SlowGuess/NeurIPS_2023_OCR huggingface dataset:
MCD/MDN/Gauss. In Appendix A.3 we compare HopCPT to non-CP architectures that can provide prediction interval estimates. Specifically, we use Monte Carlo Dropout (MCD; Gal & Ghahramani, 2016), a Mixture Density Network (MDN; Bishop, 1994), and a Gauss model (i.e., an MDN with a single Gaussian density). An LSTM serves as backbone for all models. Therefore, there is no distinction between the predictor model and the uncertainty model. To allow for a fair comparison to CP approaches, the models are trained on both the predictor training data and the calibration data combined. None of these approaches are CP methods, and MCD was specifically devised to estimate only epistemic uncertainty. Nevertheless, we consider these methods as useful comparisons, since they are widely used for uncertainty quantification. The models were trained for 150 epochs with AdamW (Loshchilov & Hutter, 2019) (using its standard parameters:  $\beta_{1} = 0.9$ ,  $\beta_{2} = 0.999$ ,  $\delta = 0.001$ ). We chose this number to ensure that the loss and model selection metrics already converged. Throughout the training, we validated after every epoch and selected the model that best fulfilled the model selection criteria described above. The learning rate was part of the hyperparameter search.
Global LSTM. We conducted an individual hyperparameter tuning of the global LSTM prediction model for each dataset. Table 3 shows the hyperparameters used for the grid search. Similar to the LSTM uncertainty models, we trained for 150 epochs with an AdamW optimizer (Loshchilov & Hutter, 2019) and validated after each epoch. We optimized for the mean squared error and selected the model with the smallest validation mean squared error.
Table 3: Parameters used in the hyperparameter search of the global LSTM prediction model.
Parameter	Value
Learning Rate	0.005, 0.001, 0.0001
Dropout	0, 0.1, 0.25
Batch Size	512, 256
Hidden Size	64, 128, 256
A.2 SPCI Retraining
As our results show, SPCI is very competitive (see, for example, Section 3.2). Xu & Xie (2022b) did, however, design SPCI so that its random forest regressor is retrained after each prediction step. This design choice is computationally prohibitive for larger datasets. To nevertheless allow a comparison against SPCI on the larger datasets, we modified the algorithm so that the retraining is skipped. Experiments on the smallest dataset (Table 6) show only small performance decrease with the modified algorithm. One limitation of the adapted algorithm is, however, that a strong shift in the error distribution(s) would potentially require a retraining on the new distribution. A viable proxy to detect such a change in our setting is the coverage performance of standard CP. The reason for this is that standard CP predictions are solely based on the calibration data and can thus not account for shifts. In the experiments (Section 3 and Appendix A.3), standard CP achieves good coverage with only one exception (see Section 3). Hence, we decided to include SPCI (in the modified version) to enable a comparison to it on larger datasets.
Table 4: Performance of SPCI with an input window of length 100 and length 25 on the solar datasets. The differences in performance are small compared to the differences across methods (Section 3.2). The error term represents the standard deviation over repeated runs.
Data	FC	Window	100	25
Solar Y	Forest	Δ Cov	0.045±0.000	0.047±0.000
PI-Width	97.2±0.1	97.8±0.3
Winkler	1.26±0.00	1.25±0.00
LGBM	Δ Cov	0.045±0.000	0.047±0.000
PI-Width	96.6±0.2	97.6±0.1
Winkler	1.26±0.00	1.26±0.00
Ridge	Δ Cov	0.031±0.000	0.030±0.000
PI-Width	112.9±0.2	113.7±0.1
Winkler	1.40±0.00	1.42±0.00
LSTM	Δ Cov	0.018±0.000	0.018±0.000
PI-Width	22.5±0.0	22.2±0.0
Winkler	0.41±0.00	0.40±0.00
Solar Small	Forest	Δ Cov	-0.064±0.002	-0.024±0.000
PI-Width	38.8±0.4	43.2±0.2
Winkler	1.82±0.01	1.53±0.00
LGBM	Δ Cov	-0.052±0.002	-0.027±0.001
PI-Width	37.4±0.2	43.3±0.1
Winkler	1.84±0.01	1.63±0.00
Ridge	Δ Cov	-0.055±0.002	-0.057±0.001
PI-Width	51.9±0.2	52.8±0.2
Winkler	1.91±0.02	1.83±0.00
Figure 4: Time series of the prediction error  $\epsilon$  for the ridge regression model on the Sap flow dataset. The time series spans the calibration  $(t < 0)$  and test  $(t \geq 0)$  data. The red line (fitted by the least squares method) shows a strong trend in the error distribution.
