Conformal prediction provides distribution-free prediction intervals but relies on exchangeability, an assumption often violated in economic forecasting because of covariate shift, concept drift, local heterogeneity and latent regimes. We propose Dynamic Regime-Aware Conformal Prediction (DRACP), which combines density-ratio, localized kernel and probabilistic regime-aware weighting with a self-tuning online significance controller in a unified weighted conformal calibration framework. We distinguish three theoretical results: finite-sample validity under oracle importance weights, a coverage-gap bound for estimated weights with rates in effective sample size, and deterministic or regret guarantees for the online controller. We evaluate DRACP against six baselines on 48 real forecasting series covering euro-area and EU-27 HICP inflation, US macroeconomic and energy indicators, and daily financial series. Recent online methods (FACI, strongly-adaptive online conformal prediction and conformal PID) were verified against the authors' implementations. DRACP is not the most efficient method: strongly-adaptive online conformal prediction achieves the best interval score and intervals about 20% narrower. Instead, DRACP provides the most reliable calibration, achieving coverage closest to the nominal 0.90 (0.890), never falling below 0.80 on any series, maintaining the best coverage at all forecast horizons, and performing best during the 2021-2023 inflation surge. The strongly-adaptive method undercovers on 20 of 48 series versus 10 for DRACP. DRACP therefore offers a principled trade-off between calibration and efficiency, favoring reliable coverage when prediction intervals must satisfy coverage standards. An ablation study shows that the online controller and conditional-scale normalization provide most of the performance gain, whereas the weighting components make a smaller contribution.
Figures & tables
Figure 1: The DRACP pipeline at a single forecast origin t . A standard point forecaster produces errors on past periods; four weighting mechanisms score how relevant each past period is to the present; the interval is the weighted quantile of those errors. A controller adjusts the working significance level from realised coverage. All quantities are computed from information available at t .
Symbol
Meaning
(Xt,Yt)
covariates and outcome at time t
μ,σ
point forecaster and conditional scale estimate
Si
scaled conformity score ∣Yi−μ(Xi)∣/σ(Xi)
Dt
rolling calibration buffer of size m
wi,t,wˉi,t
composite and normalized calibration weights
λ
temporal decay factor
Table 1: Notation.
Figure 2: The DRACP calibration step. Four weighting components—temporal decay, density-ratio correction for covariate shift, an ESS-adaptive local kernel, and regime-similarity weighting—are combined into a single normalized weight vector that defines a weighted conformal radius qt and interval Ct . An online controller adapts the significance level αt from realized coverage, closing the loop.
Group
# series
Source
Content
HICP inflation
28
Eurostat
monthly annual inflation, euro area and EU-27
US macro & energy
13
FRED, UCI
activity, prices, money, sentiment, electricity
Financial (daily)
7
FRED
rates, FX, commodities, equity, volatility
Table 2: The 48 real forecasting series, by group.
Role
Model
Target
Configuration
Point forecaster μ
Hist. gradient boosting
Y
learning rate 0.05 ; 150 – 200 iterations
Conditional scale σ
Random forest
∣Y−μ(X)∣
150 trees; min. leaf size 8
Density ratio rt
Logistic regression
recent vs. older
standardised inputs; clipped to [rmin,rmax]
Regime posterior π
Gaussian mixture
covariates
K components, full covariance, causal fit
Feature scaling
Standardisation
covariates
used for kernel distances
Table 3: Models used in the experiments. The point forecaster is shared by every method; the nuisance estimators are specific to DRACP.
Method
Score rank
# best
Mean cov.
Mean ∣cov−0.90∣
Width rank
SAOCP
2.17
33
0.868
0.046
3.50
FACI
2.71
5
0.883
0.031
3.92
DRACP
3.15
5
0.890
0.027
5.69
Adaptive
4.38
0
0.884
0.031
5.52
C-PID
4.52
3
0.825
0.083
3.88
Rolling
5.52
0
0.841
0.078
3.44
Table 4: Overall comparison across the 48 real series: average interval-score rank (1 = best), number of series where best, mean empirical coverage against a nominal 0.90 , mean absolute coverage deviation, and average width rank. Ordered by score rank.
