Beyond the Graph: An Adaptive Meta-Learner Fuses Explainability, Weather, and Dynamics for Robust Bus ETA Prediction

Pratham Payra; Jagadish

cs.LGSep 14, 2026

Beyond the Graph: An Adaptive Meta-Learner Fuses Explainability, Weather, and Dynamics for Robust Bus ETA Prediction

Authors: Pratham Payra, Jagadish

Organizations: SQC and OR division, Indian Statistical Unit, BT Road, Kolkata, 700108, West Bengal, India. · SQC and OR unit, Indian Statistical Unit, 8th Mile, Mysore Road, Bangalore, 560059, Karnataka, India.

Abstract

Accurate bus Estimated Time of Arrival (ETA) prediction is vital for urban mobility, passenger satisfaction, and transit efficiency, yet existing models falter against nonlinear spatiotemporal dynamics, data sparsity, and factors such as weather. This paper proposes HYB(nm), an adaptive hybrid ensemble framework that dynamically fuses five complementary models - a historical baseline (MST-AV), periodical temporal pattern analysis (GDRN-DFT), Koopman Neural Operators for nonlinear dynamics (KOOP-NET), weather-integrated feature-engineered neural networks (FENN), and real-time graph convolutional networks (MGCN) - via a meta-learner attuned to real-time context. Evaluated on GPS and weather data from three Kolkata bus routes comprising more than 4,000 trips, the framework leverages the individual strengths of its components (for example, the low-latency explainability of MST-AV, the weather resilience of FENN, and the network-dynamics capture of MGCN) to deliver the superior robustness of HYB(2), state-of-the-art accuracy rivalling leading graph neural networks, and balanced trade-offs in stability and efficiency across prediction horizons and operating conditions. The extensible HYB(k) architecture equips transit agencies with flexible tools, ranging from economical single models to tailored high-fidelity hybrids, advancing predictive, equitable urban transport.

Figures & tables

Limitations / Gaps	Methods Used	Addressing the Limitation
Lack of spatial discretisation	MST-AV with grid approximation	Continuous GPS trajectories are discretised using $50\,\text{m}\times 50\,\text{m}$ grids with median referencing, enabling graph-based modelling and noise reduction.
Absence of non-linear temporal modelling	Koopman-based KNN	Non-linear bus-speed dynamics are captured via linear evolution in a lifted latent space using the Koopman operator.
No incorporation of weather effects	FENN with feature engineering	Real-time weather variables from OpenWeatherMap are encoded to improve contextual forecasting.
Inadequate modeling of spatio-temporal periodicity	GDRN-DFT	Graph diffusion combined with DFT captures daily and weekly traffic cycles effectively.
Poor handling of missing data	GDRN with Bi-LSTM	Spatial diffusion and bidirectional temporal modeling enable effective imputation of missing speeds.
Lack of real-time message passing	Masked GCN	Observed speeds are propagated across the graph in real time to complete network-wide estimates.

Table 1: Summary of Limitations in Prior Studies and Mitigation Strategies Employed by the Proposed Framework

Figure 1: Overview of the proposed methodology

Figure 2: Selected routes for ETA computation

	Routes
Metric	Route 1	Route 2	Route 3
	(DN2/1: Shapoorji–Dakshineswar)	(KB22: Shapoorji–Kolkata Station)	(KB16: Shapoorji–Bangoor)
Total Sessions	1,251	1,829	953
Total GPS Points	45,321	62,553	35,268
Distance (km)	$26$	$22$	$19$
Number of Stops	$13$	$11$	$10$

Table 2: Specific Route GPS data and Statistics

Month	% of Total Sessions	% of Total GPS Points
Aug 2024	16%	15%
Sep 2024	17%	18%
Oct 2024	15%	16%
Nov 2024	13%	13%
Dec 2024	12%	12%
Jan 2025	11%	11%

Table 3: Table shows an even distribution across the study period, with slight weighting toward monsoon months (Aug–Oct) for higher collection frequency due to weather covariate focus. Total: 100%.