Table 5: Performance of SPCI with an input window of length 100 and length 25 on the different non-solar datasets. The differences in performance are small compared to the differences across methods (Section 3.2). The error term represents the standard deviation over repeated runs.
Data	FC	Window	100	25
Air 10 PM	Forest	Δ Cov	0.008±0.000	0.008±0.000
PI-Width	118.5±0.0	118.5±0.1
Winkler	2.23±0.00	2.23±0.00
LGBM	Δ Cov	0.024±0.000	0.023±0.000
PI-Width	113.2±0.2	113.2±0.1
Winkler	1.94±0.00	1.94±0.00
Ridge	Δ Cov	0.024±0.000	0.024±0.000
PI-Width	93.9±0.0	93.8±0.0
Winkler	1.52±0.00	1.52±0.00
LSTM	Δ Cov	0.010±0.000	0.010±0.000
PI-Width	62.3±0.1	62.3±0.0
Winkler	1.21±0.00	1.21±0.00
Air 25 PM	Forest	Δ Cov	-0.009±0.000	-0.009±0.000
PI-Width	81.5±0.0	81.4±0.0
Winkler	2.02±0.00	2.02±0.00
LGBM	Δ Cov	0.002±0.001	0.001±0.000
PI-Width	73.3±0.0	73.3±0.0
Winkler	1.84±0.00	1.83±0.00
Ridge	Δ Cov	0.010±0.000	0.010±0.000
PI-Width	65.5±0.0	65.4±0.1
Winkler	1.40±0.00	1.40±0.00
LSTM	Δ Cov	-0.017±0.000	-0.017±0.000
PI-Width	32.4±0.0	32.3±0.0
Winkler	0.93±0.00	0.92±0.00
Sap flow	Forest	Δ Cov	0.007±0.000	0.007±0.000
PI-Width	1743.1±1.4	1742.7±1.2
Winkler	0.59±0.00	0.59±0.00
LGBM	Δ Cov	0.003±0.000	0.003±0.000
PI-Width	1581.9±0.7	1583.6±1.3
Winkler	0.49±0.00	0.49±0.00
Ridge	Δ Cov	-0.241±0.000	-0.242±0.000
PI-Width	2061.5±1.4	2066.8±1.2
Winkler	2.52±0.00	2.52±0.00
LSTM	Δ Cov	0.004±0.000	0.006±0.000
PI-Width	628.6±0.8	631.9±0.2
Winkler	0.24±0.00	0.24±0.00
Table 6: Performance of the original SPCI algorithm (Retrain) and the modified version (No Retrain) on the Solar (3M) dataset. Performance in terms of the evaluation metrics is similar compared to the differences between methods (Section 3.2). The computational demand of the modified version is considerably lower. The error term represents the standard deviation over repeated runs.
