Structure-Aware Graph Abstention for Reliable Selective Forecasting

Jianxiang Xie; Belal Alsinglawi

cs.LGOct 6, 2026

Structure-Aware Graph Abstention for Reliable Selective Forecasting

Authors: Jianxiang Xie, Belal Alsinglawi

Organizations: University of New South Wales · Zayed University

Abstract

Selective forecasting abstains on high-risk test windows under a retained-coverage budget. Existing gates such as TEM (Brusokas et al., 2025) score each forecast as a whole; for multivariate outputs, trajectories can look plausible while violating dependencies among variables. We treat instance-level plausibility and relational consistency as distinct reliability axes and operationalize the latter via a learned sparse graph and a Dirichlet-style structural energy E_struct, trained with error-weighted graph regularization and score-error alignment. On seven long-horizon benchmarks and four backbones, structural gating often reduces selective MSE versus TEM at matched coverage, with the largest gains where cross-variable structure appears more informative in our benchmarks; gains are not universal, indicating a complementary abstention signal. Table 1 is a Protocol A ranking diagnostic (seed 2024); three-seed deployable Protocol B on an aligned subset is in Table 3 (full validation-to-test grids: Appendix A).

Figures & tables

Figure 1: Overview: backbone $\hat{Y}$ ; baseline TEM energy $E_{\mathrm{TEM}}$ (prior pipeline); our graph head yields $A$ and $E_{\mathrm{struct}}(\hat{Y},A)$ . Under matched test-split quantile gating, we obtain comparable selective evaluations for graph-only vs. TEM -only gates.

Dataset	Backbone	$\rho=0.5$				$\rho=0.7$				$\rho=0.9$				Full
		TEM	Graph	I% $\uparrow$	S% $\uparrow$	TEM	Graph	I% $\uparrow$	S% $\uparrow$	TEM	Graph	I% $\uparrow$	S% $\uparrow$	Full
ETTh1	Autoformer	0.4453	0.4209	+5.5	+4.1	0.4403	0.4091	+7.1	+6.8	0.4388	0.4250	+3.1	+3.1	0.4388
	FEDformer	0.4030	0.3893	+3.4	+3.9	0.4063	0.3828	+5.8	+5.5	0.4051	0.3956	+2.3	+2.3	0.4049
	PatchTST	0.3837	0.3547	+7.6	+7.6	0.3824	0.3650	+4.6	+4.9	0.3839	0.3780	+1.5	+1.5	0.3838
	TimesNet	0.3716	0.3484	+6.2	+10.2	0.3740	0.3685	+1.5	+5.0	0.3862	0.3916	-1.4	-1.0	0.3879
ETTh2	Autoformer	0.2587	0.2448	+5.4	+18.7	0.2665	0.2685	-0.8	+10.8	0.2882	0.2907	-0.9	+3.5	0.3011

Table 1: (Protocol A; ranking diagnostic.) Test selective prediction MSE across datasets and backbones (lower is better). Thresholds are test-split empirical

\rho

-quantiles of each raw score ( TEM vs. structural), not validation-calibrated deployment thresholds (Protocol B: Table 5 ). Graph is bolded when lower than TEM at the same

\rho

. I%

\uparrow

(\text{TEM}-\text{Graph})/\text{TEM}\times 100

. S%

\uparrow

(\text{Full}-\text{Graph}_{\rho})/\text{Full}\times 100

. Positive is better for both.

Panel of Figure 2 — Figure 2: Case study (coverage $0.80$ ). Percentile ranks on the test split (lower $\Rightarrow$ more likely accepted). Dashed lines mark $80\%$ nominal coverage via test-split score quantiles under the same protocol as Table 1 . Left: ranks of $E_{\mathrm{TEM}}$ vs. $E_{\mathrm{struct}}$ ; upper-left marks relational misses ( TEM accepts, graph rejects). Right: test MSE by acceptance region; relational misses skew higher-error than mutual acceptance.

Figure 3: A representative relational miss (test index 9451): TEM accepts at $0.80$ target coverage while the structural gate rejects. Heatmap: channel-wise squared error over the horizon (log-scaled).

		Weather			Electricity			ETTh2
Backbone	Ablation	$0.5$	$0.7$	$0.9$	$0.5$	$0.7$	$0.9$	$0.5$	$0.7$	$0.9$
PatchTST	Full (A0)	20.70	18.10	8.20	15.10	11.10	2.50	24.10	21.30	8.20
	w/o align (A1)	-8.13	-2.04	0.78	13.30	9.74	2.12	-4.76	1.70	4.45
	w/o err.-wt (A2)	10.16	12.51	7.88	12.87	5.54	0.80	9.17	9.15	5.38
	Corr. gate (A3)	-7.30	-3.97	-0.81	13.61	10.19	1.75	4.29	6.86	1.88
Autoformer	Full (A0)	20.50	15.70	6.30	9.80	5.40	2.00	18.70	10.80	3.50

Table 2: Graph-only ablation ( PatchTST , Autoformer ; tau_mode = quantile , Protocol A). Test relative improvement

(1-\mathrm{MSE}_{\mathrm{sel}}/\mathrm{MSE}_{\mathrm{orig}})\times 100

c\in\{0.5,0.7,0.9\}

on Weather, Electricity, ETTh2. A0 full; A1 w/o alignment; A2 w/o error weighting; A3 correlation-graph gate. Full (A0) matches Table 1 S%

\uparrow

; bold: best per column within each backbone block.

