Neuralized Multi-Wavelet Decomposition for Time Series Classification and Forecasting

Xiaohan Jiang; Jingyuan Wang; Jiahao Ji; Yongyao Wang; Chen Yang; Junjie Wu

cs.LGSep 24, 2026

Neuralized Multi-Wavelet Decomposition for Time Series Classification and Forecasting

Authors: Xiaohan Jiang, Jingyuan Wang, Jiahao Ji, Yongyao Wang, Chen Yang, Junjie Wu

Organizations: School of Computer Science and Engineering, Beihang University, Beijing, China · School of Computer Science and Engineering and the School of Economics and Management, Beihang University, Beijing, China · School of Computer Science and Engineering, Beihang University, Beijing, China, and are also with the MOE Engineering Research Center of Advanced Computer Application Technology, Beihang University, China · School of Economics and Management, Beihang University, Beijing 100191, China, and the Key Laboratory of Data and Decision Intelligence (Beihang University), Ministry of Industry and Information Technology, Beijing 100191, China

Abstract

Time series analysis is fundamental in domains such as finance, healthcare, and meteorology. Real-world time series often exhibit multiscale characteristics shaped by diverse latent factors, resulting in intricate temporal patterns and rich frequency structures. However, existing approaches typically focus on either frequency-domain decomposition or time-domain pattern extraction in isolation, neglecting their joint structure. This decoupled modeling limits representation expressiveness and undermines performance in tasks requiring simultaneous temporal and spectral reasoning. To address this gap, we propose m-WCN, a novel end-to-end deep learning framework that neuralizes multi-wavelet decomposition for joint extraction of temporal patterns and frequency components. By approximating the classical GHM multi-wavelet transform with trainable convolutional operators and enforcing orthogonality constraints, m-WCN produces interpretable multi-resolution representations. Built on this foundation, we introduce two task-specific architectures: TFBC for time series classification, which boosts discriminative features across frequency scales, and FTB for forecasting, which ensembles frequency-aware predictors. Extensive experiments on 64 UCR datasets and seven public forecasting benchmarks demonstrate the effectiveness of our approach. Built on the neuralized m-WCN, our TFBC and FTB outperform various baseline models across diverse datasets, achieving average improvements of 19.97% in classification and 19.92% in forecasting tasks.

Figures & tables

Notations	Description
$\bm{s}=(s_{0},\ldots,s_{T})$	The input series for m-WCN and TSC.
$\bm{c}=(c_{1},\ldots,c_{M})$	The one-hot category label for TSC.
$\bm{s}_{t}=(s_{t-L},\ldots,s_{t})$	The input series for m-WCN and TSF.
$\bm{z}_{t}=(s_{t+1},\ldots,s_{t+L^{\prime}})$	The series to be predicted for TSF.
$\bm{X}^{h_{n}},n\in\{1,\ldots,N\}$	The $n$ -th high frequency component of $\bm{s}$ .
$\bm{X}^{l_{n}},\;n\in\{1,\ldots,N\}$	The $n$ -th low frequency component of $\bm{s}$ .

TABLE I: Notations used in multi-wavelet decomposition and m-WCN.

Fig. 1: Comparison of single-wavelet and multi-wavelet decomposition (Pat: Pattern). ${\bf A1}$ and ${\bf A2}$ denote the temporal patterns associated with the morning peak and the evening peak, respectively. Single-wavelet decomposition processes the mixed patterns ${\bf A1}$ and ${\bf A2}$ together, whereas multi-wavelet decomposition first separates them through pre-filtering and then performs frequency decomposition on the separated pattern components.

Fig. 2: Illustration of the m-WCN framework. Dec: Decomposition. Conv: Convolution.

\downarrow 2

denotes the downsampling operation with a sample rate of 2.

*

is the convolutional operator.

Fig. 3: The TFBC network for classification: A three-level example.

Fig. 4: The FTB model for forecasting: A three-level example. FRP: Frequency Representation Pre-training.

Dataset	miniRocket	Hivecote2	TS-Chief	OS-CNN	TimeURL	MILLET	mWDN	TFBC	Dataset	miniRocket	Hivecote2	TS-Chief	OS-CNN	TimeURL	MILLET	mWDN	TFBC
Comput/1	0.268	0.240	0.300	0.293	0.264	0.240	0.360	0.220	CrkX/4	0.179	0.172	0.179	0.145	0.267	0.154	0.216	0.168
ElecD/1	0.258	0.274	0.284	0.276	0.273	0.271	0.342	0.235	CrkY/4	0.172	0.154	0.200	0.133	0.264	0.149	0.172	0.141
LrgKA/1	0.125	0.080	0.232	0.104	0.087	0.096	0.152	0.067	CrkZ/4	0.172	0.141	0.169	0.137	0.254	0.136	0.162	0.169
RefrD/1	0.520	0.448	0.429	0.497	0.379	0.491	0.493	0.424	Haptics/4	0.471	0.445	0.471	0.490	0.451	0.513	0.461	0.427
ScrnT/1	0.539	0.429	0.501	0.474	0.467	0.405	0.480	0.461	ISkat/4	0.524	0.456	0.473	0.571	0.556	0.533	0.566	0.445
SmKA/1	0.173	0.163	0.176	0.279	0.233	0.227	0.221	0.168	ToeS1/4	0.039	0.035	0.031	0.046	0.066	0.039	0.031	0.026

TABLE II: Classification performance comparison on 64 UCR time series datasets regarding error rate. The Dataset column is in the form of “Abbr/Type ID”, where the dataset’s full name and corresponding type are in Sec. 1.1 of SM. The best result is bold, while the second-best result is underlined.