Figure 3: Graph Extraction Process (see Algorithm 1 .)

Figure 4: GDRN-DFT Diagram.(For training : Algorithm 4 and Computation of real time ETA : Algorithm 5 )

Figure 5: KNN Diagram. (see for training : Algorithm 6 and real time ETA computation : Algorithm 7 .)

Figure 6: MGCN diagram.(Algorithm 10 for training and Algorithm 11 for real time ETA computation.)

Figure 7: Hybrid ETA diagram.(Algorithm 12 for training and Algorithm 13 for real time ETA computation)

		Normal Conditions					Bus Stop Groups				Extreme Weather
		(Actual ETA ranges(min))									(Actual ETA ranges(min))
Model	Metric	0-10	10-25	25-45	45+	Overall	1-2	3-4	5-6	7+	0-10	10-25	25-45	45+	Overall
MST-AV	MAE	4.2	5.9	8.2	10.3	8.0	4.7	6.8	8.9	11.1	6.0	7.2	8.9	11.4	8.8
	MAPE	60.9	35.7	25.6	17.4	21.1	44.8	28.9	20.4	18.2	75.9	40.1	28.7	20.4	25.1
	RMSE	5.4	7.2	9.8	12.7	9.6	6.0	8.4	10.7	13.6	7.1	8.6	10.4	13.1	10.2
GDRN-DFT	MAE	3.4	5.6	6.1	7.2	5.9	4.1	6.9	7.3	8.6	5.4	7.0	7.9	8.8	8.0

Table 4: Route 1: Model performance values of MAE, MAPE %, RMSE of individual and hybridization models for route-1.

Figure 8: The figure displays a 3x3 grid of line plots showing the average performance (MAE, MAPE, RMSE) of six models across Routes 1, 2, and 3 for Normal Conditions, Bus Stop Groups, and Extreme Weather, highlighting fluctuations across ETA ranges and bus stop groups.

Model	Metric	Route 1					Route 2					Route 3
Model	Metric	0–10	10–25	25–45	45+	Overall	0–10	10–25	25–45	45+	Overall	0–10	10–25	25–45	45+	Overall
HYB(1)	MAE	1.4	4.1	6.2	7.4	4.3	1.6	5.0	8.0	9.1	5.4	1.5	4.6	7.2	8.4	5.0
	MAPE	20.5	24.2	18.8	10.9	13.2	24.4	31.3	22.4	13.0	15.9	22.9	29.0	20.5	12.3	14.7
	RMSE	1.8	5.3	7.6	8.5	5.6	2.2	6.5	9.5	10.6	7.1	2.0	6.0	8.7	9.7	6.1
HYB(2)
Proposed Model	MAE	1.3	4.3	6.5	7.3	4.5	1.7	5.3	8.1	9.0	5.7	1.6	4.8	7.4	8.3	4.8

Table 5: Comparison of MAE, MAPE (%), and RMSE for HYB models across ETA ranges in Route 1, Route 2, and Route 3.(Lowest values per route are in bold.)

Figure 9: Comparative performance of five hybrid ETA prediction models (HYB(1)–HYB(5)) across three bus routes (rows) and three error metrics (columns). Each subplot depicts the variation of prediction errors across ETA ranges (0–10, 10–25, 25–45, and 45+ minutes). Metrics include Mean Absolute Error (MAE, minutes), Mean Absolute Percentage Error (MAPE, %), and Root Mean Squared Error (RMSE, minutes). Distinct markers, colors, and line styles represent different hybrid configurations, where the meta-learner $f_{NN}$ integrates predictions from a varying number of base models.

