FreSia: Frequency-Semantic Instantiation and Alignment for Multivariate Time Series Analysis

Yubo Wang; Hui He; Hezhe Qiao; Guoqing Ji; Zhendong Niu

cs.AIOct 5, 2026

FreSia: Frequency-Semantic Instantiation and Alignment for Multivariate Time Series Analysis

Authors: Yubo Wang, Hui He, Hezhe Qiao, Guoqing Ji, Zhendong Niu

Organizations: School of Computer Science and Technology, Beijing Institute of Technology, No.5 Zhongguancun South Street, Beijing, 100081, China · School of Computing and Information Systems, Singapore Management University, 80 Stamford Road, 178902, Singapore

Abstract

Large Language Models (LLMs) have shown strong potential in multivariate time series forecasting and anomaly detection. Existing studies predominantly inject temporal information into LLMs via direct numerical tokenization or heuristic textual descriptions. However, LLMs still face difficulty in perceiving the underlying structural patterns of numerical time series, particularly the seasonal and trend components obscured by discrete numerical tokens. To bridge this gap, we propose FreSia, a frequency-aware framework that establishes an effective alignment between the semantic space of LLMs and the frequency space of time series. Specifically, FGPrompt, a Frequency-Guided Prompt mechanism within FreSia, distills the frequency-domain structures of time series and projects them into prompts tailored to the semantic space of LLMs. Furthermore, we introduce a Global-driven Context Learning (GCL) component, which uses a global CLS-driven probe to generate global context to bridge the time-frequency domain gap and fuse the multi-modal information. Experiments on eight forecasting benchmarks show that FreSia achieves average improvements of 13.48% and 8.06% in MSE and MAE, respectively.

Figures & tables

Panel of Figure 1 — Figure 1 : (a) Representation gap between raw numerical prompts and frequency-guided semantic prompts. (b) Top-K frequency components explain a large portion of spectral energy across eight benchmark datasets. This motivates the use of dominant frequency components as compact structural primitives for frequency-guided semantic instantiation.

Figure 2 : Framework of FreSia . ❶ Frequency-Guided Prompt (FGPrompt) injects frequency-domain information into prompts and the frequency pool. ❷ TSEncoder captures temporal dependencies and generates a CLS representation. ❸ Global-driven Context Learning (GCL) bridges the time-frequency domain gap and fuses multi-modal information.

Property	Numerical Range	Linguistic Descriptor
Trend ( $\hat{k}$ )	$\hat{k}>0.5$	strong upward trend
	$0.1<\hat{k}\leq 0.5$	slight upward trend
	$-0.1\leq\hat{k}\leq 0.1$	stable trend
	$-0.5\leq\hat{k}<-0.1$	slight downward trend
	$\hat{k}<-0.5$	strong downward trend
Volatility ( $v$ )	$v>1.0$	high volatility

Table 1: Mapping rules for linguistic descriptors in Frequency-Guided Prompts.

Models		FreSia		LangTime		TimeCMA		Time-LLM		UniTime
Metrics		MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETTm1	96	0.310	0.348	0.319	0.348	0.312	0.351	0.359	0.381	0.322	0.363
	192	0.352	0.374	0.368	0.375	0.361	0.378	0.383	0.393	0.366	0.387
	336	0.378	0.396	0.413	0.402	0.392	0.401	0.416	0.414	0.398	0.407
	720	0.430	0.427	0.487	0.439	0.453	0.438	0.483	0.449	0.454	0.440
	Avg	0.367	0.386	0.397	0.391	0.380	0.392	0.410	0.409	0.385	0.399

Table 2 : Results of multivariate time series forecasting. The input sequence length is 36 for the ILI dataset and 96 for others. The best results are in red and the second best are blue.

Models		FreSia		OFA		iTransformer		TimesNet		RLinear
Metrics		MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETTm1	96	0.310	0.348	0.335	0.369	0.334	0.368	0.338	0.375	0.355	0.376
	192	0.352	0.374	0.374	0.385	0.377	0.391	0.374	0.387	0.387	0.392
	336	0.378	0.396	0.407	0.406	0.426	0.420	0.410	0.411	0.424	0.415
	720	0.430	0.427	0.469	0.442	0.491	0.459	0.478	0.450	0.487	0.450
	Avg	0.367	0.386	0.396	0.401	0.407	0.410	0.400	0.406	0.413	0.408

Table 3 : Results of multivariate time series forecasting. The input sequence length is 36 for the ILI dataset and 96 for others. The best results are in red and the second best are blue.

Dataset	Metric	FreSia	TSINR	FPT	TimesNet	Anomaly.	LightTS	DLinear
SMD	P	88.56	83.09	87.27	87.98	78.72	87.04	87.27
	R	84.07	80.46	81.08	81.54	65.43	78.39	80.99
	F1	86.25	81.76	84.06	84.64	71.46	82.49	84.01
PSM	P	97.61	99.21	98.55	98.51	98.76	98.29	98.66
	R	96.22	89.37	95.79	96.27	83.25	93.60	94.70
	F1	96.91	94.04	97.15	97.38	90.35	95.89	96.64

Table 4 : Results of anomaly detection. The input sequence length is 100. The best results are in red and the second best are blue.

Figure 3 : Sensitivity analysis of frequency pool size.