FC	α	UC	No Retrain	Retrain
Forest	0.05	Δ Cov	-0.074±0.001	-0.049±0.001
PI-Width	58.0±0.2	62.4±0.0
Winkler	2.62±0.03	2.28±0.01
0.10	Δ Cov	-0.064±0.002	-0.045±0.003
PI-Width	38.8±0.4	41.6±0.0
Winkler	1.82±0.01	1.67±0.00
0.15	Δ Cov	-0.050±0.002	-0.038±0.003
PI-Width	26.8±0.2	28.9±0.0
Winkler	1.44±0.02	1.35±0.00
LGBM	0.05	Δ Cov	-0.062±0.001	-0.061±0.001
PI-Width	56.1±0.4	58.2±0.0
Winkler	2.65±0.03	2.51±0.01
0.10	Δ Cov	-0.052±0.002	-0.049±0.001
PI-Width	37.4±0.2	39.8±0.0
Winkler	1.84±0.01	1.78±0.01
0.15	Δ Cov	-0.042±0.001	-0.037±0.001
PI-Width	26.9±0.3	28.3±0.0
Winkler	1.47±0.00	1.42±0.00
Ridge	0.05	Δ Cov	-0.063±0.001	-0.056±0.002
PI-Width	67.3±0.2	68.7±0.0
Winkler	2.47±0.01	2.25±0.01
0.10	Δ Cov	-0.055±0.002	-0.048±0.001
PI-Width	51.9±0.2	54.2±0.0
Winkler	1.91±0.02	1.77±0.01
0.15	Δ Cov	-0.039±0.003	-0.036±0.000
PI-Width	43.1±0.3	45.0±0.0
Winkler	1.59±0.00	1.50±0.00
Figure 5: Visualization of the 30 highest association weights that the Hopfield network places on previous time steps. HopCPT retrieves similar error values when predicting at a time of high error, and it retrieves similar, previous small errors when predicting at a time step with small error.
A.3 Additional Results
Tables 17-23 show the results for miscoverage levels  $\alpha \in \{0.05, 0.10, 0.15\}$  for all evaluated combinations of datasets and predictors. Results for individual time series of the datasets are uploaded to the code repository (Appendix J).
CMAL. For the Streamflow dataset, we additionally compare to CMAL, which is the state-of-the-art non-CP uncertainty estimation technique in the respective domain (Klotz et al., 2022). CMAL is a mixture density network (Bishop, 1994) based on an LSTM that predicts the parameters of asymmetric Laplacian distributions. As our experiments use the same dataset, we adopt the hyperparameters from Klotz et al. (2022) but lower the learning rate to 0.0001 because we train CMAL on more training samples (18 years, i.e., the training and calibration period combined, which allows for a fair comparison against the CP methods). Despite the fact that CMAL is not based on the CP paradigm, it achieves good coverage results, however, at the cost of wide prediction intervals. We argue that the Winkler score is low because CMAL considers all (and therefore also uncovered) samples during the optimization while CP methods do not consider them at all — therefore, the distance of these points to the interval might be smaller (see Table 23).
Error trend. Figure 4 shows the prediction errors of the ridge regression model on the Sap flow dataset. The errors exhibit a strong trend and shift towards a negative expectation value.
Association weights. Figure 5 investigates the association patterns of HopCPT and shows its capabilities to focus on time steps from similar error regimes. The depicted time step in a regime with negative errors retrieves time steps with primarily negative errors and, likewise, the time step in a regime with positive errors retrieves time steps with primarily positive errors of the same magnitude.
Non-CP methods We also compare HopCPT to non-CP methods to evaluate its performance within the broader field of uncertainty estimation for time series. Specifically, we compare against a Monte Carlo Dropout (MCD); a Mixture Density Network (MDN), and a Gauss model (i.e., an MDN with a single Gaussian component), with an LSTM backbone (see Appendix A.1 for more details about the models and their hyperparameters). This comparison shows that, although the non-CP methods often lead to very narrow prediction intervals, they fail to provide approximate coverage on most experiments (Table 7).
A.4 Local Coverage
Standard CP only provides marginal coverage guarantees and a constant prediction interval width (Vovk et al., 2005). In time series prediction tasks, this can lead to bad local coverage (Lei & Wasserman, 2014). To evaluate whether the coverage approximately holds also locally, we evaluated  $\Delta$  Cov on windows of size  $k$ . To avoid compensation of negative  $\Delta$  Cov in some windows by other windows with positive  $\Delta$  Cov, we upper-bounded each window's  $\Delta$  Cov by zero before averaging over the bounded coverage gaps — i.e., we calculated  $\frac{1}{W}\sum_{w=1}^{W}-\Delta\operatorname{Cov}_{w}^{\top 0}$ , with
$$
\Delta \operatorname {C o v} _ {w} ^ {\top 0} = \left\{ \begin{array}{l l} \Delta \operatorname {C o v} _ {w} & \text {i f} \Delta \operatorname {C o v} _ {w} \leq 0, \\ 0 & \text {e l s e}, \end{array} \right. \tag {10}
$$
where  $\Delta \operatorname{Cov}_w$  is the  $\Delta$  Cov within window  $w$ .
