Hybrid VMD+LSTM Model

• The paper introduces a decomposition-first architecture where VMD separates a time series into intrinsic mode functions before LSTM-based forecast aggregation.
• VMD accurately decomposes non-stationary signals by solving a constrained variational problem, ensuring narrowband mode separation and reduced mode mixing.
• LSTM modules are applied individually to normalized, sliding-window framed IMFs, enabling precise reconstruction of forecasts through additive aggregation.

Hybrid VMD+LSTM models are hybrid sequence-modeling architectures that combine variational mode decomposition (VMD) with long short-term memory (LSTM) networks. Their defining pattern is decomposition first, recurrent modeling second: a non-stationary signal is decomposed into band-limited intrinsic mode functions (IMFs), the resulting sub-signals are forecast or classified with LSTM-based modules, and the component outputs are then aggregated or otherwise fused into a final prediction. In the Bitcoin formulation, VMD is applied to the original Bitcoin price series, each IMF is modeled by an LSTM network, and the individual forecasts are summed to reconstruct the forecast of the original series; related work preserves the same decomposition-first logic while varying the recurrent backbone, feature packing, and reconstruction strategy (Boadi, 11 Sep 2025, Li, 18 Aug 2025, Zhang et al., 2020, Cheng et al., 1 Sep 2025).

1. Canonical architecture and model family

In its most direct form, the hybrid VMD+LSTM model consists of three stages: VMD decomposition of a real-valued time series into $K$ IMFs, LSTM-based learning on the decomposed components, and forecast aggregation back to the original signal domain. The Bitcoin study states this pipeline explicitly: first, VMD decomposes the original Bitcoin price series into IMFs; each IMF is then modeled using an LSTM network; the individual forecasts from the IMFs are aggregated to produce the final prediction of the original Bitcoin price series (Boadi, 11 Sep 2025).

The same architectural motif appears in several adjacent designs. Li et al. use a sliding-window VMD front end and then feed the $K=5$ IMFs as separate channels into a stacked LSTM for financial forecasting (Li, 18 Aug 2025). Zhang et al. embed VMD in a decomposition-reconstruction-ensemble framework, where VMD produces sub-signals, a CNN reconstructs additional sub-signals, and an LSTM forecasts from both decomposed and reconstructed inputs (Zhang et al., 2020). The medical BiLSTM-AM-VMD framework applies VMD as a front-end “signal decomposition” layer to each real-valued feature time-series and then processes the expanded representation with BiLSTM and multi-head attention (Cheng et al., 1 Sep 2025). The SS-LSTM model combines spatial smoothing, Sequential General VMD, and LSTM regressors, again preserving the idea that recurrent prediction is performed on decomposed or dynamically extracted components rather than on the raw series alone (Liu et al., 2024).

This suggests that “hybrid VMD+LSTM model” denotes an architectural family rather than a single canonical network. What remains invariant across the family is the use of VMD to factor a complex signal into narrower-band components before recurrent learning.

2. Variational mode decomposition as the front end

The VMD stage decomposes a real signal $f(t)$ into $K$ band-limited IMFs $u_k(t)$, each associated with an estimated center frequency $\omega_k$. In the Bitcoin study, the constrained variational problem is written as

$$\min_{\{u_k\},\{\omega_k\}} \sum_{k=1}^{K} \Bigl\|\partial_t\bigl[\bigl(\delta(t)+\tfrac{j}{\pi t}\bigr)*u_k(t)\bigr] e^{-j\omega_k t}\Bigr\|_2^2 \quad\text{s.t.}\quad \sum_{k=1}^K u_k(t) = f(t).$$

The corresponding augmented Lagrangian is

$$\mathcal{L}(\{u_k\},\{\omega_k\},\lambda) = \alpha\sum_{k=1}^K \Bigl\|\partial_t\bigl[\bigl(\delta+\tfrac{j}{\pi t}\bigr)*u_k\bigr] e^{-j\omega_k t}\Bigr\|_2^2 +\Big\|\;f-\sum_{k=1}^K u_k\Big\|_2^2 +\left\langle \lambda(t),\,f(t)-\sum_{k=1}^K u_k(t)\right\rangle,$$

where $\alpha>0$ is a quadratic penalty parameter and $\lambda(t)$ is the Lagrange multiplier enforcing reconstruction (Boadi, 11 Sep 2025).

In the Fourier domain, the alternating-direction updates are given by

$K=5$0

$K=5$1

$K=5$2

where $K=5$3 is a dual ascent step size (Boadi, 11 Sep 2025).

Mode-number selection is application-dependent. For Bitcoin, the normalized closing-price series with 2,863 daily points from 2017–2025 was fed into VMD, and the residual energy ratio

$K=5$4

was computed for $K=5$5 to $K=5$6. Beyond $K=5$7, the residual energy fell below $K=5$8 and plateaued, and the authors chose $K=5$9 as a compromise between decomposition accuracy and computational cost. The penalty parameter $f(t)$0 and dual-step $f(t)$1 were set to their typical defaults in the Dragomiretskiy–Zosso implementation, for example $f(t)$2 and $f(t)$3, which are reported to ensure narrowband mode separation without mode mixing (Boadi, 11 Sep 2025).

Other studies use different choices. Li et al. set $f(t)$4 and $f(t)$5 in a sliding-window financial setting (Li, 18 Aug 2025). Zhang et al. decompose each series into $f(t)$6 modes using vmdpy with default $f(t)$7 and $f(t)$8 (Zhang et al., 2020). The SG-VMD formulation is explicitly designed to overcome the need to know $f(t)$9 in advance by extracting modes one at a time (Liu et al., 2024). A plausible implication is that the VMD front end is not a fixed preprocessing block but a tunable modeling choice with direct consequences for downstream recurrent learning.

