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KReF: Training-Free Retrieval for Long-Term Time-Series Forecasting and Predictive Uncertainty
Yang Zhang, Rui Su
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Summary
The paper introduces KReF, a training-free retrieval framework for long-term time-series forecasting (LTSF). KReF treats retrieved historical lookback-future pairs as a query-local empirical predictive distribution. It uses handcrafted statistics or frozen Random Fourier Features to embed lookbacks, retrieves similar historical pairs, and derives point forecasts, quantiles, and uncertainty intervals (CRPS, IS90) from the similarity-weighted distribution. KReF also employs a lookback-conditioned probability-integral-transform (PIT) map and validation-selected expansion/shrinkage rates to adapt interval boundaries. It outperforms trained baselines and other training-free methods across six LTSF benchmarks.
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KReF β evaluatedon β ETT
confidence 95% Β· We evaluate on six LTSF benchmarks: the four ETT datasets
KReF β optimizes β CRPS
confidence 95% Β· KReF obtains the lowest CRPS in all 12 dataset-embedding settings
KReF β solves β Long-Term Time Series Forecasting
confidence 95% Β· We propose KReF, a training-free retrieval framework that treats retrieved historical futures as a querylocal empirical predictive distribution.
KReF β evaluatedon β Exchange
confidence 90% Β· Weather and Exchange in the Autoformer benchmark collection
KReF β evaluatedon β Weather
confidence 90% Β· Weather and Exchange in the Autoformer benchmark collection
KReF β optimizes β IS90
confidence 90% Β· KReF obtains the lowest CRPS in all 12 dataset-embedding settings and the lowest IS90 in 9 settings.
KReF β uses β Random Fourier Features
confidence 90% Β· KReF embeds each lookback using handcrafted statistics or frozen random Fourier features
KReF β uses β PIT
confidence 90% Β· KReF further uses the observed query lookback to construct a probability-integral-transform map
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Abstract
Abstract:Probabilistic long-term time-series forecasting commonly relies on trained models. Training-free conformal methods typically construct intervals around a pre-existing point forecaster and do not natively represent a complete predictive distribution; sequential variants additionally suffer from increasingly delayed feedback at long horizons. We propose KReF, a training-free retrieval framework that treats retrieved historical futures as a querylocal empirical predictive distribution. After robust preprocessing, KReF embeds each lookback using handcrafted statistics or frozen random Fourier features and retrieves similar historical lookback-future pairs. Their similarity weights directly define predictive masses, quantiles, CRPS, and a weighted-mean point forecast. KReF further uses the observed query lookback to construct a probability-integral-transform map and applies validation-selected expansion and shrinkage rates to adapt interval boundaries. Across six LTSF benchmarks and four horizons, KReF obtains the lowest CRPS in all 12 dataset-embedding settings and the lowest IS90 in 9 settings. Without gradient-based fitting, its point forecasts also match or surpass trained baselines on two of six datasets. An archive-oracle analysis further reveals substantial headroom under finer horizon- and channel-wise routing. These results establish retrieval as a useful and underexplored inductive bias for LTSF.
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- Source: https://arxiv.org/abs/2608.06748v1
- Canonical: https://arxiv.org/abs/2608.06748v1
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KReF: Training-Free Retrieval for Long-Term Time-Series Forecasting and Predictive Uncertainty Yang Zhang and Rui Su Abstract Probabilistic long-term time-series forecasting commonly re- lies on trained models. Training-free conformal methods typ- ically construct intervals around a pre-existing point fore- caster and do not natively represent a complete predictive dis- tribution; sequential variants additionally suffer from increas- ingly delayed feedback at long horizons. We propose KReF, a training-free retrieval framework that treats retrieved his- torical futures as a query-local empirical predictive distribu- tion. After robust preprocessing, KReF embeds each look- back using handcrafted statistics or frozen random Fourier features and retrieves similar historical lookback-future pairs. Their similarity weights directly define predictive masses, quantiles, CRPS, and a weighted-mean point forecast. KReF further uses the observed query lookback to construct a prob- ability-integral-transform map and applies validation-se- lected expansion and shrinkage rates to adapt interval bound- aries. Across six LTSF benchmarks and four horizons, KReF obtains the lowest CRPS in all 12 dataset-embedding settings and the lowest IS90 in 9 settings. Without gradient-based fit- ting, its point forecasts also match or surpass trained base- lines on two of six datasets. An archive-oracle analysis fur- ther reveals substantial headroom under finer horizon- and channel-wise routing. These results establish retrieval as a useful and underexplored inductive bias for LTSF. 1. Introduction Probabilistic forecasting methods in time series forecasting are commonly implemented through trained parametric models or probabilistic prediction heads. Training-free con- formal methods provide an alternative, but they typically calibrate intervals around a pre-existing point forecaster ra- ther than natively representing a complete predictive distri- bution. Sequential methods such as ACI (Gibbs and CandΓ¨s 2021) additionally rely on error feedback that becomes in- creasingly delayed in long-horizon multi-step forecasting. Meanwhile, most LTSF point forecasters also depend on it- erative parameter optimization and task-specific loss func- tions. These limitations motivate a unified question: can we construct a training-free forecasting mechanism that inde- pendently produces both point and distributional predictions? Our key observation is that historical futures already con- stitute candidate outcomes for the query future. Given his- torical lookback-future pairs, lookbacks similar to the cur- rent query identify a local collection of plausible future tra- jectories. Their weighted empirical distribution directly pro- vides predictive probability masses, quantiles, and predic- tion intervals, while their weighted aggregation provides a point forecast. This view motivates KReF, a training-free re- trieval framework for multivariate LTSF. KReF robustly preprocesses each lookback, embeds it through two embedding schemes and retrieves the nearest historical lookback-future pairs. Similarity weights over the retrieved futures define the query-local empirical predictive distribution and its weighted-mean point forecast. Within the same uncertainty construction, KReF uses the observed query lookback to form a lookback-conditioned PIT proba- bility map and validation-selected expansion and shrinkage rates to adapt interval boundaries. KReF therefore operates as a standalone point and probabilistic forecaster, but does not require its uncertainty representation to be tied to its in- ternal KNN mean. Our contributions are as follows: β’ We introduce a training-free retrieval framework in which weighted historical futures form a query-local empirical predictive distribution, enabling point forecasts, predic- tive quantiles, and direct CRPS evaluation. β’ Across six benchmarks, KReF achieves the lowest CRPS in all 12 dataset-embedding settings and the lowest IS90 in 9 settings, while its training-free point forecasts surpass trained baselines on two datasets. Analysis further ex- poses the difficulty of delayed online calibration in LTSF, while an archive-oracle study reveals substantial head- room for finer-grained retrieval. 