State-Matched Sequence Conformal Prediction

Ruihang Han; Li-Hsiang Lin

stat.MLOct 14, 2026

State-Matched Sequence Conformal Prediction

Authors: Ruihang Han, Li-Hsiang Lin

Organizations: Department of Mathematics and Statistics, Georgia State University, Atlanta, GA 30303

Abstract

Sequential conformal prediction needs to decide which past errors should calibrate uncertainty for the current target. Recency and online error feedback are useful when recent observations remain relevant, but the recent past can be a poor reference when comparable states recur at nonadjacent times, with the recurrence pattern generally unknown. We propose State-Matched Sequence Conformal Prediction (SMSCP), which constructs an explicit one-sided local-linear state from pre-target information, uses state similarity to select a matched reference set of past calibration scores, and then applies the ordinary unweighted conformal order statistic. In contrast to existing approaches, SMSCP uses state similarity to identify calibration observations relevant to the target score quantile. This motivates a theoretical analysis of whether nearby states have comparable target score quantiles under temporal dependence. We establish control of the resulting conformal cutoff and characterize predictive information not captured by the chosen state. Simulations show that state matching is most effective when score-relevant states recur across nonadjacent times, while revealing settings in which its advantage weakens or reverses. Across solar irradiance and bike-demand benchmarks, SMSCP improves interval score over recency, weighted, adaptive, and HopCPT comparators, and achieves coverage closer to the nominal 0.90 level than ResCP on five of six benchmarks.

Figures & tables

		SMSCP absolute metrics			$\Delta_{\mathrm{IS}}$ versus SMSCP
Design	Size	Cov.	Width	IS	Rolling	Recent- $k$	Tuned-Recent	Tuned-NexCP	ACI-tuned	ECI
HomoSin	S	0.928	2.727	2.946	+0.201	+0.339	+0.217	+0.203	+0.212	+0.611
HomoSin	L	0.933	2.625	2.817	+0.315	+0.414	+0.317	+0.317	+0.319	+0.527
DualFreq	S	0.930	3.312	3.555	+0.332	+0.486	+0.359	+0.337	+0.342	+0.680
DualFreq	L	0.939	3.160	3.340	+0.573	+0.718	+0.573	+0.575	+0.575	+0.770
HeteroSin	S	0.905	3.568	4.206	-0.019	+0.147	+0.006	-0.015	+0.005	+0.366

Table 1: Core non-neural simulation results at nominal

1-\alpha=0.90

. The small (S) and large (L) splits are

(100,100,100,100)

and

(300,300,300,300)

, respectively, with

B=500

paired Monte Carlo replications per cell. The first three numeric columns report SMSCP coverage, mean width, and mean interval score. The remaining columns report

\Delta_{\mathrm{IS}}=\mathrm{IS}_{\mathrm{comparator}}-\mathrm{IS}_{\mathrm{SMSCP}}

; positive values favor SMSCP . Pointwise paired 95% Monte Carlo confidence intervals are reported in Appendix B.1 .

	SMSCP absolute metrics			Non-neural $\Delta_{\mathrm{IS}}$						HopCPT $\Delta_{\mathrm{IS}}$
Station	Cov.	Width	IS	Rolling	Recent- $k$	Tuned-Recent	Tuned-NexCP	ACI-tuned	ECI	Median	Mean
Alachua	0.910	264.0	361.0	+32.7	+33.4	+32.7	+32.7	+30.4	+30.0	+30.9	+130.6
Phoenix	0.920	188.2	252.8	+31.5	+31.5	+31.5	+29.9	+30.4	+23.4	+21.1	+145.8
Seattle	0.914	208.4	286.2	+32.4	+27.3	+27.0	+22.9	+19.9	+19.1	+23.0	+29.2
Houston	0.914	249.3	332.9	+30.2	+31.7	+29.5	+30.2	+28.7	+28.5	+24.5	+24.8
Minneapolis	0.909	214.3	295.6	+22.1	+19.7	+19.6	+17.5	+16.8	+13.1	+15.3	+45.2