Table 8 shows the results of this evaluation for window sizes  $k \in \{10,20,50\}$  and miscoverage level  $\alpha = 0.1$ . Depending on the dataset, most often either HopCPT or SPCI perform best. The overall rankings are only partly similar to the evaluation of the marginal coverage (Table 1 and Appendix A.3). Especially standard CP, which achieves competitive results in marginal coverage, falls short in this comparison. Only for Solar (3M), where the approach achieves a high marginal  $\Delta$  Cov, it preserves the local coverage best. Note that this comes with the drawback of very wide prediction intervals, i.e., bad efficiency. Overall, the results show that HopCPT and other time series CP methods improve the local coverage in non-exchangeable time series settings, compared to standard CP.
A.5 Negative Results
We also tested training with more involved training procedures than directly using the mean squared error. However, based on our empirical findings, the vanilla method with mean squared error outperforms more complex approaches. In the following, we briefly describe those additional configurations (so that potential future research can avoid these paths):
- Pinball Loss. We tried to train the MHN with a pinball loss (the original use seems to stem from Fox & Rubin, 1964, we are however not aware of the naming origin) in many different variations (see points that follow), but consistently got worse results than with the mean squared error.
- Inspired by the ideas lined out by Tagasovska & Lopez-Paz (2019) we tried to use the miscoverage level as an additional input to the network, while also parameterizing the loss with it.
- We tried to use the pinball loss to predict multiple miscoverage values at the same time to get a more informed representation of the distribution we want to approximate.
- Softmax Probabilities. We tried to use the softmax output of the MHN to directly estimate probabilities and use a maximum likelihood procedure. This did train, but it did not produce good results.
A.6 Computational Resources
We used different hardware setups in our experiments, however, most of them were executed on a machine with an Nvidia P100 GPU and a Xeon E5-2698 CPU. The runtime differs greatly between different dataset sizes. We report the approximate run times for a single experiment (i.e., one seed, all evaluated coverage levels, not including training the prediction model  $\hat{\mu}$ ):
1. HopCPT: Solar (3Y) 5h, Solar (1Y) 1h, Solar (3M) 7min, Air Quality (10PM) 3h, Air Quality (25PM) 3h, Sap flow 2.5h, Streamflow 12-20h
2. SPCI: (adapted): Solar (3Y) 5-10 days, Solar (1Y) 10-30h, Solar (3M) 45min, Air Quality (10PM) 13-30h, Air Quality (25PM) 13-30h, Sap flow 20-50h, Streamflow 12-17 days
3. EnbPI: all datasets under 6h
4. NexCP: all datasets under 45min
5. AdaptiveCI: all datasets under 45min
6. Standard CP: all datasets under 45min
Table 8: Negative average of zero-upper-bounded  $\Delta$  Cov of rolling windows with miscoverage level  $\alpha = 0.1$ . We evaluate each dataset-predictor combination at the windows sizes  $k \in \{10,20,50\}$ .