2. Related Work Retrieval, analog forecasting, and random features. An- alog forecasting retrieves historical states and uses their ob- served successors as forecasts; analog ensembles extend this principle to probabilistic weather prediction (Delle Mona- che et al. 2013). Recent retrieval-based time-series methods instead pass retrieved references to trained synthesis or dif- fusion models (J. Liu et al. 2024). Similarity is computed from handcrafted summaries or frozen Random Fourier Fea- tures, which approximate shift-invariant kernels (Rahimi and Recht 2007). Uncertainty Estimation and Calibration. Conformal pre- diction provides marginally valid intervals under exchange- ability (Vovk et al. 2005; Lei et al. 2018), while sequential extensions adapt to temporal shift through online miscover- age control (Xu and Xie 2021; Gibbs and CandΓ¨s 2021). C- PID, ECI and COP refine this controller view (Angelopou- los et al. 2023; Wu et al. 2025; Hu et al. 2026), and Hallberg SzabadvΓ‘ry (2024) extends ACI to multi-step forecasts us- ing lead-specific states. These methods calibrate intervals around a given point predictor and receive increasingly de- layed feedback at longer leads. Nearest-neighbor uncertainty methods use KNN either as the underlying conformal predictor or to localize calibration samples (Papadopoulos et al. 2011; Guan 2023; Tajmouati et al. 2024), while a recent preprint further studies online localized calibration for heterogeneous sequential data (Lai and Raskutti 2026). These methods calibrate residuals or fit- ted distributions around a KNN point forecast. KReF instead treats retrieved multi-step futures as the query-local predic- tive distribution itself and reweights its probability masses using the observed query lookback. Predictive-distribution recalibration has likewise composed a base CDF with a PIT probability map (Kuleshov et al. 2018), with Cal-PIT learn- ing covariate-dependent maps for instance-wise calibration (Dey et al. 2025). KReF adopts this probability-map princi- ple but constructs its query-specific map without fitting a calibrator or observing the query future. Long-term time-series forecasting. LTSF models span patch- and variate-based Transformers such as PatchTST (Nie et al. 2023) and iTransformer (Y. Liu et al. 2024), mul- tiscale mixers such as TimeMixer++ (S. Wang et al. 2025), and efficient frequency or linear architectures such as DLin- ear (Zeng et al. 2023), FilterTS (Y. Wang et al. 2025) and MixLinear (Ma et al. 2026). Recent methods exploit addi- tional structure: CometNet guides learned experts with con- textual motifs (Wang et al. 2026), whereas CGTFra com- bines frequency resampling, dynamic graphs, and infor- mation-theoretic alignment for inter-series dependency modeling (Yu et al. 2026). All remain trained parametric forecasters; KReF directly reuses retrieved historical futures without parameter optimization. 3. Method Task Formulation Let ν 1:ν βν νΓνΆ denote a multivariate time series with C variables. Given a lookback length L and a prediction hori- zon H, each forecasting instance consists of a lookback win- dow ν ν βν νΏΓνΆ and its corresponding future trajectory ν ν β ν ν»ΓνΆ . The goal of long-term time-series forecasting is to predict ν ν from ν ν . We additionally consider probabilistic forecasting. For each horizon-channel coordinate, the target is a conditional predictive distribution νΉ ν,β,ν ( ν£ ) =νν(ν ν,β,ν β€ν£ β£ β£ ν ν ). from which a point forecast, predictive quantiles, and a nominal-νΌ prediction interval νΌ ν,β,ν ( Ξ± ) =[l ν,β,ν ( Ξ± ) ,ν’ ν,β,ν ( Ξ± ) ] can be derived. We seek empirical coordinate-wise coverage close to νΌ: 1 | β° | β νΌ ( ν,β,ν ) ββ° ν ν,β,ν βνΌ ν,β,ν ( Ξ± ) βΞ±, where β° denotes valid test sample-horizon-channel coordinates. Robust Window Preprocessing Before constructing lookback embeddings, we apply a ro- bust preprocessing procedure to reduce the effect of extreme values and unstable per-sample normalization. This step has two components: quantile clipping and safe normalization. First, for each channel, we estimate lower and upper clip- ping thresholds from the training split as empirical q and (1- q) quantiles. The lookback and future values used as re- trieval database elements are clipped to this range. Second, each clipped lookback window is normalized per sample and per channel. Let ΞΌ ν βν 1ΓνΆ and Ο ν βν 1ΓνΆ de- note the channel-wise mean and standard deviation of the clipped lookback ν ν . Standard per-sample z-score normali- zation can be unstable when a channel is nearly constant: a tiny standard deviation may amplify small deviations into very large normalized values. To avoid this, we use a safe scale νΜ ν . For each sample and channel, νΜ ν,ν = ν ν,ν ,νν ν ν,ν > ν 1,νν‘βννν€νν ν where ν is a small threshold. The normalized lookback is then ν₯Μ ν = ( ν ν βΞΌ ν ) /νΜ ν . The corresponding future trajectory is represented in the same normalized coordinate system, ν Μ ν = ( ν ν βΞΌ ν ) /νΜ ν,ν At prediction time, after obtaining a normalized fore- cast(ν Μ Μ ν ), we map it back to the original scale by ν Μ ν =ν Μ Μ ν β νΜ ν +ΞΌ ν . Lookback Embedding The embedding function Ο ( β ) maps a robustly normalized lookback window ν₯Μ ν to a vector representation used for nearest-neighbor retrieval. We study two training-free em- beddings: handcrafted statistical embeddings and Random Fourier Feature embeddings. Handcrafted Statistical Embedding The handcrafted embedding summarizes each channel of the normalized lookback using a fixed set of descriptive sta- tistics. For each channel, we construct an 11-dimensional summary vector consisting of: the last value ν₯Μ νΏ,ν ; means over the last 24 and 96 steps; the 96-step standard deviation and interquartile range; least-squares slopes over the last 24 and 96 steps; the last 96-step z-score; the difference between the last value and the 24-step mean; the correlation between the most recent 24-step segment and its preceding 24-step segment; and the value at lag 24. The slope over a trailing window of length w is ν ν€,ν = β( νβνΜ ) (ν₯Μ νΏβν€+ν,ν βν₯Μ ν€,ν Μ ) ν€ ν=1 β( νβνΜ ) 2 ν€ ν=1 For each channel c, let ν 1 ,...,ν ν denote the fixed sum- mary statistics. The handcrafted embedding is obtained by concatenating these statistics across all channels: Ο stat ( ν ν ) =concat c=1 C [ν 1 (ν Μ ν,:,ν ),...,ν ν (ν Μ ν,:,ν )] This embeddingβs purpose is to expose interpretable sim- ilarity dimensions such as local level, volatility, trend, and short-term deviation. Random Fourier Feature Embedding Handcrafted statistics are interpretable, but they compress the lookback window into a small set of summaries and may discard fine-grained temporal structure. To preserve a richer nonlinear notion of similarity over the full lookback trajec- tory, we also use Random Fourier Features (RFF). Let ν§ ν =vec(ν Μ ν )βν νΏνΆ be the flattened normalized look- back. We sample a frozen random matrix νβν νΏνΆΓν· and phase vector νβν ν· , where ν νν βΌν ( 0,Ο rff β2 ) , ν ν βΌUniform ( 0,2Ο ) The RFF embedding is Ο rff ( ν ν ) = β 2/ν·cos ( ν§ ν ν+ν ) By the random feature approximation of shift-invariant kernels, inner products in this feature space approximate Gaussian-kernel similarity between normalized lookback trajectories. In our method, W and b are sampled once and kept fixed. The bandwidth Ο rff is selected using the median- distance heuristic over historical lookbacks. Each embedding can be used independently; we also eval- uate a lightweight stacking variant that combines their pre- dictions using validation data. Figure 1: Overview of KReF. Historical lookback-future pairs are embedded and retrieved to support point forecasting and uncertainty estimation. KNN Forecasting Given an embedding function Ο ( β ) , we construct a retrieval database from historical lookback-future pairs. For each da- tabase instance j, we