Table 2: Solar irradiance benchmark at nominal

1-\alpha=0.90

. For the six non-neural comparators,

\Delta_{\mathrm{IS}}=\mathrm{IS}_{\mathrm{comparator}}-\mathrm{IS}_{\mathrm{SMSCP}}

is the paired mean difference on the fixed station sequence; all 30 corresponding pointwise 7-day moving-block-bootstrap 95% intervals are strictly positive. The last two columns report the median and mean HopCPT-minus- SMSCP interval-score differences across 50 neural initializations on the same fixed station sequence; this is initialization variability, not time-series sampling uncertainty. Width and interval score are in W/m 2 .

Appendix figures & tables15 assets

Supplementary material from the paper’s appendix.

Appendix

Design	Method	Coverage	Mean width	Mean $\mathrm{IS}_{\alpha}$
HomoSin	SMSCP	0.928	2.727	2.946
	Rolling	0.901	2.823	3.147
	Recent-k	0.921	3.010	3.286
	Tuned-Recent	0.903	2.843	3.163
	Tuned-NexCP	0.904	2.836	3.149
	ACI-tuned	0.902	2.834	3.159

Appendix

Table 3: Complete non-neural core simulation results for the small split

(100,100,100,100)

B=500

, nominal

1-\alpha=0.90

Design	Method	Coverage	Mean width	Mean $\mathrm{IS}_{\alpha}$
HomoSin	SMSCP	0.933	2.625	2.817
	Rolling	0.901	2.807	3.132
	Recent-k	0.920	2.966	3.231
	Tuned-Recent	0.901	2.809	3.133
	Tuned-NexCP	0.902	2.815	3.134
	ACI-tuned	0.902	2.813	3.135

Appendix

Table 4: Complete non-neural core simulation results for the large split

(300,300,300,300)

B=500

, nominal

1-\alpha=0.90

Design	Size	Rolling	Recent- $k$	Tuned-Recent	Tuned-NexCP	ACI-tuned	ECI
HomoSin	S	+0.201 [+0.188,+0.213]	+0.339 [+0.325,+0.354]	+0.217 [+0.204,+0.230]	+0.203 [+0.191,+0.215]	+0.212 [+0.200,+0.225]	+0.611 [+0.574,+0.649]
HomoSin	L	+0.315 [+0.308,+0.323]	+0.414 [+0.406,+0.422]	+0.317 [+0.309,+0.324]	+0.317 [+0.310,+0.324]	+0.319 [+0.311,+0.326]	+0.527 [+0.519,+0.535]
DualFreq	S	+0.332 [+0.315,+0.348]	+0.486 [+0.468,+0.504]	+0.359 [+0.342,+0.377]	+0.337 [+0.320,+0.353]	+0.342 [+0.325,+0.359]	+0.680 [+0.636,+0.724]
DualFreq	L	+0.573 [+0.565,+0.581]	+0.718 [+0.710,+0.726]	+0.573 [+0.565,+0.581]	+0.575 [+0.567,+0.584]	+0.575 [+0.567,+0.583]	+0.770 [+0.762,+0.779]
HeteroSin	S	-0.019 [-0.034,-0.003]	+0.147 [+0.128,+0.166]	+0.006 [-0.011,+0.023]	-0.015 [-0.031,+0.000]	+0.005 [-0.011,+0.022]	+0.366 [+0.330,+0.402]
HeteroSin	L	+0.129 [+0.121,+0.137]	+0.183 [+0.173,+0.192]	+0.131 [+0.124,+0.139]	+0.131 [+0.124,+0.139]	+0.136 [+0.128,+0.143]	+0.434 [+0.423,+0.444]

Appendix

Table 5: Paired interval-score comparisons for the core non-neural simulation panel. Each cell is

\Delta_{\mathrm{IS}}=\mathrm{IS}_{\mathrm{comparator}}-\mathrm{IS}_{\mathrm{SMSCP}}

followed by its pointwise 95% paired Monte Carlo confidence interval. Positive values favor SMSCP .

