuously updated, we use the version containing observations through April 30, 2020\.

Air Quality\. Originally used in the KDD Cup 2018 forecasting competition, this dataset contains hourly air\-quality measurements from 59 monitoring stations: 35 in Beijing and 24 in London\. The observations span January 1, 2017 to March 31, 2018, and cover PM2\.5, PM10, NO2, CO, O3, and SO2, yielding 270 time series indexed by city, station, and pollutant\. Leading missing values are filled with zeros, and subsequent missing values are imputed using the last observation carried forward \(LOCF\) strategy\.

Temperature\. This dataset contains 32,072 daily records of temperature observations and rainfall forecasts collected by the Australian Bureau of Meteorology from 422 weather stations across Australia between May 2, 2015 and April 26, 2017\. Missing values are replaced with zeros\. We use the mean temperature variable in our experiments\.

Rain\. This dataset shares the same source as Temperature\. We use the rainfall variable to construct the time series\.

NN5\. Originally used in the NN5 forecasting competition, this dataset contains 111 time series of daily cash withdrawals from automated teller machines \(ATMs\) in the United Kingdom\. Missing values are imputed using the median of observations falling on the same day of the week within each series\.

Fred\-MD\. This dataset contains 107 monthly macroeconomic time series obtained from the Federal Reserve’s FRED\-MD database\. Following common practice, we apply differencing and logarithmic transformations as appropriate for each series\.

Exchange\. This dataset contains daily exchange\-rate time series for eight currencies\.

We additionally evaluate unseen\-domain generation on the following two datasets, which are excluded from model training\.

Stock\. This dataset contains daily stock prices for Google \(GOOG\), listed on the NASDAQ stock exchange\.

Web\. Originally used in the Kaggle Wikipedia Web Traffic Forecasting competition, this dataset contains 145,063 daily time series of Wikipedia page views spanning July 1, 2015 to September 10, 2017\. Missing values are replaced with zeros\.

## Appendix BBaseline Details

We compare HALO with the following time series generation baselines, covering adversarial, variational, diffusion\-based, and discrete\-token generative approaches\. Table[3](https://arxiv.org/html/2610.27000#A2.T3)summarizes the code repositories of the baseline methods\.

Table 3\.Code repositories of the baseline methods\. TimeVQVAE\-C shares the implementation of TimeVQVAE\.#### TimeMAR\([Xu et al\., 2026](https://arxiv.org/html/2610.27000#bib.bib18)\)\.

TimeMAR is a two\-stage generative framework that combines multi\-scale discrete representations with autoregressive modeling\. It employs a dual\-path VQ\-VAE to encode decomposed trend and seasonal components into discrete tokens at multiple temporal resolutions, followed by a Transformer that generates these tokens in a coarse\-to\-fine manner\. Coarse seasonal representations further guide the reconstruction of fine\-grained seasonal patterns\. Following the experimental protocol of\([Huang et al\., 2025](https://arxiv.org/html/2610.27000#bib.bib34)\), we adapt it to use domain labels as prompts\.

#### TimeDP\([Huang et al\., 2025](https://arxiv.org/html/2610.27000#bib.bib34)\)\.

TimeDP is a diffusion\-based framework for multi\-domain time series generation\. It learns a shared set of semantic prototypes to represent temporal patterns and employs a prototype assignment module to construct domain prompts from input time series\. These prompts condition the diffusion model, enabling domain\-specific generation using a small set of examples from the target domain\.

#### TimeGAN\([Yoon et al\., 2019](https://arxiv.org/html/2610.27000#bib.bib24)\)\.

TimeGAN combines adversarial learning with a supervised temporal objective in a learned embedding space\. Its embedding and recovery networks provide latent representations for generation, while the supervised objective encourages the model to preserve temporal dependencies between consecutive latent states\.

#### GT\-GAN\([Jeon et al\., 2022](https://arxiv.org/html/2610.27000#bib.bib27)\)\.

GT\-GAN is an adversarial framework designed to synthesize both regularly and irregularly sampled time series\. It integrates neural ordinary and controlled differential equations with continuous\-time flow processes to capture temporal dynamics under different sampling patterns\.

#### TimeVAE\([Desai et al\., 2021](https://arxiv.org/html/2610.27000#bib.bib25)\)\.

TimeVAE is a variational autoencoder for multivariate time series generation\. It combines continuous latent representations with an interpretable decoder that explicitly models level, trend, and seasonal components, allowing generated sequences to reflect these temporal structures\.

#### TimeVQVAE\([Lee et al\., 2023](https://arxiv.org/html/2610.27000#bib.bib28)\)\.

TimeVQVAE adopts a two\-stage framework based on vector quantization and bidirectional prior modeling\. It separates time\-frequency representations into low\- and high\-frequency components and encodes them into discrete tokens\. Bidirectional Transformer priors learn the token distributions through masked modeling, enabling generation via iterative masked decoding\.

#### TimeVQVAE\-C\([Lee et al\., 2023](https://arxiv.org/html/2610.27000#bib.bib28);[Huang et al\., 2025](https://arxiv.org/html/2610.27000#bib.bib34)\)\.