store its lookback embedding Ο(ν ν ) and its normalized future trajectory ν Μ ν . For a query look- back ν ν , we compute its embedding Ο ( ν ν ) and retrieve the K database instances with the largest cosine similarity, sim ( ν,ν ) = ν ( ν ν ) β€ ν(ν ν ) |Ο ( ν ν ) | 2 |Ο(ν ν )| 2 Let ν νΎ (ν) denote the set of retrieved neighbors. We con- vert similarities into softmax weights, ν€ νν = exp ( sim ( ν,ν ) /Ο ) β exp ( sim ( ν,ν ) /ν ) νβν νΎ ( ν ) , νβν νΎ (ν) where ν is a temperature parameter. Smaller ν produces sharper weights and makes the prediction rely more heavily on the nearest neighbors; larger ν averages over a broader local neighborhood. The normalized forecast is the weighted average of the retrieved future trajectories: ν Μ Μ ν = β ν€ νν ν Μ ν Finally, the prediction is mapped back to the original scale using the query lookbackβs normalization statistics: ν Μ ν =ν Μ Μ ν βνΜ ν +ΞΌ ν The hyperparameters used, such as K, ν, and the clipping quantile, can be selected on the validation split. Because the method contains no iterative optimization tied to a particular lookback length, it can also be extended to multiple lookback lengths without retraining. Given a set of lookback lengths β, we obtain one retrieval forecast ν Μ ν (νΏ) for each νΏββ, and combine them by validation-weighted averaging, ν Μ ν =βν€ νΏ ν Μ ν (νΏ) νΏββ ,ν€ νΏ = exp(βνΎ β νΈ νΏ val ) β exp(βΞ³ β νΈ νΏ β² val ) νΏ β² ββ , where νΈ νΏ val is the validation error of lookback length L. In the offline setting, the retrieval database contains his- torical training and validation instances. In the online setting, the database is updated over time with previously observed test instances. We use a pessimistic online protocol: a test instance is added to the database only after its full future horizon becomes observable. This avoids leaking unavaila- ble future values while allowing the retrieval database to adapt to temporal distribution shift. Uncertainty Estimation The retrieved futures naturally define a query-local empiri- cal predictive distribution. For query i, let ν νΎ (ν) denote its retrieved neighbors and ν€ νν the similarity weights defined above. At horizon-channel coordinate (h,c), the base empir- ical CDF in the normalized space is νΉ ν,β,ν B ( ν§ ) =βν€ νν νβν νΎ (ν) I(ν Μ ν,β,ν β€ν§) For nominal coverage νΌ, let ν Ξ± = 1βΞ± 2 , The corresponding base interval is: νΏ ν,β,ν B ( Ξ± ) =(νΉ ν,β,ν B ) β1 ( ν Ξ± ) ,ν ν,β,ν B ( Ξ± ) =(νΉ ν,β,ν B ) β1 ( 1βν Ξ± ) These weighted quantiles describe the local variation among the retrieved futures, but their probability masses may be miscalibrated when the query occupies a systemati- cally different position from its neighbors. We therefore use the fully observed query lookback to construct a query-spe- cific probability-integral-transform adjustment. Lookback-conditioned PIT reweighting. For every valid lookback coordinate (l, c), we compute the weighted midrank of the query value among the re- trieved lookbacks:ν ν,l,ν = β ν€ νν [I(ν Μ ν,ν,ν <ν Μ ν,ν,ν )+ νβν νΎ (ν) 1 2 I(ν Μ ν,ν,ν =ν Μ ν,ν,ν )] Pooling these ranks over all valid lookback-channel coor- dinates gives the empirical mid-CDF. νΊ ν ( ν’ ) = 1 | ν± νΎ | β[I(ν ν,l,ν <ν’)+ 1 2 I(ν ν,l,ν =ν’)] ( ν,ν ) βν± νΎ where ν± νΎ is the set of valid lookback-channel coordinates. Intuitively, νΊ ν summarizes where the current lookback lies within its retrieved neighborhood. We transfer this rank in- formation to every future coordinate by composing the base future CDF with νΊ ν :νΉ ν,β,ν LB ( ν§ ) =νΊ ν (νΉ ν,β,ν B ( ν§ ) ). For a discrete implementation, sort the retrieved future values at (h,c) and define their cumulative base masses as νΆ ν,ν,β,ν = β ν€ ν ( ν ) ν ν=1 ,νΆ ν,0,β,ν =0. The PIT-adjusted mass assigned to the k-th sorted future is then ν€Μ ν(ν),β,ν =νΊ ν (νΆ ν,ν,β,ν )βνΊ ν (νΆ ν,νβ1,β,ν ) Thus, the support remains the retrieved historical futures, while their probability masses become conditional on the position of the current lookback. The adjusted masses therefore provide predictive quan- tiles and a native empirical CRPS: νΆν νν(νΉ ν,β,ν LB ,ν Μ ν,β,ν )= β ν€Μ νν,β,ν |ν Μ ν,β,ν βν Μ ν,β,ν | ν β 1 2 β ν€Μ νν,β,ν ν€Μ νν,β,ν |ν Μ ν,β,ν βν Μ ν,β,ν | ν,ν The KNN point forecast remains the similarity-weighted mean defined previously; PIT reweighting is used only for uncertainty estimation. Validation-selected interval adjustment. Let νΏ νΏν΅ (νΌ) and ν νΏν΅ (νΌ) denote the lower and upper quan- tiles of νΉ νΏν΅ (νΌ). Although Lookback-PIT can move the in- terval in a query-dependent direction, its displacement may be insufficient under strong distribution shift or unneces- sarily aggressive when the base interval is already conserva- tive. We therefore introduce an expansion rate ν ν β₯1 and a shrinkage rate 0β€ν ν β€1. Define β ν,β,ν β =νΏ ν,β,ν LB βνΏ ν,β,ν B ,β ν,β,ν + =ν ν,β,ν LB βν ν,β,ν B , The final normalized interval is: νΏ ν,β,ν =νΏ ν,β,ν ν΅ + ν ν β ν,β,ν β ,β ν,β,ν β <0 ν ν β ν,β,ν β ,β ν,β,ν β β₯0 ν ν,β,ν =ν ν,β,ν ν΅ + ν ν β ν,β,ν + ,β ν,β,ν + <0 ν ν β ν,β,ν + ,β ν,β,ν + β₯0 Thus, outward movements are amplified by ν ν , whereas inward movements are moderated by ν ν . Large expansion rates can occasionally produce extreme intervals when the base weighted-quantile interval is narrow. We therefore de- fine an optional trust-region safeguard. Let ν ν,β,ν B = ν ν,β,ν B βνΏ ν,β,ν B 2 , be the base half-width. For a clipping factor ν , the reported bounds are νΏ ν,β,ν =maxνΏ ν,β,ν ,νΏ ν,β,ν B β ( ν β1 ) ν ν,β,ν B ν ν,β,ν =minν ν,β,ν ,ν ν,β,ν B + ( ν β1 ) ν ν,β,ν B For every coverage level νΌ, we select (ν ν ,ν ν ) on the val- idation split. Validation origins are forecast using a frozen train-only retrieval archive. For each candidate pair, let νΜ νΌ β and νΜ νΌ + be the empirical lower- and upper-tail miss rates. We minimize ν₯ Ξ± = | νΜ νΌ + βν Ξ± | + | νΜ νΌ β βν Ξ± | . Ties are re- solved by smaller average width. Finally, all interval boundaries are mapped back to the original scale using the query lookbackβs normalization sta- tistics. Although KReF's uncertainty estimator does not re- quire a point forecast, its predictive distribution can be paired with either an external point forecaster or KReF's own weighted mean or median. 4. Experiments Experimental Setup Datasets. We evaluate on six LTSF benchmarks: the four ETT datasets (ETTh1, ETTh2, ETTm1, ETTm2) from the ETT collection (Zhou et al. 2021), Weather and Exchange (Exch.) in the Autoformer benchmark collection (Wu et al. 2021). The Exchange dataset was originally introduced for multivariate forecasting by Lai et al. (2018). We follow the standard chronological splits used in the LTSF literature: 6:2:2 for the ETT datasets and 7:1:2 for Weather and Ex- change. ECL and Traffic are excluded because their dimen- sionality makes H=720 retrieval exceed our 24 GB memory budget. Forecasting settings. We use a fixed lookback length L=96 and prediction horizons Hβ96,192,336,720 . All channels are predicted jointly. All metrics are reported in globally standardized data space, following the standard LTSF eval- uation convention. Method instantiation. We evaluate the two training-free embeddings introduced in the Method section, together with their lightweight validation-based stacking. The RFF em- bedding uses an output dimension of 512. Hyperparameters. The Exchange-rate dataset uses an em- pirical q-quantile of 0; the other datasets use an empirical q- quantile of 0.01. The original unclipped future values are kept for evaluation and prediction interval calibration. K and ν are selected on the validation split by minimizing valida- tion MSE. We grid-search the number of neighbors νΎβ 20,50,100,200,500,1000 and the softmax temperature νβ 0.05,0.1,0.3,0.5,1,5 and report the online pessimis- tic variant. The interval layer selects an expansion rate ν ν β 1,1.25,1.5,2,3 and a shrinkage rate ν ν β 0,0.25,0.5,0.75,1, no clipping safeguard is used in the main experiments. Selection