Figure 1: Direct comparison with HopCPT across the ten simulation cells. Points show

\Delta_{\mathrm{IS}}=\mathrm{IS}_{\mathrm{HopCPT}}-\mathrm{IS}_{\mathrm{SMSCP}}

and bars show pointwise 95% paired Monte Carlo confidence intervals across 50 independent data realizations after averaging the three fixed HopCPT initializations within each realization. Positive values favor SMSCP .

		SMSCP			HopCPT			$\Delta_{\mathrm{IS}}$ [95% CI]
Design	Size	Cov.	Width	IS	Cov.	Width	IS
HomoSin	S	0.928	2.704	2.919	0.886	2.786	3.218	+0.299 [+0.225,+0.372]
HomoSin	L	0.933	2.613	2.807	0.892	2.782	3.289	+0.482 [+0.266,+0.699]
DualFreq	S	0.927	3.297	3.549	0.899	3.497	4.013	+0.464 [+0.296,+0.632]
DualFreq	L	0.939	3.157	3.341	0.891	3.512	4.268	+0.927 [+0.463,+1.391]
HeteroSin	S	0.901	3.548	4.238	0.880	3.419	4.341	+0.103 [-0.023,+0.228]

Appendix

Table 6: Direct simulation comparison with HopCPT. The final column is

\mathrm{IS}_{\mathrm{HopCPT}}-\mathrm{IS}_{\mathrm{SMSCP}}

with a pointwise 95% paired Monte Carlo confidence interval across 50 independent data realizations. The SMSCP metrics in this table are computed on the same 50 realizations used for the paired HopCPT comparison and therefore need not equal the

B=500

averages in Table 1 .

Design	Method	Coverage	Mean width	Mean $\mathrm{IS}_{\alpha}$
AR1	SMSCP	0.899	3.524	4.460
	Rolling	0.900	3.520	4.447
	Recent-k	0.901	3.553	4.469
	Tuned-Recent	0.900	3.535	4.458
	Tuned-NexCP	0.902	3.551	4.450
	ACI-tuned	0.901	3.550	4.463

Appendix

Table 7: Additional simulation designs under the large split

(300,300,300,300)

with

B=500

paired replications.

Station	Method	Coverage	Mean width	Mean $\mathrm{IS}_{\alpha}$	$n_{\rm eval}$
Alachua	SMSCP	0.910	264.0	361.0	22887
	Rolling	0.900	266.9	393.7	22887
	Recent-k	0.907	282.7	394.4	22887
	Tuned-Recent	0.900	266.9	393.7	22887
	Tuned-NexCP	0.900	266.9	393.7	22887
	ACI-tuned	0.901	272.2	391.4	22887

Appendix

Table 8: Solar station-level absolute results for the non-neural panel. Width and interval score are in W/m 2 .

Station	Rolling	Recent- $k$	Tuned-Recent	Tuned-NexCP	ACI-tuned	ECI
Alachua	+32.7 [+28.1,+37.2]	+33.4 [+28.8,+38.3]	+32.7 [+28.5,+36.9]	+32.7 [+28.4,+36.8]	+30.4 [+26.0,+35.0]	+30.0 [+25.0,+34.8]
Phoenix	+31.5 [+28.7,+34.1]	+31.5 [+28.4,+34.6]	+31.5 [+28.5,+34.4]	+29.9 [+27.2,+32.5]	+30.4 [+27.1,+33.4]	+23.4 [+21.0,+25.7]
Seattle	+32.4 [+28.0,+37.8]	+27.3 [+21.6,+33.2]	+27.0 [+22.3,+32.2]	+22.9 [+17.9,+27.8]	+19.9 [+15.1,+24.9]	+19.1 [+14.4,+23.4]
Houston	+30.2 [+26.0,+34.8]	+31.7 [+27.5,+36.5]	+29.5 [+25.6,+34.1]	+30.2 [+26.2,+34.6]	+28.7 [+24.9,+32.8]	+28.5 [+24.0,+32.9]
Minneapolis	+22.1 [+19.1,+25.0]	+19.7 [+16.2,+23.0]	+19.6 [+16.1,+22.8]	+17.5 [+14.1,+20.7]	+16.8 [+13.5,+20.0]	+13.1 [+9.9,+16.3]

