Multi-Order Wavelet Derivative Transform for Deep Time Series Forecasting

TL;DR

The multi-order Wavelet Derivative Transform (WDT) grounded in the WT is introduced, enabling the extraction of time-aware patterns spanning both the overall trend and subtle fluctuations, and achieves state-of-the-art forecasting accuracy while retaining high computational efficiency.

Abstract

In deep time series forecasting, the Fourier Transform (FT) is extensively employed for frequency representation learning. However, it often struggles in capturing multi-scale, time-sensitive patterns. Although the Wavelet Transform (WT) can capture these patterns through frequency decomposition, its coefficients are insensitive to change points in time series, leading to suboptimal modeling. To mitigate these limitations, we introduce the multi-order Wavelet Derivative Transform (WDT) grounded in the WT, enabling the extraction of time-aware patterns spanning both the overall trend and subtle fluctuations. Compared with the standard FT and WT, which model the raw series, the WDT operates on the derivative of the series, selectively magnifying rate-of-change cues and exposing abrupt regime shifts that are particularly informative for time series modeling. Practically, we embed the WDT into a multi-branch framework named WaveTS, which decomposes the input series into multi-scale time-frequency coefficients, refines them via linear layers, and reconstructs them into the time domain via the inverse WDT. Extensive experiments on ten benchmark datasets demonstrate that WaveTS achieves state-of-the-art forecasting accuracy while retaining high computational efficiency.

Figures (14)

• Figure 1: (a)–(d) Distinct temporal variations may yield identical frequency spectra after applying the Fourier transform, revealing the spectral ambiguity. (e)(f) The Fourier-derivative makes the spectrum stationary yet discards macro-trend information. (e)(g)(h) The Wavelet Derivative Transform (WDT) retains those trends while offering complementary detail. (i) Relative to wavelet coefficients in the standard wavelet transform (blue curve), WDT coefficients pinpoint abrupt regime shifts (red curve), yielding a sharper cue for non-stationary dynamics.
• Figure 2: The main architecture of our proposed WaveTS. For each branch where $n$ starts from 1 to $N$, first-order to $N$-th order of Wavelet Derivative Transforms (WDT) are applied to capture multi-scale representations, facilitating more effective time series modeling. LH and LL are frequency coefficients that capture representations at different scales, corresponding to micro and macro patterns, respectively.
• Figure 3: Model performance variations under different scales and orders in the WDT module. Results are collected from forecasting 96 experiments, each using its optimal input length.
• Figure 4: Ablation studies of the WDT on PEMS datasets. FourierTS and WaveletTS denote the substitution of the WDT with the Discrete Fourier Transform and the Discrete Wavelet Transform based on the WaveTS.
• Figure 5: Visualization of Approximation Coefficients (LL) and Detail Coefficients (LHs) in the WDT. The data are collected from the second-order derivation (order=2) at a wavelet decomposition scale of 2 (scale=2). We select the "OT" variate with the input length is 720 for each dataset.

Theorems & Definitions (6)

• Definition 3.1: Wavelet Derivative Transform (WDT)
• Definition 3.2: Inverse Wavelet Derivative Transform (iWDT)
• Lemma A.1
• proof
• Theorem A.2: Energy Conservation of WDT
• proof