Sap flow	Air 25 PM	Air 10 PM	Solar Small	Solar 1Y	Solar 3Y	DataFCk	HopCPTSPCIEnbPINexCP	CP
Ridge	LGBM	Forest	Ridge	LGBM	Forest	Ridge	LGBM	Forest	Ridge	LGBM	Forest	Ridge	LGBM	Forest	Ridge	LGBM	Forest	Ridge	LGBM	Forest	Ridge	LGBM	Forest	Ridge	LGBM	Forest	Ridge	LGBM	Forest	Ridge	LGBM	Forest	Ridge	LGBM	Forest
10	.097	.286	10	.097	.286	10	.097	.286	10	.097	.286	10	.097	.286	10	.097	.286	10	.097	.286	10	.097	.286	10	.097
20	.088	.279	.088	.279	.088	.279	.088	.279	.088	.279	.088	.279	.088	.279	.088	.279	.088	.279	.088	.279	.088	.278	.088	.278	.088	.278	.088	.278	.088	.278	.088	.278	.088	.278	.088	.278	.088	.278	.088	.278	.088	.279	.088	.279	.088	.279	.088	.279	.088	.279	.088	.279	.088	.279	.088	.279	.088	.279	.088	.271	.071	.071	.071	.071	.071	.071	.071	.071	.071	.071	.071	.071	.071	.071	.071	.071	.071	.071	.071	.071	.102	.086	.404
B Theoretical Background
We denote the full dataset as  $D = (D_{1},\ldots ,D_{t + 1})$ , where  $D_{i} = (X_{i},Y_{i})$  is the  $i$ -th datapoint. Since we are in the split conformal setting (Vovk et al., 2005), the full dataset is subdivided into two parts: the calibration and the test data. The prediction model  $\hat{\mu}$  is already given in this setting and not selected based on  $D$ . The prediction interval  $\widehat{C}_t^\alpha$  for split conformal prediction is given by
$$
\widehat {C} _ {t} ^ {\alpha} \left(X _ {t + 1}\right) = \hat {\mu} \left(X _ {t + 1}\right) \pm \mathrm {Q} _ {1 - \alpha} \left(\sum_ {i = 1} ^ {t} \frac {1}{t + 1} \cdot \delta_ {R _ {i}} + \frac {1}{t + 1} \cdot \delta_ {+ \infty}\right), \tag {11}
$$
where  $R_{i} = |Y_{i} - \hat{\mu}(X_{i})|$ ,  $\delta_{c}$  is a point mass at  $c$ , and  $\mathbf{Q}_{\tau}$  is the  $\tau$ -quantile of a distribution. Vovk et al. (2005) show that this interval provides a marginal coverage guarantee in case of exchangeable data:
Theorem B.1 (Split Conformal Prediction (Vovk et al., 2005)).
If the data points  $(X_{1},Y_{1}),\ldots ,(X_{t},Y_{t}),(X_{t + 1},Y_{t + 1})$  are exchangeable then the split conformal prediction set defined in (11) satisfies
$$
\Pr \left\{Y _ {t + 1} \in \widehat {C} _ {t} ^ {\alpha} \left(X _ {t + 1}\right) \right\} \geq 1 - \alpha .
$$
This seminal result cannot be directly applied to our setting, because time series are often non-exchangeable (as we discuss in the introduction) and models often exhibit locally specific errors.
The remainder of this section is split into two parts. First, we discuss the theoretical connections between the MHN retrieval and CP through the lens of Foygel Barber et al. (2022). Second, we adapt the theory of Xu & Xie (2022a) to be applicable to our regime-based setting.
For the MHN retrieval, we evoke the results of Foygel Barber et al. (2022), who introduce the concept of non-exchangeable split conformal prediction. Instead of treating each calibration point equally, they assign an individual weight for each point, to account for non-exchangeability. Formally, we denote the weight for the  $i$ -th point by  $w_{i} \in [0,1]$ ,  $i = 1,\dots,t$ ; and the respective normalized weights are defined as
$$
\tilde {w} _ {i} = \frac {w _ {i}}{w _ {1} + \cdots + w _ {t} + 1}, i = 1, \dots , t, \text {a n d} \tilde {w} _ {t + 1} = \frac {1}{w _ {1} + \cdots + w _ {t} + 1}. \tag {12}
$$
The prediction interval for non-exchangeable split conformal setting  $\widehat{C}_t^\alpha$  is given by
$$
\widehat {C} _ {t} ^ {\alpha} \left(X _ {t + 1}\right) = \hat {\mu} \left(X _ {t + 1}\right) \pm \mathrm {Q} _ {1 - \alpha} \left(\sum_ {i = 1} ^ {t} \tilde {w} _ {i} \cdot \delta_ {R _ {i}} + \tilde {w} _ {t + 1} \cdot \delta_ {+ \infty}\right). \tag {13}
$$
Foygel Barber et al. (2022) show that non-exchangeable split conformal prediction satisfies a similar bound as exchangeable split conformal prediction, which is extended by a term that accounts for a potential coverage gap  $\Delta$  Cov. Formally we define  $\Delta \mathrm{Cov} = \alpha -\alpha^{\star}$ , where  $\alpha$  is the specified miscoverage and  $\alpha^{\star}$  is the observed miscoverage.
