SlowGuess/NeurIPS_2023_OCR huggingface dataset:
The separation  $\Delta_{i}$  of a pattern  $x_{i}$  is its minimal dot product difference to any of the other patterns:
$$
\Delta_ {i} = \min  _ {j, j \neq i} \left(\boldsymbol {x} _ {i} ^ {T} \boldsymbol {x} _ {i} - \boldsymbol {x} _ {i} ^ {T} \boldsymbol {x} _ {j}\right). \tag {30}
$$
A pattern is well-separated from the data if  $\Delta_{i}$  is above a given threshold (specified in Ramsauer et al., 2021). If the patterns  $x_{i}$  are well-separated, the update rule Equation 29 converges to a fixed point close to a stored pattern. If some patterns are similar to one another and, therefore, not well-separated, the update rule converges to a fixed point close to the mean of the similar patterns.
The update rule of a Hopfield network thus identifies sample-sample relations between stored patterns. This enables similarity-based learning methods like nearest neighbor search (see Schäfl et al., 2022), which HopCPT leverages to learn a retrieval of samples from similar error regimes.
I Potential Social Impact
Reliable uncertainty estimates are crucial, especially for complex time-dependent environmental phenomena. However, overreliance on these estimates can be dangerous. For example, unseen regimes might not be properly predicted. A changing climate that evolves the environment beyond already seen conditions can cause new forms of error regimes which cannot be predicted reliably. As most machine learning approaches, our method requires accurately labeled training data. Incorrect labels may lead to unexpected biases and prediction errors.
J Code and Data
The code and data to reproduce all of our experiments are available at https://github.com/ml-jku/HopCPT.
Table 10: Performance of kNN compared to HopCPT for the miscoverage  $\alpha = 0.10$  on the solar datasets. The error term represents the standard deviation over repeated runs.
Data	FC	UC	kNN	HopCPT
Solar 3Y	Forest	Δ Cov	0.015	0.029±0.012
PI-Width	97.2	39.0±6.2
Winkler	1.59	0.73±0.20
LGBM	Δ Cov	0.012	0.001±0.003
PI-Width	108.4	37.7±0.7
Winkler	1.74	0.57±0.01
Ridge	Δ Cov	-0.009	0.040±0.001
PI-Width	136.9	44.9±0.5
Winkler	1.99	0.64±0.00
LSTM	Δ Cov	-0.002	0.001±0.006
PI-Width	22.9	17.9±0.6
Winkler	0.51	0.30±0.01
Solar 1Y	Forest	Δ Cov	0.026	0.047±0.004
PI-Width	66.9	28.6±1.0
Winkler	1.00	0.40±0.04
LGBM	Δ Cov	0.003	-0.003±0.009
PI-Width	72.2	40.7±2.8
Winkler	1.08	0.57±0.08
Ridge	Δ Cov	-0.007	-0.011±0.016
PI-Width	128.8	59.9±5.6
Winkler	1.63	0.92±0.12
LSTM	Δ Cov	0.000	0.028±0.010
PI-Width	16.1	16.0±0.6
Winkler	0.33	0.22±0.01
Solar Small	Forest	Δ Cov	0.034	0.008±0.006
PI-Width	35.0	38.4±3.4
Winkler	0.93	1.09±0.08
LGBM	Δ Cov	-0.022	0.007±0.012
PI-Width	47.2	48.2±1.7
Winkler	1.17	1.24±0.08
Ridge	Δ Cov	0.004	-0.016±0.022
PI-Width	49.1	78.2±24.5
Winkler	1.08	1.78±0.59
Table 11: Performance of kNN compared to HopCPT for the miscoverage  $\alpha = 0.10$  on the different non-solar datasets. The error term represents the standard deviation over repeated runs.