Model	Pros	Cons
DCRNN Li et al. (2017)	Captures directed flows; principled diffusion modeling	RNN backbone; slow training, limited parallelism
STGCN Yu et al. (2018)	Simple, efficient baseline	Weak long-range temporal modeling
GWNet (fixed A) Wu et al. (2019)	Dilated convs capture long history; strong benchmark	Complex; tuning dilations is tricky
T-GCN Zhao et al. (2020)	Simple GCN+GRU hybrid; effective on small data	RNN bottleneck; weak long-horizon
ASTGCN Zhu et al. (2020)	Temporal+spatial attention; good for periodic patterns	Computationally heavy; hard to tune
STGAT (fixed A) Huang et al. (2019)	Flexible neighbor weighting	Overfits small datasets

Table 6: Spatio-temporal GNN models with fixed adjacency: pros and cons.

Model	Metric	Route 1					Route 2					Route 3
Model	Metric	0–10	10–25	25–45	45+	Overall	0–10	10–25	25–45	45+	Overall	0–10	10–25	25–45	45+	Overall
HYB(2) (Proposed)	MAE	1.3	4.3	6.5	7.3	4.5	1.7	5.3	8.1	9.0	5.7	1.6	4.8	7.4	8.3	4.8
	MAPE	19.4	23.5	17.8	11.2	12.3	23.1	28.0	21.2	13.3	14.9	21.7	25.9	19.6	12.6	13.8
	RMSE	1.7	5.0	7.3	8.4	5.4	2.0	6.1	9.1	10.7	6.9	2.1	5.7	8.4	9.8	6.4
DCRNN	MAE	1.2	4.0	6.8	8.2	5.1	1.4	5.9	8.7	9.9	6.3	1.9	5.1	6.8	9.2	5.7
	MAPE	22.1	22.6	19.9	12.1	14.1	25.2	27.2	22.4	14.2	16.4	21.1	26.8	21.4	12.0	16.8

Table 7: Comparison of MAE, MAPE (%), and RMSE for proposed model (HYB(2)) and baseline models across ETA ranges in Route 1, Route 2, and Route 3.(ETA computation is done for benchmark models using Algorithm 14 .)

Figure 10: Comparative performance of HYB(2) (proposed) with eight models. (Benchmark models display prediction patterns similar to HYB(2), as all fixed-adjacency methods rely on the same DFT-based initialization for imputing missing data, without which evaluation under incomplete observations is not feasible.)

Model:Win rate %	MAE			MAPE			RMSE			Overall (%)
	R1	R2	R3	R1	R2	R3	R1	R2	R3
HYB(2) (Proposed)	40.0	40.0	60.0	60.0	60.0	40.0	60.0	60.0	20.0	48.9
DCRNN	40.0	20.0	20.0	20.0	20.0	20.0	40.0	0.0	0.0	20.0
STGCN	20.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	2.2
GWNet (fixed A)	0.0	40.0	0.0	20.0	0.0	60.0	0.0	40.0	40.0	22.2
T-GCN	0.0	0.0	20.0	0.0	0.0	0.0	0.0	0.0	20.0	4.4

Table 8: Win Rate Analysis—Quantifying Model Dominance Across Prediction Scenarios. This table elucidates the comparative edge of each model by computing the win rate as the percentage of five predefined ETA prediction ranges where it registers the lowest error for specified metrics (MAE, MAPE, RMSE) on each of three bus routes (R1, R2, R3); values are derived through exhaustive pairwise comparisons within each range-metric-route triplet, with overall percentages aggregated as route-averaged means to highlight robust, context-specific leadership in bus arrival forecasting.