To specify the coverage gap they use the total variation distance  $\mathrm{d}_{\mathrm{TV}}$ , which is a distance metric between two probability distributions. It is defined as
$$
\mathrm {d} _ {\mathrm {T V}} \left(P _ {1}, P _ {2}\right) = \sup  _ {A \in \mathcal {F}} | P _ {1} (A) - P _ {2} (A) |, \tag {14}
$$
where  $P_{1}$  and  $P_{2}$  are arbitrary distributions defined on the space  $(\Omega ,\mathcal{F})$
Intuitively,  $\mathrm{d}_{\mathrm{TV}}$  measures the largest distance between two distributions. It is connected to the widely used Kullback-Leibler divergence  $D_{\mathrm{KL}}$  through Pinsker's inequality (Pinsker, 1964):
$$
\mathrm {d} _ {\mathrm {T V}} \left(P _ {1}, P _ {2}\right) \leq \sqrt {\frac {1}{2} D _ {\mathrm {K L}} \left(P _ {1} \| P _ {2}\right)}. \tag {15}
$$
Given this measure, Foygel Barber et al. (2022) bound the coverage of non-exchangeable split conformal prediction as follows:
Theorem B.2 (Non-exchangeable split conformal prediction (Foygel Barber et al., 2022)). The non-exchangeable split conformal method defined in (13) satisfies
$$
\operatorname * {P r} \left\{Y _ {t + 1} \in \widehat {C} _ {t} ^ {\alpha} (X _ {t + 1}) \right\} \geq (1 - \alpha) - \sum_ {i = 1} ^ {t} \tilde {w} _ {i} \cdot \mathsf {d} _ {\mathsf {T V}} \big (R (D), R (D ^ {i}) \big),
$$
where  $R(D) = (R_{1}, R_{2}, \ldots, R_{t + 1})$  denotes the sequence of absolute residuals on  $D$  and  $D^{i}$  denotes a sequence where the test point  $D_{t + 1}$  is swapped with the  $i$ -th calibration point.
The coverage gap is represented by the rightmost term. Therefore,
$$
\Delta \operatorname {C o v} \geq - \sum_ {i = 1} ^ {t} \tilde {w} _ {i} \cdot \mathrm {d} _ {\mathsf {T V}} (R (D), R (D ^ {i})).
$$
Foygel Barber et al. (2022) prove this for the full-conformal setting, which also directly applies to the split conformal setting (as they note, too). For the sake of completeness, we state the proof of Theorem B.2 explicitly for split conformal prediction (adapted from Foygel Barber et al., 2022):
By the definition of the non-exchangeable split conformal prediction interval (13), it follows that
$$
Y _ {t + 1} \notin \widehat {C} _ {t} ^ {\alpha} \left(X _ {t + 1}\right) \Longleftrightarrow R _ {t + 1} > \mathrm {Q} _ {1 - \alpha} \left(\sum_ {i = 1} ^ {t} \tilde {w} _ {i} \cdot \delta_ {R _ {i}} + \tilde {w} _ {t + 1} \cdot \delta_ {+ \infty}\right). \tag {16}
$$
Next, we show that, for a given  $K \in \{1, \dots, t + 1\}$ ,
$$
\mathrm {Q} _ {1 - \alpha} \left(\sum_ {i = 1} ^ {t} \tilde {w} _ {i} \cdot \delta_ {R _ {i}} + \tilde {w} _ {t + 1} \cdot \delta_ {+ \infty}\right) \geq \mathrm {Q} _ {1 - \alpha} \left(\sum_ {i = 1} ^ {t + 1} \tilde {w} _ {i} \cdot \delta_ {\left(R (D ^ {K})\right) _ {i}}\right), \tag {17}
$$
by considering that
$$
\left(R \left(D ^ {K}\right)\right) _ {i} = \left\{ \begin{array}{l l} R _ {i}, & \text {i f} i \neq K \text {a n d} i \neq t + 1, \\ R _ {t + 1}, & \text {i f} i = K, \\ R _ {K}, & \text {i f} i = t + 1. \end{array} \right. \tag {18}