uses a frozen train-only archive, and all selected settings remain fixed during testing. Baselines. For uncertainty estimation, we compare only with training-free methods that require neither a loss func- tion nor iterative optimization. All methods use the same online-pessimistic KNN point forecast, so differences re- flect only their uncertainty construction. Interval evaluation baselines include a static SplitConf (SpC; Lei et al. 2018), a causally expanding online variant (OL-SpC; cf. Tajmouati et al. 2024), and asynchronous multi-step adaptations of ACI, scorecaster-free Conformal PID, and COP (MS-ACI, MS-C-PI, and MS-COP; Hallberg SzabadvΓ‘ry 2024; Angel- opoulos et al. 2023; Hu et al. 2026). CRPS baselines include horizon-channel Gaussian resid- uals, static empirical residuals, and causally expanding em- pirical residuals (Gauss, ER-S, and ER-E), following the re- sidual-bootstrap and adaptive residual-selection literature (Hyndman and Athanasopoulos 2018; Wang et al. 2022). Detailed constructions and hyperparameters are provided in Appendix A. Point Forecasting baselines include five published LTSF models that span representative architectural designs: a patch-based Transformer (PatchTST, Nie et al. 2023), a var- iate-tokenized Transformer (iTransformer, Y. Liu et al. 2024), a multiscale neural mixer (TimeMixer++, S. Wang et al. 2025), a frequency-filtering linear model (FilterTS, Y. Wang et al. 2025) and a consistency-driven inter-variate de- pendency model (CGTFra, Yu et al. 2026). For the trained point-forecasting baselines, we report the results published in their original papers under the standard LTSF protocol. Metrics. Point forecasts are evaluated with MSE and MAE; prediction intervals by coverage, width, and interval score; and predictive distributions by CRPS. Definitions of these metrics are provided in Appendix B. Implementation. Retrieval embeddings, similarity search, top-K selection, neighbor aggregation, and KReF uncer- tainty construction are executed on a single NVIDIA Ge- Force RTX 3090 GPU. Data loading and serialization are performed on CPU. Experimental Results Uncertainty Estimation Results. Table 1 reports four-hori- zon-average interval and distributional results. KReF ob- tains the lowest IS90 under both embeddings on ETTh1, ETTh2, ETTm1, and Exchange, and additionally on ETTm2 with the handcrafted embedding. OL-SpC achieves the low- est IS90 in the remaining three settings: ETTm2-RFF and both Weather embeddings. The controller-based baselines generally approach nomi- nal coverage but often require substantially wider intervals, especially on ETTm1 and Weather. Their horizon-specific states receive feedback only after the corresponding targets become observable, which can produce stale reactions and severe overexpansion at long leads. Although KReF does not uniformly attain nominal coverage, particularly on ETTm1 and Weather, it still improves coverage over SpC and remains much narrower than the controller-based meth- ods. The four-horizon average also conceals the most severe delayed-feedback case. At H=720 on Exchange, all methods under-cover, but KReF retains the highest Cov90 at 0.845 and 0.783 under handcrafted and RFF embeddings, com- pared with 0.742 and 0.681 for the strongest adaptive base- line (Appendix C1). Average coverage results at the 50% and 80% nominal levels are reported in Appendix C2. For full-distribution forecasting, KReF achieves the low- est CRPS in all 12 dataset-embedding settings, reducing CRPS by 3.8%β11.3% relative to the strongest distribu- tional baseline. ER-E improves upon ER-S in all 12 settings, confirming that causally incorporating newly observed re- siduals is useful. KReF nevertheless remains consistently stronger, suggesting that its query-specific retrieved distri- bution captures information unavailable to residual pools conditioned only on forecast coordinates and observation time. Dataset Metric Handcrafted RFF Ours SpC MS- ACI MS- C-PI MS- COP SpC- OL Ours SpC MS- ACI MS- C-PI MS- COP SpC- OL ETTh1 Cov90 0.887 0.881 0.900 0.889 0.892 0.896 0.887 0.886 0.902 0.886 0.889 0.902 Width90 2.699 2.680 4.773 16.073 14.252 2.791 2.298 2.497 4.207 13.350 11.930 2.568 IS90 3.432 3.739 5.659 17.228 15.371 3.712 2.978 3.360 4.975 14.401 12.948 3.318 ETTh2 Cov90 0.905 0.908 0.901 0.892 0.893 0.913 0.907 0.911 0.901 0.891 0.893 0.914 Width90 1.961 2.111 4.209 9.749 9.036 2.095 1.855 2.089 4.024 9.048 8.407 2.061 IS90 2.575 2.674 4.827 10.518 9.781 2.639 2.455 2.639 4.628 9.794 9.127 2.601 ETTm1 Cov90 0.884 0.906 0.900 0.918 0.918 0.908 0.868 0.906 0.900 0.919 0.919 0.908 Width90 2.224 2.545 32.359 50.434 48.344 2.532 1.939 2.489 31.775 51.873 48.651 2.466 IS90 3.020 3.446 33.275 51.172 49.072 3.428 2.791 3.374 32.688 52.583 49.353 3.351 ETTm2 Cov90 0.902 0.899 0.900 0.916 0.917 0.908 0.902 0.912 0.900 0.916 0.917 0.914 Width90 1.667 1.896 10.713 22.324 21.765 1.863 1.968 2.023 10.131 21.523 21.163 1.964 IS90 2.196 2.456 11.221 22.818 22.256 2.373 2.472 2.496 10.636 22.039 21.670 2.426 Weather Cov90 0.886 0.845 0.898 0.908 0.908 0.871 0.873 0.844 0.898 0.909 0.910 0.872 Width90 2.113 1.684 10.539 18.530 17.975 1.765 2.044 1.702 10.766 20.376 19.260 1.777 IS90 2.738 2.557 11.157 19.105 18.548 2.518 2.737 2.603 11.399 20.952 19.844 2.544 Exch. Cov90 0.851 0.761 0.843 0.843 0.845 0.811 0.824 0.739 0.829 0.832 0.834 0.797 Width90 2.619 1.790 3.459 7.620 7.319 2.192 2.463 1.663 3.347 7.428 7.151 2.089 IS90 3.496 3.745 4.787 9.197 8.876 3.760 3.470 4.127 5.016 9.187 8.899 3.982 Dataset CRPS Ours Gauss ER-S ER-E Ours Gauss ER-S ER-E ETTh1 CRPS 0.436 0.466 0.456 0.453 0.379 0.411 0.402 0.398 ETTh2 CRPS 0.282 0.305 0.309 0.297 0.271 0.294 0.299 0.288 ETTm1 CRPS 0.351 0.564 0.366 0.365 0.333 0.586 0.390 0.349 ETTm2 CRPS 0.228 0.333 0.247 0.241 0.229 0.341 0.248 0.243 Weather CRPS 0.226 0.329 0.247 0.246 0.233 0.336 0.251 0.250 Exch. CRPS 0.340 0.374 0.549 0.457 0.341 0.385 0.563 0.462 Table 1: Four-horizon average results of uncertainty estimation in LTSF. The best results of IS90 and CRPS are highlighted in bold. For IS and CRPS, lower is better, while Cov90 should be interpreted relative to the nominal level of 0.90. Point Forecasting Results. Table 2 reports MSE and MAE averaged uniformly across the four forecasting horizons. The full per-horizon results are reported in Appendix C3.We report both base retrieval embeddings and their late-fusion variant, Ours-Stack, which combines the complementary handcrafted and RFF forecasts under the same input-length protocol. We also tested the results of our method under multi-input-length fusion, but for the sake of fair compari- son, we did not include it in the main table. Relevant results can be found in Appendix C4. The proposed retrieval method outperforms all selected trained baselines on two datasets. On ETTh2, Ours-RFF achieves the best average result, with 0.299 MSE and 0.370 MAE, improving over TimeMixer++ (0.339/0.380). On ETTm2, Ours-Stack obtains 0.218 MSE and 0.311 MAE, compared with 0.269 for TimeMixer++ in MSE and 0.316 for CGTFra in MAE; the handcrafted component already reaches 0.218/0.312, while stacking slightly improves MAE. The fusion also improves or matches the stronger individual embedding on ETTh1, Weather, and Exchange, although trained models remain superior overall on ETTh1, ETTm1, Weather, and Exchange. Ablation Study. At H=96 in uncertainty estimation, Look- back-PIT generally produces sharper candidate intervals, while validation-selected rate tempering determines whether this contraction should be retained or reversed. PIT has only a marginal effect on CRPS, confirming that full- distribution performance primarily originates from the re- trieved empirical support. Freezing the retrieval archive pro- duces little change in point-forecast accuracy on ETTh2 and ETTm2, indicating that their gains do not arise from access to newly observed test futures. Detailed ablation