Compositional Spectral Prompts for LLM-based Online Time Series Forecasting

# Compositional Spectral Prompts for LLM-based Online Time Series Forecasting
Source: [https://arxiv.org/html/2609.02093](https://arxiv.org/html/2609.02093)
Conference:Proceedings of the 35th ACM International Conference on Information and Knowledge Management; November 07–11, 2026; Rome, ItalyProceedings of the 35th ACM International Conference on Information and Knowledge Management \(CIKM ’26\), November 07–11, 2026, Rome, ItalyDOI:[10\.1145/3799682\.3841092](https://doi.org/10.1145/3799682.3841092)ISBN:979\-8\-4007\-2539\-5/2026/11CCS:Computing methodologies Machine learning algorithms,Hyunchul KimAffiliation:KAIST,Daejeon,Republic of Koreaemail:[khchul@kaist\.ac\.kr](mailto:[email protected]),Jae\-Gil LeeAffiliation:KAIST,Daejeon,Republic of Koreaemail:[jaegil@kaist\.ac\.kr](mailto:[email protected])andChanyoung ParkAffiliation:KAIST,Daejeon,Republic of Koreaemail:[cy\.park@kaist\.ac\.kr](mailto:[email protected])

© cc

###### Abstract\.

To address the sequential and evolving nature of time series, the Online Time Series Forecasting \(OTSF\) task has been extensively studied in multiple domains\. Existing research focuses on adapting to non\-stationary environments by employing memory buffer\-based retrieval strategies\. However, we observe that such frameworks struggle with long\-term adaptation and fail to generalize to unseen patterns\. To this end, we introduceCoSPOT, an LLM\-based online time series forecasting framework that leverages a pre\-trained LLM as the backbone online forecaster, motivated by its strong few\-shot capabilities\. For efficient online adaptation,CoSPOTkeeps the LLM frozen and employs compositional spectral prompts grounded in frequency\-domain bases to guide the model with the overall distribution of the input, thereby substantially reducing the number of parameters updated during the online phase\. Specifically,CoSPOTdecomposes time series into frequency bases and composes the corresponding spectral basis prompts according to their amplitudes, allowing unseen patterns to be represented as new combinations of learned basis prompts\. Our extensive experiments on real\-world datasets demonstrate the superiority and practicality ofCoSPOTacross challenging online scenarios, including extended online phases and cross\-dataset settings with substantial distribution shifts\. Our code is available at[https://github\.com/seungyoon\-Choi/CoSPOT](https://github.com/seungyoon-Choi/CoSPOT)\.

###### Keywords:

Time Series Forecasting, Online Learning, Prompt Learning, Large Language Models

††cc\-license:by## 1\.Introduction

Early research in deep learning\-based time series forecasting\([Nie et al\., 2022](https://arxiv.org/html/2609.02093#bib.bib10);[Zhang and Yan, 2023](https://arxiv.org/html/2609.02093#bib.bib11);[Zhou et al\., 2022b](https://arxiv.org/html/2609.02093#bib.bib12);[Zhou et al\., 2021](https://arxiv.org/html/2609.02093#bib.bib20);[Wu et al\., 2021](https://arxiv.org/html/2609.02093#bib.bib21);[Wu et al\., 2022](https://arxiv.org/html/2609.02093#bib.bib17)\)primarily focused on batch learning methods utilizing static training and evaluation datasets\. However, given the sequential and evolving nature of time series data, shifts in underlying patterns over time are inevitable\. Consequently, traditional batch learning approaches often fail to adapt to such changes, while frequent model retraining to accommodate new patterns is both labor\-intensive and impractical for real\-world applications\. To address these challenges, online learning paradigms, which enable models to incrementally update as new data arrive in dynamic environments, have been increasingly explored\.

![Refer to caption](https://arxiv.org/html/2609.02093v1/motivation_v4.png)Figure 1\.\(a\) Performance of prior methods and our proposed method \(i\.e\.,CoSPOT\) under an extended online phase\. \(b\) Performance of prior methods andCoSPOTin a cross\-dataset scenario with distribution shifts in pattern\. \(c & d\) Performance of DSOF and LLM\-based time series forecasting methods under an extended online phase and cross\-dataset scenario\. Note that the ETTh2 dataset is used for \(a\) and \(c\)\.Existing studies on online time series forecasting \(OTSF\)\([Pham et al\., 2022](https://arxiv.org/html/2609.02093#bib.bib1);[Wen et al\., 2023](https://arxiv.org/html/2609.02093#bib.bib2);[Lau et al\., 2025](https://arxiv.org/html/2609.02093#bib.bib3)\)have focused on effective adaptation to evolving data streams\. The pioneering work, FSNet\([Pham et al\., 2022](https://arxiv.org/html/2609.02093#bib.bib1)\), addressed rapid adaptation and pattern reuse by employing lightweight per\-layer adapters and an associative memory for pattern retrieval\. Building on this, OneNet\([Wen et al\., 2023](https://arxiv.org/html/2609.02093#bib.bib2)\)introduced reinforcement learning to dynamically ensemble models based on their real\-time performance\. Furthermore, DSOF\([Lau et al\., 2025](https://arxiv.org/html/2609.02093#bib.bib3)\)identified the information leakage issue in prior setups and proposed a dual\-stream \(i\.e\., teacher\-student\) residual framework to handle delayed adaptation\. These methods share a strategy of employing memory buffer\-based retrieval to adapt to non\-stationary environments\.

Despite the recent advancements, the buffer\-based design of existing methods faces two fundamental challenges in practical OTSF scenarios\.\(1\) Inadaptability to extended online phases\.As new data is continuously streamed in online scenarios, models require sufficient capacity for long\-term adaptation to effectively learn from evolving patterns\. As shown in Figure[1](https://arxiv.org/html/2609.02093#S1.F1)\(a\), increasing the proportion of the online phase \(i\.e\., test data\) leads to significant performance deterioration in previous approaches111To create a more challenging and extended online phase within the given dataset, we deviate from the typical 20%/5%/75% split used in prior studies and instead adopt a 10%/5%/85% train/validation/test split\.\. This degradation stems from the inherent capacity constraints of memory buffers: as the online period lengthens, the increasing diversity of recurring patterns makes it difficult to explicitly retain them all in the associative memory, leading to catastrophic forgetting\.\(2\) Inadaptability to unseen patterns\.In online forecasting scenarios, previously unseen patterns \(i\.e\., distribution shifts in pattern\) may emerge during test time \(i\.e\., online phase\)\. Thus, models must possess adaptability to such shifts for effective adjustments\. Figure[1](https://arxiv.org/html/2609.02093#S1.F1)\(b\) shows experimental results obtained under a setting where the time series patterns in the training and online phases are intentionally made different\. Specifically, we used two datasets from the same domain \(i\.e\., ETTh1 and ETTh2\) and conducted experiments under two scenarios: One where model training and online updates are both performed on ETTh2 \(in blue\), and the other where models are initially trained on ETTh1 dataset, then updated online as ETTh2 data streams in \(in red\)\. We observe that existing OTSF methods degrade substantially on patterns unseen during training, as they rely on associative memory to retrieve patterns similar to the new input\. In other words, when an unprecedented pattern appears, retrieval from previously learned patterns becomes unreliable, leading to failure in adaptation under distribution shifts\.

To address the challenges of extended online phases and adaptability to unseen patterns, we proposeCoSPOT, a Large Language Model \(LLM\)\-based online time series forecasting framework built upon compositional spectral prompting\. Our design is motivated by the observation that integrating a pre\-trained LLM is pivotal for maintaining adaptability over extended online phases\. Recent studies\([Jin et al\., 2023](https://arxiv.org/html/2609.02093#bib.bib6);[Chang et al\., 2023](https://arxiv.org/html/2609.02093#bib.bib4);[Zhou et al\., 2023](https://arxiv.org/html/2609.02093#bib.bib5)\)have shown that LLMs can significantly enhance forecasting performance when aligned with time series tasks, particularly infew\-shotsetting\. Such transferability of LLMs is especially advantageous in OTSF, as it enables effective adaptation under limited data\. In Figure[1](https://arxiv.org/html/2609.02093#S1.F1)\(c\), we compare the state\-of\-the\-art OTSF model that relies solely on a time series model \(i\.e, DSOF\) against those that align the time series backbone with LLMs \(i\.e\., TimeLLM\([Jin et al\., 2023](https://arxiv.org/html/2609.02093#bib.bib6)\), LLM4TS\([Chang et al\., 2023](https://arxiv.org/html/2609.02093#bib.bib4)\), and GPT4TS\([Zhou et al\., 2023](https://arxiv.org/html/2609.02093#bib.bib5)\)\) under an extended online phase\. Surprisingly, we found that although these LLM\-based models were not originally designed for online learning222We adhere to their original training protocols and fine\-tune only the output projection in the online phase\., they demonstrate performance comparable to DSOF as the online phase lengthens\. However, despite their effectiveness in extended online phases, LLM\-based models struggle to generalize effectively to pattern shifts \(Figure[1](https://arxiv.org/html/2609.02093#S1.F1)\(d\)\), limiting their applicability in OTSF\. This deficiency underscores the need for continuous adaptation of the model’s representational capacity\. However, updating the LLM itself is impractical for online scenarios, as the large number of parameters hinders rapid adaptation\. To address this, we propose a novel prompting mechanism that alleviates this burden by keeping the LLM frozen, while leveraging a trainable prompt to guide the model’s response to evolving patterns\. To further enhance the model’s generalizability,CoSPOTgrounds its prompts in frequency\-domain representations, which we refer to as spectral prompts\. This distinguishes our approach from prior OTSF methods that rely on time\-domain representations, as frequency features are superior at capturing underlying periodic structures\([Zhou et al\., 2022b](https://arxiv.org/html/2609.02093#bib.bib12);[Yi et al\., 2023](https://arxiv.org/html/2609.02093#bib.bib13)\)\. Specifically, we decompose the input time series into frequency components using the Discrete Fourier Transform \(DFT\)\. Each universal frequency basis, which serves as a fundamental building block of time series patterns, is associated with a learnable spectral basis prompt\. These spectral basis prompts are then compositionally aggregated according to the amplitudes of the corresponding frequency components, forming an input\-specific compositional spectral prompt fed to the LLM\. This design effectively transforms the adaptation problem: instead of memorizing infinitely many specific patterns, the model represents any unseen pattern as a novel composition of spectral basis prompts that are pre\-learned and frozen during the online phase\. Consequently, our framework achieves two key advantages: \(i\)Generalization: Newly emerged patterns are represented by recombining spectral basis prompts associated with frequency bases, enabling robust adaptation to distribution shifts\. \(ii\)Efficiency: Since the spectral prompt bank and the LLM are frozen during the online phase,CoSPOTachieves inductive adaptation without the computational overhead of large\-scale parameter updates\.

In this study, we make the following contributions:

- •We identify that existing memory buffer\-based OTSF methods struggle with long\-term adaptation and fail to generalize to unseen patterns\.
- •We proposeCoSPOT, the first approach to integrate LLMs into OTSF\.CoSPOTemploys compositional spectral prompting, which constructs input\-specific prompts by composing spectral basis prompts grounded in frequency\-domain bases\.
- •Through extensive experiments under various online learning scenarios, we demonstrate thatCoSPOTconsistently outperforms state\-of\-the\-art OTSF methods with extremely few online parameter updates\.

## 2\.Related Works

Online Time Series Forecasting\.Given the evolving nature of time series data, online forecasting has gained prominence for practical applications\([Kuznetsov and Mohri, 2016](https://arxiv.org/html/2609.02093#bib.bib9);[Gultekin and Paisley, 2018](https://arxiv.org/html/2609.02093#bib.bib7);[Aydore et al\., 2019](https://arxiv.org/html/2609.02093#bib.bib8)\)\. Recently, online deep learning models have been proposed to further capture complex patterns within time series data\. FSNet\([Pham et al\., 2022](https://arxiv.org/html/2609.02093#bib.bib1)\)introduces calibration module to dynamically balance fast adaptation to recent changes with the retention of prior knowledge\. OneNet\([Wen et al\., 2023](https://arxiv.org/html/2609.02093#bib.bib2)\)incorporates reinforcement learning to model cross\-variable and cross\-time concept drifts\. Addressing the information leakage issue in previous research, DSOF\([Lau et al\., 2025](https://arxiv.org/html/2609.02093#bib.bib3)\)redefines the OTSF setting and proposes a dual\-stream mechanism to update model parameters\. Nevertheless, prior studies do not explicitly model the patterns in input signals, hindering their ability to adapt to unobserved distributions\. Additionally, their explicit storage of pattern information restricts the model’s adaptability to extended online phases\.