Method
Mean cov.
Mean ∣ dev ∣
Worst ∣ dev ∣
Min cov.
<0.85
<0.80
DRACP
0.890
0.027
0.092
0.808
10
0
FACI
0.883
0.031
0.092
0.808
12
0
Adaptive
0.884
0.031
0.105
0.795
12
1
SAOCP
0.868
0.046
0.150
0.750
20
5
Rolling
0.841
0.078
0.245
0.655
24
16
C-PID
0.825
0.083
0.245
0.655
25
20
Table 5: Calibration reliability across the 48 real series. “Worst ∣ dev ∣ ” is the largest absolute deviation from nominal on any series and “Min cov.” the lowest empirical coverage attained; the last two columns count series undercovering below 0.85 and below 0.80 . Ordered by mean deviation.
Figure 4: Average interval-score rank of each method over the 48 real series (lower is better; the dotted line marks the Nemenyi critical distance from the best method).
Figure 5: Distribution of DRACP’s interval-score rank across the 48 real series; it is best on five and in the top three on roughly half.
Group
n
SAOCP
FACI
DRACP
Adaptive
C-PID
Rolling
Split
DRACP cov.
EU HICP inflation
28
1.14
2.32
2.57
4.18
5.11
6.18
6.50
0.878
US macro & energy
13
3.15
3.69
3.62
4.46
3.92
4.85
4.31
0.908
Financial (daily)
7
4.43
2.43
4.57
5.00
3.29
4.14
4.14
0.905
Table 6: Average interval-score rank by group (1 = best), with DRACP empirical coverage in the last column. Ordered as in Table 4 .
Figure 6: Average interval-score rank by method within each of the three series groups.
Figure 7: Distribution of empirical coverage across the 48 real series, by method, ordered by mean absolute deviation from nominal. DRACP and FACI are the only methods whose lower whisker stays above 0.80 ; the strongly-adaptive procedure, which attains the best interval score, undercovers on twenty series.
Figure 8: Prediction intervals for German and Romanian HICP inflation through the 2021–23 surge (shaded). DRACP (top) widens and re-narrows around the break; FACI (bottom) adapts more slowly. Coverage and score annotations are computed from the single reproducibility run that produced these interval paths, and therefore differ slightly from the 20-seed means in E .
Figure 9: Left: interval width for Germany—DRACP expands at the break and contracts afterwards. Right: the self-tuned significance level αt absorbing residual miscoverage.
Unadjusted
BH-adjusted
Baseline
Tests
better
worse
better
worse
SAOCP
48
2
15
2
9
FACI
48
4
14
4
9
Adaptive
48
19
5
12
2
C-PID
48
10
5
4
2
Rolling
48
6
5
5
2
Table 7: Diebold–Mariano outcomes for DRACP versus each baseline over the 48 real series (synthetic scenarios excluded). “Unadjusted” counts use p<0.05 per test; “BH-adjusted” counts control the false discovery rate at 5% across all 288 tests jointly.
Series
Split
Rolling
Adaptive
FACI
SAOCP
Conf-PID
Electricity
6.6 ∗∗∗†
5.8 ∗∗∗†
6.3 ∗∗∗†
6.1 ∗∗∗†
5.8 ∗∗∗†
5.6 ∗∗∗†
Euro area HICP
1.8
1.7
3.2 ∗∗†
1.9
1.7
1.7
US retail sales
1.0
0.9
1.9
0.6
0.9
0.9
US ind. production
1.4
1.5
1.7
1.5
1.5
1.5
HICP DE
2.0 ∗
1.9
3.1 ∗∗
1.5
1.9
1.9
HICP FR
2.0 ∗
1.9
2.1 ∗
1.6
1.9
1.9
Table 8: Diebold–Mariano statistics on representative structured series (positive favours DRACP). ∗p<0.05 , ∗∗p<0.01 , ∗∗∗p<0.001 (unadjusted); † also significant after Benjamini–Hochberg control of the false discovery rate at 5% across all 288 tests jointly.
Table 10: Synthetic mean interval score (20-seed means; lowest in bold).
Figure 10: Empirical coverage across synthetic scenarios of increasing shift complexity.