and ar- chive-update results are reported in Appendix C5. Dataset Ours-HC Ours-RFF Ours-Stack CGTFra TimeMixer++ FilterTS iTrans PatchTST Mse/Mae Mse/Mae Mse/Mae Mse/Mae Mse/Mae Mse/Mae Mse/Mae Mse/Mae ETTh1 0.759/0.627 0.550/0.534 0.549/0.532 0.434/0.427 0.418/0.432 0.433/0.430 0.454/0.467 0.507/0.472 ETTh2 0.330/0.393 0.299/0.370 0.299/0.371 0.369/0.394 0.339/0.380 0.372/0.396 0.383/0.407 0.391/0.411 ETTm1 0.520/0.492 0.465/0.461 0.509/0.480 0.388/0.386 0.368/0.378 0.388/0.398 0.409/0.410 0.402/0.406 ETTm2 0.218/0.312 0.247/0.325 0.218/0.311 0.277/0.316 0.269/0.320 0.276/0.337 0.288/0.332 0.290/0.334 Weather 0.344/0.329 0.361/0.339 0.342/0.325 0.251/0.273 0.226/0.262 0.244/0.274 0.258/0.278 0.265/0.285 Exch. 0.493/0.459 0.472/0.455 0.427/0.440 0.318/0.384 0.357/0.409 0.352/0.397 0.360/0.403 0.366/0.404 Table 2: Four-horizon average point-forecasting results in LTSF. The best results of MSE and MAE are highlighted in bold. Retrieval Headroom Analysis. To distinguish the limita- tions of KReFβs whole-trajectory routing from the infor- mation available in the historical archive, we conduct an ar- chive single-neighbor oracle analysis at H=96. At each fore- cast origin, the archive contains the training and validation samples together with only earlier test futures that have al- ready been fully observed. Using the query future solely for ex-post selection, the oracle copies the MSE-nearest histor- ical block, selecting one sample for the complete trajectory, independently per horizon, or independently per channel; no averaging or fitting is performed. Across six datasets, trajec- tory routing yields a mean MSE/MAE of 0.1954/0.2940. Horizon-wise routing reduces these errors to 0.0302/0.1096, corresponding to an 84.5% MSE reduction, while channel- wise routing obtains 0.0866/0.1786, a 55.7% MSE reduction. Horizon-wise MSE decreases by 93.2%β94.2% on the four ETT datasets, whereas Exchange favors channel-wise rout- ing, which reduces MSE by 82.5%. Although non-deploya- ble and not a strict upper bound, this analysis shows that finer routing can expose historical information hidden by whole-trajectory matching and motivates blockwise and learned retrieval. Full per-dataset results and implementa- tion details are provided in Appendix E. Figure 2: Retrieval headroom under archive single- neighbor oracle routing at H=96. MSE is normalized by the best reported KReF result on each dataset, shown by the dashed line at one; lower is better. Finer horizon- or channel-wise routing reveals substantial unused historical information. 5. Discussion KReF is more broadly effective for uncertainty estimation than for point forecasting. Appendix D further analyzes the characteristics of the six datasets and demonstrates KReF's capability for uncertainty estimation under strong distribu- tional shifts; retrieved futures can still characterize local dis- persion when their mean is not the strongest point forecast. Lookback-PIT and expansion/shrinkage rates further adjust its distribution and interval, avoiding the reliance on cover- age feedback required by online calibration controllers and the poor IS90 caused by delayed feedback. This helps ex- plain its strong CRPS and IS90 results, particularly at long horizons, although lookback-to-future rank transfer may not provide a formal coverage guarantee. The main limitations are computational scale and histori- cal transferability. Storing lookbackβfuture pairs and aggre- gating neighbor futures incurs memory costs that grow with dimensionality and prediction horizon, currently preventing long-horizon experiments on ECL and Traffic. Strong dis- tribution shift can also reduce point accuracy and interval sharpness, while the absence of learned representations lim- its cross-dataset transfer. These limitations suggest scalable and hybrid extensions, including clustering or approximate disk-backed indices, re- trieval in decomposed or learned latent spaces, and block- wise, channel-adaptive, or autoregressive routing motivated by the oracle analysis. Combining retrieval with pretrained representations and developing diagnostics for when histor- ical reuse remains reliable are promising directions. 6. Conclusion We proposed a training-free retrieval framework for long- term time-series forecasting and uncertainty estimation. Re- trieved historical futures support point forecasting and de- fine a local empirical predictive distribution without a trained probabilistic head. Lookback-conditioned PIT re- weighting adapts this distribution, while validation-selected rates adjust the reported prediction intervals. 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Proceedings of the AAAI Con- ference on Artificial Intelligence, 35(12): 11106β11115. doi.org/10.1609/aaai.v35i12.17325. Appendices Appendix Overview. This supplement provides implemen- tation details, complete horizon-wise results, applicability analyses, and additional theoretical interpretation. In partic- ular, Appendix F establishes two properties of Lookback- PIT: its interval boundaries are equivalent to empirical prob- ability-level selection under the standard left generalized in- verse, and directional expansion and shrinkage can strictly reduce oracle boundary residual when inward and outward rank transfer exhibit different reliability. Clarifications and Typographical Corrections to the Main Text. For precision, we clarify four points in the main text. These corrections concern notation and implementa- tion details and do not alter any reported result or conclusion. 1. Empirical CRPS. In the empirical CRPS formula in the Method section, the pairwise-distance term contains a typo- graphical indexing error. Its distance factor should be ν Μ j,β,ν βν Μ ν,β,ν , rather than ν Μ ν,β,ν βν Μ ν,β,ν . The intended for- mula is: νΆν νν(νΉ ν,β,ν LB ,ν Μ ν,β,ν )= β ν€Μ νν,β,ν |ν Μ ν,β,ν βν Μ ν,β,ν | ν β 1 2 β ν€Μ νν,β,ν ν€Μ νν,β,ν |ν Μ j,β,ν βν Μ ν,β,ν | ν,ν .The implementation and all reported CRPS results use the correct pairwise term ν Μ j,β,ν βν Μ ν,β,ν . 2. Shrinkage-rate notation. In the Experimental Setup, the shrinkage rate is inadvertently denoted by ν e . It should be denoted by ν ν ; ν e denotes the expansion rate. This is a nota- tion-only correction, and the implementation uses separate expansion and shrinkage rates. 3. Endpoint completion and quantile interpolation. The main Method section omits one finite-sample implementa- tion conventions used by Lookback-PIT. At the cumulative- mass endpoint, we set the transformed cumulative masses to νΆ ν,νΎ,β,ν =1, while the empirical mid-CDF transformation is applied at the interior cumulative-mass knots. Prediction boundaries are obtained by linearly interpolating between adjacent retrieved support values. These conventions are used in all experiments and are formally specified in Appen- dix F.1. 