Backbone	Cells	Seeds	Wins	Mean $I\%\uparrow$
PatchTST	3	9	9/9	$+10.6\pm 12.8$
Autoformer	3	9	7/9	$+7.1\pm 17.3$

Table 3: Multiseed deployable Protocol B (aligned subset). Same three complete

(\text{dataset},\rho)

cells for both backbones: ETTh1 (

\rho{=}0.7,0.9

) and ETTh2 (

\rho{=}0.9

), with seeds

\{0,1,2024\}

(partial cells excluded).

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

		Weather			Electricity			ETTh2
Backbone	Ablation	$0.5$	$0.7$	$0.9$	$0.5$	$0.7$	$0.9$	$0.5$	$0.7$	$0.9$
PatchTST	Full (A0)	12.78	9.73	3.55	28.66	20.50	17.36	38.17	11.27	8.80
	w/o align (A1)	-9.06	-2.42	0.29	8.62	9.31	13.52	-7.09	-2.58	2.03
	w/o err.-wt (A2)	12.31	9.42	3.39	24.01	20.39	14.70	37.91	10.72	8.54
	Corr. gate (A3)	-7.51	-7.30	-3.28	14.60	14.16	14.27	10.87	-5.78	-7.65
Autoformer	Full (A0)	20.50	15.66	6.87	17.83	14.10	13.94	76.77	66.98	52.25

Appendix

Table 4: (Supplementary.) Ablation study for graph-only selective inference ( PatchTST and Autoformer ). For each target coverage, we choose energy thresholds on the validation split and apply them to select samples on the test split (complementary to the selective reporting used for Table 1 in the main text). We report test relative improvement

(1-\mathrm{MSE}_{\mathrm{sel}}/\mathrm{MSE}_{\mathrm{orig}})\times 100

c\in\{0.5,0.7,0.9\}

. Datasets: Weather, Electricity, ETTh2 (multivariate M ). A0 : full model (learned adaptive graph, alignment + error-weighted graph loss); A1 :

\lambda_{a}{=}0

(no alignment); A2 : uniform graph regularization weights (no error weighting); A3 : fixed correlation-graph gate (no learned

A

). Bold values: best relative improvement in each column within each backbone block (ties bolded together).

Dataset	Backbone	$\rho=0.5$				$\rho=0.7$				$\rho=0.9$				Full
		TEM	Graph	I% $\uparrow$	S% $\uparrow$	TEM	Graph	I% $\uparrow$	S% $\uparrow$	TEM	Graph	I% $\uparrow$	S% $\uparrow$	Full
ETTh1	Autoformer	0.4562	0.3983	+12.7	+9.2	0.4418	0.4186	+5.3	+4.6	0.4392	0.4145	+5.6	+5.5	0.4388
	FEDformer	0.4019	0.3100	+22.9	+23.4	0.4071	0.3849	+5.5	+4.9	0.4044	0.3989	+1.4	+1.5	0.4049
	PatchTST	0.3810	0.3394	+10.9	+11.6	0.3826	0.3649	+4.6	+4.9	0.3836	0.3827	+0.2	+0.3	0.3838
	TimesNet	0.3713	0.2729	+26.5	+29.6	0.3715	0.3146	+15.3	+18.9	0.3870	0.3332	+13.9	+14.1	0.3879
ETTh2	Autoformer	0.2549	0.0955	+62.5	+68.3	0.2598	0.1210	+53.4	+59.8	0.2715	0.1591	+41.4	+47.2	0.3011

Appendix

Table 5: Validation

\to

test selective prediction MSE across datasets and backbones (lower is better). For each

\rho\in\{0.5,0.7,0.9\}

, acceptance thresholds are fit as empirical

\rho

-quantiles of each raw abstention score ( TEM vs. structural) on the validation split and then applied to the test split without re-calibration (Protocol B). Graph is bolded when lower than TEM at the same

\rho

. I%

\uparrow

is relative MSE reduction vs. TEM:

(\text{TEM}-\text{Graph})/\text{TEM}\times 100

. S%

\uparrow

is relative MSE reduction vs. full-split MSE:

(\text{Full}-\text{Graph}_{\rho})/\text{Full}\times 100

. Positive is better for both. N/A indicates that the validation-derived threshold yields zero accepted test samples (coverage

=0

), so selective MSE and the derived I/S metrics are undefined.

Panel of Figure 4 — Figure 4: Autoformer: test selective MSE vs. target coverage on three datasets, using the same test-split quantile protocol as Section 5 . We compare TEM -only ( ebm_only ), graph-only ( graph_only ), and the full-coverage baseline ( orig_MSE ). Across these datasets, the structural gate typically yields substantially lower selective error at matched nominal coverage.

Panel of Figure 5 — Figure 5: PatchTST: test selective MSE vs. target coverage on three datasets, using the same test-split quantile protocol as Section 5 . We compare TEM -only ( ebm_only ), graph-only ( graph_only ), and the full-coverage baseline ( orig_MSE ).

Dataset	Method	50%	70%	90%
ETTh2	Graph (ours)	0.195	0.202	0.236
	Error predictor	0.226	0.245	0.251
	MC dropout	0.199	0.214	0.239
Electricity	Graph (ours)	0.172	0.180	0.198
	Error predictor	0.183	0.185	0.208
	MC dropout	0.174	0.187	0.202

Appendix

Table 6: (Supplementary comparison.) Test selective MSE on retained samples at nominal coverages on the test split (thresholds from test-split quantiles of raw scores; aligned with Table 1 ; lower is better). We compare our graph-only structural gate to MC dropout and an error predictor (instance-level baselines) on two representative backbones across three datasets (the TEM comparison is reported in the main table).

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