Metric	mRocket	Hivecote2	TS-Chief	OS-CNN	TimeURL	MILLET	mWDN	TFBC
Avg. Err.	0.187	0.171	0.186	0.192	0.173	0.186	0.184	0.146
Count	3	7	6	7	12	5	11	34
Rank (a)	5.11	3.53	5.00	5.34	4.17	5.00	4.63	1.88
Rank (g)	4.58	3.10	4.32	4.61	3.32	4.32	3.71	1.59

TABLE III: TSC performance summary. Avg. Err. denotes the average classification error rate. Count: winning count. Rank(a)/Rank(g): the average ranking in terms of arithmetic and geometry.

Metric	r/fp	r/wd	w/o m-WCN	w/o OR	w/o FCL	TFBC
Avg. Err.	0.162	0.170	0.294	0.156	0.154	0.146
Count	9	15	5	10	14	26
Rank(a)	3.187	3.516	5.125	2.781	2.828	2.281
Rank(g)	2.790	2.943	4.641	2.487	2.471	1.921

TABLE IV: Ablation study on the 64 UCR datasets. Avg. Err. denotes the average classification error rate. Count: winning count. Rank(a)/Rank(g): the average ranking in terms of arithmetic and geometry.

Dataset	Horizon	Metric	Auto.	Fed.	DLinear	Basis.	Path.	TimeLLM	PatchTST	FTB
ETTh1	96	MSE	0.071	0.079	0.056	0.055	0.057	0.058	0.057	0.052
	96	MAE	0.206	0.215	0.180	0.178	0.180	0.183	0.179	0.165
	192	MSE	0.114	0.104	0.071	0.072	0.075	0.072	0.076	0.064
	192	MAE	0.262	0.245	0.204	0.204	0.208	0.202	0.209	0.180
	336	MSE	0.107	0.119	0.098	0.086	0.076	0.082	0.093	0.063
	336	MAE	0.258	0.270	0.244	0.227	0.216	0.231	0.240	0.201

TABLE V: Performance comparison of TSF with different prediction lengths. The best result is bold, while the second-best result is underlined. Auto.=Autoformer, Fed.=Fedformer, Basis.=Basisformer, and Path.=PathFormer.

Metric	Auto.	Fed.	DLinear	Basis.	Path.	TimeLLM	PatchTST	FTB
Avg. MSE	0.192	0.147	0.163	0.146	0.153	0.129	0.126	0.112
Avg. MAE	0.303	0.265	0.271	0.248	0.242	0.238	0.233	0.216
Count	0	0	1	0	5	0	4	50
Rank(a)	7.42	5.61	5.55	4.87	3.72	4.49	3.17	1.17
Rank(g)	7.26	5.35	5.22	4.65	3.21	4.26	2.86	1.12

TABLE VI: TSF performance summary over seven forecasting datasets. Count denotes the number of wins including ties over all metric-horizon comparisons. Rank(a) and Rank(g) denote the arithmetic and geometric average ranks.

Horizon	96		192		336		720
Metric	MSE	MAE	MSE	MAE	MSE	MAE	MSE	MAE
FTB	0.203	0.322	0.247	0.351	0.285	0.378	0.323	0.421
r/ fp	0.215	0.352	0.266	0.386	0.288	0.359	0.342	0.432
r/ wd	0.247	0.359	0.319	0.366	0.340	0.389	0.373	0.486
w/o mwcn	0.256	0.389	0.307	0.395	0.332	0.425	0.388	0.502
w/o or	0.238	0.333	0.258	0.362	0.288	0.381	0.359	0.453

TABLE VII: Ablation study of FTB on the Electricity dataset.

Method	TT (s/epoch)	TM (MB)	IT (s)	IM (MB)
Autoformer	66.27	3848.13	22.53	738.82
Fedformer	237.78	2612.12	15.32	677.35
DLinear	7.54	21.14	4.62	19.63
Basisformer	42.49	58.47	3.52	39.15
PathFormer	58.39	2227.41	6.82	367.09
PatchTST	8.91	153.49	3.00	68.23

TABLE VIII: Average computational cost on the forecasting experiments. TT, TM, IT, and IM denote training time, training memory, inference time, and inference memory, respectively.

Fig. 5: Visualization of forecasting results. Subfigures (a) and (b) show the prediction results of FTB, while subfigures (c) and (d) show the prediction results of PatchTST on the same samples.

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Multivariate time series forecasting remains a challenge due to the complexity of local temporal dynamics and global dependencies across multiple variables. In this paper, we propose \textbf{N}eighboring \textbf{P}atching \textbf{Mixer} (\textbf{NPMixer}), a hierarchical architecture featuring a Learnable Stationary Wavelet Transform that adaptively learns filter coefficients to decompose signals into trend and detail components in a data-dependent manner. Our framework introduces a Neighboring Mixer Block that captures local temporal dynamics through a series of hierarchical MLP layers operating on non-overlapping patches. Specifically, the mixer block utilizes MLPs to learn temporal patterns within and across these patches, expanding the receptive field to capture multi-scale dependencies. A Channel-Mixing Encoder is applied to high-frequency components to learn channel correlations while preserving the stability of the underlying global trend. Extensive experiments on seven benchmark datasets demonstrate that NPMixer consistently outperforms state-of-the-art models, achieving better performance in 20 out of 28 (

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Abstract

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