Data	FC	UC	kNN	HopCPT
Air 10 PM	Forest	Δ Cov	-0.071	0.028±0.019
PI-Width	186.3	93.9±11.1
Winkler	3.81	1.50±0.09
LGBM	Δ Cov	-0.064	0.017±0.016
PI-Width	164.6	85.6±7.4
Winkler	3.42	1.45±0.06
Ridge	Δ Cov	-0.054	0.010±0.007
PI-Width	114.8	79.9±4.7
Winkler	2.39	1.35±0.06
LSTM	Δ Cov	-0.031	-0.002±0.005
PI-Width	57.2	62.7±1.5
Winkler	1.30	1.33±0.01
Air 25 PM	Forest	Δ Cov	-0.081	-0.024±0.017
PI-Width	158.4	48.1±5.6
Winkler	3.77	1.12±0.05
LGBM	Δ Cov	-0.073	-0.021±0.019
PI-Width	130.4	46.8±6.0
Winkler	3.10	1.10±0.05
Ridge	Δ Cov	-0.059	-0.005±0.026
PI-Width	97.5	49.3±10.6
Winkler	2.30	1.04±0.11
LSTM	Δ Cov	-0.033	0.007±0.008
PI-Width	34.7	40.7±4.3
Winkler	0.94	0.88±0.05
Sap flow	Forest	Δ Cov	-0.056	0.009±0.019
PI-Width	3246.1	1078.7±73.7
Winkler	1.00	0.30±0.01
LGBM	Δ Cov	-0.063	-0.016±0.013
PI-Width	2647.5	875.5±49.8
Winkler	0.88	0.27±0.00
Ridge	Δ Cov	-0.052	-0.025±0.023
PI-Width	2808.4	1639.6±93.2
Winkler	0.82	0.46±0.05
LSTM	Δ Cov	-0.026	0.004±0.004
PI-Width	719.5	594.3±7.7
Winkler	0.24	0.19±0.01
Streamflow	LSTM	Δ Cov	-0.108	0.001±0.041
PI-Width	1.70	1.39±0.17
Winkler	1.39	0.79±0.03
Table 12: Performance of the weighted sample quantile (Sample) and the weighted empirical CDF (ECDF) quantile strategies that HopCPT uses for the miscoverage  $\alpha = 0.10$  on the solar datasets. The error term represents the standard deviation over repeated runs.
Data	FC	UC	Sample	ECDF
Solar 3Y	Forest	Δ Cov	0.029±0.012	0.021±0.008
PI-Width	39.0±6.2	33.1±1.0
Winkler	0.73±0.20	0.62±0.03
LGBM	Δ Cov	0.001±0.003	-0.006±0.006
PI-Width	37.7±0.7	37.1±0.9
Winkler	0.57±0.01	0.58±0.01
Ridge	Δ Cov	0.040±0.001	0.041±0.001
PI-Width	44.9±0.5	45.0±0.5
Winkler	0.64±0.00	0.64±0.00
LSTM	Δ Cov	0.001±0.006	0.001±0.006
PI-Width	17.9±0.6	17.8±0.4
Winkler	0.30±0.01	0.30±0.00
Solar 1Y	Forest	Δ Cov	0.047±0.004	0.048±0.005
PI-Width	28.6±1.0	28.9±1.0
Winkler	0.40±0.04	0.40±0.04
LGBM	Δ Cov	-0.003±0.009	0.000±0.007
PI-Width	40.7±2.8	40.5±2.9
Winkler	0.57±0.08	0.56±0.08
Ridge	Δ Cov	-0.011±0.016	-0.006±0.013
PI-Width	59.9±5.6	57.5±2.5
Winkler	0.92±0.12	0.88±0.10
LSTM	Δ Cov	0.028±0.010	0.032±0.007
PI-Width	16.0±0.6	16.6±1.1
Winkler	0.22±0.01	0.22±0.01
Solar Small	Forest	Δ Cov	0.008±0.006	0.026±0.023
PI-Width	38.4±3.4	37.1±5.7
Winkler	1.09±0.08	0.99±0.06
LGBM	Δ Cov	0.007±0.012	0.011±0.013
PI-Width	48.2±1.7	46.3±1.3
Winkler	1.24±0.08	1.16±0.04
Ridge	Δ Cov	-0.016±0.022	0.012±0.011
PI-Width	78.2±24.5	109.0±18.6
Winkler	1.78±0.59	2.55±0.38
Table 13: Comparison of HopCPT (a) with applying the approach proposed by (Xu & Xie, 2022a,b) to search for a  $\beta$  that minimizes the prediction interval width (right column), and (b) without this approach which corresponds to  $\beta = \alpha /2$  (left column). The experiments are done with a miscoverage rate  $\alpha = 0.1$ . The error term represents the standard deviation over repeated runs.