Time Series Forecasting with LLMs\.Recent advancements in LLMs have prompted researchers to investigate their transferability to forecasting tasks in data\-sparse time series domains\. LLM4TS\([Chang et al\., 2023](https://arxiv.org/html/2609.02093#bib.bib4)\)introduces two\-stage fine\-tuning approach to leverage LLMs for time series forecasting\. GPT4TS\([Zhou et al\., 2023](https://arxiv.org/html/2609.02093#bib.bib5)\)retrains the positional embeddings and normalization layers of LLMs to preserve pre\-trained knowledge while enhancing performance on downstream tasks\. Additionally, TimeLLM\([Jin et al\., 2023](https://arxiv.org/html/2609.02093#bib.bib6)\)employs reprogramming method to align time series data with word embeddings\. Inspired by the proven adaptability of these models, we leverage LLMs to enable rapid adjustments in online scenarios\.

Prompt\-based Continual Learning\.Rehearsal\-free continual learning methods leverage the strong general representations of pre\-trained models like ViT\([Dosovitskiy et al\., 2020](https://arxiv.org/html/2609.02093#bib.bib22)\)\. By fine\-tuning only small, learnablepromptsfor each task, these methods achieve significant memory and computational efficiency, as the core model parameters remain unchanged\. VPT\([Jia et al\., 2022](https://arxiv.org/html/2609.02093#bib.bib23)\)optimizes a single prompt, L2P\([Wang et al\., 2022b](https://arxiv.org/html/2609.02093#bib.bib24)\)uses a shared pool of prompts\. S\-Prompts\([Wang et al\., 2022a](https://arxiv.org/html/2609.02093#bib.bib25)\), train a unique prompt for each individual task to address catastrophic forgetting\. However, the application of prompt learning to address distribution shifts in the time series domain remains unexplored\. Given the continuous nature of time series data, the online learning scenario is more suitable than continual learning, which assumes distinct tasks\.CoSPOTis the first to achieve an efficient and scalable prompting crucial for online learning scenarios by encoding knowledge from the underlying frequency bases\.

Frequency Analysis in Time Series Forecasting\.Due to the complex temporal variations in time series data, frequency analysis techniques such as the Discrete Fourier Transform \(DFT\) and Discrete Wavelet Transform \(DWT\) are widely used to capture recurring patterns\. DFT analyzes global frequency components, while DWT provides localized frequency information at different scales\. FEDformer\([Zhou et al\., 2022b](https://arxiv.org/html/2609.02093#bib.bib12)\)is a representative frequency\-domain forecasting model that leverages DFT and DWT through Fourier Enhanced Structure and Wavelet Enhanced Structure, respectively\. However, such frequency\-domain approaches mainly exploit frequency coefficients without learning prompt\-level knowledge associated with each frequency basis, limiting robustness to unseen patterns under distribution shifts\. In contrast,CoSPOTgrounds learnable spectral basis prompts in frequency\-domain bases and composes them according to the input spectrum, allowing unobserved patterns to be represented as new combinations of learned prompts and improving adaptability in online scenarios\.

## 3\.Preliminaries

### 3\.1\.Time Series Forecasting

Let𝐗=\(x1,…,xNd​a​t​a\)∈ℝNd​a​t​a×n\\mathbf\{X\}=\(x\_\{1\},\\ldots,x\_\{N\_\{data\}\}\)\\in\\mathbb\{R\}^\{N\_\{data\}\\times n\}be the entire time series withNd​a​t​aN\_\{data\}observations, where each observationxi∈ℝnx\_\{i\}\\in\\mathbb\{R\}^\{n\}containsnndimensions\. The dataset𝐗\\mathbf\{X\}is then partitioned intoNt​r​a​i​nN\_\{train\},Nv​a​lN\_\{val\}, and,No​n​l​i​n​eN\_\{online\}time stamps according to predefined ratios, maintaining the chronological order of the data\. Given the look\-back window of lengthLL, denoted as𝐗i−L\+1:i=\(xi−L\+1,xi−L\+2,…,xi\)\\mathbf\{X\}\_\{i\-L\+1:i\}=\(x\_\{i\-L\+1\},x\_\{i\-L\+2\},\\ldots,x\_\{i\}\), the objective of time series forecasting is to predict the followingHHsteps \(i\.e\.,𝐗i\+1:i\+H\\mathbf\{X\}\_\{i\+1:i\+H\}\), where the model’s prediction att=it=ifor the nextHHsteps is denoted by𝐗^i\+1:i\+H=\(x^i\+1,x^i\+2,…,x^i\+H\)=f\(𝐗i−L\+1:i\)\\hat\{\\mathbf\{X\}\}\_\{i\+1:i\+H\}=\(\\hat\{x\}\_\{i\+1\},\\hat\{x\}\_\{i\+2\},\\ldots,\\hat\{x\}\_\{i\+H\}\)=f\(\\mathbf\{X\}\_\{i\-L\+1:i\}\)\. The objective is to minimize the mean squared error \(MSE\) between the ground truth and the predicted outputs, i\.e\.,1H​Σh=1H​‖x^i\+h−xi\+h‖22\\frac\{1\}\{H\}\\Sigma\_\{h=1\}^\{H\}\|\|\\hat\{x\}\_\{i\+h\}\-x\_\{i\+h\}\|\|\_\{2\}^\{2\}\.

### 3\.2\.Online Time Series Forecasting

The OTSF scenario consists of two phases: training phase and online phase\. In the training phase, the entireNt​r​a​i​nN\_\{train\}time series are utilized to create\(L\+H\)\(L\+H\)sized time sequences\. The objective of the training phase is to let the model to learn the base knowledge through static batch training strategy\. The online phase follows the training phase, whereNo​n​l​i​n​eN\_\{online\}time steps are streamed sequentially with a moving window of size 1\. This mirrors real\-world scenarios, and the model is updated in real\-time\.

Objective and Evaluation Criterion\.Our ultimate goal is to accurately predict the ground truth by minimizing the cumulative MSE between the ground truth and predicted values over the entire prediction horizon ofHHsteps, usingMSEo​n​l​i​n​e\\textup\{MSE\}\_\{online\}to evaluate performance as follows:

\(1\)MSEonline\\displaystyle\\text\{MSE\}\_\{\\text\{online\}\}=1Nonline−L−H\+1\\displaystyle=\\frac\{1\}\{N\_\{\\text\{online\}\}\-L\-H\+1\}∑i=Ntrain\+Nval\+LNdata−H‖f\(𝐗i−L\+1:i\)−𝐗i\+1:i\+H‖22\.\\displaystyle\\sum\_\{i=N\_\{\\text\{train\}\}\+N\_\{\\text\{val\}\}\+L\}^\{N\_\{\\text\{data\}\}\-H\}\\left\\\|f\(\\mathbf\{X\}\_\{i\-L\+1:i\}\)\-\\mathbf\{X\}\_\{i\+1:i\+H\}\\right\\\|\_\{2\}^\{2\}\.

## 4\.Proposed Method:CoSPOT

In this section, we introduce our proposed methodCoSPOT\. The key components of this framework are as follows: \(1\) leveraging a pre\-trained LLM together with a time series backbone and textual recent information to enhance adaptability in data\-scarce online scenarios \(Sec\.[4\.1](https://arxiv.org/html/2609.02093#S4.SS1)\), and \(2\) compositional spectral prompting, which composes frequency\-grounded spectral basis prompts to robustly and efficiently adapt to pattern shifts \(Sec\.[4\.2](https://arxiv.org/html/2609.02093#S4.SS2)\)\. Overall framework ofCoSPOTis shown in Figure[2](https://arxiv.org/html/2609.02093#S4.F2)\.

### 4\.1\.Enhancing model adaptability using a pre\-trained LLM

To enable effective and rapid adaptation in data\-scarce online scenarios,CoSPOTleverages a pre\-trained LLM together with a time\-series backbone\. The LLM remains frozen throughout the training and online phases, serving as a stable knowledge source, while the time\-series backbone encodes the input sequence\. This design exploits the LLM’s transferability to support adaptation when only limited online observations are available\. However, online forecasting also requires contextual information about recent dynamics, which may not be sufficiently captured from numerical inputs alone\. To this end, we introduce a text description as an additional modality, allowing recent contextual cues to be provided to the LLM in natural language\. Given the input time series𝐗\\mathbf\{X\}, the text description \(i\.e\.,t​e​x​t𝐗text\_\{\\mathbf\{X\}\}\) contains task details, dataset information, recent values, and recent frequency information\. This text description provides short\-term contextual cues, complementing the compositional spectral prompt in Section[4\.2](https://arxiv.org/html/2609.02093#S4.SS2), which captures the overall distribution of the input\. The recent values summarize the latest time\-domain variation, while the recent frequency information captures local frequency changes near the current time point\. Although Short\-Time Fourier Transform \(STFT\) can provide localized frequency information, its fixed window size limits its ability to capture non\-stationary recent dynamics due to the trade\-off between time and frequency resolution\. Hence, we use the Discrete Wavelet Transform \(DWT\), whose adaptive time\-frequency resolution is better suited for capturing recent variations in non\-stationary signals\. Given the input time series𝐗\\mathbf\{X\}, the equation of DWT using a scaling functionϕ\\phiand a wavelet functionψ\\psiis as follows:

\(2\)𝐀j​\[k\]=∑n𝐗⁡\[n\]​ϕj,k​\[n\],𝐃j​\[k\]=∑n𝐗⁡\[n\]​ψj,k​\[n\],\\small\\mathbf\{A\}\_\{j\}\[k\]=\\sum\_\{n\}\\mathbf\{X\}\[n\]\\phi\_\{j,k\}\[n\],\\hskip 9\.24994pt\\mathbf\{D\}\_\{j\}\[k\]=\\sum\_\{n\}\\mathbf\{X\}\[n\]\\psi\_\{j,k\}\[n\],\\vskip\-9\.68745ptwhere𝐗⁡\[n\]\\mathbf\{X\}\[n\]refers to thenn\-th index in the time series𝐗\\mathbf\{X\},𝐀j​\[k\]​and​𝐃j​\[k\]\\mathbf\{A\}\_\{j\}\[k\]\\,\\text\{and\}\\,\\mathbf\{D\}\_\{j\}\[k\]refer to the approximation and detail coefficient at leveljj, respectively, andϕj,k​\[n\]​and​ψj,k​\[n\]\\phi\_\{j,k\}\[n\]\\,\\text\{and\}\\,\\psi\_\{j,k\}\[n\]are the scaling and wavelet functions at leveljj, respectively\. After the decomposition, the time series is passed through a filter bank that separates the low\-pass and high\-pass components, and downsampling is performed as follows:

\(3\)𝐀j\+1​\[k\]=∑nh⁡\[n−2​k\]​𝐀j​\[n\],𝐃j\+1​\[k\]=∑ng⁡\[n−2​k\]​𝐀j​\[n\],\\small\\mathbf\{A\}\_\{j\+1\}\[k\]=\\sum\_\{n\}h\[n\-2k\]\\mathbf\{A\}\_\{j\}\[n\],\\hskip 9\.24994pt\\mathbf\{D\}\_\{j\+1\}\[k\]=\\sum\_\{n\}g\[n\-2k\]\\mathbf\{A\}\_\{j\}\[n\],whereh⁡\[n\]​and​g​\[n\]h\[n\]\\,\\text\{and\}\\,g\[n\]refer the low\- and high\-pass filters, respectively\. Through this process, we utilize𝐀j​\[−1\]\\mathbf\{A\}\_\{j\}\[\-1\]to provide the model with the most recent frequency information wherejjis a hyperparameter\.

An example of the text description is shown in Figure[2](https://arxiv.org/html/2609.02093#S4.F2)\. The text description is first processed through the pre\-trained LLM’s tokenizer\. The resulting token IDs are then passed through the frozen LLM’s input embedding layer to retrieve their corresponding token embeddings\. This sequence of text token embeddings is the resulting embedding \(denoted as𝒯𝐗\\mathcal\{T\}\_\{\\mathbf\{X\}\}\) that is used as input for the final prediction\. Providing text descriptions enriches data in data\-scarce online scenarios, offering recent information that aids effective adaptation, all without requiring additional training\.