3. LSTM modeling of decomposed components

In the Bitcoin model, each IMF $K$0 is preprocessed independently by Min–Max normalization to $K$1 and sliding-window supervised framing with look-back $K$2 days. The network structure for each IMF consists of one LSTM layer with hidden dimension $K$3, not explicitly stated and described as typically 50–100 units, followed by a dense output layer with linear activation producing a scalar one-step forecast. Training uses Mean Squared Error (MSE) loss, the Adam optimizer, 20 epochs, and batch size 32 (Boadi, 11 Sep 2025).

The LSTM cell is described by the standard gating equations

$K$4

$K$5

$K$6

After training, each IMF model produces a 30-day forecast $K$7, and the final reconstructed forecast is

$K$8

A final inverse Min–Max scaling returns predictions to USD (Boadi, 11 Sep 2025).

This per-IMF design is not the only way to couple VMD to recurrent learning. In the SW-VMD-LSTM framework, each of the $K$9 IMFs is treated as a separate channel in a $u_k(t)$0 input sequence, and the model uses 3 stacked LSTM layers with 128 memory cells, dropout $u_k(t)$1, and $u_k(t)$2 regularization with coefficient $u_k(t)$3 (Li, 18 Aug 2025). In VMD-CNN-LSTM, the decomposed modes $u_k(t)$4 are concatenated with CNN-reconstructed sub-signals $u_k(t)$5 and then passed through a stacked LSTM and fully connected output layer (Zhang et al., 2020). This suggests two major recurrent design patterns within the literature: independent per-mode experts with additive reconstruction, and joint sequence learners operating on stacked decomposed channels.

4. Data handling, normalization, and hyperparameterization

The Bitcoin implementation uses daily closing prices from August 11, 2017 to June 13, 2025, yielding 2,863 observations. Missing values are dropped via pandas’ dropna(), and the data are normalized with Min–Max scaling to $u_k(t)$6. The critical implementation settings reported are $u_k(t)$7, residual-energy threshold $u_k(t)$8, $u_k(t)$9 and $\omega_k$0 as per standard VMD defaults, look-back $\omega_k$1, Adam optimizer, MSE loss, 20 epochs, batch size 32, forecast horizon 30 days ahead, and aggregation by simple summation of IMF forecasts followed by inverse scaling (Boadi, 11 Sep 2025).

The sliding-window financial framework makes different preprocessing choices. Raw features include daily closing price $\omega_k$2, return $\omega_k$3, turnover, and pre-close; the window length is $\omega_k$4 trading days with step size 1 day, VMD is applied to each window, and the IMF channels are normalized by Z-score over the training set and clipped to $\omega_k$5. The data split is sequential rather than cross-validated: the last 60 days are used as the test set, the previous 60 days as the validation set, and the remaining $\omega_k$6–$\omega_k$7 days as training data, explicitly to avoid look-ahead bias (Li, 18 Aug 2025).

Hyperparameter search procedures also vary. The medical BiLSTM-AM-VMD model tunes $\omega_k$8, $\omega_k$9, hidden size $$\min_{\{u_k\},\{\omega_k\}} \sum_{k=1}^{K} \Bigl\|\partial_t\bigl[\bigl(\delta(t)+\tfrac{j}{\pi t}\bigr)*u_k(t)\bigr] e^{-j\omega_k t}\Bigr\|_2^2 \quad\text{s.t.}\quad \sum_{k=1}^K u_k(t) = f(t).$$0, number of heads, dropout, and related settings with PSO, using population $$\min_{\{u_k\},\{\omega_k\}} \sum_{k=1}^{K} \Bigl\|\partial_t\bigl[\bigl(\delta(t)+\tfrac{j}{\pi t}\bigr)*u_k(t)\bigr] e^{-j\omega_k t}\Bigr\|_2^2 \quad\text{s.t.}\quad \sum_{k=1}^K u_k(t) = f(t).$$1, iterations $$\min_{\{u_k\},\{\omega_k\}} \sum_{k=1}^{K} \Bigl\|\partial_t\bigl[\bigl(\delta(t)+\tfrac{j}{\pi t}\bigr)*u_k(t)\bigr] e^{-j\omega_k t}\Bigr\|_2^2 \quad\text{s.t.}\quad \sum_{k=1}^K u_k(t) = f(t).$$2, and $$\min_{\{u_k\},\{\omega_k\}} \sum_{k=1}^{K} \Bigl\|\partial_t\bigl[\bigl(\delta(t)+\tfrac{j}{\pi t}\bigr)*u_k(t)\bigr] e^{-j\omega_k t}\Bigr\|_2^2 \quad\text{s.t.}\quad \sum_{k=1}^K u_k(t) = f(t).$$3 (Cheng et al., 1 Sep 2025). In contrast, the Bitcoin model relies on residual-energy analysis for mode selection and default VMD parameters for $$\min_{\{u_k\},\{\omega_k\}} \sum_{k=1}^{K} \Bigl\|\partial_t\bigl[\bigl(\delta(t)+\tfrac{j}{\pi t}\bigr)*u_k(t)\bigr] e^{-j\omega_k t}\Bigr\|_2^2 \quad\text{s.t.}\quad \sum_{k=1}^K u_k(t) = f(t).$$4 and $$\min_{\{u_k\},\{\omega_k\}} \sum_{k=1}^{K} \Bigl\|\partial_t\bigl[\bigl(\delta(t)+\tfrac{j}{\pi t}\bigr)*u_k(t)\bigr] e^{-j\omega_k t}\Bigr\|_2^2 \quad\text{s.t.}\quad \sum_{k=1}^K u_k(t) = f(t).$$5 (Boadi, 11 Sep 2025). A plausible implication is that reported performance depends not only on the hybrid architecture but also on the data-engineering and hyper