Data	FC	UC	β = α/2	With β search
Solar 3Y	Forest	Δ Cov	0.029±0.012	0.010±0.018
PI-Width	39.0±6.2	41.0±7.6
Winkler	0.73±0.20	0.86±0.22
LGBM	Δ Cov	0.001±0.003	-0.013±0.003
PI-Width	37.7±0.7	36.8±6.2
Winkler	0.57±0.01	0.62±0.12
Ridge	Δ Cov	0.040±0.001	0.021±0.015
PI-Width	44.9±0.5	50.0±28.3
Winkler	0.64±0.00	0.77±0.43
LSTM	Δ Cov	0.001±0.006	-0.013±0.007
PI-Width	17.9±0.6	16.2±0.5
Winkler	0.30±0.01	0.31±0.00
Air 10 PM	Forest	Δ Cov	0.028±0.019	0.007±0.028
PI-Width	93.9±11.1	83.4±11.2
Winkler	1.50±0.09	1.49±0.07
LGBM	Δ Cov	0.017±0.016	-0.000±0.016
PI-Width	85.6±7.4	77.3±6.0
Winkler	1.45±0.06	1.41±0.04
Ridge	Δ Cov	0.010±0.007	-0.001±0.007
PI-Width	79.9±4.7	74.6±1.6
Winkler	1.35±0.06	1.32±0.02
LSTM	Δ Cov	-0.002±0.005	-0.006±0.007
PI-Width	62.7±1.5	60.9±2.3
Winkler	1.33±0.01	1.33±0.02
Sap flow	Forest	Δ Cov	0.009±0.019	-0.044±0.025
PI-Width	1078.7±73.7	857.2±56.7
Winkler	0.30±0.01	0.30±0.01
LGBM	Δ Cov	-0.016±0.013	-0.095±0.007
PI-Width	875.5±49.8	690.9±31.0
Winkler	0.27±0.00	0.28±0.01
Ridge	Δ Cov	-0.025±0.023	-0.050±0.020
PI-Width	1639.6±93.2	1389.0±128.0
Winkler	0.46±0.05	0.42±0.03
LSTM	Δ Cov	0.004±0.004	-0.003±0.003
PI-Width	594.3±7.7	557.8±8.9
Winkler	0.19±0.01	0.19±0.00
Table 14: Performance of the weighted sample quantile (Sample) and the weighted empirical CDF (ECDF) quantile strategies that HopCPT uses for the miscoverage  $\alpha = 0.10$  on the different non-solar datasets. The error term represents the standard deviation over repeated runs.
Data	FC	UC	Sample	ECDF
Air 10 PM	Forest	Δ Cov	0.028±0.019	0.027±0.021
PI-Width	93.9±11.1	93.4±11.0
Winkler	1.50±0.09	1.50±0.09
LGBM	Δ Cov	0.017±0.016	0.014±0.015
PI-Width	85.6±7.4	84.5±8.3
Winkler	1.45±0.06	1.43±0.09
Ridge	Δ Cov	0.010±0.007	0.013±0.010
PI-Width	79.9±4.7	79.3±4.6
Winkler	1.35±0.06	1.34±0.05
LSTM	Δ Cov	-0.002±0.005	-0.005±0.010
PI-Width	62.7±1.5	65.6±20.1
Winkler	1.33±0.01	1.35±0.15
Air 25 PM	Forest	Δ Cov	-0.024±0.017	-0.024±0.019
PI-Width	48.1±5.6	47.7±4.5
Winkler	1.12±0.05	1.11±0.01
LGBM	Δ Cov	-0.021±0.019	-0.023±0.019
PI-Width	46.8±6.0	46.6±6.1
Winkler	1.10±0.05	1.10±0.05
Ridge	Δ Cov	-0.005±0.026	-0.007±0.027
PI-Width	49.3±10.6	49.1±10.7
Winkler	1.04±0.11	1.04±0.11
LSTM	Δ Cov	0.007±0.008	-0.002±0.015
PI-Width	40.7±4.3	35.0±2.1
Winkler	0.88±0.05	0.82±0.05
Sap flow	Forest	Δ Cov	0.009±0.019	0.011±0.019
PI-Width	1078.7±73.7	1082.3±74.0
Winkler	0.30±0.01	0.30±0.01
LGBM	Δ Cov	-0.016±0.013	-0.017±0.013
PI-Width	875.5±49.8	875.6±49.5
Winkler	0.27±0.00	0.27±0.00
Ridge	Δ Cov	-0.025±0.023	-0.035±0.046
PI-Width	1639.6±93.2	1634.0±90.2
Winkler	0.46±0.05	0.50±0.13
LSTM	Δ Cov	0.004±0.004	0.005±0.003
PI-Width	594.3±7.7	592.0±9.7
Winkler	0.19±0.01	0.19±0.00
Streamflow	LSTM	Δ Cov	0.001±0.041	0.028±0.002
PI-Width	1.39±0.17	1.49±0.02
Winkler	0.79±0.03	0.77±0.01
Table 15: Performance of the CP algorithms HopCPT, SPCI, EnbPI, and NexCP, each combined with AdaptiveCI, for the miscoverage  $\alpha = 0.10$  on the solar datasets. Bold numbers correspond to the best result for the respective metric in the experiment (PI-Width and Winkler score). The error term represents the standard deviation over repeated runs.  $\dagger$  combined with AdaptiveCI.