![Refer to caption](https://arxiv.org/html/2609.02093v1/architecture.png)Figure 2\.Overall model framework\. Given the input time series𝐗i−L\+1:i\\mathbf\{X\}\_\{i\-L\+1:i\}, the aligned embedding \(i\.e\.,emb𝐗i−L\+1:iemb\_\{\\mathbf\{X\}\_\{i\-L\+1:i\}\}\), embedded text description \(i\.e\.,𝒯𝐗i−L\+1:i\\mathcal\{T\}\_\{\\mathbf\{X\}\_\{i\-L\+1:i\}\}\), and compositional spectral prompt \(i\.e\.,𝒫𝐗i−L\+1:i\\mathcal\{P\}\_\{\\mathbf\{X\}\_\{i\-L\+1:i\}\}\) are provided as input to the LLM\. The representation computed by the LLM is passed through the output projection layer to produce the final prediction, i\.e\.,𝐗^i\+1:i\+H\\mathbf\{\\hat\{X\}\}\_\{i\+1:i\+H\}\.
### 4\.2\.Robust Adaptation via Compositional Spectral Prompting

While the frozen LLM integrated with text descriptions provides a strong foundation for adaptability, it still faces challenges when encountering completely unseen patterns that deviate significantly from the patterns learned during the training phase\. However, updating the LLM itself is impractical for online scenarios, as the large number of parameters hinders rapid adaptation\. Therefore, we need an efficient mechanism to provide the frozen model with inductive bias about these evolving structures\. To address this, we introducecompositional spectral prompting, which provides the frozen model with reliable guidance on the overall data distribution\. Here,spectralindicates that the prompts are grounded in the frequency\-domain structure of time series\. We leverage the frequency domain because, compared with the time domain, it more effectively isolates underlying pattern components that are often entangled in raw temporal signals, making it well suited for representing evolving time\-series patterns333Section[5\.3\.5](https://arxiv.org/html/2609.02093#S5.SS3.SSS5)elaborates on the comparative advantages of the frequency\-domain approach, highlighting its enhanced suitability for robust pattern representation compared to the time\-domain approach\.\. Instead of directly storing complex patterns, which renders prior memory\-based methods\([Pham et al\., 2022](https://arxiv.org/html/2609.02093#bib.bib1);[Wen et al\., 2023](https://arxiv.org/html/2609.02093#bib.bib2);[Lau et al\., 2025](https://arxiv.org/html/2609.02093#bib.bib3)\)unreliable when facing unseen patterns,CoSPOTdecomposes time series into frequency bases and represents each input as a composition of learnable spectral basis prompts\. This design allows newly emerging patterns to be expressed through new combinations of pre\-learned prompts, without updating the LLM during the online phase\. Specifically, we use the DFT to decompose the input time series𝐗\\mathbf\{X\}of lengthLL\. The DFT converts the sequence from the time domain to the frequency domain, while its inverse \(IDFT\) converts it back\. Their expressions are as follows:

\(4\)ℱ\(k\)=DFT\(𝐗\)=∑n=0L−1𝐗\[n\]exp\(−i2​π​k​nL\),k=0,1,…,L−1,\\small\\mathcal\{F\}\(k\)=DFT\(\\mathbf\{X\}\)=\\sum\_\{n=0\}^\{L\-1\}\\mathbf\{X\}\[n\]\\textup\{exp\}\\Big\(\-i\\frac\{2\\pi kn\}\{L\}\\Big\),\\hskip 9\.24994ptk=0,1,\\ldots,L\-1,\(5\)𝐗\[n\]=IDFT\(ℱ\)=1L∑k=0L−1ℱ\(k\)exp\(i2​π​k​nL\),n=0,1,…,L−1,\\small\\mathbf\{X\}\[n\]=IDFT\(\\mathcal\{F\}\)=\\frac\{1\}\{L\}\\sum\_\{k=0\}^\{L\-1\}\\mathcal\{F\}\(k\)\\textup\{exp\}\\Big\(i\\frac\{2\\pi kn\}\{L\}\\Big\),\\hskip 9\.24994ptn=0,1,\\ldots,L\-1,whereℱ\\mathcal\{F\}refers to the frequency spectrum of the input andiirepresents the imaginary unit\. From the perspective of frequency basis, assuming thatLLis even, both the DFT and IDFT can be represented usingL2\+1\\frac\{L\}\{2\}\+1orthogonal cosine\-based frequency basis because the DFT of a real\-valued signal exhibits Hermitian symmetry\. Thus, the IDFT can be rewritten as follows:

\(6\)𝐗⁡\[n\]\\displaystyle\\mathbf\{X\}\[n\]=1L​∑k=0L2\(𝐑k⋅cos⁡\(2​π​k​nL−ϕ\)\)\\displaystyle=\\frac\{1\}\{L\}\\sum\_\{k=0\}^\{\\frac\{L\}\{2\}\}\\Big\(\\mathbf\{R\}\_\{k\}\\cdot\\cos\\Big\(\\frac\{2\\pi kn\}\{L\}\-\\phi\\Big\)\\Big\)=1L∑k=0L2\(𝐑k⋅freqk\),n=0,1,…,L−1\.\\displaystyle=\\frac\{1\}\{L\}\\sum\_\{k=0\}^\{\\frac\{L\}\{2\}\}\\Big\(\\mathbf\{R\}\_\{k\}\\cdot\\text\{freq\}\_\{k\}\\Big\),\\hskip 9\.24994ptn=0,1,\\ldots,L\-1\.
wherefreqk\\text\{freq\}\_\{k\}and𝐑k∈ℝ\\mathbf\{R\}\_\{k\}\\in\\mathbb\{R\}denote the basis of thekk\-th frequency and its amplitude, respectively\.

Building on this formulation, which reconstructs the time series as a weighted combination of frequency bases, we introduce a spectral prompt bank that associates each frequency basis with a learnable spectral basis prompt\. Specifically, each spectral basis prompt encodes the characteristic temporal pattern corresponding to its frequency basis\. Let𝐏=\[𝐩0,…,𝐩L2\]∈ℝ\(L2\+1\)×d\\mathbf\{P\}=\[\\mathbf\{p\}\_\{0\},\\ldots,\\mathbf\{p\}\_\{\\frac\{L\}\{2\}\}\]\\in\\mathbb\{R\}^\{\(\\frac\{L\}\{2\}\+1\)\\times d\}denote the spectral prompt bank, where𝐩k\\mathbf\{p\}\_\{k\}represents the learnable spectral basis prompt corresponding to thekk\-th frequency basis\. However, learning knowledge for all frequency bases is not effective in capturing the overall pattern of the given time series\. That is, high frequencies represent rapidly oscillating periodicities compared to low frequencies, and therefore, they do not capture the overall pattern information\. Hence, in time series analysis, high frequencies are often treated as noise, which is why low\-pass filtering techniques\([Zhou et al\., 2022a](https://arxiv.org/html/2609.02093#bib.bib18);[Xu et al\., 2023](https://arxiv.org/html/2609.02093#bib.bib19)\)are widely used\. Accordingly, to effectively capture the overall pattern while removing noise, we introduce a hyperparameterγ∈\[0,1\]\\gamma\\in\[0,1\]to eliminate the high\-frequency bases, i\.e\.,𝐏l​o​w=\[𝐩0,…,𝐩⌈L2⋅γ⌉\]∈ℝ⌈\(L2⋅γ\+1\)⌉×d\\mathbf\{P\}\_\{low\}=\[\\mathbf\{p\}\_\{0\},\\ldots,\\mathbf\{p\}\_\{\\left\\lceil\\frac\{L\}\{2\}\\cdot\\gamma\\right\\rceil\}\]\\in\\mathbb\{R\}^\{\\left\\lceil\(\\frac\{L\}\{2\}\\cdot\\gamma\+1\)\\right\\rceil\\times d\}\. We then construct an input\-specific compositional spectral prompt by weighting each spectral basis prompt according to the amplitude of its corresponding frequency component, which provides the LLM with guidance on the overall distribution \(i\.e\., overall pattern\) of the input:

\(7\)𝒫𝐗=Concat​\(𝐑0⋅𝐩0,𝐑1⋅𝐩1,…,𝐑⌈L2⋅γ⌉⋅𝐩⌈L2⋅γ⌉\)∈ℝ⌈\(L2⋅γ\+1\)⌉×d,\\mathcal\{P\}\_\{\\mathbf\{X\}\}=\\textup\{Concat\}\\Big\(\\mathbf\{R\}\_\{0\}\\cdot\\mathbf\{p\}\_\{0\},\\mathbf\{R\}\_\{1\}\\cdot\\mathbf\{p\}\_\{1\},\\ldots,\\mathbf\{R\}\_\{\\left\\lceil\\frac\{L\}\{2\}\\cdot\\gamma\\right\\rceil\}\\cdot\\mathbf\{p\}\_\{\\left\\lceil\\frac\{L\}\{2\}\\cdot\\gamma\\right\\rceil\}\\Big\)\\in\\mathbb\{R\}^\{\\left\\lceil\(\\frac\{L\}\{2\}\\cdot\\gamma\+1\)\\right\\rceil\\times d\},\\vskip\-2\.15277ptwhere𝒫𝐗\\mathcal\{P\}\_\{\\mathbf\{X\}\}refers the compositional spectral prompt of the input time series𝐗\\mathbf\{X\}to be provided to the model\. Through compositional spectral prompting, the model explicitly captures pattern\-level information under continuous distribution shifts\. Even when previously unseen patterns emerge in the online phase, the model can effectively represent and adapt to them by compositionally recombining the knowledge encoded in spectral basis prompts during the training phase, without requiring any additional training\.444In Section[5\.3\.4](https://arxiv.org/html/2609.02093#S5.SS3.SSS4), we observe that the frequency basis\-driven prompt bank, trained only during the training phase, can effectively adapt to unseen patterns that emerge in the online phase\.Moreover, since newly emerging patterns can be expressed as compositions of a finite set of spectral basis prompts, the model maintains its memory efficiency without degradation\.

### 4\.3\.Overall Framework

Figure[2](https://arxiv.org/html/2609.02093#S4.F2)shows the overall framework, and the pseudo code can be found in Algorithm[1](https://arxiv.org/html/2609.02093#alg1)\.

Align Module\.In OTSF scenarios, effectively aligning continuous time series data with discrete token\-based LLMs is crucial yet challenging\. Since pre\-trained LLMs lack inherent knowledge of time series patterns, prior studies\([Jin et al\., 2023](https://arxiv.org/html/2609.02093#bib.bib6);[Chang et al\., 2023](https://arxiv.org/html/2609.02093#bib.bib4);[Zhou et al\., 2023](https://arxiv.org/html/2609.02093#bib.bib5)\)focus on aligning two modalities \(i\.e\., time series and language\) to leverage the knowledge within LLMs, enabling accurate, data\-efficient, and task\-agnostic forecasting\. As the goal of this study is to enhance the LLM’s ability in the online scenario, not to focus on the align module itself, we utilize a pre\-developed align module\([Jin et al\., 2023](https://arxiv.org/html/2609.02093#bib.bib6);[Chang et al\., 2023](https://arxiv.org/html/2609.02093#bib.bib4)\)\. The align module aligns the representation of the time series computed by the time series backbone \(i\.e\.,e​m​b𝐗=b⁡\(𝐗\)emb\_\{\\mathbf\{X\}\}=b\(\\mathbf\{X\}\), whereb⁡\(⋅\)b\(\\cdot\)is the time series backbone\) with the natural language modality, and outputs the resulting aligned embeddings \(i\.e\.,e​m​b𝐗a​l​i​g​nemb^\{align\}\_\{\\mathbf\{X\}\}\)\.

Utilizing a pre\-trained LLM\.The input to the LLM is formed by concatenating the compositional spectral prompt \(i\.e\.,𝒫\\mathcal\{P\}\), embedded text description \(i\.e\.,𝒯\\mathcal\{T\}\), and aligned time series embedding \(i\.e\.,e​m​ba​l​i​g​nemb^\{align\}\)\. Within the text description, we utilize special tokens \(e\.g\.,\[START PROMPT\]and\[END PROMPT\]\) as delimiters, enabling the LLM to clearly discern the boundaries between these heterogenous modalities\. The concatenated embedding sequence is input to the LLM’s transformer layers, all of which remain frozen\. The last hidden state representation of the LLM serves as the time series representation, which is flattened and linearly projected to generate the final forecast\.

Training Phase\.The training phase serves to learn the overall base knowledge before entering the online phase\. Therefore, during the training phase, except for the pre\-trained LLM, we train the time series backbone network, align module, spectral prompt bank, and output projection layer by minimizing the following objective:

\(8\)ℒt​r​a​i​n​i​n​g=1Nt​r​a​i​n−L−H\+1∑i=LNt​r​a​i​n−H\|\|f\(𝐗i−L\+1:i\)−𝐗i\+1:i\+H\|\|22\.\\mathcal\{L\}\_\{training\}=\\frac\{1\}\{N\_\{train\}\-L\-H\+1\}\\sum\_\{i=L\}^\{N\_\{train\}\-H\}\|\|f\(\\mathbf\{X\}\_\{i\-L\+1:i\}\)\-\\mathbf\{X\}\_\{i\+1:i\+H\}\|\|\_\{2\}^\{2\}\.\\vskip\-4\.30554ptwheref⁡\(⋅\)f\(\\cdot\)is the overall framework ofCoSPOT\.