TimeVQVAE\-C denotes the class\-conditional variant of TimeVQVAE\. It retains the time\-frequency tokenization framework while conditioning the bidirectional token prior on class labels, enabling the generation of time series associated with a specified class\.

## Appendix CMore Implementation Details

We provide detailed hyperparameter settings for ORBIT\. For the first\-stage FH\-VQ\-VAE, we report the number and dimensionality of encoder\-produced continuous latent representations, the frequency\-band partitioning scheme, the token budget for each frequency band, and the codebook size\. Table[4](https://arxiv.org/html/2610.27000#A3.T4)presents the detailed hyperparameter settings\. For the second\-stage FBAR, we report its backbone width, number of Transformer layers, number of attention heads, and hidden dimensionality of the feed\-forward network \(FFN\)\. Detailed hyperparameter settings are presented in Table[5](https://arxiv.org/html/2610.27000#A3.T5)\. Both stages are optimized using the AdamW optimizer\([Kingma and Ba, 2014](https://arxiv.org/html/2610.27000#bib.bib32)\)with batch size 256\.

Table 4\.Detailed hyperparameter settings of FH\-VQ\-VAE\.MM,dd,BB,nbn\_\{b\}, andKKdenote the number of continuous latents produced by the encoder, the hidden dimensionality, the number of frequency bands, the number of quantization tokens allocated to thebb\-th frequency band, and the total number of codewords in the codebook, respectively\.Table 5\.Detailed hyperparameter settings of FBAR\.DD,L​a​y​e​rLayer,H​e​a​dHead, andF​N​NFNNdenote the backbone width, number of Transformer layers, number of attention heads, and hidden dimensionality of the feed\-forward network, respectively\.
## Appendix DFurther Visualizations

Figure 7\.t\-SNE visualization of real and synthetic time series samples on the Traffic, Taxi, Pedestrian, and Fred\-MD benchmarks with a sequence length of 24\.Figure 8\.t\-SNE visualization of real and synthetic time series samples on the Traffic, Taxi, Pedestrian, and Fred\-MD benchmarks with a sequence length of 96\.Figure 9\.t\-SNE visualization of real and synthetic time series samples on the Traffic, Taxi, Pedestrian, and Fred\-MD benchmarks with a sequence length of 168\.Figures[7](https://arxiv.org/html/2610.27000#A4.F7)\-[9](https://arxiv.org/html/2610.27000#A4.F9)present additional visualizations of generated time series on other benchmarks with sequence lengths of 24, 91, and 168\. As observed, real and synthetic samples exhibit substantial overlap in their primary distribution regions, with no pronounced global shifts\. Notably, several peripheral clusters, in addition to the central region, contain both real and synthetic samples, suggesting that ORBIT captures diverse sample patterns rather than merely reproducing dominant distribution modes\.

## Appendix EAdditional Experimental Results

In this section, we provide additional experimental results covering sampling temperature sensitivity and a case study on next frequency\-band prediction\.

Sensitivity to Sampling Temperature\.As shown in Figures[10](https://arxiv.org/html/2610.27000#A5.F10)and[11](https://arxiv.org/html/2610.27000#A5.F11), increasing the temperature generally improves distribution matching, although the optimal setting varies across datasets and metrics\. The lowest mean MMD occurs at temperatures between 0\.9 and 1\.0 on six of the eight datasets, while the lowest mean K\-L occurs in this range on seven datasets\. Solar and Temperature exhibit consistent improvements in both metrics as the temperature increases\. However, Wind and Fred\-MD achieve their lowest MMD at temperatures of 0\.8 and 0\.7, respectively, whereas their K\-L is minimized at 1\.0\. Taxi exhibits a distinct trade\-off: its MMD decreases with increasing temperature, while its K\-L reaches its minimum near 0\.7 before increasing\. These results support 0\.9\-1\.0 as a favorable operating range for most evaluated datasets, while demonstrating that no single temperature uniformly optimizes both metrics\. Notably, ORBIT achieves state\-of\-the\-art overall generation performance without requiring careful tuning of the sampling temperature, which further highlights the robustness and effectiveness of our framework\.

Figure 10\.Sensitivity analysis of MMD to sampling temperature across eight benchmark datasets\.Figure 11\.Sensitivity analysis of K\-L to sampling temperature across eight benchmark datasets\.Case Study: Next Frequency\-Band Prediction\.To provide a more intuitive understanding of the next frequency\-band prediction process, we conduct a case study\. Figure[12](https://arxiv.org/html/2610.27000#A5.F12)illustrates the generation process on the Electricity and Taxi benchmarks\. We observe that ORBIT first establishes a coarse\-grained global temporal structure at the initial frequency level\. As the generation progressively proceeds toward higher frequency bands, increasingly fine\-grained local temporal details are incrementally incorporated into the generated sequence\.

Figure 12\.Autoregressive generation process guided by next frequency\-band prediction on the Electricity and Taxi benchmarks with a sequence length of 24\.