Base forecaster
RMSE
DRACP rk
SAOCP rk
FACI rk
Δ vs best (%)
DRACP cov.
SAOCP cov.
ridge ar
1.00
3.52
2.69
3.58
+6.5
0.901
0.891
mlp
1.26
4.73
2.08
2.40
+36.5
0.906
0.883
random forest
1.62
3.31
1.98
2.75
+27.2
0.889
0.866
gradient boosting
2.03
3.10
2.21
2.69
+18.8
0.890
0.863
naive
3.67
2.75
2.19
3.14
+13.3
0.892
0.872
Table 11: Comparison repeated with five base forecasters (48 series). “RMSE” is the median point-forecast RMSE relative to the best model on the same series ( 1.00= most accurate). Ranks are average interval-score ranks among the seven conformal methods. “ Δ vs best” is the mean per-series percentage difference in interval score between DRACP and the best baseline on that series (positive is worse for DRACP).
Interval-score rank
Coverage
Horizon
SAOCP
FACI
DRACP
Adapt.
C-PID
Roll.
Split
DRACP
SAOCP
FACI
h=1
2.19
2.73
3.10
4.38
4.52
5.52
5.56
0.890
0.868
0.883
h=3
2.46
2.73
4.06
4.25
4.29
4.88
5.33
0.887
0.839
0.870
h=6
3.42
2.92
4.52
4.52
4.44
4.11
4.07
0.873
0.804
0.844
h=12
4.50
3.33
4.92
4.56
4.75
3.05
2.89
0.842
0.786
0.822
Table 12: Average interval-score rank by forecast horizon over the seven methods (48 series, 10 seeds); ranks sum to 28 in every row and are comparable across rows. The last three columns are empirical coverage against a nominal 0.90 for the three leading methods.
Figure 11: Empirical coverage against forecast horizon over the 48 real series. All methods lose coverage as the horizon grows; DRACP loses it most slowly, and its margin over the strongly-adaptive procedure widens from 2.2 coverage points at h=1 to 5.6 at h=12 .
Table 14: Sensitivity to the number of latent regimes K on the combined-shift benchmark (nominal coverage 0.90 ). “ESS” is the mean effective calibration sample size.
Variant
Δ score (%)
Worse on
Mean ∣cov−0.90∣
Full DRACP
+0.0
0%
0.027
– density ratio
+0.7
50%
0.028
– regime weighting
+0.7
50%
0.027
– localization
+0.2
62%
0.025
– temporal decay
+4.6
71%
0.037
– adaptive control
+11.8
92%
0.051
Table 15: Component ablation on the real panel (48 series, 20 seeds). “ Δ score” is the mean per-series percentage change in interval score relative to full DRACP (positive is worse); “worse on” is the share of series on which removal degrades the score. The last row replaces the fitted conditional scale with a constant, isolating the contribution of scale normalisation from that of the weights and the controller.
Appendix figures & tables15 assets
Supplementary material from the paper’s appendix.
Appendix
Series
Source
Freq.
Transform
Test
Test period
Euro area HICP
Eurostat
M
rate (level)
73
2019-12–2025-12
HICP AT
Eurostat
M
rate (level)
84
2019-01–2025-12
HICP BE
Eurostat
M
rate (level)
84
2019-01–2025-12
HICP BG
Eurostat
M
rate (level)
82
2019-03–2025-12
HICP CY
Eurostat
M
rate (level)
84
2019-01–2025-12
HICP CZ
Eurostat
M
rate (level)
84
2019-01–2025-12
Appendix
Table 16: The 48 real forecasting series, in three blocks: HICP inflation, US macroeconomic and energy, daily financial. Frequency: M monthly, D daily (business days), H hourly. “Transform” is the modelling target; “Test” the number of sequential one-step predictions. The test period is reported at the resolution of the series: calendar month for monthly series, calendar day for the daily financial series, and date and hour for the hourly electricity series.
