QGAPHEnsemble Adaptive Weighting

• QGAPHEnsemble Adaptive Weighting is a dynamic ensemble framework that integrates quantum-enhanced LSTM base models with an online error-driven weighting rule.
• It employs hybrid quantum genetic algorithm–particle swarm optimization and nested Bayesian optimization to fine-tune hyperparameters and boost predictive accuracy.
• Empirical evaluations on extensive weather data demonstrate improved MAPE and reliable, real-time forecasting over classical and single-model approaches.

QGAPHEnsemble Adaptive Weighting is a dynamic ensemble learning methodology for model combination, centered on quantum-enhanced LSTM base learners and a data-driven online weighting rule. This approach is designed to improve predictive accuracy and reliability in settings such as short-term weather forecasting by adaptively favoring the best-performing expert at each timestep. The QGAPHEnsemble framework interleaves advanced optimization strategies—including a hybrid quantum genetic algorithm–particle swarm optimizer (QGA-PSO) and Bayesian optimization—with an explicit, temporally-adapted weighting vector whose updates are grounded in recent error history. Adaptive weighting, as implemented in QGAPHEnsemble, connects to broader theoretical frameworks for ensemble learning, including operator-agnostic axiomatic approaches, probabilistic weighting priors, and calibration-aware uncertainty quantification.

1. QGAPHEnsemble Architecture and Base Learners

QGAPHEnsemble integrates $M$ quantum LSTM (QLSTM) base models, each producing a one-step forecast $\hat y_m(t)$ at time $t$. The core architectural unit is the GenHybQLSTM, whose cell operations replace classical LSTM linear maps with parametrized variational quantum circuits (VQC), using RX/RZ gates and entangling layers with trainable rotation angles. At each timestep, the QLSTM cell computes the forget, input, candidate, and output activations as: $$\begin{aligned} f_t &= \sigma(\mathrm{VQC}_1(x_t, h_{t-1})) \ i_t &= \sigma(\mathrm{VQC}_2(x_t, h_{t-1})) \ \tilde{c}_t &= \tanh(\mathrm{VQC}_3(x_t, h_{t-1})) \ o_t &= \sigma(\mathrm{VQC}_4(x_t, h_{t-1})) \end{aligned}$$

$$c_t = f_t \odot c_{t-1} + i_t \odot \tilde{c}_t, \qquad h_t = \mathrm{VQC}_5(o_t \odot \tanh(c_t)), \qquad y_t = \mathrm{VQC}_6(o_t \odot \tanh(c_t))$$

Stacking 2–3 such QLSTM layers, the final output is projected to the scalar forecast. Hyperparameters such as number of qubits, learning rate, batch size, and circuit depth are subject to global optimization (Sen et al., 18 Jan 2025).

2. Adaptive Weighting Rule

Ensemble prediction is constructed as

$\hat y(t) = \sum_{m=1}^M w_m(t) \hat y_m(t)$

subject to $0 \leq w_m(t) \leq 1$ and $\sum_{m=1}^M w_m(t) = 1$. The weights $w_m(t)$ are continuously updated via a gradient-like online rule reflecting recent performance. For each base model $m$ at update step $\hat y_m(t)$0, the exponentially-weighted cumulative error is

$\hat y_m(t)$1

with forgetting factor $\hat y_m(t)$2 (typically $\hat y_m(t)$3) and rolling window $\hat y_m(t)$4. The update increment is inverse-error normalized: $\hat y_m(t)$5

$\hat y_m(t)$6

Weights are renormalized so that their sum is unity. This rule incrementally shifts ensemble emphasis toward the model(s) with the lowest recent absolute error (Sen et al., 18 Jan 2025).