UC	HopCPT†	SPCI†	EnbPI†	NexCP†	CopulaCPTS‡	CP/CF-RNN‡
Solar 3Y	Forest	Δ Cov	0.004±0.001	0.002±0.000	-0.001	-0.000	0.000	0.000
PI-Width	29.8±0.7	97.6±0.1	155.7	170.8	168.6	168.5
Winkler	0.59±0.03	1.69±0.00	2.43	2.54	2.52	2.52
LGBM	Δ Cov	-0.001±0.000	0.002±0.000	-0.001	-0.000	0.000	0.000
PI-Width	39.9±7.0	95.8±0.1	155.0	164.8	162.3	162.4
Winkler	0.62±0.10	1.70±0.00	2.49	2.57	2.54	2.54
Ridge	Δ Cov	0.002±0.000	-0.000±0.000	-0.006	0.000	0.000	-0.000
PI-Width	36.0±0.3	110.6±0.1	166.3	177.3	172.8	172.4
Winkler	0.62±0.00	1.77±0.00	2.28	2.63	2.61	2.58
LSTM	Δ Cov	0.000±0.000	0.001±0.000	-0.002	-0.000	0.001	0.001
PI-Width	17.8±0.2	25.5±0.0	30.6	30.6	28.8	28.8
Winkler	0.30±0.00	0.59±0.00	0.63	0.62	0.63	0.63
Solar 1Y	Forest	Δ Cov	0.007±0.001	0.011±0.000	-0.002	-0.001	0.002	0.003
PI-Width	21.5±1.2	72.6±0.1	109.5	126.7	128.9	129.5
Winkler	0.40±0.02	1.14±0.00	1.63	1.85	1.84	1.84
LGBM	Δ Cov	0.000±0.001	0.005±0.000	-0.001	-0.000	0.002	0.003
PI-Width	41.6±1.9	69.2±0.2	106.2	117.1	119.1	119.4
Winkler	0.56±0.04	1.15±0.00	1.63	1.78	1.77	1.77
Ridge	Δ Cov	-0.003±0.005	0.004±0.000	-0.002	-0.000	-0.000	-0.000
PI-Width	61.5±9.9	98.9±0.2	167.8	174.8	174.9	175.6
Winkler	0.87±0.11	1.32±0.00	2.01	2.08	2.08	2.08
LSTM	Δ Cov	0.001±0.000	0.003±0.000	-0.001	-0.001	0.001	0.001
PI-Width	14.3±0.7	18.9±0.0	21.8	20.4	19.9	20.0
Winkler	0.22±0.01	0.40±0.00	0.42	0.41	0.41	0.41
Solar Small	Forest	Δ Cov	0.006±0.003	-0.023±0.002	-0.003	-0.014	-0.007	-0.009
PI-Width	33.6±2.9	64.7±0.7	92.9	127.3	131.1	128.1
Winkler	1.01±0.04	1.95±0.01	2.38	3.27	3.25	3.23
LGBM	Δ Cov	-0.001±0.002	-0.015±0.001	-0.005	-0.005	-0.008	-0.010
PI-Width	44.8±1.0	60.9±0.6	96.4	135.7	128.3	125.0
Winkler	1.18±0.04	1.95±0.01	2.54	3.32	3.33	3.31
Ridge	Δ Cov	0.008±0.004	-0.015±0.001	-0.003	0.008	0.008	0.006
PI-Width	104.7±19.8	69.3±0.7	102.5	97.7	96.2	93.0
Winkler	2.44±0.42	1.92±0.01	2.56	2.75	2.75	2.72
Table 16: Performance of the CP algorithms HopCPT, SPCI, EnbPI, and NexCP, each combined with AdaptiveCI, for the miscoverage  $\alpha = 0.10$  on the different non-solar datasets. Bold numbers correspond to the best result for the respective metric in the experiment (PI-Width and Winkler score). The error term represents the standard deviation over repeated runs.  $\dagger$  combined with AdaptiveCI.