Appendix

Table 9: Solar paired interval-score differences. Each cell is comparator-minus- SMSCP mean interval score followed by the 7-day moving-block-bootstrap pointwise 95% interval based on 500 bootstrap replications. All 30 intervals are strictly positive.

	SMSCP			HopCPT across initializations
Station	Cov.	Width	IS	Mean cov.	Median cov.	Mean width	Median width	Mean IS	Median IS
Alachua	0.910	264.0	361.0	0.844	0.898	244.8	261.0	491.6	391.9
Phoenix	0.920	188.2	252.8	0.849	0.901	196.0	205.9	398.6	273.9
Seattle	0.914	208.4	286.2	0.881	0.895	206.4	205.8	315.4	309.2
Houston	0.914	249.3	332.9	0.897	0.897	239.2	239.1	357.7	357.4
Minneapolis	0.909	214.3	295.6	0.867	0.894	200.2	206.1	340.8	310.9

Appendix

Table 10: Solar comparison with HopCPT across 50 neural initializations on each fixed station sequence. Mean and median refer to variability over initialization, not repeated time-series samples.

SMSCP absolute metrics			$\Delta_{\mathrm{IS}}$ versus SMSCP
Cov.	Width	IS	Rolling	Recent- $k$	Tuned-Recent	Tuned-NexCP	ACI-tuned	ECI	HopCPT
0.894	326.4	419.6	+315.4	+284.7	+296.2	+276.3	+286.9	+199.5	+48.2

Appendix

Table 11: Bike Sharing benchmark on the common set of 4169 test targets at nominal

1-\alpha=0.90

. The first three columns report SMSCP absolute metrics. The remaining columns report

\Delta_{\mathrm{IS}}=\mathrm{IS}_{\mathrm{comparator}}-\mathrm{IS}_{\mathrm{SMSCP}}

. For the seven comparators listed here, all corresponding pointwise paired 7-day moving-block-bootstrap 95% intervals are strictly positive; full absolute results and intervals are reported in Appendix B.6 . HopCPT uses the targetwise mean over 50 neural initializations before time-series bootstrap inference.

Method	Coverage	Mean width	Mean $\mathrm{IS}_{\alpha}$
SMSCP	0.894	326.4	419.6
Rolling	0.868	397.3	734.9
Recent- $k$	0.907	502.4	704.3
Tuned-Recent	0.899	494.3	715.8
Tuned-NexCP	0.912	507.4	695.9
ACI-tuned	0.906	501.9	706.4

Appendix

Table 12: Bike Sharing benchmark: absolute results on the common set of 4169 test targets. HopCPT metrics are averaged across 50 neural initializations within each target.

Comparator	$\Delta$ Coverage [95% CI]	$\Delta$ Width [95% CI]	$\Delta_{\mathrm{IS}}$ [95% CI]
Rolling	-0.0261 [-0.0368,-0.0175]	+70.8 [+44.3,+95.7]	+315.4 [+302.3,+344.7]
Recent- $k$	+0.0127 [+0.0024,+0.0227]	+176.0 [+163.8,+190.4]	+284.7 [+274.6,+306.4]
Tuned-Recent	+0.0048 [-0.0038,+0.0133]	+167.9 [+145.2,+190.1]	+296.2 [+285.8,+312.8]
Tuned-NexCP	+0.0178 [+0.0089,+0.0297]	+181.0 [+168.2,+203.3]	+276.3 [+262.4,+304.5]
ACI-tuned	+0.0120 [+0.0029,+0.0232]	+175.5 [+161.2,+192.3]	+286.9 [+277.7,+307.2]
ECI	+0.0091 [-0.0005,+0.0233]	+107.2 [+97.2,+126.2]	+199.5 [+190.1,+220.1]

Appendix

Table 13: Bike Sharing paired 7-day moving-block-bootstrap comparisons,

B=500

. Differences are comparator-minus- SMSCP on the same 4169 targets. Positive interval-score differences favor SMSCP .