$$
In the case of  $K = t + 1$ , this holds trivially since both sides are equivalent. In the case of  $K \leq t$ , we can rewrite the left-hand side of inequality (17) as
$$
\begin{array}{l} \sum_ {i = 1} ^ {t} \tilde {w} _ {i} \cdot \delta_ {R _ {i}} + \tilde {w} _ {t + 1} \cdot \delta_ {+ \infty} \\ = \sum_ {i = 1, \dots , t; i \neq K} \tilde {w} _ {i} \cdot \delta_ {R _ {i}} + \tilde {w} _ {K} (\delta_ {R _ {K}} + \delta_ {+ \infty}) + (\tilde {w} _ {t + 1} - \tilde {w} _ {K}) \delta_ {+ \infty}. \\ \end{array}
$$
The right-hand side can be similarly represented as
$$
\begin{array}{l} \sum_ {i = 1} ^ {t + 1} \tilde {w} _ {i} \cdot \delta_ {(R (D ^ {K})) _ {i}} = \sum_ {i = 1, \dots , t; i \neq K} \tilde {w} _ {i} \cdot \delta_ {R _ {i}} + \tilde {w} _ {K} \delta_ {R _ {t + 1}} + \tilde {w} _ {t + 1} \delta_ {R _ {K}} \\ = \sum_ {i = 1, \dots , t; i \neq K} \tilde {w} _ {i} \cdot \delta_ {R _ {i}} + \tilde {w} _ {K} (\delta_ {R _ {K}} + \delta_ {R _ {t + 1}}) + (\tilde {w} _ {t + 1} - \tilde {w} _ {K}) \delta_ {R _ {K}}. \\ \end{array}
$$
By the definition of the normalized weights (12) and the assumption that  $w_{K} \in [0, 1]$ , it holds that  $\tilde{w}_{t+1} \geq \tilde{w}_{K}$ . Applying this to the rewritten forms of the left- and right-hand sides, we can see that inequality (17) holds.
Substituting the right hand side of (16) by using (17), we see that
$$
Y _ {t + 1} \notin \widehat {C} _ {t} ^ {\alpha} (X _ {t + 1}) \Rightarrow R _ {t + 1} > \mathrm {Q} _ {1 - \alpha} \left(\sum_ {i = 1} ^ {t + 1} \tilde {w} _ {i} \cdot \delta_ {(R (D ^ {\kappa})) _ {i}}\right),
$$
or equivalently by using (18):
$$
Y _ {t + 1} \notin \widehat {C} _ {t} ^ {\alpha} \left(X _ {t + 1}\right) \Rightarrow (R \left(D ^ {K}\right)) _ {K} > \mathrm {Q} _ {1 - \alpha} \left(\sum_ {i = 1} ^ {t + 1} \tilde {w} _ {i} \cdot \delta_ {\left(R \left(D ^ {K}\right)\right) _ {i}}\right). \tag {19}
$$
For the next step we use the definition of the set of so-called strange points  $S$ :
$$
\mathcal {S} (\epsilon) = \left\{i \in [ t + 1 ]: \epsilon_ {i} > \mathrm {Q} _ {1 - \alpha} \left(\sum_ {j = 1} ^ {t + 1} \tilde {w} _ {j} \cdot \delta_ {\epsilon_ {j}}\right) \right\}. \tag {20}
$$
A strange point  $s \in S$  therefore refers to a data point with residuals that are outside the prediction interval. Given the definition of  $S$ , we can directly see that
$$
\sum_ {i \in \mathcal {S} (\epsilon)} \tilde {w} _ {i} \leq \alpha \text {f o r a l l} \epsilon \in \mathbb {R} ^ {t + 1}, \tag {21}
$$
Combining (19) and (20) we see that non-coverage of  $Y_{t + 1}$  implies strangeness of point  $K$ :
$$
Y _ {t + 1} \notin \widehat {C} _ {t} ^ {\alpha} \left(X _ {t + 1}\right) \Rightarrow K \in \mathcal {S} \left(R \left(D ^ {K}\right)\right). \tag {22}
$$
Given this, we arrive at the final derivation of the coverage bound: