DecoKAN: Interpretable Crypto Forecasting

• DecoKAN is an interpretable multivariate time series forecasting framework designed for cryptocurrency markets, integrating frequency decoupling and nonlinear modeling.
• The framework employs multi-level discrete wavelet transforms for precise frequency separation and Kolmogorov-Arnold Network mixers for hierarchical spline-based function approximation.
• Its symbolic analysis pipeline prunes model complexity and extracts explicit formulas, achieving state-of-the-art accuracy and transparent, human-readable output.

DecoKAN is an interpretable multivariate time series forecasting framework tailored for the unique dynamical properties of cryptocurrency markets. The methodology leverages a multi-level Discrete Wavelet Transform (DWT) for nonparametric frequency decoupling and Kolmogorov-Arnold Network (KAN) mixers for hierarchical, spline-based nonlinear function approximation. The architecture is augmented by a symbolic analysis pipeline for output sparsification, pruning, and analytical formula extraction, producing transparent, human-readable representations of learned patterns. DecoKAN demonstrably closes the gap between predictive performance and interpretability, achieving state-of-the-art accuracy (lowest MSE across BTC, ETH, XMR) while generating explicit symbolic forecasts, thereby supporting trustworthy decision-making in volatile financial contexts (Gao et al., 23 Dec 2025).

1. Multi-Level Discrete Wavelet Transform: Mathematical Formulation and Frequency Separation

DecoKAN begins with a multi-level DWT decomposition of the input time series. Let $X_L \in \mathbb{R}^{C \times L}$ denote the normalized multivariate input over a look-back window $L$. A compactly supported orthogonal wavelet (Daubechies 4) is selected; the mother scaling function $\phi(t)$ and corresponding wavelet $\psi(t)$ satisfy the two-scale equations: $$\phi(t) = \sqrt{2}\sum_{n}h[n]\phi(2t-n), \quad \psi(t) = \sqrt{2}\sum_{n}g[n]\phi(2t-n)$$ where $h[n]$ and $g[n]$ are the db4-specific low-pass and high-pass filter coefficients.

One DWT level on a 1D series $x$ yields approximation coefficients $a_k = \sum_n h[n] x[2k-n]$ and detail coefficients $d_k = \sum_n g[n] x[2k-n]$. Multi-level ($m$) DWT recursively applies this procedure to approximations, yielding $$\left(X_{A_m}, X_{D_m}, \dots, X_{D_1}\right) = \mathrm{DWT}(X_L, \psi, m)$$, as described in Eq. (3), with each coefficient series of dimension $C \times L_i$. The inverse DWT reconstructs predictions via upsampling and convolution with transposed filters: $$Y = \mathrm{IDWT}(Y_{A_m}, Y_{D_m}, \dots, Y_{D_1})$$ (cf. Eq. (14)).

Frequency separation is structurally enforced: the $m$-th approximation $X_{A_m}$ models slow-varying, low-frequency socio-economic trends, while $X_{D_1}, \dots, X_{D_m}$ capture increasingly coarse, high-frequency market oscillations (speculative volatility). Figure 1 illustrates this orthogonal decomposition.

2. Kolmogorov-Arnold Network Mixers: Spline-Based Modeling and Hierarchical Resolution

Each decomposed coefficient series $X_{w_i}$ is routed to a dedicated "Resolution Branch" comprising KAN Mixer blocks. The KANLinear module employs learnable univariate activations: $$\phi(x) = w_b b(x) + w_s \sum_{j=0}^{G+k-1} c_j B_j(x)$$ as per Eq. (11); $b(x)$ is a fixed base (SiLU), $B_j(x)$ are B-spline basis functions of order $k$ across a grid of size $G$, and $w_b, w_s, c_j$ are optimized parameters. This parametrization is a Kolmogorov-Arnold universal approximator with intrinsic interpretability due to its explicit spline expansion.

Each Resolution Branch embeds its input with overlapping patches and applies two KAN Mixer blocks (Eqs. (4)-(6)). A standard block consists of:

1. Temporal KAN along the patch axis ($N_i$),
2. Feature KAN along the embedding axis ($d$), with LayerNorm and residual connections (Eqs. (7)-(10)). Crucially, per-branch modeling circumvents cross-frequency attention, thereby eliminating spectral entanglement and facilitating transparent decomposition.