Series
Split
Rolling
Adaptive
FACI
SAOCP
C-PID
DRACP
Euro area HICP
23.39
19.13
12.53
10.52
19.23
18.71
9.69
HICP AT
25.80
21.57
11.84
11.25
21.61
21.46
11.29
HICP BE
22.55
19.96
13.97
11.51
20.01
19.77
10.13
HICP BG
7.78
7.24
6.52
6.62
7.27
7.03
4.58
HICP CY
9.11
9.14
8.66
8.60
9.27
8.88
7.89
HICP CZ
31.46
28.07
14.34
13.48
28.12
27.85
8.68
Appendix
Table 17: Mean interval score per series (lower is better; best per row in bold).
Series
Split
Rolling
Adaptive
FACI
SAOCP
C-PID
DRACP
Euro area HICP
0.603
0.712
0.795
0.795
0.712
0.726
0.806
HICP AT
0.607
0.750
0.833
0.821
0.750
0.762
0.845
HICP BE
0.571
0.762
0.821
0.810
0.762
0.762
0.847
HICP BG
1.000
1.000
0.976
0.976
1.000
1.000
0.959
HICP CY
0.857
0.857
0.905
0.881
0.857
0.857
0.900
HICP CZ
0.524
0.738
0.833
0.810
0.738
0.738
0.915
Appendix
Table 18: Empirical coverage per series (nominal 0.90 ; closest per row in bold).
Series
Split
Rolling
Adaptive
FACI
SAOCP
C-PID
DRACP
Euro area HICP
2.13
5.68
10.28
8.12
5.68
5.90
7.39
HICP AT
1.38
5.19
9.04
7.43
5.19
5.27
8.47
HICP BE
1.62
5.62
10.71
7.55
5.62
5.77
7.52
HICP BG
7.78
7.24
6.43
6.55
7.27
7.03
4.38
HICP CY
5.90
6.22
6.91
6.57
6.23
6.36
6.41
HICP CZ
1.46
7.11
11.39
10.13
7.10
7.24
7.98
Appendix
Table 19: Mean interval width per series (narrowest per row in bold).
Series
Split
Rolling
Adaptive
FACI
SAOCP
Conf-PID
Electricity
6.6 †
5.8 †
6.3 †
6.1 †
5.8 †
5.6 †
Euro area HICP
1.8
1.7
3.2 †
1.9
1.7
1.7
Fed funds (d)
14.4 †
14.4 †
9.8 †
9.7 †
14.4 †
14.4 †
HICP AT
1.7
1.5
1.1
-0.0
1.5
1.5
HICP BE
1.7
1.8
3.4 †
1.4
1.8
1.8
HICP BG
5.9 †
4.9 †
3.5 †
3.7 †
5.0 †
4.4 †
Appendix
Table 20: Complete Diebold–Mariano matrix: DRACP versus each baseline on every one of the 48 real series (positive favours DRACP). ∗ significant at the unadjusted 5% level; † significant after Benjamini–Hochberg control of the false discovery rate at 5% across all 288 tests jointly (threshold p≤0.0073 ). Entries marked † are a subset of those marked ∗ .
Scenario
Split
Rolling
Adaptive
FACI
DRACP
Stationary
3.78
3.83
3.83
3.82
4.08
Covariate
3.79
3.82
3.81
3.81
4.05
Conditional
4.81
7.65
8.37
8.29
8.97
Abrupt
8.15
12.55
12.87
12.86
13.78
Gradual
8.67
10.40
10.25
10.26
10.92
Regime
6.79
6.81
6.79
6.70
7.00
Appendix
Table 21: Synthetic mean interval width (20-seed means; narrowest in bold).
Figure 12: Per-series mean interval score (means over 20 repeated fits, 95% paired moving-block bootstrap whiskers) for a representative subset of structured (top rows) and financial (bottom row) series. Full set produced by reporting.mc_report .
Figure 14: Diagnostics. Top: estimated density-ratio contribution and posterior regime probabilities. Bottom: the online significance level αt and the effective calibration sample size versus bandwidth.
Figure 15: Further diagnostics: estimated regime transition matrix, localization weights vs. distance, and the normal Q–Q plot and histogram of standardized residuals.
Figure 16: Hyperparameter sensitivity of DRACP (combined-shift scenario): localization bandwidth, temporal forgetting factor, and density-ratio clipping.