Online Phase\.In the online phase, only the output projection layer is tuned to match the streaming data distribution, while all other parameters, including the pre\-trained LLM, time series backbone network, align module, and spectral prompt bank, remain frozen\.555We emphasize thatCoSPOTis an efficient framework despite utilizing an LLM, as the number of parameters updated during the online phase is extremely small\. Please see Section[5\.3\.3](https://arxiv.org/html/2609.02093#S5.SS3.SSS3)for a detailed analysis\.According to\([Lau et al\., 2025](https://arxiv.org/html/2609.02093#bib.bib3)\), when the prediction horizonHHis greater than 1, calculating the loss using the ground truth for allHHtime steps \(𝐗i\+1:i\+H\\mathbf\{X\}\_\{i\+1:i\+H\}\) at each moving window step for model updates leads to information leakage\. To avoid this, the moving window should be extended toHHsteps rather than updating at each step, which however introduces an update delay and hinders effective adaptation\. Hence, we employ a pseudo\-labeling technique while keeping updates at each step\. When calculating the loss for the model’s output att=it=ito update the model, only the ground truth for the immediate next time point, i\.e\.,𝐗i\+1\\mathbf\{X\}\_\{i\+1\}is used, and pseudo\-labels are generated and utilized for the remaining time steps \(i\.e\.,𝐗~i\+2:i\+H\\mathbf\{\\tilde\{X\}\}\_\{i\+2:i\+H\}\)\. We propagate the representation of the time series backbone network \(i\.e\.,b⁡\(⋅\)b\(\\cdot\)\) through a linear layer \(i\.e\.,l⁡\(⋅\)l\(\\cdot\)\) to project it into the output space and obtain the pseudo\-label\. This linear layer is utilized while being frozen during the online phase, and is trained along with Equation[8](https://arxiv.org/html/2609.02093#S4.E8)during the training phase using MSE loss\. Therefore, Equation[8](https://arxiv.org/html/2609.02093#S4.E8)is modified as follows:

\(9\)ℒtraining\+=1Ntrain−L−H\+1\\displaystyle\\mathcal\{L\}^\{\+\}\_\{\\text\{training\}\}=\\frac\{1\}\{N\_\{\\text\{train\}\}\-L\-H\+1\}∑i=LNtrain−H\(‖f\(𝐗i−L\+1:i\)−𝐗i\+1:i\+H‖22\\displaystyle\\sum\_\{i=L\}^\{N\_\{\\text\{train\}\}\-H\}\\Big\(\\left\\\|f\(\\mathbf\{X\}\_\{i\-L\+1:i\}\)\-\\mathbf\{X\}\_\{i\+1:i\+H\}\\right\\\|\_\{2\}^\{2\}\+‖l\(b\(𝐗i−L\+1:i\)\)−𝐗i\+1:i\+H‖22\)\.\\displaystyle\+\\left\\\|l\\\!\\left\(b\(\\mathbf\{X\}\_\{i\-L\+1:i\}\)\\right\)\-\\mathbf\{X\}\_\{i\+1:i\+H\}\\right\\\|\_\{2\}^\{2\}\\Big\)\.
wherel⁡\(⋅\)l\(\\cdot\)denotes the pseudo\-label projection layer which projects the time series representation into the output space\. During the online phase, we use the frozenb⁡\(⋅\)b\(\\cdot\)andl⁡\(⋅\)l\(\\cdot\), which are trained in the training phase, to generate pseudo\-labels:𝐗~i\+2:i\+H=l\(b\(𝐗i−L\+1:i\)\)\[1:\]\\mathbf\{\\tilde\{X\}\}\_\{i\+2:i\+H\}=l\(b\(\\mathbf\{X\}\_\{i\-L\+1:i\}\)\)\[1:\]\. These pseudo\-labels are then employed to tune the model\. To mitigate the impact of prediction errors and pseudo\-labels as the forecast horizon extends from the current observation, we apply a geometric decay factorδ∈\[0,1\]\\delta\\in\[0,1\]to the online loss as follows:

\(10\)ℒonline\\displaystyle\\mathcal\{L\}\_\{\\text\{online\}\}=1Nonline−L−H\+1∑i=Ntrain\+Nval\+LNdata−H\\displaystyle=\\frac\{1\}\{N\_\{\\text\{online\}\}\-L\-H\+1\}\\sum\_\{i=N\_\{\\text\{train\}\}\+N\_\{\\text\{val\}\}\+L\}^\{N\_\{\\text\{data\}\}\-H\}\(1H​∑h=1Hδh−1​‖𝐗^i\+h−𝐗¯i\+h‖22\)\.\\displaystyle\\Big\(\\frac\{1\}\{H\}\\sum\_\{h=1\}^\{H\}\\delta^\{h\-1\}\\left\\\|\\hat\{\\mathbf\{X\}\}\_\{i\+h\}\-\\bar\{\\mathbf\{X\}\}\_\{i\+h\}\\right\\\|\_\{2\}^\{2\}\\Big\)\.
where𝐗¯i\+1:i\+H=Concat\(𝐗i\+1,𝐗~i\+2:i\+H\)\\mathbf\{\\overline\{X\}\}\_\{i\+1:i\+H\}=\\textup\{Concat\}\(\\mathbf\{X\}\_\{i\+1\},\\mathbf\{\\tilde\{X\}\}\_\{i\+2:i\+H\}\)is a concatenated sequence of the ground truth and pseudo\-labels\. Finally, the model’s online performance is evaluated using Equation[1](https://arxiv.org/html/2609.02093#S3.E1)\.

Pseudocode\.Algorithm[1](https://arxiv.org/html/2609.02093#alg1)describes the training and online phases ofCoSPOT\.

Algorithm 1Pseudocode for training and online phases ofCoSPOT1:Input:Time Series

𝐗∈ℝNd​a​t​a\\mathbf\{X\}\\in\\mathbb\{R\}^\{N\_\{data\}\}, Pre\-trained LLM, Batch Size, Align Module

2:Output:Time Series Backbone Network

b⁡\(⋅\)b\(\\cdot\), Spectral Prompt Bank

𝐏\\mathbf\{P\}, Output Projection Layer

3:

4:\# Training Phase

5:Freeze the Pre\-trained LLM

6:fori in range\(

Nt​r​a​i​nBatch Size\\tfrac\{N\_\{train\}\}\{\\textup\{Batch Size\}\}\)do

7:

𝐗\(i\)∈ℝBatch Size×L\\mathbf\{X\}^\{\(i\)\}\\in\\mathbb\{R\}^\{\\textup\{Batch Size\}\\times L\}⊳\\trianglerightii\-th batch for batch training

8:

9:\# Aligning

10:

e​m​b𝐗\(i\)=b⁡\(𝐗\(i\)\)emb\_\{\\mathbf\{X\}^\{\(i\)\}\}=b\(\\mathbf\{X\}^\{\(i\)\}\)
11:Generate

e​m​b𝐗\(i\)a​l​i​g​nemb\_\{\\mathbf\{X\}^\{\(i\)\}\}^\{align\}using the Align Module

12:

13:\# Text Description

14:Generate

𝒯𝐗\(i\)\\mathcal\{T\}\_\{\\mathbf\{X\}^\{\(i\)\}\}using

t​e​x​t𝐗\(i\)text\_\{\\mathbf\{X\}^\{\(i\)\}\}
15:

16:\# Compositional Spectral Prompt

17:Decompose

𝐗\(i\)\\mathbf\{X\}^\{\(i\)\}via DFT into bases and amplitudes⊳\\trianglerightEq\.[6](https://arxiv.org/html/2609.02093#S4.E6)

18:Compose

𝒫𝐗\(i\)\\mathcal\{P\}\_\{\\mathbf\{X\}^\{\(i\)\}\}from spectral basis prompts⊳\\trianglerightEq\.[7](https://arxiv.org/html/2609.02093#S4.E7)

19:

20:Feed

\[𝒫𝐗\(i\);𝒯𝐗\(i\);e​m​b𝐗\(i\)a​l​i​g​n\]\[\\mathcal\{P\}\_\{\\mathbf\{X\}^\{\(i\)\}\};\\mathcal\{T\}\_\{\\mathbf\{X\}^\{\(i\)\}\};emb\_\{\\mathbf\{X\}^\{\(i\)\}\}^\{align\}\]into the LLM

21:The Pre\-trained LLM’s representation is projected to obtain

𝐗^\(i\)\\hat\{\\mathbf\{X\}\}^\{\(i\)\}
22:

23:Train all parameters except the Pre\-trained LLM⊳\\trianglerightEq\.[9](https://arxiv.org/html/2609.02093#S4.E9)

24:endfor

25:

26:\# Online Phase

27:Freeze all parameters except those in the Output Projection Layer

28:fori in range\(

No​n​l​i​n​eN\_\{online\}\)do

29:

𝐗\(i\)∈ℝL\\mathbf\{X\}^\{\(i\)\}\\in\\mathbb\{R\}^\{L\}⊳\\trianglerightii\-th data instance

30:

31:\# Aligning

32:

e​m​b𝐗\(i\)=b⁡\(𝐗\(i\)\)emb\_\{\\mathbf\{X\}^\{\(i\)\}\}=b\(\\mathbf\{X\}^\{\(i\)\}\)
33:Generate

e​m​b𝐗\(i\)a​l​i​g​nemb\_\{\\mathbf\{X\}^\{\(i\)\}\}^\{align\}using the Align Module

34:

35:\# Text Description

36:Generate

𝒯𝐗\(i\)\\mathcal\{T\}\_\{\\mathbf\{X\}^\{\(i\)\}\}using

t​e​x​t𝐗\(i\)text\_\{\\mathbf\{X\}^\{\(i\)\}\}
37:

38:\# Compositional Spectral Prompt

39:Decompose

𝐗\(i\)\\mathbf\{X\}^\{\(i\)\}via DFT into bases and amplitudes⊳\\trianglerightEq\.[6](https://arxiv.org/html/2609.02093#S4.E6)

40:Compose

𝒫𝐗\(i\)\\mathcal\{P\}\_\{\\mathbf\{X\}^\{\(i\)\}\}from spectral basis prompts⊳\\trianglerightEq\.[7](https://arxiv.org/html/2609.02093#S4.E7)

41:

42:Feed

\[𝒫𝐗\(i\);𝒯𝐗\(i\);e​m​b𝐗\(i\)a​l​i​g​n\]\[\\mathcal\{P\}\_\{\\mathbf\{X\}^\{\(i\)\}\};\\mathcal\{T\}\_\{\\mathbf\{X\}^\{\(i\)\}\};emb\_\{\\mathbf\{X\}^\{\(i\)\}\}^\{align\}\]into the LLM

43:The Pre\-trained LLM’s representation is projected to obtain

𝐗^\(i\)\\hat\{\\mathbf\{X\}\}^\{\(i\)\}
44:

45:Update the Output Projection Layer using

ℒo​n​l​i​n​e\\mathcal\{L\}\_\{online\}⊳\\trianglerightEq\.[10](https://arxiv.org/html/2609.02093#S4.E10)

46:endfor

47:

48:Evaluate the online phase⊳\\trianglerightEq\.[1](https://arxiv.org/html/2609.02093#S3.E1)

## 5\.Experiments

Datasets\.Following prior studies\([Pham et al\., 2022](https://arxiv.org/html/2609.02093#bib.bib1);[Wen et al\., 2023](https://arxiv.org/html/2609.02093#bib.bib2);[Lau et al\., 2025](https://arxiv.org/html/2609.02093#bib.bib3)\), we evaluate our method on five widely used time series forecasting benchmarks from various domains, splitting each into training, validation, and testing sets with a 20%, 5%, and 75% ratio, respectively\.ETT666https://github\.com/zhouhaoyi/ETDatasetcontains electricity load and oil temperature data collected at 15\-minute \(ETTm1, ETTm2\) and hourly \(ETTh1, ETTh2\) intervals, each with 6 covariates\.Weather777https://www\.ncei\.noaa\.gov/data/local\-climatological\-data/consists of 21 meteorological variables recorded at hourly intervals, covering diverse climate features such as temperature, humidity, and wind speed\.ECL888https://archive\.ics\.uci\.edu/ml/datasets/ElectricityLoadDiagrams20112014comprises hourly electricity consumption records from 321 clients over two years\.Traffic999https://pems\.dot\.ca\.gov/contains occupancy rate measurements from 862 freeway sensors in the San Francisco Bay Area, recorded at 5\-minute intervals\.Exchange Rate101010https://github\.com/laiguokun/multivariate\-time\-series\-datacontains daily exchange\-rate records across multiple countries\. Compared with the other benchmarks, Exchange Rate exhibits less pronounced periodicity and is more influenced by random economic and market events, providing a challenging testbed for evaluating robustness beyond strongly seasonal patterns\.

Baselines\.We utilize various deep learning\-based time series forecasting models as baselines, categorized into four groups\.Static forecasting models:DLinear\([Zeng et al\., 2023](https://arxiv.org/html/2609.02093#bib.bib14)\)decomposes time series into trend and seasonal components and applies linear layers to each component\.PatchTST\([Nie et al\., 2022](https://arxiv.org/html/2609.02093#bib.bib10)\)applies patch\-based Transformer encoding to preserve local temporal patterns\.iTransformer\([Liu et al\., 2023](https://arxiv.org/html/2609.02093#bib.bib15)\)attends along the feature dimension to capture cross\-variable interactions\.TimeMixer\([Wang et al\., 2024](https://arxiv.org/html/2609.02093#bib.bib26)\)introduces a multiscale\-mixing architecture that decomposes and mixes seasonal and trend components across different sampling scales\.Time series foundation models:Chronos\-2\([Ansari et al\., 2025](https://arxiv.org/html/2609.02093#bib.bib27)\)is a time series foundation model that supports zero\-shot univariate, multivariate, and covariate\-informed forecasting through a unified architecture\.TimesFM\([Das et al\., 2023](https://arxiv.org/html/2609.02093#bib.bib28)\)is a decoder\-only foundation model pre\-trained on large\-scale time series corpora for zero\-shot forecasting\.LLM\-based models:LLM4TS\([Chang et al\., 2023](https://arxiv.org/html/2609.02093#bib.bib4)\)aligns time series representations with pre\-trained LLMs through a two\-stage fine\-tuning strategy for forecasting\.GPT4TS\([Zhou et al\., 2023](https://arxiv.org/html/2609.02093#bib.bib5)\)repurposes frozen GPT\-style Transformers by tuning only lightweight components for time series prediction\.Time\-LLM\([Jin et al\., 2023](https://arxiv.org/html/2609.02093#bib.bib6)\)reprograms time series into text prototypes with a Prompt\-as\-Prefix strategy\.Online time series forecasting \(OTSF\) models:FSNet\([Pham et al\., 2022](https://arxiv.org/html/2609.02093#bib.bib1)\)captures short\- and long\-term patterns via dual fast and slow learners\.OneNet\([Wen et al\., 2023](https://arxiv.org/html/2609.02093#bib.bib2)\)adaptively weights an online ensemble based on recent performance\.DSOF\([Lau et al\., 2025](https://arxiv.org/html/2609.02093#bib.bib3)\)introduces a dual\-stream framework that updates parameters through distinct short\- and long\-term temporal contexts\.

Implementation Details\.Consistent with previous studies\([Pham et al\., 2022](https://arxiv.org/html/2609.02093#bib.bib1);[Wen et al\., 2023](https://arxiv.org/html/2609.02093#bib.bib2);[Lau et al\., 2025](https://arxiv.org/html/2609.02093#bib.bib3)\), we set prediction lengthHHto 1, 24, and 48, with a lookback lengthLLof 96\. We utilize PatchTST\([Nie et al\., 2022](https://arxiv.org/html/2609.02093#bib.bib10)\)as the time series backbone networkb⁡\(⋅\)b\(\\cdot\)and Llama\-7B\([Touvron et al\., 2023](https://arxiv.org/html/2609.02093#bib.bib16)\)as the default LLM unless stated otherwise\. The align module of Time\-LLM\([Jin et al\., 2023](https://arxiv.org/html/2609.02093#bib.bib6)\)is used as the default align module\. The evaluation metrics include mean square error \(MSE\) and mean absolute error \(MAE\)\. Our method is implemented with Python 3\.11 and PyTorch 2\.2\.2\. We use the AdamW optimizer, training for 10 epochs during the training phase and performing one\-step updates per data instance during the online phase\. Key hyperparameters are set as follows:γ=0\.3\\gamma=0\.3\(Eq\.[7](https://arxiv.org/html/2609.02093#S4.E7)\) for low\-pass filtering, wavelet decomposition levelj=2j=2\(Eq\.[3](https://arxiv.org/html/2609.02093#S4.E3)\), andδ=0\.8\\delta=0\.8\(Eq\.[10](https://arxiv.org/html/2609.02093#S4.E10)\) to assign stronger supervision to near\-future values\. All experiments are conducted on a 48GB NVIDIA RTX A6000\.

Table 1\.Comparison of MSE and MAE results in OTSF for predicting 1, 24, and 48 prediction horizon \(i\.e\.,HH\) with a lookback lengthL=96L=96\(Best: bold red, the second\-best: underlined in blue\)\.DLinearPatchTSTiTransformerTimeMixerChronos\-2TimesFMLLM4TSGPT4TSTime\-LLMFSNetOneNetDSOFCoSPOTHHMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEETTh110\.5020\.6090\.7790\.8280\.9930\.9760\.5570\.7060\.5790\.6810\.6210\.7081\.4361\.0581\.5391\.1411\.3821\.07613\.263\.4414\.0231\.9560\.8020\.8570\.4250\.601242\.3331\.4273\.7971\.8493\.0281\.6902\.7191\.5282\.4841\.3962\.4181\.4553\.1551\.6233\.0281\.6402\.9471\.41719\.334\.0979\.0012\.8012\.3111\.3721\.2170\.993482\.8021\.5535\.1352\.0663\.9981\.9293\.2741\.6992\.7481\.4582\.9941\.6306\.2562\.3126\.7382\.3975\.9972\.04925\.824\.88112\.213\.3344\.2331\.9731\.4821\.073ETTh210\.4680\.6430\.8950\.9240\.8740\.8880\.4390\.5630\.4760\.5900\.5970\.5720\.8110\.8750\.8270\.8290\.8100\.83610\.783\.0832\.6341\.5310\.5490\.5370\.3980\.423242\.1871\.3384\.8852\.0022\.6881\.4511\.9381\.2961\.3431\.0561\.4191\.0913\.0231\.5382\.8871\.5092\.3811\.50317\.994\.0417\.0192\.3382\.1791\.4100\.9320\.607482\.3491\.4126\.3682\.3493\.7681\.5412\.2161\.3982\.1031\.2502\.1081\.2525\.4112\.0265\.5102\.2375\.2312\.08722\.794\.6739\.9303\.0194\.0031\.9911\.2450\.699ETTm110\.1320\.3120\.1350\.3520\.1830\.3970\.1370\.3410\.1450\.2240\.1730\.2600\.3710\.4890\.3870\.5220\.3090\.5060\.1900\.2310\.1540\.2710\.0960\.1420\.0830\.128240\.6180\.7711\.1021\.0291\.1011\.0090\.6020\.7260\.7170\.6000\.9120\.7310\.7020\.7070\.7110\.7410\.6710\.7951\.5201\.2021\.1031\.0030\.4120\.4520\.3720\.403480\.8290\.8942\.0831\.2431\.2981\.0390\.8110\.8821\.0520\.7811\.1520\.8900\.9350\.8460\.9330\.8860\.8970\.8362\.2831\.3901\.4921\.210\.5590\.5400\.4510\.515ETTm210\.1110\.3030\.1130\.2960\.1370\.3500\.1140\.3260\.1070\.2570\.1260\.3050\.3020\.5250\.3210\.5360\.1870\.3330\.1270\.3260\.1110\.2970\.0660\.2190\.0520\.208240\.5890\.6671\.0350\.9170\.7990\.8530\.5250\.6880\.6470\.7740\.6240\.7100\.6310\.7140\.5870\.7060\.4990\.6241\.1800\.9760\.6030\.7160\.3480\.5190\.2870\.517480\.8090\.7941\.8021\.2021\.0030\.9500\.7840\.8540\.8150\.8230\.8460\.8800\.7750\.8330\.7130\.7490\.5890\.7071\.8471\.1940\.8430\.8810\.4120\.6280\.3480\.539WTH10\.3590\.4990\.5210\.6910\.4910\.6070\.3380\.5430\.2050\.3520\.3600\.4000\.4890\.6590\.4930\.6020\.4880\.6490\.7310\.7540\.5010\.5360\.2980\.3980\.0520\.114241\.2191\.0081\.5041\.2081\.4821\.0171\.2081\.0020\.4360\.5600\.5370\.5331\.3111\.0411\.2871\.0340\.9990\.8951\.7561\.2181\.3771\.0230\.6770\.6490\.1100\.165481\.7411\.2472\.0011\.2571\.6991\.1461\.7331\.2160\.7100\.6430\.7310\.6551\.6101\.1691\.6511\.1851\.4831\.0782\.6201\.6092\.7031\.5910\.9190\.8130\.2150\.297ECL12\.9111\.6174\.2791\.9681\.8971\.2322\.7081\.5491\.0960\.8461\.1010\.8496\.3542\.3716\.3842\.3276\.0452\.35531116\.1829\.884\.994\.6392\.1410\.1600\.2202413\.213\.45615\.613\.7514\.0121\.8037\.2092\.5831\.2310\.8091\.2110\.9016\.7112\.5396\.8132\.2116\.3172\.41342819\.6883\.279\.114\.5512\.0720\.2630\.2644825\.984\.90615\.883\.8494\.8772\.0089\.2432\.8401\.6001\.0651\.5541\.0477\.3422\.5967\.5232\.4747\.3212\.55746919\.65144\.8911\.435\.8192\.3550\.3440\.317Traffic10\.2980\.4890\.2790\.4980\.2390\.4580\.2800\.5090\.5800\.5620\.6620\.6140\.6020\.6760\.6530\.6980\.5110\.6150\.6120\.7120\.2590\.3810\.2730\.3710\.2110\.259240\.6560\.7040\.5910\.6680\.4580\.6460\.6610\.7000\.6360\.7520\.6670\.7960\.6970\.7350\.7110\.8030\.6360\.7470\.7590\.8110\.5810\.5890\.3690\.3860\.3070\.281480\.7910\.8680\.6330\.7560\.5170\.6900\.7590\.8280\.8000\.8680\.7320\.8560\.7240\.8010\.7830\.8240\.6840\.7740\.8140\.8310\.7010\.6330\.3810\.3890\.3570\.391Exchange10\.5790\.5610\.8870\.8420\.8360\.8140\.4320\.5570\.5020\.6390\.5310\.6490\.8470\.8200\.8590\.8270\.7780\.7820\.4600\.5780\.2750\.4240\.2430\.4030\.1530\.256241\.4270\.9951\.2731\.0281\.0470\.9230\.9800\.8901\.1310\.9631\.0740\.9361\.2831\.0431\.3321\.0541\.2951\.0381\.2500\.9181\.1290\.8631\.0390\.9390\.8030\.618482\.4731\.2732\.7111\.5472\.2791\.4101\.7481\.1221\.6931\.2011\.5761\.1552\.3711\.4402\.1621\.3701\.8841\.2732\.3791\.2421\.5071\.0281\.3741\.0721\.1000\.843

Table 2\.Ablation studies of each component ofCoSPOT\(MSE\)\.ETTh2ETTm1WTHECLTraffic𝒫\\mathcal\{P\}𝒯\\mathcal\{T\}1244812448124481244812448\(1\)✗✗0\.9273\.1214\.7900\.2110\.5610\.6660\.2380\.5950\.7673\.4673\.4793\.6880\.5240\.7670\.804\(2\)✓✗0\.4991\.7232\.0310\.0960\.3880\.5010\.1030\.2270\.2950\.9911\.3211\.3890\.2760\.3920\.399\(3\)✗✓0\.7182\.3212\.8890\.1220\.4810\.5440\.1790\.4310\.6012\.1312\.7732\.9080\.4110\.6030\.640\(4\)\-1✓✓0\.3980\.9321\.2450\.0830\.3720\.4510\.0520\.1100\.2150\.1600\.2630\.3440\.2110\.3070\.357\(4\)\-2✓✓\(STFT\)0\.3811\.2411\.4400\.0880\.3750\.4720\.0500\.1360\.2490\.2180\.3370\.4720\.2080\.3360\.379

### 5\.1\.Overall Performance

The experimental results on eight datasets are summarized in Table[1](https://arxiv.org/html/2609.02093#S5.T1)\. The reported results represent averages over three runs\. We make the following key observations:\(1\)In settings without information leakage , FSNet and OneNet, which are designed for OTSF, underperform static models such as DLinear, iTransformer, and TimeMixer\. This implies that FSNet and OneNet mainly exploit the leaked data on which they are trained—leading to rapid convergence—rather than truly acquiring the underlying structure of newly arriving patterns or learning how to adapt to them\.\(2\)Time series foundation models, i\.e\., Chronos\-2 and TimesFM, underperformCoSPOT\. Although these models exhibit strong zero\-shot forecasting ability from large\-scale pre\-training, they lack an explicit mechanism to adapt to evolving target distributions in online scenarios\. In contrast,CoSPOTcombines compositional spectral prompts with text descriptions, providing both distribution\-aware guidance and recent contextual information for more effective adaptation\.\(3\)LLM\-based models, i\.e\., LLM4TS, GPT4TS, and Time\-LLM, substantially underperformCoSPOT\. This gap arises because their alignment modules cannot handle continuous distribution shifts, leading to a breakdown in modality alignment\. Conversely,CoSPOTsuccessfully adapts to these shifts using compositional spectral prompts and text descriptions, thereby preserving the crucial alignment between the LLM and the time series backbone\.\(4\)While the state\-of\-the\-art OTSF method, DSOF, demonstrates strong performance compared with other baselines, it significantly underperformsCoSPOT\. This demonstrates that while the dual\-stream framework of DSOF prevents update delays, allowing the model to adapt quickly without information leakage, its emphasis on rapid convergence to incoming data limits its ability to capture underlying patterns, leaving the model poorly equipped to adapt when data are scarce\.\(5\)CoSPOTdemonstrates robust performance across multiple datasets by combining the adaptability of a pre\-trained LLM with compositional spectral prompts, which provide explicit guidance on the underlying time\-series patterns under continuous distribution shifts\.