Model	Optimal Configuration	Performance	Rationale	Not Studied
MST-AV	Grid size = 50 mtr $\times$ 50 mtr	MAE $\approx$ 8.0–8.4 min, Time $\approx$ 85.6–59.2 ms	Finer grids (50 $\times$ 50) capture localised traffic variations in dense urban areas, improving path accuracy, but scaling to 100 $\times$ 100 reduces MAE variance across routes by averaging over more historical traces per cell. This range avoids the computational explosion of sub-50 m bins while outperforming coarser ones (e.g., 1000 $\times$ 1000) that lose route-specific details, prioritising low latency for battery-constrained apps.	Only average speed is taken for each grid. No other central tendency study (e.g., mode, median) is done.
GDRN-DFT	$\Delta=7$ days, $\delta=15$ min, $\tau=2$ , $d=16$	MAE = 5.936 min, Time = 1.149 s	This configuration strikes an optimal balance by capturing weekly traffic cycles ( $\Delta=7$ days) without excessive historical data overhead, while a moderate sampling interval ( $\delta=15$ min) aligns with typical GPS update rates to reduce noise. The diffusion lag $\tau=2$ enables effective spatial smoothing of sparse signals, and $d=16$ provides sufficient Fourier detail for periodic patterns without amplifying high-frequency artifacts, minimising both error and computation.	$\epsilon_{g}=1.5$ km/hr, $\tau_{f}=\tau_{b}$ taken as $\tau$ the same for forward and backward diffusion.
KNN	$\delta=5$ min, $d=5$ , $D=12$	MAE = 8.842 min, Time = 1.7 s	A finer sampling ( $\delta=5$ min) preserves short-term dynamics in urban bus routes, where rapid changes occur, while $d=5$ embedding dimension captures essential nonlinearities in speed profiles without overparameterisation. Lifting to $D=12$ in the Koopman space linearises dynamics efficiently, avoiding the quadratic time scaling of higher $D$ values, thus optimizing for real-time inference on edge devices.	Number of intermediate layers for encoder = decoder = 2; $p$ , $q$ (encoder) and $p^{\prime}$ , $q^{\prime}$ (decoder) nodes optimized via Ray Tune (not manually studied); $\lambda_{k}=1$ (regularization studied between prediction and reconstruction loss components). Activation function used: Swish.Swish provides the smooth differentiability required by Koopman theory while avoiding the gradient saturation of Tanh, enabling better propagation through the encoder-decoder architecture.
FENN	$\delta=10$ min, $b=6$ , $l=3$ , $k=7$	MAE = 8.271 min, Time = 0.791 s	The 10-min interval balances temporal granularity with feature stability, incorporating weather and location without aliasing. A compact sinusoidal basis ( $b=6$ ) encodes diurnal cycles succinctly, reducing dimensionality, while 3 layers ( $l=3$ ) provide depth for feature interactions without vanishing gradients. Sequence length $k=7$ aligns with hourly patterns, ensuring predictive power while keeping the model lightweight for mobile deployment.	Number of hidden layers studied in a shallow range, but number of nodes in hidden layers not studied manually; it was optimised via Ray Tune.Swish (Activation for hidden layers) offers a good balance between non-saturation and smooth gradients, enhancing feature interaction learning while maintaining computational efficiency.
MGCN	$\Delta T=7$ days, $\delta_{g}=10$ min, $L=3$	MAE = 7.39 min, Time = 1.039 s	Weekly history ( $\Delta T=7$ days) encompasses recurring patterns like rush hours, enhancing imputation accuracy for masked graph nodes. Graph sampling at $\delta_{g}=10$ min maintains resolution for spatial dependencies, and 3 layers ( $L=3$ ) propagate information across the MST-augmented graph sufficiently to leverage neighbors without over-smoothing, which occurs at $L=4$ and degrades performance on sparse bus networks.	No regularisation between L1 and L2, so $\lambda_{m}=0$ (no smoothness enforced for DFT estimates).

Table 9: Optimal Hyperparameters and Ablation Insights Across Models. Only for models MGCN and MST-AV the performance for different hyperparameters are given in Appendix. 12 , 13

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

		Normal Conditions					Bus Stop Groups				Extreme Weather
Model	Metric	0-10	10-25	25-45	45+	Overall	1-2	3-4	5-6	7+	0-10	10-25	25-45	45+	Overall
MST-AV	MAE	5.0	7.0	10.2	13.4	9.8	5.6	8.8	11.4	14.1	7.4	9.4	10.2	14.6	11.1
	MAPE	67.0	38.9	28.8	20.0	23.4	49.1	33.2	23.3	20.7	85.0	46.3	30.9	23.3	28.4
	RMSE	6.6	8.6	12.5	16.9	11.9	7.3	11.2	14.0	17.6	9.0	11.5	12.1	17.1	13.1
GDRN-DFT	MAE	3.9	6.8	7.1	9.3	7.0	5.5	8.5	9.5	10.3	6.8	8.3	9.6	11.2	9.2
	MAPE	45.9	34.4	19.2	13.1	15.9	43.0	30.8	19.2	15.6	71.4	36.5	25.1	16.7	22.9