4. Meaning of training-free uncertainty estimation. The statement that the uncertainty baselines require neither a loss function nor iteration should be interpreted as stating that they do not require gradient-based training of a parametric predictive model. KReF and the uncertainty baselines are not hyperparameter-free: they use validation-based selec- tion of their respective hyperparameters, as described in the Experimental Setup and Appendix A. Their online variants may also update calibration states sequentially, but none trains a parametric forecasting model through gradient back- propagation. Appendix-A: Interval baselines. All interval methods use the same online-pessimistic KNN point forecasts. For mul- tistep multivariate prediction, each method maintains a sep- arate state for every horizon-channel coordinate. Feedback for a forecast issued at origin i and lead h is released only when its target becomes observable, preventing future leak- age. Static Split Conformal estimates a finite-sample abso- lute-residual quantile from validation data. Online Split Conformal maintains a causally expanding residual pool and inserts each test residual only after the corresponding hori- zon-specific target has arrived. MS-ACI applies the published multistep diagonal-feed- back construction with the practical largest-observed-score clipping rule. We select ν from 0.1,0.05,0.01,0.005. MS- C-PI uses the scorecaster-free P+I controller, searching νβ 1,0.5,0.1,0.05, ( νΆ ν νν‘ ,νΎ νΌ ) β ( 1,200 ) , ( 1,100 ) , ( 5,10 ) , with a score-range window of 100. MS-COP uses its ECDF correction with window 100, Ξ» ν‘ /Ξ· ν‘ =0.5, and Ξ·β 1,0.5,0.1,0.05. For each method, one global configuration is selected using causal IS90 on a chronological validation suffix and is then frozen throughout testing; adaptive states remain horizon-channel specific. Distributional Baselines. Gaussian-Res centers a Gaussian distribution at the shared KNN point forecast and estimates its scale separately for every horizon-channel coordinate. ER-Static treats signed validation residual trajectories as equally weighted empirical samples. ER-Expanding begins with causally observable validation residuals and inserts each test residual after its target becomes available, without discarding earlier residuals. Each residual trajectory remains an empirical sample rather than being averaged. Appendix-B: Uncertainty Metrics. All metrics are aver- aged over valid (sample, horizon, channel) elements at nom- inal level Ξ± (Ξ±=0.9 in the main text). Coverage is the empirical fraction of targets inside the in- terval, cov = mean 1[lo β€ y β€ hi], and should match Ξ±. Interval score (Gneiting and Raftery 2007) combines sharpness and calibration:νΌν= ( βνβνν ) +( 2 1βνΌ β [( ννβ ν¦ ) + + ( ν¦ββν ) + ] , rewarding narrow intervals while penal- izing targets that fall outside; we report its mean. CRPS scores full predictive distribution νΉ:νΆν νν ( νΉ,ν¦ ) = β« (νΉ ( ν§ ) β1 [ ν§β₯ν¦ ] ) 2 νν§. We therefore report CRPS only for KReF, Gaussian-Res, and the empirical-residual distributions. Split Conformal, Online Split Conformal, MS-ACI, MS-C-PI, and MS-COP specify interval boundaries but do not natively assign prob- ability mass over possible outcomes. KReF computes CRPS from its PIT-reweighted retrieved futures; Gaussian-Res uses its fitted Gaussian distribution; ER-Static and ER-Ex- panding use their corresponding empirical residual distribu- tions. Appendix-C1: Per-horizon interval quality. Table 3 re- ports Cov90 and IS90 separately at every prediction horizon. KReF achieves the lowest CRPS in all 48 dataset-embed- ding-horizon settings, showing that its average CRPS ad- vantage is not driven by a small number of favorable hori- zons. For IS90, KReF is strictly best in 34 settings and tied for best in one additional setting. Interval coverage is less uniform, ranging from 0.783 (Exchange RFF on Horizon 720) to 0.917 (ETTh2 RFF on Horizon 720) across the 48 KReF configurations. This dis- tinction is important: a lower CRPS indicates a better com- plete predictive distribution but does not imply exact cali- bration at every individual nominal level, while the expan- sion/shrinkage rate hyperparameters selected on the valida- tion set to minimize the coverage loss rate may fail due to validation-test set drift, resulting in over-coverage/under- coverage. At H=720 on Exchange, all methods under-cover, yet KReF retains Cov90 of 0.845 and 0.783 under hand- crafted and RFF embeddings, compared with 0.742 and 0.681 for MS-ACI. KReF also obtains the lowest IS90 in both settings. In contrast, the adaptive controllers often at- tain near-nominal average coverage elsewhere only through substantially expanded intervals, illustrating the difficulty of horizon-delayed feedback in LTSF. Emb Da- taset H Cov90 IS90 Ours SpC SpC- OL MS- ACI MS- C-PI MS- COP Ours SpC SpC- OL MS- ACI MS- C-PI MS- COP Handcrafted ETTh1 96 0.895 0.900 0.905 0.903 0.901 0.901 3.012 3.298 3.303 3.709 4.996 4.708 192 0.891 0.892 0.902 0.902 0.897 0.900 3.209 3.518 3.511 4.682 10.048 9.159 336 0.885 0.877 0.895 0.901 0.891 0.892 3.435 3.757 3.727 5.909 19.754 16.862 720 0.875 0.854 0.882 0.894 0.869 0.873 4.071 4.385 4.309 8.335 34.114 30.753 ETTh2 96 0.898 0.912 0.915 0.903 0.903 0.905 2.075 2.294 2.243 3.431 4.031 4.183 192 0.903 0.912 0.914 0.902 0.903 0.904 2.379 2.529 2.498 4.220 7.043 6.524 336 0.906 0.906 0.909 0.899 0.894 0.896 2.684 2.677 2.659 5.267 11.860 10.599 720 0.913 0.904 0.912 0.899 0.867 0.869 3.162 3.198 3.157 6.389 19.137 17.819 ETTm1 96 0.887 0.911 0.912 0.900 0.926 0.926 2.623 3.012 2.994 16.052 37.625 37.362 192 0.884 0.910 0.911 0.900 0.926 0.926 2.887 3.303 3.290 29.431 41.160 41.580 336 0.885 0.905 0.908 0.900 0.918 0.918 3.151 3.567 3.561 36.857 46.301 47.221 720 0.880 0.897 0.903 0.900 0.902 0.902 3.420 3.902 3.866 50.759 79.603 70.123 ETTm 2 96 0.893 0.890 0.901 0.900 0.919 0.918 1.741 2.061 1.985 5.405 7.357 6.159 192 0.900 0.899 0.907 0.900 0.919 0.921 2.001 2.316 2.219 8.029 11.954 11.796 336 0.907 0.903 0.910 0.900 0.917 0.918 2.307 2.556 2.451 11.410 20.393 20.349 720 0.910 0.905 0.912 0.899 0.909 0.910 2.734 2.894 2.836 20.040 51.568 50.721 Weather 96 0.888 0.852 0.875 0.900 0.916 0.916 2.177 1.957 1.936 6.290 6.911 6.739 192 0.879 0.848 0.872 0.899 0.915 0.915 2.529 2.367 2.325 8.922 12.788 12.941 336 0.885 0.845 0.870 0.898 0.908 0.909 2.977 2.727 2.667 12.445 22.178 21.194 720 0.890 0.838 0.868 0.894 0.891 0.893 3.268 3.175 3.143 16.972 34.542 33.318 Exchan 96 0.878 0.835 0.871 0.894 0.906 0.907 1.597 1.597 1.663 2.761 4.580 4.423 192 0.851 0.797 0.849 0.885 0.895 0.898 2.363 2.493 2.590 4.114 7.385 7.293 336 0.832 0.741 0.812 0.852 0.866 0.869 3.258 3.801 3.791 5.113 13.110 12.466 720 0.845 0.673 0.711 0.742 0.705 0.706 6.765 7.088 6.997 7.162 11.713 11.322 Random Fourier Features ETTh1 96 0.891 0.907 0.909 0.903 0.901 0.901 2.506 2.909 2.907 3.335 4.312 4.070 192 0.893 0.897 0.907 0.903 0.893 0.898 2.745 3.139 3.119 4.200 8.471 7.765 336 0.890 0.881 0.900 0.903 0.886 0.890 3.016 3.400 3.353 5.212 16.297 14.621 720 0.873 0.860 0.890 0.897 0.865 0.868 3.645 3.991 3.895 7.154 28.524 25.335 ETTh2 96 0.903 0.914 0.917 0.903 0.903 0.905 2.028 2.274 2.218 3.460 4.055 4.244 192 0.903 0.912 0.914 0.903 0.901 0.903 2.288 2.511 2.477 4.082 6.771 6.351 336 0.904 0.908 0.910 0.899 0.893 0.895 2.524 2.646 2.625 4.911 10.778 9.617 720 0.917 0.911 0.916 0.900 0.865 0.867 2.982 3.125 3.084 6.061 17.570 16.296 ETTm1 96 0.871 0.914 0.913 0.901 0.927 0.926 2.493 2.967 2.943 15.077 37.943 37.650 192 0.870 0.910 0.911 0.900 0.928 0.928 2.637 3.237 3.220 28.349 44.021 40.935 336 0.870 0.905 0.907 0.900 0.918 0.919 2.884 3.487 3.477 37.172 48.599 54.028 720 0.862 0.896 0.902 0.900 0.903 0.904 3.151 3.806 3.766 50.154 79.769 64.801 E TTm2 96 0.887 0.894 0.902 0.900 0.919 0.919 1.815 2.086 2.017 5.297 6.948 5.801 192 0.900 0.912 0.914 0.900 0.918 0.921 2.287 2.360 2.279 7.596 11.480 11.191 336 0.906 0.921 0.920 0.900 0.918 0.918 2.637 2.603 2.520 10.648 20.848 20.980 720 0.914 0.920 0.919 0.899 0.909 0.911 3.147 2.935 2.888 19.005 48.878 48.708 Weather 96 0.871 0.849 0.874 0.900 0.919 0.919 2.100 2.031 1.992 6.366 7.874 7.483 192 0.864 0.847 0.873 0.899 0.916 0.916 2.527 2.389 2.338 8.598 14.285 13.471 336 0.871 0.844 0.871 0.898 0.909 0.910 2.634 2.764 2.679 13.020 25.014 23.151 720 0.885 0.837 0.868 0.894 0.893 0.894 3.688 3.227 3.167 17.613 36.635 35.269 Exchan 96 0.865 0.833 0.867 0.894 0.906 0.907 1.711 1.681 1.738 2.900 4.587 4.474 192 0.831 0.796 0.849 0.886 0.889 0.892 2.337 2.600 2.647 4.195 7.832 7.869 336 0.816 0.741 0.817 0.854 0.856 0.859 3.592 3.916 3.792 5.041 12.629 11.948 720 0.783 0.587 0.654 0.681 0.677 0.676 6.240 8.313 7.750 7.931 11.701 11.303 Table 3: Per-horizon uncertainty-estimation results in LTSF. The best results of IS90 are highlighted in bold. Appendix-C2: Table 4 reports four-horizon-average cover- age at nominal levels of 50% and 80%. Cov50 ranges from 0.421 to 0.505, while Cov80 ranges from 0.730 to 0.806. Thus, several central