Sap flow	Air 25 PM	Air 10 PM
LSTM	Ridge	LGBM	Forest	LSTM	Ridge	LGBM	Forest	LSTM	Ridge	LGBM	Forest	LSTM	Ridge	LGBM	Forest	LSTM
Δ Cov	-0.006±0.002	-0.121±0.000	-0.007±0.000	-0.006	-0.004±0.000	-0.006	-0.006	-0.006	-0.006	-0.006	-0.006	-0.006	-0.006	-0.006	-0.006	-0.006
Δ Cov	-0.001±0.003	-0.001±0.003	-0.001±0.003	-0.001±0.003	-0.001±0.003	-0.001±0.003	-0.001±0.003	-0.001±0.003	-0.001±1.00	-0.001±1.00	-0.001±1.00	-0.001±1.00	-0.001±1.00	-0.001±1.00	-0.001±1.00	-0.001±1.00
Δ Cov	-0.001±0.003	-0.001±0.003	-0.001±0.003	-0.001±0.003	-0.001±0.003	-0.001±0.003	-0.001±0.003	-0-0.006	-0.006	-0.006	-0.006	-0.006	-0.006	-0.006	-0.006	-0.006
Δ Cov	-0.001±0.003	-0.001±0.003	-0.002±0.003	-0.002±0.003	-0.002±0.003	-0.002±0.003	-0.002±0.003	-0.002±0.003	-0.002±0.003	-0.002±0.003	-1.00	-1.00	-1.00	-1.00	-1.00	-1.00
Δ Cov	-0.002±0.003	-0.002±0.003	-0.002±0.003	-0.002±0.003	-0.002±0.003	-0.002±0.003	-0.002±0.003	-0.002±1.00	-0.002±1.00	-0.002±1.00	-1.00	-1.00	-1.00	-1.00	-1.00	-1.00
Δ Cov	-0.002±0.003	-0.002±0.003	-0.002±0.003	-0.002±0.003	-0.003±1.00	-0.003±1.00	-0.003±1.00	-0.003±1.00	-0.003±1.00	-0.003±1.00	-1.00	-1.00	-1.00	-1.00	-1.00	-1.00
Δ Cov	-0.003±1.00	-0.003±1.00	-0.003±1.00	-0.003±1.00	-0.003±1.00	-0.003±1.00	-0.003±1.00	-0.003±1.00	-0.003±1.00	-0-0.006	-0.006	-0.006	-0.006	-0.006	-0.006	-0.006
Δ Cov	-0.004±1.00	-0.004±1.00	-0.004±1.00	-0.004±1.00	-0.004±1.00	-0.004±1.00	-0.004±1.00	-0.004±1.00	-0.004±1.00	-0-0.006	-0.006	-0.006	-0.006	-0.006	-0.006	-0.006
Δ Cov	-0.005±1.00	-0.005±1.00	-0.005±1.00	-0.005±1.00	-0.005±1.00	-0.005±1.00	-0.005±1.00	-0.005±1.00	-0.005±1.00	-0-0.006	-0.006	-0.006	-0.006	-0.006	-0.006	-0.006
Δ Cov	-0.011±1.00	-0.011±1.00	-0.011±1.00	-0.011±1.00	-0.011±1.00	-0.011±1.00	-0.011±1.00	-0.011±1.00	-0.011±1.00	-0-0.006	-0.006	-0.006	-0.006	-0.006	-0.006	-0.006
Δ Cov	-0.021±1.00	-0.021±1.00	-0.021±1.00	-0.021±1.00	-0.021±1.00	-0.021±1.00	-0.021±1.00	-0.021±1.00	-0.021±1.00	-0-0.006	-0.006	-0.006	-0.006	-0.006	-0.006	-0.006
Table 17: Performance of the evaluated CP algorithms on the Solar (3Y) datasets for the miscoverage levels  $\alpha \in \{0.05,0.10,0.15\}$ . The column FC specifies the prediction algorithm used for the experiment (Forest: Random Forest, LGBM: LightGBM, Ridge: Ridge Regression, LSTM: LSTM neural network). Bold numbers correspond to the best result for the respective metric in the experiment (PI-Width and Winkler score), given that  $\Delta \mathrm{Cov} \geq -0.25\alpha$ , i.e., the specific algorithm reached at least approximate coverage (the result is grayed out otherwise). The error term represents the standard deviation over repeated runs (results without an error term are from deterministic models).
LSTM	Ridge	LGBM	Forest	FC
0.15	0.10	0.05	0.15	0.10	0.05	0.15	0.10	0.05	0.15	0.10	0.05	0.15
Δ Cov	-0.003±0.005	0.004±0.000	-0.003±0.005	0.004±0.000	-0.003±0.005	-0.003±0.005	-0.003±0.005	-0.003±0.005	-0.003±0.005	-0.003±0.005	-0.003±0.005	-0.003±0.005
Δ Cov	0.001±0.006	0.014±0.000	-0.009±0.000	0.009±0.000	-0.009±0.000	-0.009±0.000	-0.009±0.000	-0.009±0.000	-0.009±0.000	-0.009±0.000	-0.009±0.000	-0.009±0.
Δ Cov	17.9±0.6	27.8±0.0	24.6±0.2	28.2±0.1	31.9±0.3	33.0±0.3	34.7±0.2	34.7±0.2	34.7±0.2	34.7±0.2	34.7±0.2	34.7±0.2
Winkler	0.38±0.01	0.87±0.00	0.90	0.87	0.87	0.96	0.97	0.97	0.97	0.97	0.97	0.97
Table 18: Performance of the evaluated CP algorithms on the Solar (1Y) dataset for the miscoverage levels  $\alpha \in \{0.05,0.10,0.15\}$ . The column FC specifies the prediction algorithm used for the experiment (Forest: Random Forest, LGBM: LightGBM, Ridge: Ridge Regression, LSTM: LSTM neural network). Bold numbers correspond to the best result for the respective metric in the experiment (PI-Width and Winkler score), given that  $\Delta \mathrm{Cov} \geq -0.25\alpha$ , i.e., the specific algorithm reached at least approximate coverage (the result is grayed out otherwise). The error term represents the standard deviation over repeated runs (results without an error term are from deterministic models).