Figure 17: Left: coverage across scenarios of increasing shift complexity. Right: distribution of interval widths by method under combined shift.
Figure 18: Computational cost of DRACP: runtime per prediction versus calibration-buffer size (left) and the O(Bd) memory footprint (right).
Figure 19: A synthetic regime-switching series with annotated regimes (left) and DRACP prediction intervals on the combined-shift scenario (right).
Figure 20: DRACP prediction intervals on electricity demand (left) and euro-area inflation (right).
Recent advances in uncertainty quantification for time series forecasting show that conformal prediction can provide reliable prediction intervals, yet standard conformal methods are often inefficient under temporal dependence, drift, and heterogeneous error behavior. Existing methods typically either update miscoverage rates over time or learn unconstrained calibration weights, without explicitly separating two central sources of nonstationarity: smoothly drifting error distributions and co-existing distinct error regimes. We introduce RareCP, a regime-aware retrieval method for adaptive conformal time series prediction. RareCP learns local calibration representations through a mixture of cosine-attention experts that each capture distinct error regimes, while a compact hypernetwork adapts the kernel parameters to track temporal drift. Given a new forecasting context, RareCP retrieves the top-k most relevant calibration examples, assigns similarity weights, and forms a weighted conformal quantile over their signed residuals, yielding asymmetric prediction intervals. The adaptive kernel is trained using a smooth interval score objective, with a parameter-space anchor to a lightweight teacher kernel to preserve stable local representations. On the GIFT-Eval benchmark, RareCP improves interval efficiency over recent conformal baselines and foundation model uncertainty estimates while maintaining empirical coverage. Ablations confirm that regime-specific experts, drift-adaptive kernels, sparse retrieval, and teacher anchoring each contribute to the final performance.
Manuel Heurich, Maximilian Granz, Tim Landgraf
Institute for Computer Science Freie Universität Berlin Arnimallee 7 14195 Berlin
We present Distribution-aware Conformal Prediction (DCP), a unified framework integrating probabilistic predictors like Monte Carlo dropout, deep ensembles, and quantile regression with score-agnostic conformal calibration to produce valid and efficient prediction intervals. Leveraging a numerical inversion approach to construct interval bounds, DCP accommodates arbitrary combinations of distribution generating predictors and nonconformity scores. Benchmark analysis on synthetic and real-world time series data demonstrate DCP's ability to adaptively calibrate prediction intervals under varying uncertainty regimes. Crucially, DCP's modular design facilitates plug-and-play experimentation with different predictor-score pairings, quantitatively supported by a newly introduced modified Winkler score that balances validity and efficiency by explicitly penalizing undercoverage. While DCP generalizes and extends existing approaches like Conformalized Quantile Regression and Conformalized Monte Carlo, its modular design allows further extensions, setting a foundation for advancing uncertainty quantification in dynamic environments and high-risk applications.
Daniel Schweizer, Peter Kuhn, Jayant Sharma +3
Fraunhofer Institute for Highspeed Dynamics, Ernst-Mach-Institut, EMI Freiburg, Germany
Adaptive Conformal Inference (ACI) provides distribution-free prediction intervals with asymptotic coverage guarantees for time series under distribution shift. However, ACI only adapts the quantile threshold -- it cannot shift the interval center. When a base forecaster develops persistent bias after a regime change, ACI compensates by widening intervals symmetrically, producing unnecessarily conservative bands. We propose Bias-Corrected ACI (BC-ACI), which augments standard ACI with an online exponentially weighted moving average (EWM) estimate of forecast bias. BC-ACI corrects nonconformity scores before quantile computation and re-centers prediction intervals, addressing the root cause of miscalibration rather than its symptom. An adaptive dead-zone threshold suppresses corrections when estimated bias is indistinguishable from noise, ensuring no degradation on well-calibrated data. In controlled experiments across 688 runs spanning two base models, four synthetic regimes, and three real datasets, BC-ACI reduces Winkler interval scores by 13--17% under mean and compound distribution shifts (Wilcoxon p < 0.001) while maintaining equivalent performance on stationary data (ratio 1.002x). We provide finite-sample analysis showing that coverage guarantees degrade gracefully with bias estimation error.