Datasets	ETTh2		ETTm2		ECL
Metrics	MSE	MAE	MSE	MAE	MSE	MAE
w/o LLM	0.357	0.388	0.288	0.331	0.186	0.271
w/o TSE	0.393	0.417	0.321	0.358	0.382	0.424
w/o PE	0.353	0.387	0.279	0.325	0.177	0.274
w/o CA	0.356	0.390	0.291	0.333	0.179	0.268
w/o CLS	0.356	0.388	0.282	0.327	0.179	0.276

Table 5 : Ablation studies of FreSia

Dataset	FreSia		TimeCMA-style		w/o Freq. Path.		w/o LLM
	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
Weather	0.236	0.263	0.237	0.264	0.238	0.264	0.252	0.273
Exchange	0.339	0.394	0.383	0.417	0.392	0.420	0.404	0.425
ILI	1.563	0.779	1.577	0.775	1.601	0.782	1.696	0.787
Avg.	0.713	0.479	0.732	0.485	0.743	0.489	0.784	0.495

Table 6: Disentangling the effects of frequency-domain information and LLM semantic encoding.

Figure 4 : Effect of mixed prompting strategy on Exchange.

Panel of Figure 5 — Figure 5 : Attention difference analysis of CLS-driven calibration.

Panel of Figure 6 — Figure 6 : ETTh1 frequency pool analysis case1.

Panel of Figure 7 — Figure 7 : ETTh1 frequency pool analysis case2.

Figure 9 : Comparison with different losses.

Model	Test Time (s)	GPU Memory (MiB)	Params (M)
TimeCMA	16.78	2690	29.53
OFA	1697.66	916	3.92
iTransformer	129.98	2248	4.96
TimeKAN	44.32	3556	0.044
FreSia	11.29	2936	28.29

Table 7: Efficiency comparison. We report test time, peak GPU memory usage, and model parameters under the same evaluation setting.

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	Time Residual	Time-Freq. Residual	Freq. Residual
SMD	0.863	0.392	0.162
PSM	0.969	0.875	0.781
SWaT	0.943	0.913	0.846
MSL	0.856	0.171	0.080
SMAP	0.787	0.641	0.637
Avg.	0.884	0.598	0.501

Appendix

Table 1: F1 comparison of anomaly scoring rules. Time Residual denotes the standard reconstruction-error score used by FreSia. Time-Freq. Residual combines time-domain and frequency-domain residuals with equal weight. Freq. Residual uses only the frequency residual.

Models		FreSia		TimeKAN		PaiFilter		TexFilter		FredFormer
Metrics		MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
ETTm1	96	0.310	0.348	0.322	0.361	0.318	0.358	0.321	0.361	0.328	0.363
	192	0.352	0.374	0.357	0.383	0.364	0.383	0.367	0.387	0.367	0.382
	336	0.378	0.396	0.383	0.402	0.396	0.406	0.401	0.409	0.395	0.403
	720	0.429	0.427	0.446	0.436	0.456	0.444	0.477	0.448	0.454	0.440
	Avg	0.367	0.386	0.377	0.395	0.384	0.398	0.392	0.401	0.386	0.397

Appendix

Table C.1 : Comparison with frequency-domain baselines. The input sequence length is 36 for the ILI dataset and 96 for others. The best results are in red and the second best are blue.

Variant	ETTm2		ECL		Avg.
Variant	MSE	MAE	MSE	MAE	MSE	MAE
Adaptive-Quantile	0.276	0.321	0.177	0.274	0.227	0.298
Continuous	0.274	0.320	0.178	0.274	0.226	0.297
Freq-only	0.276	0.322	0.179	0.276	0.228	0.299
FreSia	0.274	0.320	0.175	0.271	0.224	0.296

Appendix

Table D.1: Robustness to auxiliary trend/volatility mappings. The results are averaged over all horizons.

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Recent advances in Large Language Models (LLMs) have spurred cross-modal solutions for time-series forecasting. However, existing methods rely heavily on textual prompts for modality alignment-introducing nontrivial computational overhead and failing to leverage the rich spectral dynamics inherent in time-series data. To enable prompt-free, frequency-aware adaptation of frozen LLMs, we propose FM-LLM (Frequency-Enhanced Mixture-of-Experts for adapting LLMs to Time Series Forecasting), an autoregressive framework grounded in constrained asymmetric coupling. A Fourier Analysis Network (FAN)-based spectral token aligner injects structured harmonic representations directly into the frozen LLM with numerical compatibility. An asymmetric Mixture-of-Experts (MoE) decoder enforces role separation: shared experts with lightweight FAN layers reconstruct the global periodic backbone, while routed experts-restricted to standard FFNs-specialize in modeling non-periodic residual dynamics. A time-frequency hybrid loss function jointly optimizes temporal accuracy and spectral consistency, mitigating error accumulation during long-horizon autoregressive rollouts. Evaluated across eleven public benchmarks, FM-LLM achieves state-of-the-art performance on 59 out of 78 evaluation metrics. Compared to the strongest autoregressive LLM-based baseline, it delivers average improvements of 5.3% in MSE and 5.6% in MAE, with maximum gains reaching 8.0% for MSE and 8.4% for MAE. FM-LLM also demonstrates robust transferability, maintaining superior performance in 10% few-shot and zero-shot forecasting scenarios.

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