Dynamic Dual-Feedback Conformal Inference for Time Series Forecasting

Songlin Du; Ling Luo; Uwe Aickelin

stat.MLOct 19, 2026

Dynamic Dual-Feedback Conformal Inference for Time Series Forecasting

Authors: Songlin Du, Ling Luo, Uwe Aickelin

Organizations: Faculty of Engineering and Information Technology, School of Computing and Information Systems, University of Melbourne

Abstract

Quantifying uncertainty in time series forecasting is particularly demanding because sequential data exhibit temporal dependence and are prone to distributional changes. Conformal inference has emerged as a powerful uncertainty quantification approach through the construction of prediction sets. Recent advances have introduced online conformal methods that adaptively adjust prediction thresholds through feedback mechanisms. However, the existing feedback mechanism typically relies solely on miscoverage indicators (actual feedback), whether the true label falls within the interval at each time step, while overlooking the empirical prediction threshold (estimated feedback) that is derived from the oracle conformal method. In this paper, we propose Dynamic Dual-feedback Conformal Inference (DDCI), which incorporates a dual-feedback mechanism consisting of actual feedback and estimated feedback. The former drives the primary adjustment of the intervals based on true observations, while the latter dampens excessive expansions or contractions by leveraging empirical thresholds from conformal inference during updates. By balancing these two signals, DDCI achieves more stable and narrower prediction intervals in sequential settings while maintaining near-nominal coverage in practice.

Figures & tables

Panel of Figure 1 — Figure 1: Functional shape comparison of the actual feedback $\tfrac{|x|}{2B}\tanh(cx)$ (blue) and the estimated feedback $(1-\frac{|x|}{2B})\tanh(cx)$ (orange) with $c=0.2$ and $B=5$ or $30$ . The two functions are plotted on the same axis to highlight their distinct regimes of dominance: the former grows in magnitude and dominates for large |x|, whereas the latter exhibits attenuation and dominates near the origin.

Method	Synthetic Data 1			Synthetic Data 2			Synthetic Data 3
Method	Coverage (%)	Avg. width	Winkler Score	Coverage (%)	Avg. width	Winkler Score	Coverage (%)	Avg. width	Winkler Score
OGD	$90.0\pm 0.12$	$11.01\pm 0.50$	$12.82\pm 0.54$	$89.9\pm 0.17$	$17.28\pm 7.59$	$28.13\pm 18.54$	$85.0\pm 4.78$	$371.43\pm 350.57$	$506.07\pm 541.8$
SF-OGD	$90.7\pm 0.16$	$75.75\pm 0.11$	$84.93\pm 0.14$	$90.3\pm 0.23$	$76.83\pm 1.50$	$86.57\pm 2.44$	$89.4\pm 0.98$	$183.13\pm 138.25$	$213.06\pm 162.6$
decay-OGD	$89.7\pm 0.82$	$21.34\pm 2.10$	$23.32\pm 2.44$	$84.3\pm 4.90$	$44.05\pm 34.24$	$56.04\pm 47.25$	$59.6\pm 22.53$	$419.60\pm 307.13$	$2202.6\pm 2148.2$
PID	$89.9\pm 0.11$	$13.25\pm 1.18$	$15.19\pm 1.33$	$89.8\pm 0.08$	$18.04\pm 6.53$	$25.74\pm 11.62$	$89.8\pm 0.13$	$124.17\pm 100.70$	$144.22\pm 115.73$
ECI	$90.0\pm 0.10$	$13.07\pm 1.39$	$15.07\pm 1.53$	$90.0\pm 0.12$	$19.95\pm 7.15$	$30.58\pm 11.92$	$81.9\pm 17.84$	$88.85\pm 54.96$	$119.52\pm 61.22$

Table 1: Results on the three synthetic datasets generated under the Prophet model with nominal level 10%. All experiments are repeated 5 times, and the table reports the mean and standard deviation. The best result (shortest width) is marked in bold, and the second-best result is marked with an underline.

Dataset	Method	Prophet Model			AR Model			Theta Model			Transformer
Dataset	Method	Coverage (%)	Avg. width	Winkler Score	Coverage (%)	Avg. width	Winkler Score	Coverage (%)	Avg. width	Winkler Score	Coverage (%)	Avg. width	Winkler Score
Google	ACI	90.0	$\infty$	$\infty$	89.8	$\infty$	$\infty$	90.5	$\infty$	$\infty$	90.2	$\infty$	$\infty$
	OGD	89.7	57.60	74.74	90.7	33.76	70.96	89.9	31.49	47.88	90.1	109.27	148.76
	SF-OGD	89.6	58.92	77.52	89.9	28.31	41.93	90.0	34.04	51.01	90.1	88.30	114.34
	decay-OGD	89.9	77.23	92.36	90.2	46.54	58.27	90.2	55.33	67.89	89.9	120.25	144.38
	PID	89.9	56.13	71.18	89.8	64.19	83.58	89.3	75.71	97.37	89.8	77.36	106.78

Table 2: The experimental results on three datasets (Google, Amazon, Microsoft).