intervals are systematically under-cov- ered even though the 90% intervals are generally better cal- ibrated. Validation selection frequently reaches the maxi- mum expansion candidate of 3.0 at the 50% level, indicating that the compact search grid constrains some interval adjust- ments. Dataset HC Cov50 HC Cov80 RFF Cov50 RFF Cov80 ETTh1 0.4748 0.7741 0.4210 0.7688 ETTh2 0.5049 0.7993 0.4749 0.8058 ETTm1 0.4764 0.7791 0.4598 0.7533 ETTm2 0.4623 0.7903 0.4607 0.7956 Exchange 0.4740 0.7362 0.4609 0.7297 Weather 0.4982 0.7882 0.4887 0.7807 Table 4: Four-horizon average coverage results at the 50% and 80% nominal levels. We therefore conduct a uniform post-hoc sensitivity ex- periment over all six datasets and both embeddings. The ex- pansion candidate grid is extended to 1, 2, 3, 5, 8, ..., 2584, while the shrinkage grid remains unchanged. Because an un- restricted expansion grid can produce extreme sample-level intervals, we apply the same trust-region factorν =8 to every dataset, embedding, horizon, and nominal level. The ex- panded grid and clipping factor participate in validation se- lection and are then frozen during testing. These settings are used only for sensitivity analysis and do not replace the compact-grid main results. As shown in Table 5, the expanded configuration changes the central-interval calibration pattern substantially. Aver- age Cov50 increases from 0.471 to 0.508, eliminating sev- eral severe under-coverage cases, although Exchange-RFF becomes over-covered at 0.610. Consequently, the mean ab- solute Cov50 deviation changes only slightly, from 0.029 to 0.032. At the 80% level, average coverage improves from 0.775 to 0.795 and the mean absolute deviation decreases from 0.026 to 0.024. The improvement is especially clear on Exchange, where Cov80 increases from 0.736/0.730 to 0.789/0.787 under handcrafted/RFF embeddings. In contrast, ETTh2 becomes over-covered at 0.848/0.842, illustrating that greater expansion capacity does not transfer uniformly. The fixed trust region keeps the resulting widths finite: Width50 ranges from 0.505 to 1.955 and Width80 from 1.259 to 4.226. This avoids the pathological widths ob- served under unrestricted expansion, but the comparatively wide ETTh2 and Exchange intervals show that clipping can- not by itself resolve validation-to-test mismatch. Datasets Emb Cov50 Wid50 Cov80 Wid80 ETTh1 HC 0.4748 1.0152 0.7741 1.9562 RFF 0.4871 1.4621 0.7759 2.4339 ETTh2 HC 0.5268 0.6899 0.8480 3.0826 RFF 0.5184 0.7141 0.8417 2.7728 ETTm1 HC 0.4764 0.7683 0.7774 2.1054 RFF 0.4652 0.6964 0.7434 1.6440 ETTm2 HC 0.5372 0.7578 0.8193 2.4481 RFF 0.4665 0.5051 0.8081 1.9612 Exch. HC 0.5469 1.6147 0.7888 4.2264 RFF 0.6098 1.9551 0.7872 4.1625 Weather HC 0.5030 0.5140 0.7978 1.9220 RFF 0.4875 0.5148 0.7798 1.2594 Table 5: Four-horizon-average sensitivity results using an expansion-rate candidate grid extending to 2584 and a fixed trust-region factor ν =8. It should be noted that although the experiments in Ap- pendix C2 include 85 combinations of expansion and con- traction rates, the overall hyperparameter grid search is ac- tually very fast because the intermediate results of the com- putation on the validation set can be saved. Besides, the fi- nite rate grid is an implementation choice rather than an in- trinsic requirement of KReF. An exact breakpoint- or order- statistic-based selector could therefore replace the manually specified grid without repeating retrieval. Exact validation optimization, however, is not guaranteed to transfer better under validationβtest shift. In the expanded-grid sensitivity experiment, the additional expansion capacity changes Ex- change-RFF Cov50 from 0.461 to 0.610. The result does not establish that an exact selector necessarily overfits, but they demonstrate that greater validation-time flexibility can am- plify calibration mismatch on the test distribution. We there- fore retain a coarse, reliability-constrained rate grid as a form of regularization rather than pursuing the exact empir- ical validation optimum. A further extension could allow signed rates, which re- flect an inward PIT proposal into an outward correction, or vice versa. Such a generalization would require explicit non- crossing constraints and is left for future work. Appendix-C3: Per-Horizon Point-Forecasting Re- sults.Table 6 reports MSE and MAE separately at each pre- diction horizon. The four-horizon averages in the main text do not arise from an isolated short-horizon gain. On ETTh2, the KReF variants achieve the lowest MSE at all four hori- zons: Ours-Stack obtains 0.232 at H=96, Ours-RFF/Stack obtain 0.278 at H=192, Ours-RFF obtains 0.307 at H=336, and Ours-RFF/Stack obtain 0.378 at H=720. They also achieve the lowest MAE through H=336, while TimeMixer++ is slightly better in MAE at H=720. On ETTm2, Ours-Stack or Ours-HC attains the lowest MSE at every horizon, from 0.158 at H=96 to 0.294 at H=720; Ours-Stack further achieves the lowest MAE at the longer horizons H=336 and H=720. Thus, the main-table gains on ETTh2 and ETTm2 persist across forecasting lengths, especially in MSE. On the remaining datasets, the per-horizon results show that trained baselines outperform KReF at most horizons, confirming that KReF gains are da- taset-dependent rather than being obscured by horizon aver- aging. Da- taset H Ours-HC Ours-RFF Ours- Stack CGTFra TimeMixer++ FilterTS iTrans PatchTST Metrics Mse/Mae Mse/Mae Mse/Mae Mse/Mae Mse/Mae Mse/Mae Mse/Mae Mse/Mae ETTh1 96 0.648/0.561 0.443/0.465 0.439/0.461 0.372/0.387 0.361/0.403 0.374/0.391 0.386/0.405 0.460/0.447 192 0.708/0.599 0.497/0.503 0.493/0.498 0.424/0.418 0.416/0.441 0.424/0.421 0.441/0.512 0.477/0.429 336 0.764/0.632 0.554/0.538 0.551/0.534 0.473/0.443 0.430/0.434 0.464/0.441 0.487/0.458 0.546/0.496 720 0.917/0.716 0.705/0.632 0.713/0.637 0.473/0.464 0.467/0.451 0.470/0.466 0.503/0.491 0.544/0.517 ETTh2 96 0.246/0.339 0.235/0.326 0.232/0.324 0.288/0.336 0.276/0.328 0.290/0.338 0.297/0.349 0.308/0.355 192 0.296/0.373 0.278/0.356 0.278/0.356 0.364/0.384 0.342/0.379 0.374/0.390 0.380/0.400 0.393/0.405 336 0.333/0.398 0.307/0.376 0.310/0.379 0.410/0.422 0.346/0.398 0.406/0.420 0.428/0.432 0.427/0.436 720 0.446/0.461 0.378/0.422 0.378/0.424 0.414/0.433 0.392/0.415 0.418/0.437 0.427/0.445 0.436/0.450 ETTm1 96 0.421/0.437 0.361/0.396 0.406/0.424 0.315/0.344 0.310/0.334 0.321/0.360 0.334/0.368 0.352/0.374 192 0.484/0.470 0.437/0.446 0.492/0.469 0.366/0.372 0.348/0.362 0.363/0.382 0.390/0.393 0.374/0.387 336 0.553/0.508 0.501/0.480 0.539/0.493 0.398/0.395 0.376/0.391 0.395/0.403 0.420/0.420 0.421/0.414 720 0.621/0.552 0.563/0.522 0.600/0.532 0.472/0.435 0.440/0.423 0.474/0.446 0.491/0.459 0.462/0.449 ETTm2 96 0.159/0.266 0.165/0.268 0.158/0.264 0.171/0.249 0.170/0.245 0.172/0.255 0.180/0.264 0.183/0.270 192 0.190/0.293 0.218/0.305 0.190/0.293 0.238/0.293 0.229/0.291 0.237/0.299 0.250/0.309 0.255/0.314 336 0.229/0.322 0.272/0.341 0.229/0.322 0.300/0.333 0.303/0.343 0.299/0.398 0.311/0.348 0.309/0.347 720 0.295/0.368 0.333/0.384 0.294/0.367 0.397/0.391 0.373/0.399 0.397/0.394 0.412/0.407 0.412/0.404 Weather 96 0.242/0.258 0.263/0.274 0.239/0.257 0.152/0.190 0.155/0.205 0.162/0.207 0.174/0.214 0.186/0.227 192 0.317/0.305 0.317/0.317 0.309/0.302 0.203/0.239 0.201/0.245 0.209/0.252 0.221/0.254 0.234/0.265 336 0.366/0.347 0.384/0.353 0.359/0.344 0.257/0.279 0.237/0.265 0.263/0.294 0.278/0.296 0.284/0.301 720 0.451/0.405 0.478/0.411 0.459/0.398 0.338/0.334 0.312/0.334 0.344/0.344 0.358/0.347 0.356/0.349 Exchange 96 0.117/0.231 0.119/0.241 0.114/0.231 0.083/0.202 0.085/0.214 0.081/0.199 0.086/0.206 0.088/0.205 192 0.208/0.322 0.215/0.327 0.204/0.318 0.173/0.296 0.175/0.313 0.171/0.294 0.177/0.299 0.176/0.299 336 0.393/0.452 0.401/0.457 0.388/0.450 0.324/0.412 0.316/0.420 0.321/0.409 0.331/0.417 0.301/0.397 720 1.255/0.832 1.154/0.795 1.001/0.759 0.668/0.619 0.851/0.689 0.837/0.688 0.847/0.691 0.901/0.714 Table 6: Per-horizon point estimates