Appendix

Table 10: Route 2 : Performance analysis (MAE %; MAPE %, RMSE)

		Normal Conditions					Bus Stop Groups				Extreme Weather
Model	Metric	0-10	10-25	25-45	45+	Overall	1-2	3-4	5-6	7+	0-10	10-25	25-45	45+	Overall
MST-AV	MAE	4.70	6.64	9.26	11.68	9.11	5.37	7.80	10.26	12.84	6.97	8.40	10.42	13.40	10.38
	MAPE	68.2	40.2	28.9	19.7	24.0	51.2	33.2	23.5	21.1	88.2	46.8	33.6	24.0	29.6
	RMSE	6.04	8.10	11.07	14.40	10.93	6.86	9.64	12.34	15.73	8.25	10.03	12.18	15.40	12.03
GDRN-DFT	MAE	3.81	6.30	6.89	8.16	6.72	4.69	7.92	8.41	9.95	6.27	8.16	9.25	10.34	9.44
	MAPE	47.6	35.0	20.0	12.9	16.6	41.9	31.7	19.2	16.4	73.7	38.8	26.5	17.3	25.1

Appendix

Table 11: Route 3 : Performance analysis (MAE %; MAPE %, RMSE)

$\Delta T_{m}$ (days)	$L_{m}=1$				$L_{m}=2$				$L_{m}=3$				$L_{m}=4$
	5 min	10 min	15 min	20 min	5 min	10 min	15 min	20 min	5 min	10 min	15 min	20 min	5 min	10 min	15 min	20 min
1	11.5 0.271	11.83 0.9	12.234 1.8	12.65 2.808	10.769 0.305	10.96 0.908	11.41 1.78	11.75 2.77	10.332 0.295	10.58 1.009	10.978 1.93	11.39 2.937	10.665 0.406	10.863 1.059	11.34 1.98	11.75 2.99
5	9.195 0.271	8.893 0.94	9.473 1.78	9.949 2.76	8.633 0.33	8.257 0.88	8.93 1.792	9.276 2.839	8.144 0.33	7.787 0.988	8.53 1.943	8.89 2.91	8.308 0.37	8.02 1.101	8.74 1.963	9.1 2.97
7	8.2 0.292	8.98 0.85	9.188 1.76	9.558 2.824	7.86 0.29	8.13 0.91	8.632 1.81	8.991 2.83	7.83 0.28	7.39 1.039	8.537 1.861	8.65 2.86	8.02 0.361	8.002 1.1	8.74 1.971	8.92 3.011
10	9.03 0.31	8.75 0.889	9.59 1.766	10.03 2.79	8.456 0.33	8.334 0.86	8.978 1.845	9.38 2.75	8.22 0.253	8.184 0.979	8.89 1.92	9.22 2.936	8.4 0.431	8.52 1.06	9.1 2.05	9.464 3.004

Appendix

Table 12: Hyperparameter study of MGCN on MAE (Min) and Computation Time (s) with varying history length

\Delta T_{m}

, sampling interval

\delta_{m}

, and number of layers

L_{m}

. The optimal point is highlighted.

Metric	Unit(mtr x mtr)	25x25	50x50	100x100	250x250	500x500	1000x1000
MAE	(minutes)	8.3	8.0	8.4	10.8	15.0	18.8
Avg. Computation Time	(ms)	120.4	85.6	59.2	57.8	51.4	48.2

Appendix

Table 13: Performance Metrics of MST-AV Across Different Grid Sizes.

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