Figure 2: Visualisation of comparison results between ECI (single feedback only) and DDCI (proposed,

\epsilon=0.2

) on Google stock dataset under Prophet model. The blue crosses and red points represent the miscovered points of each method.

Method	GOOGL	AMZN	MSFT	Delhi	Electricity
PID	8.55	6.94	1.02	1.05	0.0645
ECI	9.66	7.78	0.81	0.43	0.0628
DDCI ( $c=0$ ; no additional feedback)	8.32	6.64	0.63	0.32	0.0622
DDCI ( $\epsilon=0$ ; actual feedback only)	9.93	8.05	0.68	0.31	0.0617
DDCI ( $\epsilon=0.2$ ; dual feedback)	6.89	5.35	0.54	0.28	0.0603

Table 3: Stability evaluation (MASD,

\downarrow

) and ablation study on the five public datasets under Prophet.

Figure 3: The influence of different smoothing parameter

\epsilon

on the size of prediction interval.

Figure 4: The influence of different smoothing parameter

\epsilon

on the coverage rate.

Appendix figures & tables16 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	Method	Prophet Model			AR Model			Theta Model			Transformer
Dataset	Method	Coverage (%)	Avg. width	Winkler Score	Coverage (%)	Avg. width	Winkler Score	Coverage (%)	Avg. width	Winkler Score	Coverage (%)	Avg. width	Winkler Score
Delhi Temp.	ACI	91.0	$\infty$	$\infty$	90.0	$\infty$	$\infty$	90.2	$\infty$	$\infty$	90.3	$\infty$	$\infty$
	OGD	90.4	7.54	9.78	90.1	6.82	9.11	90.0	6.36	8.48	89.9	11.08	13.15
	SF-OGD	90.0	10.63	13.24	90.1	10.92	13.72	90.1	11.33	13.99	90.0	10.93	13.64
	decay-OGD	90.1	11.24	13.03	90.2	10.52	12.25	90.2	10.95	12.85	89.7	14.43	16.18
	PID	89.9	10.87	13.36	89.7	12.28	15.11	89.7	12.49	15.13	89.9	11.57	14.15

Appendix

Table 4: The experimental results on Delhi Temperature and Electricity Demand datasets.

Figure 5: Comparison of empirical coverage rate, mean interval width, and Winkler Score across a wide range of significance levels ( $\alpha$ = 0.05–0.9) for four online conformal inference methods: DDCI, ECI, PID, and OGD.

Method	$\alpha=0.20$			$\alpha=0.10$			$\alpha=0.05$
Method	Coverage (%)	Avg. width	Winkler Score	Coverage (%)	Avg. width	Winkler Score	Coverage (%)	Avg. width	Winkler Score
OGD	79.7	33.70	44.27	89.6	55.15	67.57	94.3	77.24	92.63
SF-OGD	79.7	44.51	58.31	89.5	61.47	73.94	94.5	79.11	92.74
decay-OGD	80.1	46.92	57.98	89.9	97.22	107.79	90.9	107.32	159.64
PID	80.3	36.45	47.97	89.8	52.56	56.02	94.8	54.65	66.58
ECI	80.8	35.39	45.85	90.1	47.00	59.74	94.8	60.10	73.81

Appendix

Table 5: Results under different significance levels

\alpha

on Amazon stock dataset.

Figure 6: Coverage result for AR model.

Figure 7: Prediction sets result for AR model.

Figure 8: Coverage result for Prophet model.

Figure 9: Prediction sets result for Prophet model.

Figure 10: Coverage result for Theta model.

Figure 11: Prediction sets result for Theta model.

Figure 12: Coverage result for Transformer model.

Figure 13: Prediction sets result for Transformer model.

Figure 14: The influence of different window length on Google and Amazon datasets.

Figure 15: The influence of different $d$ on the Google dataset under the Prophet model.

Figure 16: The influence of different

c

on the Google dataset under the Prophet model.

Figure 17: Visualisation of the proposed DDCI method with different

\epsilon

(0.0 and 0.8 respectively) under AR model.

Forecaster	Dataset	PID	ECI	DDCI ( $c=0$ )	DDCI ( $\epsilon=0$ )	DDCI
AR	GOOGL	1.3204	0.5630	0.4707	0.4710	0.4294
	AMZN	1.1762	0.8711	0.7260	0.7227	0.6330
	MSFT	0.8166	0.5351	0.4100	0.4034	0.3768
	Delhi	1.0344	0.4004	0.2860	0.2757	0.2654
	Electricity	10.2395	4.0433	4.0455	4.0011	3.9913
Theta	GOOGL	1.6048	1.3182	1.1391	1.1356	1.0094

Appendix

Table 6: Additional stability results under AR, Theta, and Transformer. Electricity values are displayed after multiplication by

10^{3}

. Bold and underline denote the lowest and second-lowest unrounded values, respectively.

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