LSTM	Ridge	LGBM	Forest	FCα
0.15	0.10	0.05	0.15	0.15	0.10	0.05
Δ Cov	0.028±0.010	0.018±0.000	-0.018	-0.001	0.010	0.013	0.013	0.040
Δ Cov	0.029±0.017	0.029±0.000	-0.014	-0.001	0.010	0.013	0.013	0.040
Δ Cov	0.029±0.017	0.029±0.000	-0.014	-0.001	0.010	0.013	0.013	0.040
Table 19: Performance of the evaluated CP algorithms on the Solar (3M) dataset for the miscoverage levels  $\alpha \in \{0.05,0.10,0.15\}$ . The column FC specifies the prediction algorithm used for the experiment (Forest: Random Forest, LGBM: LightGBM, Ridge: Ridge Regression). Bold numbers correspond to the best result for the respective metric in the experiment (PI-Width and Winkler score), given that  $\Delta \mathrm{Cov} \geq -0.25\alpha$ , i.e., the specific algorithm reached at least approximate coverage (the result is grayed out otherwise). The error term represents the standard deviation over repeated runs (results without an error term are from deterministic models).
FC	α	HopCPT	SPCI	EnbPI	NexCP	CopulaCPTS	CP/CF-RNN	AdaptiveCI
Forest	0.05	Δ Cov	-0.002±0.005	-0.074±0.001	-0.022	-0.013	-0.022	-0.028
PI-Width	47.5±5.0	57.9±0.3	110.7	160.2	162.0	155.0
Winkler	1.29±0.11	2.62±0.02	2.92	3.70	3.95	4.04
0.10	Δ Cov	0.008±0.006	-0.064±0.002	-0.022	-0.021	-0.027	-0.025
PI-Width	38.4±3.4	38.8±0.3	86.0	122.9	110.4	111.4
Winkler	1.09±0.08	1.82±0.01	2.54	3.28	3.47	3.48
0.15	Δ Cov	0.020±0.010	-0.052±0.002	-0.028	-0.019	-0.020	-0.017
PI-Width	32.6±2.8	26.7±0.3	64.8	91.0	77.5	79.6
Winkler	0.99±0.07	1.45±0.02	2.40	2.91	2.99	2.98
LGBM	0.05	Δ Cov	0.002±0.008	-0.063±0.001	-0.024	-0.014	-0.018	-0.019	-0.000
PI-Width	59.7±3.4	56.3±0.3	111.7	161.7	164.5	162.1	61.8
Winkler	1.49±0.15	2.66±0.02	3.10	3.87	4.13	4.18	1.45
0.10	Δ Cov	0.007±0.012	-0.052±0.002	-0.022	-0.020	-0.022	-0.021	-0.004
PI-Width	48.2±1.7	37.5±0.3	84.3	119.9	107.6	106.9	49.2
Winkler	1.24±0.08	1.84±0.01	2.67	3.35	3.52	3.53	1.26
0.15	Δ Cov	0.005±0.016	-0.043±0.002	-0.024	-0.019	-0.013	-0.011	-0.007
PI-Width	41.4±1.3	26.9±0.2	60.8	85.1	75.8	77.5	44.1
Winkler	1.11±0.07	1.47±0.01	2.48	2.93	2.98	2.97	1.22
Ridge	0.05	Δ Cov	-0.017±0.014	-0.064±0.001	-0.021	0.004	0.009	-0.002
PI-Width	92.7±32.4	67.2±0.3	110.1	144.8	156.1	133.6
Winkler	1.99±0.69	2.49±0.02	3.07	3.82	3.78	3.75
0.10	Δ Cov	-0.016±0.022	-0.056±0.002	-0.020	0.007	0.014	-0.003
PI-Width	78.2±24.5	51.8±0.3	84.9	85.2	88.6	78.1
Winkler	1.78±0.59	1.91±0.01	2.66	2.85	2.83	2.81
0.15	Δ Cov	-0.009±0.021	-0.039±0.002	-0.017	0.003	0.014	0.003
PI-Width	67.4±19.0	43.2±0.2	71.6	70.9	71.9	70.7
Winkler	1.61±0.51	1.59±0.00	2.29	2.34	2.34	2.33
Table 20: Performance of the evaluated CP algorithms on the Air Quality (10PM) dataset for the miscoverage levels  $\alpha \in \{0.05, 0.10, 0.15\}$ . The column FC specifies the prediction algorithm used for the experiment (Forest: Random Forest, LGBM: LightGBM, Ridge: Ridge Regression, LSTM: LSTM neural network). Bold numbers correspond to the best result for the respective metric in the experiment (PI-Width and Winkler score), given that  $\Delta \mathrm{Cov} \geq -0.25\alpha$ , i.e., the specific algorithm reached at least approximate coverage (the result is grayed out otherwise). The error term represents the standard deviation over repeated runs (results without an error term are from deterministic models).
LSTM	Ridge	LGBM	Forest
α	0.15	0.10	0.05	0.15	0.10	0.05	0.15	0.10	0.05	0.15	0.10	0.05	0.15	0.10
Δ Cov	-0.002±0.005	0.017±0.000	0.010±0.000	0.003±0.000	-0.021	-0.002	-0.021	-0.002	-0.021	-0.002	-0.021	-0.002	-0.021	-0.002
Δ Cov	-0.002±0.005	0.017±0.000	0.010±0.000	0.003±0.000	-0.021	-0.002	-0.021	-0.002	-0.021	-0.002	-0-0.002	-0.021	-0.002	-0.021
Δ Cov	-0.002±0.005	0.017±0.000	0.010±0.000	0.003±0.000	-0.021	-0.002	-0.021	-0.002	-0.021	-0.002	-0 0.002	-0.021	-0.002	-0.021
Δ Cov	-0.002±0.005	0.017±0.000	0.010±0.000	0.003±0.000	-0.021	-0.002	-0.002	-0.002	-0.002	-0.002	-0.002	-0.002	-0.002	-0.002
Δ Cov	-0.002±0.005	0.017±0.000	0.010±0.000	0.003±0.000	-0.021	-0.002	-0.021	-0.002	-0.002	-0.002	-0.002	-0.002	-0.002	-0.002
Δ Cov	-0.002±0.005	0.017±0.000	0.010±0.000	0.003±0	-0.021	-0.002	-0.021	-0.002	-0.002	-0.002	-0.002	-0.002	-0.002	-0.002
Δ Cov	-0.002±0.005	0.017±0.0	0.010±0.0	0.003±0	-0.021	-0.002	-0.021	-0.002	-0.002	-0.002	-0.002	-0.002	-0.002	-0.002
Δ Cov	-0.002±0.005	0 0.17±0.0	0.017±0.0	0.003±0	-0.021	-0.002	-0.021	-0.002	-0.002	-0.002	-0.002	-0.002	-0.002	-0.002
Δ Cov	-0.002±0.00	0.17±0.0	0.017±0.0	0.003±0	-0.021	-0.002	-0.021	-0.002	-0.002	-0.002	-0.002	-0.002	-0.002	-0.002
Delta Cov	-0.002±0.00	0.17±0.0	0.017±0.0	0.003±0	-0.021	-0.002	-0.021	-0.002	-0.002	-0.002	-0.002	-0.002	-0-0.002	-0.002
Δ Cov	-0.002±0.00	0.17±0.0	0.017±0.0	0.003±0	-0.021	-0.002	-0.021	-0.002	-0.002	-0.002
Δ Cov	-0.002±0.00	0.17±0.0	0.017±0.0	0.003±0	-0.021	-0.002	-0.021	-0.002	-0.002	-0.002
Δ Cov	-0.17±0.1	0.17±0.1	0.17±0.1	0.17±0.1	-0.17±0.1	-0.17±0.1	-0.17±0.1	-0.17±0.1	-0.17±0.1	-0.17±0.1
Δ Cov	-0.17±0.1	0.17±0.1	0.17±0.1	0.17±0.1	-0.17±0.1	-0.17±0.1	-0.17±0.1	-0.17±0.1	-0.17±0.1	-0-0.17
Δ Cov	-0.17±0.1	0.17±0.1	0.17±0.1	0.17±0.1	-0.17±0.1	-0.17±0.1	-0.17±0.1	-0.17±0.1	-0.17±0.1	-0 -1.7
Δ Cov	-0.17±0.1	0.17±0.1	0.17±0.1	0.17±0.1	-0.17±0.1	-0.17±0.1	-0.17±0.1	-0.17±0.1	-0.17±0.1	-1.7
Δ Cov	-0.17±0.1	0.17±0.1	0.17±0.1	0.17±0.1	-0.17±0.1	-0.17±0.1	-0.17±0.1	-0.17±0.1	-0.17±0.1
Δ Cov	-0.17±0.1	0.17±0.1	0.17±0.1	0.17±0.1	-0.17±0.1	-0.17±0.1	-0.17±0.1	-0.17±0.1	-0.17±0.1
Table 21: Performance of the evaluated CP algorithms on the Air Quality (25PM) dataset for the miscoverage levels  $\alpha \in \{0.05, 0.10, 0.15\}$ . The column FC specifies the prediction algorithm used for the experiment (Forest: Random Forest, LGBM: LightGBM, Ridge: Ridge Regression, LSTM: LSTM neural network). Bold numbers correspond to the best result for the respective metric in the experiment (PI-Width and Winkler score), given that  $\Delta \mathrm{Cov} \geq -0.25\alpha$ , i.e., the specific algorithm reached at least approximate coverage (the result is grayed out otherwise). The error term represents the standard deviation over repeated runs (results without an error term are from deterministic models).