Table 3\.Comparison of MSE and MAE results across various datasets under scenarios with an extended online phase, where the train/valid/test split is set to 10%/5%/85%\.ETTh2ETTm1WTHECLTraffic1244812448124481244812448FSNetMSE15\.23124\.59128\.9350\.3992\.4525\.1131\.2212\.3414\.512299\.34435\.12458\.981\.2121\.7511\.999MAE3\.8024\.7585\.1790\.5161\.3652\.0611\.0041\.3302\.04115\.30117\.85920\.4231\.0011\.2321\.353OneNetMSE4\.2409\.34213\.5820\.2781\.4281\.6770\.8131\.5352\.99130\.09182\.29142\.2730\.5310\.7780\.933MAE2\.0093\.0063\.4850\.4271\.0741\.2040\.8011\.0981\.5995\.2858\.04110\.7970\.5880\.7420\.905DSOFMSE0\.8512\.8895\.0280\.1730\.5610\.7790\.3130\.7111\.1374\.8985\.1736\.0920\.4990\.7110\.793MAE0\.8721\.4992\.0420\.3970\.6680\.8460\.5390\.8031\.0062\.1342\.2042\.3680\.6560\.7590\.801CoSPOTMSE0\.4120\.9781\.3050\.0870\.4110\.4870\.0510\.1030\.2230\.1790\.2910\.3620\.4850\.3560\.379MAE0\.4340\.6130\.7300\.1940\.6010\.6370\.2060\.2910\.4520\.4030\.5090\.5810\.4340\.4810\.578

### 5\.2\.Ablation Study

To assess the impact of the compositional spectral prompt \(i\.e\.,𝒫\\mathcal\{P\}\) and text description \(i\.e\.,𝒯\\mathcal\{T\}\) inCoSPOT, Table[2](https://arxiv.org/html/2609.02093#S5.T2)presents ablation studies across five cases, including the vanillaCoSPOT\(Row \(4\)\-1\), with key observations as follows:\(1\)Introducing the compositional spectral prompt is helpful \(Row \(1\) vs\. \(2\)\)\. Given the sequential nature of time\-series data, in which continuous distribution shifts are inevitable, providing the model with compositional spectral prompts that capture these dynamics is highly effective\. Specifically,CoSPOTcaptures overall time\-series patterns by learning spectral basis prompts grounded in decomposed frequency bases\. This enables effective adaptation to unseen patterns by compositionally recombining the learned spectral basis prompts, resulting in nearly a 50% improvement in MSE\. We further analyze the advantage of frequency\-domain prompting over time\-domain prompting in Section[5\.3\.5](https://arxiv.org/html/2609.02093#S5.SS3.SSS5)\.\(2\)Providing recent pattern information to the LLM in the form of text descriptions is effective \(Row \(1\) vs\. \(3\)\)\. The text description serves as a practical auxiliary modality for online forecasting, as it injects dynamically changing contextual information into the LLM beyond raw numerical observations, enriching data\-scarce online scenarios without additional training and enabling effective adaptation\.\(3\)Leveraging both compositional spectral prompts and text description together can yield synergistic effects \(Row \(2&3\) vs\. \(4\)\-1\)\. Through compositional spectral prompts, the overall pattern of the given time series is captured, while text description provides information on recent patterns in the time and frequency domains, enabling the model to effectively adapt to recent patterns without being hindered by distribution shifts\.\(4\)Utilizing DWT instead of STFT is more effective for providing recent pattern information from a frequency perspective \(Row \(4\)\-1 vs\. \(4\)\-2\)\. STFT struggles with temporal resolution due to its use of a fixed window size, whereas DWT adapts the window size based on the frequency of the time series, making it more effective in capturing non\-stationary signals\. This results in comparable performance in relatively easy tasks with a prediction horizon of 1\. However, for more challenging tasks with longer horizons, where capturing the underlying recent patterns is crucial, the use of DWT proves to be more effective\.

Table 4\.Comparison of MSE and MAE results in a cross\-dataset scenario, where different datasets from the same ETT domain are used in the training and online phases to induce distribution shifts, with the prediction horizon \(i\.e\.,HH\) set to 1\.T​r​a​i​n​i​n​gTrainingrefers to the dataset used in the training phase, whileO​n​l​i​n​eOnlinerefers to the dataset used in the online phase\.TrainingOnlineFSNetOneNetDSOFCoSPOTMSEMAEMSEMAEMSEMAEMSEMAEETTh1→\\rightarrowETTh213\.6643\.1244\.0251\.8620\.9930\.8060\.4270\.453→\\rightarrowETTm21\.8151\.1321\.7321\.0160\.7230\.7020\.1780\.319ETTh2→\\rightarrowETTh112\.6673\.0113\.8331\.8080\.9010\.8490\.3990\.531→\\rightarrowETTm21\.7911\.0291\.4891\.0220\.8180\.7440\.2110\.359ETTm1→\\rightarrowETTh215\.3713\.5713\.9891\.6101\.0030\.9010\.5210\.621→\\rightarrowETTm21\.3351\.0030\.9890\.8610\.5420\.5360\.1280\.257ETTm2→\\rightarrowETTh214\.9893\.1994\.1381\.7341\.2110\.9450\.5130\.616→\\rightarrowETTm11\.2991\.0391\.3150\.9460\.5170\.6190\.1320\.263

### 5\.3\.Further Analysis

#### 5\.3\.1\.Robustness to the extension of the online phase\.

In Table[3](https://arxiv.org/html/2609.02093#S5.T3), we analyze the model’s robustness when the online phase is extended\. Specifically, we evaluate performance by adjusting the OTSF train/valid/test split from 20%/5%/75% to 10%/5%/85%, extending the online phase\. Prior methods \(i\.e\., FSNet, OneNet, and DSOF\) experience significant performance degradation compared to the results in Table[1](https://arxiv.org/html/2609.02093#S5.T1)for the following two reasons: \(1\) They are incapable of storing all the recurring patterns in an associative memory as the online phase is extended, and \(2\) the continuous occurrence of unseen patterns prevents the model from maintaining adaptability\. In contrast,CoSPOT, thanks to the rich knowledge and transferability of the pre\-trained LLM, effectively maintains adaptability even in data\-scarce online scenarios, demonstrating performance comparable to the results in Table[1](https://arxiv.org/html/2609.02093#S5.T1)\. We argue that adaptability to an extended online phase is enhanced by two key designs: \(i\) compositional spectral prompting, which represents evolving patterns using a finite set of spectral basis prompts, and \(ii\) a text description of recent patterns provided to the LLM, which compensates for data scarcity and enables efficient adaptation without additional training\.

#### 5\.3\.2\.Robustness to distribution shifts\.

Table[4](https://arxiv.org/html/2609.02093#S5.T4)shows the cross\-dataset experiments using the ETT datasets\. ETTh1 and ETTh2 are hourly measurements from two different electricity transformers, while ETTm1 and ETTm2 are their corresponding 15\-minute\-resolution variants\. Therefore, each pair shares the same sampling resolution but exhibits different temporal patterns due to differences in the underlying transformers\. To induce distribution shifts, we deliberately use different datasets for the training and online phases\. The model learns the base knowledge from the training data and adapts to the streaming online data, with the prediction horizon set to 1\. We observe thatCoSPOToutperforms all baselines across the 8 scenarios\. FSNet, OneNet, and DSOF rely on associative memory to store recurring patterns and adapt by retrieving similar ones\. This approach fails when unseen patterns arise, as no meaningful associations can be found, leading to substantial performance degradation \(compare with the results in Table[1](https://arxiv.org/html/2609.02093#S5.T1)\)\. In contrast,CoSPOTrepresents patterns through compositional spectral prompts by learning a spectral basis prompt for each frequency basis, rather than directly storing the time series patterns\. As a result, even unseen patterns can be reliably expressed as new compositions of spectral basis prompts, making the model robust to distribution shifts and allowing its performance to remain largely consistent with the results in Table[1](https://arxiv.org/html/2609.02093#S5.T1)\.

#### 5\.3\.3\.Online Updating Cost\.

To analyze the efficiency ofCoSPOT, we compare the number of updated parameters during the online phase and runtime statistics with those of existing online time series forecasting models\.

Table 5\.The number of parameters updated during the online phase of OTSF models for prediction horizon 1 on the ETTh2 dataset\.FSNetOneNetDSOFCoSPOT2,037,1151,018,0451,236,349897

##### Parameters\.

Table[5](https://arxiv.org/html/2609.02093#S5.T5)reports the number of parameters updated during the online phase \(i\.e\., those that are not frozen\) for each OTSF model in experiments using the ETTh2 dataset\.CoSPOTupdates significantly fewer parameters compared to the baselines, indicating its efficiency\. This demonstrates thatCoSPOTcan leverage model adaptability to achieve strong performance \(see Table[1](https://arxiv.org/html/2609.02093#S5.T1)\) with fewer updated parameters\. Updating many parameters during the online phase can make models vulnerable to distribution shifts and cause them to forget the base knowledge learned during training phase\. In this regard,CoSPOTeffectively and efficiently adapts to new data without forgetting previously learned knowledge by updating only a small subset of parameters\.

Table 6\.Comparison of runtime statistics betweenCoSPOTand existing OTSF methods for the scenario with a prediction horizon of 1 on each dataset, utilizing the total Training Phase Duration \(sec\), the total Online Phase Duration \(sec\), and the Inference Latency \(sec/itr\), defined as the time required per update\.MetricETTh1ETTh2ETTm1ETTm2WTHECLTrafficFSNetTraining Phase Duration \(sec\)275274801830775398159Online Phase Duration \(sec\)3413321,0241,019998463351Inference Latency \(sec/itr\)0\.0310\.0300\.0220\.0220\.0250\.0230\.026OneNetTraining Phase Duration \(sec\)5515401,6211,6331,503848303Online Phase Duration \(sec\)7016902,2372,2412,079994741Inference Latency \(sec/itr\)0\.0630\.0610\.0500\.0500\.0510\.0500\.053DSOFTraining Phase Duration \(sec\)6226131,9041,9361,789994379Online Phase Duration \(sec\)7337052,5882,5592,3711,201855Inference Latency \(sec/itr\)0\.0640\.0620\.0580\.0560\.0570\.0590\.062CoSPOTTraining Phase Duration \(sec\)4,6504,65014,42814,50113,7867,2492,788Online Phase Duration \(sec\)7877412,9382,9732,8411,389931Inference Latency \(sec/itr\)0\.0680\.0650\.0640\.0660\.0680\.0650\.067

##### Runtime Comparison\.

Table[6](https://arxiv.org/html/2609.02093#S5.T6)presents the runtime statistics for each dataset in the scenario where the prediction horizon is 1\. We utilize the total Training Phase Duration \(sec\), the total Online Phase Duration \(sec\), and the Inference Latency \(sec/itr\), defined as the time required per update\. OTSF requires a model to be sufficiently pre\-trained on initial data \(i\.e\., training phase\) and then continually adapted to the subsequent data stream \(i\.e\., online phase\)\. Consequently, resource optimization must prioritize the cost\-sensitive online phase over the training phase\. The duration of the online phase and the inference latency forCoSPOTrequire only slightly more time compared to existing OTSF models\. This demonstrates thatCoSPOTcontinuously adapts to new online distributions with significantly less parameter tuning \(see Table[5](https://arxiv.org/html/2609.02093#S5.T5)\) by simultaneously leveraging the LLM’s superior transferability and the distribution guidance provided by compositional spectral prompts\. WhileCoSPOTincurs a higher initial cost for training duration, this cost is justifiable as the training phase is less resource\-sensitive and prioritizes the sufficient acquisition of base knowledge\. In summary,CoSPOTachieves significantly superior performance \(please refer to Table[1](https://arxiv.org/html/2609.02093#S5.T1)\) in the cost\-sensitive online phase with only a comparable cost, despite its larger initial overhead during the initial training\.