3. Hyperparameter Optimization Strategies

Two global optimization strategies are implemented for QGAPHEnsemble's base models:

• Hybrid Quantum Genetic Algorithm–Particle Swarm Optimization (GenHybQLSTM): Each set of hyperparameters is encoded in a quantum chromosome. The QGA phase iterates via quantum rotations guided by the validation loss gradient, generating an optimal seed. A PSO phase further refines real-valued hyperparameters using a standard velocity-update rule across particles.
• Nested Bayesian Optimization (BO-QEnsemble): Each base QLSTM is individually optimized via a Gaussian process surrogate with a squared-exponential kernel and Expected Improvement acquisition. Multidimensional BO sweeps over joint configurations (top $\hat y_m(t)$7 per base model), and the final ensemble is selected by lowest joint MSE using the adaptive weighting rule (Sen et al., 18 Jan 2025).

4. Evaluation Metrics and Empirical Results

Training minimizes mean squared error (MSE); early stopping and reporting rely also on mean absolute percentage error (MAPE): $\hat y_m(t)$8 Empirical studies on 96,432 weather observations show that QLSTM-based architectures outperform classical LSTM, GRU, and ANN baselines, with GenHybQLSTM (QGA-PSO + adaptive weights) surpassing single QLSTM by $\hat y_m(t)$9 in MAPE, and the BO-QEnsemble variant superior still (see Table 1) (Sen et al., 18 Jan 2025).

Model	MSE (×10⁻³)	MAPE (%)
LSTM	2.51	1.99
GRU	2.47	1.98
ANN	2.73	2.09
DE-ANN	1.62	1.15
Single QLSTM	1.24	1.12
GenHybQLSTM (QGA-PSO + AW)	1.08	0.92
BO-QEnsemble (BO + AW)	0.99	0.91

MAPE reduction by GenHybQLSTM over single QLSTM is significant ($t$0), and BO-QEnsemble's increment over GenHybQLSTM is marginal but consistent ($t$1).

5. Relation to Adaptive Ensemble Learning Theories

The adaptive weighting in QGAPHEnsemble relates to a wider literature on probabilistic and operator-agnostic ensemble combination. Frameworks such as dependent tail-free process priors (Liu et al., 2018, Liu et al., 2019) generalize adaptive weighting by parametrizing the weight vector over input space $t$2 via Gaussian processes, with softmax transforms ensuring normalization: $t$3 This stochastic process approach captures both model-selection uncertainty (variance in $t$4) and predictive uncertainty, and admits full Bayesian or variational posterior inference with calibration-aware objectives such as KL+CRPS or composite-divergence loss.

Recent advancements provide an axiomatic, operator-agnostic framework for adaptive weighting in multi-model ensembles. Core axioms—normalization, bounded influence, regularity, and ordinal safety monotonicity—guarantee the existence of broad families of weighting operators. Product-structure normalization allows modular, multi-scale composition of token-, task-, and context-level weights, each of which can be implemented and optimized independently before aggregation. These principles are applicable to various ensemble settings, including online, spatiotemporal, and safety-constrained environments (Flouro et al., 25 Jan 2026).

6. Calibration and Uncertainty Quantification

Well-calibrated ensembles require proper uncertainty assessment for both weights and predictions. Probabilistic adaptive-weight models (not used in QGAPHEnsemble, but foundational in theory) use Gaussian process posteriors to capture the distribution over weights, which is critical for honest coverage in prediction intervals and robust decision-making in regions of base-model disagreement or data sparsity. Calibration diagnostics—such as coverage probability curves and average CRPS—quantify the frequentist validity of uncertainty estimates (Liu et al., 2019, Liu et al., 2018). QGAPHEnsemble itself maintains a deterministic (gradient-like) online weighting rule, but the principle of adaptively tracking local best predictors is congruent with the probabilistic frameworks.

7. Significance and Practical Implications

QGAPHEnsemble Adaptive Weighting demonstrates the efficacy of streaming, performance-coupled model combination for time-evolving forecasting tasks. Empirical hyperparameter optimization, adaptive emphasis on the most accurate base models, and avoidance of ad hoc or static weighting distinguish this paradigm. Broader theoretical frameworks confirm the flexibility and robustness of such adaptive approaches, establish convergence and stability guarantees, and enable domain-specific adaptations (e.g., for safety or distribution shift) with principled operator design (Flouro et al., 25 Jan 2026). This suggests adaptive weighting rules as a canonical template for ensemble learning with heterogeneous, non-stationary, or safety-critical data.