FC	α	HopCPT	SPCI	EnbPI	NexCP	CopulaCPTS	CP/CF-RNN	AdaptiveCI
Forest	0.05	Δ Cov	-0.033±0.013	-0.015±0.000	-0.065	-0.009	-0.018	-0.031
PI-Width	58.7±6.9	102.6±0.1	211.3	283.9	271.4	249.8
Winkler	1.51±0.04	2.67±0.00	5.01	4.71	6.47	6.59
0.10	Δ Cov	-0.024±0.017	-0.009±0.000	-0.079	-0.007	-0.025	-0.042
PI-Width	48.1±5.6	81.5±0.0	177.3	235.6	212.6	203.5
Winkler	1.12±0.05	2.02±0.00	4.31	4.05	4.94	4.98
0.15	Δ Cov	-0.014±0.019	-0.000±0.001	-0.084	-0.005	-0.029	-0.045
PI-Width	41.4±4.9	70.8±0.0	153.0	206.9	185.5	179.0
Winkler	0.94±0.05	1.71±0.00	3.88	3.67	4.23	4.25
LGBM	0.05	Δ Cov	-0.031±0.014	-0.007±0.000	-0.054	-0.006	-0.020	-0.032	-0.001
PI-Width	57.2±7.8	93.5±0.1	176.4	239.5	212.7	196.0	208.8
Winkler	1.48±0.05	2.45±0.00	4.30	4.35	5.47	5.58	3.99
0.10	Δ Cov	-0.021±0.019	0.002±0.000	-0.063	-0.007	-0.027	-0.042	-0.001
PI-Width	46.8±6.0	73.3±0.0	142.1	182.2	154.5	143.9	163.3
Winkler	1.10±0.05	1.83±0.00	3.54	3.55	4.09	4.14	3.35
0.15	Δ Cov	-0.010±0.022	0.010±0.001	-0.066	-0.006	-0.029	-0.045	-0.001
PI-Width	40.2±5.1	62.4±0.0	118.8	147.6	124.2	117.1	133.3
Winkler	0.92±0.05	1.55±0.00	3.07	3.08	3.41	3.43	2.99
Ridge	0.05	Δ Cov	-0.018±0.018	-0.001±0.000	-0.043	-0.005	-0.002	-0.004
PI-Width	60.2±13.1	84.0±0.1	132.5	177.6	180.5	177.1
Winkler	1.35±0.12	1.82±0.00	3.11	3.31	3.82	3.85
0.10	Δ Cov	-0.005±0.026	0.011±0.000	-0.051	-0.006	-0.002	-0.004
PI-Width	49.3±10.6	65.5±0.0	106.2	134.3	127.4	125.3
Winkler	1.04±0.11	1.40±0.00	2.54	2.66	2.92	2.93
0.15	Δ Cov	0.008±0.032	0.021±0.000	-0.053	-0.007	-0.001	-0.003
PI-Width	42.6±9.3	55.6±0.0	88.7	109.7	102.5	101.6
Winkler	0.89±0.10	1.20±0.00	2.21	2.30	2.46	2.46
LSTM	0.05	Δ Cov	0.005±0.005	-0.015±0.000	-0.023	-0.003	-0.015	-0.021
PI-Width	57.1±5.6	44.9±0.0	50.8	56.0	48.4	46.1
Winkler	1.19±0.08	1.29±0.00	1.33	1.27	1.39	1.40
0.10	Δ Cov	0.007±0.008	-0.017±0.000	-0.028	-0.003	-0.019	-0.025
PI-Width	40.7±4.3	32.4±0.0	35.9	38.6	34.0	32.8
Winkler	0.88±0.05	0.93±0.00	0.97	0.94	0.99	0.99
0.15	Δ Cov	0.005±0.011	-0.016±0.000	-0.029	-0.003	-0.019	-0.025
PI-Width	32.5±3.7	26.1±0.0	28.4	30.2	26.8	26.1
Winkler	0.74±0.04	0.76±0.00	0.79	0.77	0.80	0.81
Table 22: Performance of the evaluated CP algorithms on the Sap flow dataset for the miscoverage levels  $\alpha \in \{0.05,0.10,0.15\}$ . The column FC specifies the prediction algorithm used for the experiment (Forest: Random Forest, LGBM: LightGBM, Ridge: Ridge Regression, LSTM: LSTM neural network). Bold numbers correspond to the best result for the respective metric in the experiment (PI-Width and Winkler score), given that  $\Delta \mathrm{Cov} \geq -0.25\alpha$ , i.e. the specific algorithm reached at least approximate coverage (the result is grayed out otherwise). The error term represents the standard deviation over repeated runs (results without an error term are from deterministic models).