Table 7\.Average cosine similarity between prompts in the prompt bank, generated by three strategies, for the prediction horizon1 in the ETTh2 dataset\.𝐏l​o​w2\\mathbf\{P\}^\{2\}\_\{low\}𝐏l​o​w3\\mathbf\{P\}^\{3\}\_\{low\}Similarity with𝐏l​o​w1\\mathbf\{P\}^\{1\}\_\{low\}0\.7375\-0\.1827

#### 5\.3\.4\.Generalizability of Compositional Spectral Prompting\.

In Table[7](https://arxiv.org/html/2609.02093#S5.T7), to demonstrate that the composition of spectral basis prompts can effectively represent unseen patterns in the online phase, even with learning solely from the training phase, we compare three Spectral Prompt Banks \(i\.e\.,𝐏l​o​w1\\mathbf\{P\}^\{1\}\_\{low\},𝐏l​o​w2\\mathbf\{P\}^\{2\}\_\{low\}, and𝐏l​o​w3\\mathbf\{P\}^\{3\}\_\{low\}\) generated using the following strategies:

- •𝐏l​o​w1\\mathbf\{P\}^\{1\}\_\{low\}\(Optimal\): The prompt bank is trained during the training phase and further trained during the online phase\.
- •𝐏l​o​w2\\mathbf\{P\}^\{2\}\_\{low\}\(CoSPOT\): The prompt bank is trained during the training phase, and frozen during the online phase\.
- •𝐏l​o​w3\\mathbf\{P\}^\{3\}\_\{low\}: The prompt bank is randomly generated\.

The cosine similarity between the optimal Prompt Bank 1 \(i\.e\.,𝐏l​o​w1\\mathbf\{P\}^\{1\}\_\{low\}\) and each of Prompt Bank 2 \(i\.e\.,𝐏l​o​w2\\mathbf\{P\}^\{2\}\_\{low\}\) and Prompt Bank 3 \(i\.e\.,𝐏l​o​w3\\mathbf\{P\}^\{3\}\_\{low\}\) are shown in Table[7](https://arxiv.org/html/2609.02093#S5.T7)\.𝐏l​o​w1\\mathbf\{P\}^\{1\}\_\{low\}is the optimal prompt bank that can be obtained when unseen patterns \(i\.e\., patterns in online phase\) are included in the training\.𝐏l​o​w2\\mathbf\{P\}^\{2\}\_\{low\}shows a high degree of similarity to the𝐏l​o​w1\\mathbf\{P\}^\{1\}\_\{low\}, both in absolute terms and especially when compared to𝐏l​o​w3\\mathbf\{P\}^\{3\}\_\{low\}\. This indicates that, even without additional learning during the online phase, compositional spectral prompting can effectively represent incoming new patterns by recombining knowledge learned from frequency bases\.

#### 5\.3\.5\.Comparison of frequency\- and time\-domain prompting\.

To demonstrate that compositional spectral prompting effectively captures underlying time\-series patterns, we compare its frequency\-domain construction with an alternative time\-domain prompting strategy\. The time\-domain strategy replaces frequency bases with representation\-space prototypes derived from the time\-series backbone\. During training, input sequences are encoded by the pre\-trained time\-series backbone and clustered in the representation space, where each cluster centroid is treated as a distributional prototype and assigned a learnable prompt\. During the online phase, each incoming instance is encoded into the same representation space, matched to the nearest prototype, and guided by the corresponding prompt\. Thus, the time\-domain strategy retrieves a prompt based on representation similarity, whereas our frequency\-domain construction composes spectral basis prompts according to the amplitudes of decomposed frequency components\. The frequency\- and time\-domain prompts are trained under two distinct settings:

- •S1S\_\{1\}\(Optimal\): The prompt bank is trained during the training phase and further trained during the online phase\.
- •S2S\_\{2\}\(Freezing\): The prompt bank is trained during the training phase, and frozen during the online phase\.

Based on these settings, we define four variations of prompt banks as follows:

- •𝐏f​r​e​qS1\\mathbf\{P\}^\{S\_\{1\}\}\_\{freq\}and𝐏f​r​e​qS2\\mathbf\{P\}^\{S\_\{2\}\}\_\{freq\}: Frequency basis\-driven prompt banks generated under theS1S\_\{1\}andS2S\_\{2\}settings, respectively\.
- •𝐏t​i​m​eS1\\mathbf\{P\}^\{S\_\{1\}\}\_\{time\}and𝐏t​i​m​eS2\\mathbf\{P\}^\{S\_\{2\}\}\_\{time\}: Cluster\-based time\-domain prompt banks generated under theS1S\_\{1\}andS2S\_\{2\}settings, respectively\.

Table 8\.Performance and the total time \(in seconds\) required for the online phase of frequency\- and time\-domain prompting\.𝐏f​r​e​qS1\\mathbf\{P\}^\{S\_\{1\}\}\_\{freq\}𝐏f​r​e​qS2\\mathbf\{P\}^\{S\_\{2\}\}\_\{freq\}𝐏t​i​m​eS1\\mathbf\{P\}^\{S\_\{1\}\}\_\{time\}𝐏t​i​m​eS2\\mathbf\{P\}^\{S\_\{2\}\}\_\{time\}Metric12448124481244812448MSE0\.3720\.8591\.1480\.3980\.9321\.2450\.3951\.1011\.2630\.6351\.5971\.924MAE0\.4020\.5820\.6590\.4230\.6070\.6990\.5981\.0010\.9230\.7561\.0761\.273Time9361,0281,3307418118371,0311,1411,438843931958

Table 9\.Cosine similarity between finalized prompts ofS1S\_\{1\}andS2S\_\{2\}across frequency and time domains\.Sim​\(𝐏f​r​e​qS1,𝐏f​r​e​qS2\)\\textup\{Sim\}\(\\mathbf\{P\}^\{S\_\{1\}\}\_\{freq\},\\mathbf\{P\}^\{S\_\{2\}\}\_\{freq\}\)Sim​\(𝐏t​i​m​eS1,𝐏t​i​m​eS2\)\\textup\{Sim\}\(\\mathbf\{P\}^\{S\_\{1\}\}\_\{time\},\\mathbf\{P\}^\{S\_\{2\}\}\_\{time\}\)Cosine Similarity0\.73750\.2481

Table[8](https://arxiv.org/html/2609.02093#S5.T8)presents a comparative analysis of the performance of four variants and their cumulative online phase duration for prediction lengths \{1, 24, 48\} on the ETTh2 dataset\. Across both domains, continuously updating prompts during the online phase \(i\.e\.,S1S\_\{1\}\) yields superior performance, albeit at the cost of reduced efficiency\. In contrast, when prompts are learned only during the training phase and kept frozen during online inference \(i\.e\.,S2S\_\{2\}\), the time\-domain prompting strategy exhibits a substantial performance degradation\. This indicates that the time\-domain approach, which relies on clustering representations to characterize data distributions, fails to yield a representative prompt for the data distribution when shifts occur\. Conversely, frequency\-domain prompting remains robust, as it represents emerging distributions as combinations of underlying frequency bases, thereby preserving performance even in the presence of distributional shifts\. Additionally, we confirmed this difference from the perspective of the learned prompt itself\. In table[9](https://arxiv.org/html/2609.02093#S5.T9), we compared the similarity between the prompt finalized after the online phase was completed inS1S\_\{1\}and the prompt finalized after the end of the training phase inS2S\_\{2\}, across both domains\. Compared to the time domain, the frequency domain prompt exhibits a significantly higher similarity between the learned prompts in both cases\. This demonstrates that the effective learning of the underlying basis during the training phase alone enables the model to adequately handle the shift without additional learning in the online phase\.

#### 5\.3\.6\.Sensitivity Analysis\.

We present a sensitivity analysis for the hyperparametersγ\\gamma\(in Equation[7](https://arxiv.org/html/2609.02093#S4.E7)\) andδ\\delta\(in Equation[10](https://arxiv.org/html/2609.02093#S4.E10)\) utilized inCoSPOT\.

##### Hyperparameterγ\\gamma

To analyze the sensitivity ofCoSPOTto the hyperparameterγ\\gamma, which is used to filter for significant low\-frequency information when constructing compositional spectral prompts, we conduct an experiment presented in Figure[3](https://arxiv.org/html/2609.02093#S5.F3)\. Using the ETTh2 and ETTm1 datasets, we vary theγ\\gammaparameter in Equation[7](https://arxiv.org/html/2609.02093#S4.E7)across values of 0\.3, 0\.5, 0\.8, and 1, and observe the corresponding MSE for prediction horizons of 1, 24, and 48\. On both datasets, optimal performance is achieved whenγ=0\.3\\gamma=0\.3\. Performance degrades as more high\-frequency components are retained, with the most significant drop observed whenγ=1\\gamma=1\. This indicates that when constructing compositional spectral prompts from frequency components to capture the overall distribution of the time series data, the high\-frequency components are largely irrelevant to the overall distribution and instead act as noise\.

![Refer to caption](https://arxiv.org/html/2609.02093v1/gamma.png)Figure 3\.Sensitivity analysis ofγ\\gammain Equation[7](https://arxiv.org/html/2609.02093#S4.E7)\. MSE results are reported on \(a\) ETTh2 and \(b\) ETTm1 forH∈\{1,24,48\}H\\in\\\{1,24,48\\\}andγ∈\{0\.3,0\.5,0\.8,1\}\\gamma\\in\\\{0\.3,0\.5,0\.8,1\\\}\.![Refer to caption](https://arxiv.org/html/2609.02093v1/delta.png)Figure 4\.Sensitivity analysis of the geometric decay factorδ\\deltain Equation[10](https://arxiv.org/html/2609.02093#S4.E10)\. MSE results are reported on \(a\) ETTh2 and \(b\) ETTm1 forH∈\{24,48\}H\\in\\\{24,48\\\}andδ∈\{0\.3,0\.5,0\.8,1\}\\delta\\in\\\{0\.3,0\.5,0\.8,1\\\}\.
##### Geometric Decay Factorδ\\delta

To investigate the sensitivity ofCoSPOTto the geometric decay factorδ\\delta, which is utilized to reduce the influence of pseudo\-label unreliability and prediction errors for timestamps distant from the current observation when the prediction horizon is greater than 1, we conduct an experiment presented in Figure[4](https://arxiv.org/html/2609.02093#S5.F4)\. Using the ETTh2 and ETTm1 datasets, we vary theδ\\deltaparameter in Equation[10](https://arxiv.org/html/2609.02093#S4.E10)across values of 0\.3, 0\.5, 0\.8, and 1 and observe the corresponding MSE for prediction horizons of 24 and 48\. On both datasets, optimal performance is achieved whenδ\\deltais around 0\.8\. Aδ\\deltavalue of 1 which signifies no decay effect, leads to a performance drop\. This is due to the negative influence of unreliable pseudo\-labels generated by the time series backbone and the less accurate predictions for distant timestamps\. Therefore, to robustly adapt to continuous distribution shifts, we utilize an appropriate geometric decay factor to mitigate noise during the model’s training process\. We observed that a similar value ofδ\\delta\(i\.e\.,δ=0\.8\\delta=0\.8\) is consistently effective across various datasets\.

## 6\.Conclusion

In this paper, we present the first LLM\-based OTSF framework, calledCoSPOT, which excels in both continuous distribution shifts and extended online scenarios\. We devise a compositional spectral prompting strategy that captures the overall distribution \(i\.e\., overall pattern\) of each input by recombining learnable spectral basis prompts according to its frequency\-domain structure\. These distribution\-aware prompts guide the pre\-trained LLM to adapt its rich knowledge and transferability to continuously shifting online distributions, enabling efficient rapid adaptation in data\-scarce scenarios\. Furthermore, text descriptions containing recent pattern information enrich the limited online data without requiring additional training\.CoSPOTdemonstrates promising performance across various real\-world datasets and exhibits robust performance against distribution shifts and the extension of the online phase, highlighting its applicability to real\-world online forecasting scenarios\.

###### Acknowledgements\.

This work was supported by Institute of Information & communications Technology Planning & Evaluation \(IITP\) grant funded by the Korea government\(MSIT\) \(RS\-2022\-II220157\), National Research Foundation of Korea\(NRF\) grant funded by the Korea government\(MSIT\) \(RS\-2024\-00406985\), and National Research Foundation of Korea\(NRF\) funded by Ministry of Science and ICT \(RS\-2022\-NR068758\)\.