Comparator	$\Delta_{\mathrm{IS}}$	7-day 95% CI	14-day 95% CI
Rolling	+315.4	[+302.3,+344.7]	[+298.4,+348.1]
Recent- $k$	+284.7	[+274.6,+306.4]	[+274.2,+310.0]
Tuned-Recent	+296.2	[+285.8,+312.8]	[+284.9,+313.9]
Tuned-NexCP	+276.3	[+262.4,+304.5]	[+266.2,+307.5]
ACI-tuned	+286.9	[+277.7,+307.2]	[+276.3,+309.4]
ECI	+199.5	[+190.1,+220.1]	[+191.9,+220.6]

Appendix

Table 14: Bike Sharing block-length sensitivity for paired interval-score comparisons. Differences are comparator-minus- SMSCP on the same 4169 targets. The point estimate

\Delta_{\mathrm{IS}}

is unchanged across block lengths; only the moving-block-bootstrap 95% interval changes.

	SMSCP			ResCP			$\Delta_{\mathrm{IS}}$ [95% CI]
Dataset	Cov.	Width	IS	Cov.	Width	IS
Alachua	0.910	264.0	361.0	0.886	202.4	334.0	-27.0 [-31.9,-21.8]
Phoenix	0.920	188.2	252.8	0.883	90.8	188.5	-64.4 [-69.9,-58.4]
Seattle	0.914	208.4	286.2	0.885	142.8	242.5	-43.7 [-49.0,-38.9]
Houston	0.914	249.3	332.9	0.867	165.8	297.3	-35.7 [-41.6,-30.3]
Minneapolis	0.909	214.3	295.6	0.889	150.8	251.7	-43.9 [-48.2,-39.3]

Appendix

Table 15: Real-data comparison with ResCP at nominal

1-\alpha=0.90

. The last column reports

\Delta_{\mathrm{IS}}=\mathrm{IS}_{\mathrm{ResCP}}-\mathrm{IS}_{\mathrm{SMSCP}}

with its pointwise paired 7-day moving-block-bootstrap 95% interval (

B=500

). Negative values indicate lower interval score for ResCP. Coverage entries are empirical point estimates and are interpreted relative to the nominal level. Width and interval score are in the response units of the corresponding dataset and should not be compared in magnitude across datasets.

Design	Size	SMSCP median sec.	HopCPT median train+infer sec.	HopCPT train-time IQR
HomoSin	S	1.42	20.51	30.11
HomoSin	L	3.49	26.17	13.34
DualFreq	S	1.41	20.64	1.46
DualFreq	L	4.64	29.03	2.94
HeteroSin	S	1.06	18.55	4.42
HeteroSin	L	4.72	28.54	1.63

Appendix

Table 16: Execution diagnostics from the simulation runs. HopCPT times are per neural initialization; SMSCP times are the median wall time for the full SMSCP path per data realization. These are not controlled cross-method runtime benchmarks.

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Online conformal prediction must balance fast adaptation to distribution shift against stable coverage: feedback-driven methods react quickly but become volatile, while strongly discounted Bayesian methods lag and inflate intervals at tight coverage. We introduce State-Adaptive Bayesian Conformal Prediction (SA-BCP), which forms the predictive quantile as a gated convex combination of long-term temporal inertia and local spatial evidence from a kernel density estimate, controlled by a single interpretable evidence threshold

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K^*_{\mathrm{MSE}}=α(1-α)/M^{\mathcal{T}}

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M^{\mathcal{T}}

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K

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and, for a segmented variant, a sublinear dynamic-regret bound under sublinearly many (

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3\times

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