3. Symbolic Analysis Pipeline: Sparsification, Pruning, Symbolization

Symbolic interpretability is realized during and after training. Regularization loss (Eq. (12)) enforces coefficient sparsity and entropy minimization: $$L_{\rm reg} = \lambda_1 \sum_{\text{edges}} |\phi|_{1} + \lambda_2 \sum_{\text{edges}} S(\phi)$$ where $S(\phi) = -\sum_j p_j \log p_j$, $p_j = |c_j|/\sum_k |c_k|$. This bias towards low-entropy spline weights results in concise activations.

Edge pruning (Algorithm 1, Phase 2) removes connections with activation $L_2$-norm below threshold $\tau$. Surviving edges are symbolized via regression over a function library (polynomial, sinusoidal, $\tanh$ forms), producing maximal $R^2$ closed forms $f(x) \approx \phi(x)$. Typical examples include quartic polynomials (e.g., $$f(x) = -0.076x^4 + 0.216x^3 + 0.252x^2 - 1.370x - 0.116$$, $R^2 = 0.9968$) and sinusoidal patterns (e.g., $0.923\sin(1.348x + 0.695) + 0.801\cos(2.624x)$), facilitating explicit explanation of high-frequency phenomena.

The pruning statistics reveal structural separation: for ETH, the approximation branch pruned just 4.79% of edges, versus 76.28% in the detail branch (Table IV).

4. Model Architecture and Training Procedure

The overall computational flow (cf. Fig. 2) is:

• RevIN normalization of $X_L$
• Multi-level DWT $(\rightarrow \{X_{A_m}, X_{D_m}, \dots\})$
• Parallel KAN Resolution Branches (two Mixer blocks/branch)
• Head linear layers produce predicted coefficients
• IDWT and RevIN denormalization yield the final forecast $X_T$

Loss minimization employs

$$L_{\rm total} = L_{\rm forecast} + \gamma L_{\rm reg}, \qquad L_{\rm forecast} = \mathrm{MSE}(X_T, X_{\rm true})$$

with $\gamma = 10^{-5}$. Optimization uses Adam over a parameter grid (learning rates $[10^{-4}, 5 \times 10^{-4}]$), with 30 training epochs (see Table II).

Hyperparameters (see Table II) include look-back $L=96$, DWT levels $m \in \{1,2\}$, KAN spline grid $G=5$, order $k=3$, patch size $P \in \{8,16\}$, embedding $d \in \{64,128,256\}$, resulting in 0.11–0.18M parameters and 0.0073 GFLOPs at $T=96$. Training time is 12.6s/epoch, inference ≈5s; B-spline ops induce training overhead, while inference remains real-time capable.

5. Experimental Evaluation: Quantitative Results, Ablations

Extensive benchmarking on BTC, ETH, and XMR datasets across various forecasting horizons ($T=\{24,48,96,168\}$) demonstrate DecoKAN's superior mean squared error (MSE) and mean absolute error (MAE) metrics (Table III). Results include:

Dataset	DecoKAN MSE	WPMixer MSE	TimeFilter MSE
BTC	0.136	0.160	0.146
ETH	0.100	0.118	0.160
XMR	0.219	0.235	0.233

DecoKAN ranks lowest in 27/32 crypto cases ("1stCount"); paired t-tests yield $p<0.05$ significance versus WPMixer and TimeFilter. Per-branch error ablation (not tabulated) indicates the Detail Branch contributes $\sim70\%$ of forecasting variance during volatility spikes.

6. Interpretability: Symbolic Case Studies and Market Mapping

Case study on ETH (Fig. 5) traces the lifecycle from learned spline $\phi$ to pruned formulas $f$, with high-frequency detail branches governing rapid price swings and approximation branches preserving global trend structure.

Symbolic regression outputs exhibit financial domain relevance: quartic polynomials encode momentum decay, sinusoids reproduce weekly/daily cycles, and $\tanh$ captures rally saturation effects (Table V). The resulting formula set forms an auditable analytic toolkit for market practitioners.

This suggests that DecoKAN's separation and symbolization pipeline materially assists in post-hoc logic auditing, bridging the interpretability-performance gap typical of conventional black-box deep learning time series models. Its utility for risk management and systematic trading is substantiated by explicit mapping from learned formulas to identifiable market patterns.

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DecoKAN synthesizes multi-level DWT-based frequency decoupling and interpretable KAN-based nonlinear modeling, augmented by rigorous symbolic regression, to enable not only predictive state-of-the-art performance on challenging cryptocurrency data but also unprecedented transparency in the model's internal logic and decision pathways (Gao et al., 23 Dec 2025).