## GenAI Disclosure

We acknowledge the limited use of LLMs \(e\.g\., GPT\-5 and Claude\) for \(1\) improving the grammar, clarity, and stylistic variation of this paper, as well as reducing its length to comply with page limits, and \(2\) minor code refactoring and debugging for plotting and visualization\. All AI\-assisted revisions and code modifications were carefully reviewed and validated by the authors\. The core ideas, methodology, experiments, and interpretations presented in this work are entirely original contributions of the authors\.

## References

- Ansariet al\.\(2025\)A\. F\. Ansari, O\. Shchur, J\. Küken, A\. Auer, B\. Han, P\. Mercado, S\. S\. Rangapuram, H\. Shen, L\. Stella, X\. Zhang,et al\.Chronos\-2: from univariate to universal forecasting\.arXiv preprint arXiv:2510\.15821\.Cited by:[§5](https://arxiv.org/html/2609.02093#S5.p2.1)\.
- Aydoreet al\.\(2019\)S\. Aydore, T\. Zhu, and D\. P\. FosterDynamic local regret for non\-convex online forecasting\.Advances in neural information processing systems32\.Cited by:[§2](https://arxiv.org/html/2609.02093#S2.p1.1)\.
- Changet al\.\(2023\)C\. Chang, W\. Peng, and T\. ChenLlm4ts: two\-stage fine\-tuning for time\-series forecasting with pre\-trained llms\.CoRR\.Cited by:[§1](https://arxiv.org/html/2609.02093#S1.p4.1),[§2](https://arxiv.org/html/2609.02093#S2.p2.1),[§4\.3](https://arxiv.org/html/2609.02093#S4.SS3.p2.1),[§5](https://arxiv.org/html/2609.02093#S5.p2.1)\.
- Daset al\.\(2023\)A\. Das, W\. Kong, R\. Sen, and Y\. ZhouA decoder\-only foundation model for time\-series forecasting\.arXiv preprint arXiv:2310\.10688\.Cited by:[§5](https://arxiv.org/html/2609.02093#S5.p2.1)\.
- Dosovitskiyet al\.\(2020\)A\. Dosovitskiy, L\. Beyer, A\. Kolesnikov, D\. Weissenborn, X\. Zhai, T\. Unterthiner, M\. Dehghani, M\. Minderer, G\. Heigold, S\. Gelly,et al\.An image is worth 16x16 words: transformers for image recognition at scale\.arXiv preprint arXiv:2010\.11929\.Cited by:[§2](https://arxiv.org/html/2609.02093#S2.p3.1)\.
- Gultekin and Paisley \(2018\)S\. Gultekin and J\. PaisleyOnline forecasting matrix factorization\.IEEE Transactions on Signal Processing67\(5\),pp\. 1223–1236\.Cited by:[§2](https://arxiv.org/html/2609.02093#S2.p1.1)\.
- Jiaet al\.\(2022\)M\. Jia, L\. Tang, B\. Chen, C\. Cardie, S\. Belongie, B\. Hariharan, and S\. LimVisual prompt tuning\.InEuropean conference on computer vision,pp\. 709–727\.Cited by:[§2](https://arxiv.org/html/2609.02093#S2.p3.1)\.
- Jinet al\.\(2023\)M\. Jin, S\. Wang, L\. Ma, Z\. Chu, J\. Y\. Zhang, X\. Shi, P\. Chen, Y\. Liang, Y\. Li, S\. Pan,et al\.Time\-llm: time series forecasting by reprogramming large language models\.arXiv preprint arXiv:2310\.01728\.Cited by:[§1](https://arxiv.org/html/2609.02093#S1.p4.1),[§2](https://arxiv.org/html/2609.02093#S2.p2.1),[§4\.3](https://arxiv.org/html/2609.02093#S4.SS3.p2.1),[§5](https://arxiv.org/html/2609.02093#S5.p2.1),[§5](https://arxiv.org/html/2609.02093#S5.p3.1)\.
- Kuznetsov and Mohri \(2016\)V\. Kuznetsov and M\. MohriTime series prediction and online learning\.InConference on Learning Theory,pp\. 1190–1213\.Cited by:[§2](https://arxiv.org/html/2609.02093#S2.p1.1)\.
- Lauet al\.\(2025\)Y\. A\. Lau, Z\. Shao, and D\. YeungFast and slow streams for online time series forecasting without information leakage\.InThe Thirteenth International Conference on Learning Representations,Cited by:[§1](https://arxiv.org/html/2609.02093#S1.p2.1),[§2](https://arxiv.org/html/2609.02093#S2.p1.1),[§4\.2](https://arxiv.org/html/2609.02093#S4.SS2.p1.1),[§4\.3](https://arxiv.org/html/2609.02093#S4.SS3.p5.2),[§5](https://arxiv.org/html/2609.02093#S5.p1.1),[§5](https://arxiv.org/html/2609.02093#S5.p2.1),[§5](https://arxiv.org/html/2609.02093#S5.p3.1)\.
- Liuet al\.\(2023\)Y\. Liu, T\. Hu, H\. Zhang, H\. Wu, S\. Wang, L\. Ma, and M\. LongItransformer: inverted transformers are effective for time series forecasting\.arXiv preprint arXiv:2310\.06625\.Cited by:[§5](https://arxiv.org/html/2609.02093#S5.p2.1)\.
- Nieet al\.\(2022\)Y\. Nie, N\. H\. Nguyen, P\. Sinthong, and J\. KalagnanamA time series is worth 64 words: long\-term forecasting with transformers\.arXiv preprint arXiv:2211\.14730\.Cited by:[§1](https://arxiv.org/html/2609.02093#S1.p1.1),[§5](https://arxiv.org/html/2609.02093#S5.p2.1),[§5](https://arxiv.org/html/2609.02093#S5.p3.1)\.
- Phamet al\.\(2022\)Q\. Pham, C\. Liu, D\. Sahoo, and S\. C\. HoiLearning fast and slow for online time series forecasting\.arXiv preprint arXiv:2202\.11672\.Cited by:[§1](https://arxiv.org/html/2609.02093#S1.p2.1),[§2](https://arxiv.org/html/2609.02093#S2.p1.1),[§4\.2](https://arxiv.org/html/2609.02093#S4.SS2.p1.1),[§5](https://arxiv.org/html/2609.02093#S5.p1.1),[§5](https://arxiv.org/html/2609.02093#S5.p2.1),[§5](https://arxiv.org/html/2609.02093#S5.p3.1)\.
- Touvronet al\.\(2023\)H\. Touvron, T\. Lavril, G\. Izacard, X\. Martinet, M\. Lachaux, T\. Lacroix, B\. Rozière, N\. Goyal, E\. Hambro, F\. Azhar,et al\.Llama: open and efficient foundation language models\.arXiv preprint arXiv:2302\.13971\.Cited by:[§5](https://arxiv.org/html/2609.02093#S5.p3.1)\.
- Wanget al\.\(2024\)S\. Wang, H\. Wu, X\. Shi, T\. Hu, H\. Luo, L\. Ma, J\. Y\. Zhang, and J\. ZhouTimemixer: decomposable multiscale mixing for time series forecasting\.arXiv preprint arXiv:2405\.14616\.Cited by:[§5](https://arxiv.org/html/2609.02093#S5.p2.1)\.
- Wanget al\.\(2022a\)Y\. Wang, Z\. Huang, and X\. HongS\-prompts learning with pre\-trained transformers: an occam’s razor for domain incremental learning\.Advances in Neural Information Processing Systems35,pp\. 5682–5695\.Cited by:[§2](https://arxiv.org/html/2609.02093#S2.p3.1)\.
- Wanget al\.\(2022b\)Z\. Wang, Z\. Zhang, C\. Lee, H\. Zhang, R\. Sun, X\. Ren, G\. Su, V\. Perot, J\. Dy, and T\. PfisterLearning to prompt for continual learning\.InProceedings of the IEEE/CVF conference on computer vision and pattern recognition,pp\. 139–149\.Cited by:[§2](https://arxiv.org/html/2609.02093#S2.p3.1)\.
- Wenet al\.\(2023\)Q\. Wen, W\. Chen, L\. Sun, Z\. Zhang, L\. Wang, R\. Jin, T\. Tan,et al\.Onenet: enhancing time series forecasting models under concept drift by online ensembling\.Advances in Neural Information Processing Systems36,pp\. 69949–69980\.Cited by:[§1](https://arxiv.org/html/2609.02093#S1.p2.1),[§2](https://arxiv.org/html/2609.02093#S2.p1.1),[§4\.2](https://arxiv.org/html/2609.02093#S4.SS2.p1.1),[§5](https://arxiv.org/html/2609.02093#S5.p1.1),[§5](https://arxiv.org/html/2609.02093#S5.p2.1),[§5](https://arxiv.org/html/2609.02093#S5.p3.1)\.
- Wuet al\.\(2022\)H\. Wu, T\. Hu, Y\. Liu, H\. Zhou, J\. Wang, and M\. LongTimesnet: temporal 2d\-variation modeling for general time series analysis\.arXiv preprint arXiv:2210\.02186\.Cited by:[§1](https://arxiv.org/html/2609.02093#S1.p1.1)\.
- Wuet al\.\(2021\)H\. Wu, J\. Xu, J\. Wang, and M\. LongAutoformer: decomposition transformers with auto\-correlation for long\-term series forecasting\.Advances in neural information processing systems34,pp\. 22419–22430\.Cited by:[§1](https://arxiv.org/html/2609.02093#S1.p1.1)\.
- Xuet al\.\(2023\)Z\. Xu, A\. Zeng, and Q\. XuFITS: modeling time series with10​k10kparameters\.arXiv preprint arXiv:2307\.03756\.Cited by:[§4\.2](https://arxiv.org/html/2609.02093#S4.SS2.p5.1)\.
- Yiet al\.\(2023\)K\. Yi, Q\. Zhang, W\. Fan, S\. Wang, P\. Wang, H\. He, N\. An, D\. Lian, L\. Cao, and Z\. NiuFrequency\-domain mlps are more effective learners in time series forecasting\.Advances in Neural Information Processing Systems36,pp\. 76656–76679\.Cited by:[§1](https://arxiv.org/html/2609.02093#S1.p4.1)\.
- Zenget al\.\(2023\)A\. Zeng, M\. Chen, L\. Zhang, and Q\. XuAre transformers effective for time series forecasting?\.InProceedings of the AAAI conference on artificial intelligence,Vol\.37,pp\. 11121–11128\.Cited by:[§5](https://arxiv.org/html/2609.02093#S5.p2.1)\.
- Zhang and Yan \(2023\)Y\. Zhang and J\. YanCrossformer: transformer utilizing cross\-dimension dependency for multivariate time series forecasting\.InThe eleventh international conference on learning representations,Cited by:[§1](https://arxiv.org/html/2609.02093#S1.p1.1)\.
- Zhouet al\.\(2021\)H\. Zhou, S\. Zhang, J\. Peng, S\. Zhang, J\. Li, H\. Xiong, and W\. ZhangInformer: beyond efficient transformer for long sequence time\-series forecasting\.InProceedings of the AAAI conference on artificial intelligence,Vol\.35,pp\. 11106–11115\.Cited by:[§1](https://arxiv.org/html/2609.02093#S1.p1.1)\.
- Zhouet al\.\(2022a\)T\. Zhou, Z\. Ma, Q\. Wen, L\. Sun, T\. Yao, W\. Yin, R\. Jin,et al\.Film: frequency improved legendre memory model for long\-term time series forecasting\.Advances in neural information processing systems35,pp\. 12677–12690\.Cited by:[§4\.2](https://arxiv.org/html/2609.02093#S4.SS2.p5.1)\.
- Zhouet al\.\(2022b\)T\. Zhou, Z\. Ma, Q\. Wen, X\. Wang, L\. Sun, and R\. JinFedformer: frequency enhanced decomposed transformer for long\-term series forecasting\.InInternational conference on machine learning,pp\. 27268–27286\.Cited by:[§1](https://arxiv.org/html/2609.02093#S1.p1.1),[§1](https://arxiv.org/html/2609.02093#S1.p4.1),[§2](https://arxiv.org/html/2609.02093#S2.p4.1)\.
- Zhouet al\.\(2023\)T\. Zhou, P\. Niu, L\. Sun, R\. Jin,et al\.One fits all: power general time series analysis by pretrained lm\.Advances in neural information processing systems36,pp\. 43322–43355\.Cited by:[§1](https://arxiv.org/html/2609.02093#S1.p4.1),[§2](https://arxiv.org/html/2609.02093#S2.p2.1),[§4\.3](https://arxiv.org/html/2609.02093#S4.SS3.p2.1),[§5](https://arxiv.org/html/2609.02093#S5.p2.1)\.