in LTSF results. The best results are highlighted in bold. Appendix-C4: Multi-Sequence-Length Softmax.We fur- ther evaluate a multi-sequence-length variant that aggre- gates predictions from lookback lengths 48, 96, 192, 336, 720. As shown in Table 7, this variant improves 14/18 re- sults at prediction length 96 and 14/18 results at prediction length 720. The gains are especially clear on Weather, where the stacked result improves from 0.239 to 0.185 at horizon 96. These results suggest that different lookback lengths capture complementary recurrence patterns, which can be exploited without training additional models. Exchange un- der long-horizon forecasting is the main exception, where multi-sequence aggregation degrades performance, espe- cially after validation-fitted stacking. This indicates that adaptive lookback aggregation can be sensitive under distri- bution shift. Da- taset Embed- ding/model Multi input 96 Orig- inal 96 Multi input 720 Orig- inal 720 ETT h1 Handcrafted 0.627 0.648 0.889 0.917 RFF 0.437 0.443 0.677 0.706 Stacking 0.439 0.439 0.661 0.713 ETT h2 Handcrafted 0.239 0.246 0.420 0.447 RFF 0.228 0.235 0.374 0.378 Stacking 0.233 0.232 0.393 0.378 ETT m1 Handcrafted 0.389 0.421 0.568 0.621 RFF 0.338 0.362 0.495 0.563 Stacking 0.335 0.406 0.497 0.600 ETT m2 Handcrafted 0.153 0.159 0.274 0.295 RFF 0.152 0.165 0.265 0.333 Stacking 0.151 0.158 0.269 0.294 Weat her Handcrafted 0.186 0.242 0.353 0.462 RFF 0.198 0.269 0.363 0.478 Stacking 0.185 0.239 0.345 0.459 Ex- chang Handcrafted 0.114 0.113 1.699 1.255 RFF 0.130 0.119 1.859 1.154 Stacking 0.110 0.114 4.479 1.005 Table 7: Point-forecasting results of multi-input-length ag- gregation at prediction horizons 96 and 720. Appendix-C5: Uncertainty Component Ablation. We evaluate three nested variants at prediction length 96. Base- WQ directly forms intervals and predictive masses from the retrieval weights. Lookback-PIT uses the observed query lookback to produce query-specific probability reweighting and candidate bounds. Full additionally applies the valida- tion-selected expansion and shrinkage rates to temper the PIT-induced bound displacement. The two calibration components play complementary roles. Lookback-PIT generally produces sharper candidate intervals, reducing the average 90% width from 1.485 to 1.394, although this sharpening can also reduce coverage. Rate tempering determines how much of this contraction should be retained and how strongly outward adjustments should be amplified. Among the three configurations for which Base-WQ already attains Cov90 above 0.89, Full maintains approximately nominal coverage between 0.887 and 0.896 while reducing IS90 in all three cases, by 0.033 on average. Among the remaining nine initially under-cov- ered configurations, Full increases Cov90 in every case, raising their average coverage from 0.866 to 0.884. Thus, the combined method preserves useful sharpening when the base distribution is already sufficiently calibrated, while preferentially widening intervals when the retrieved distri- bution is under-dispersed. PIT reweighting has only a marginal effect on CRPS: its average value changes from 0.2406 for Base-WQ to 0.2410, with individual differences no larger than 0.0019. This indi- cates that KReFβs CRPS advantage primarily originates from the locally retrieved empirical distribution. Lookback- PIT and rate tempering instead act mainly as query-specific interval adaptation mechanisms. Their heterogeneous ef- fects are consistent with lookback-to-future rank transfer be- ing an approximate, dataset-dependent relationship rather than a universal assumption. Dataset Emb Metric Base +PIT Full ETTh1 HC Cov90 0.894 0.876 0.896 IS90 3.047 2.989 3.012 RFF Cov90 0.899 0.839 0.891 IS90 2.529 2.546 2.507 ETTh2 HC Cov90 0.885 0.874 0.898 IS90 2.075 2.078 2.075 RFF Cov90 0.888 0.861 0.904 IS90 2.021 1.996 2.028 ETTm1 HC Cov90 0.905 0.863 0.887 IS90 2.665 2.561 2.623 RFF Cov90 0.838 0.817 0.871 IS90 2.461 2.437 2.493 ETTm2 HC Cov90 0.878 0.856 0.893 IS90 1.723 1.742 1.741 RFF Cov90 0.864 0.848 0.887 IS90 1.754 1.779 1.815 Exch. HC Cov90 0.876 0.878 0.878 IS90 1.584 1.597 1.597 RFF Cov90 0.856 0.851 0.865 IS90 1.644 1.635 1.711 Weather HC Cov90 0.872 0.871 0.889 IS90 1.708 1.769 2.177 RFF Cov90 0.837 0.845 0.871 IS90 1.777 1.772 2.100 Table 8: Uncertainty-component ablation at H=96. Base denotes Base-WQ, +PIT adds Lookback-PIT, and Full ad- ditionally applies the interval adjustment. Effect of Test-Time Archive Updates. The main experi- ments use the online-pessimistic protocol to align the point and uncertainty pipelines. To quantify the contribution of archive adaptation, we additionally evaluate a static offline variant whose retrieval database contains only training and validation instances and remains fixed throughout testing. As shown in Table 9, online updates have little effect on ETTh2 and ETTm2, where KReF obtains its strongest point- forecasting results. The larger improvement on ETTm1 does not change the method ranking. These results indicate that KReFβs two favorable comparisons are primarily attributa- ble to retrieval from pre-test history rather than access to newly observed test futures. Da- taset HC Online HC Offline ΞM SE RFF Online RFF Offline ΞMS E ETT h1 0.759/ 0.627 0.778/ 0.637 +2.5 % 0.550/ 0.534 0.570/ 0.551 +3.6 3% ETT h2 0.330/ 0.393 0.334/ 0.395 +1.2 1% 0.299/ 0.370 0.302/ 0.373 +1.0 % ETT m1 0.520/ 0.492 0.549/ 0.510 +5.5 7% 0.465/ 0.461 0.514/ 0.491 +10. 54% ETT m2 0.218/ 0.312 0.219/ 0.313 +0.4 6% 0.247/ 0.325 0.249/ 0.326 +0.8 1% Wea ther 0.344/ 0.329 0.348/ 0.333 +1.1 6% 0.361/ 0.339 0.361/ 0.341 β0.0 % Exch 0.493/ 0.459 0.494/ 0.461 +0.0 2% 0.472/ 0.455 0.472/ 0.455 β0.0 % Table 9: Online uses the pessimistic update protocol; Of- fline freezes the retrieval archive after the validation split. Each cell reports MSE/MAE averaged over four horizons. Appendix-D: What Makes KReF Work? For each bench- mark we measure STL seasonal strength (Cleveland et al. 1990), noise-to-signal ratio (Hyndman and Athanasopoulos 2018), the trainβtest Wasserstein-1 shift (PeyrΓ© and Cuturi 2019) and a similarity-rank ratio that splits each query's K neighbors into similarity quintiles and reports MSE(least- similar)/MSE(most-similar); a value below 1 means the most-similar neighbors are no betterβslightly worseβthan distant ones. (Exact definitions are provided in the code sup- plement.) The point-forecasting results indicate that the applicabil- ity of direct retrieval depends jointly on predictability and temporal stability rather than on any single diagnostic. KReF performs particularly well on ETTh2 and ETTm2, which exhibit low residual noise-to-signal ratios and mod- erate distribution shift. Exchange is even cleaner, but com- bines the weakest seasonal structure with the largest shift, making historical futures less transferable despite favorable local similarity. These observations help explain why re- trieval can be highly effective on selected datasets without yielding uniform improvements over trained forecasters. The distributional results are substantially broader than the point-forecasting gains. KReF achieves the lowest CRPS across all evaluated datasets, embeddings, and horizons, in- cluding settings in which its retrieved mean is not the strong- est point forecast. The component ablation shows that this advantage is already present in the base locally retrieved em- pirical distribution, while Lookback-PIT changes CRPS only marginally. Interval estimation exhibits a different pat- tern. Across the six benchmarks, KReFβs IS90 advantage does not vary monotonically with seasonality, noise-to-signal ra- tio, or similarity rank. It obtains the lowest average IS90 for both embeddings on ETTh1, ETTh2, ETTm1, and Exchange, and for the handcrafted embedding on ETTm2. These da- tasets span seasonal-strength values from 0.118 to 0.519, noise-to-signal ratios from 0.045 to 0.378, and similarity ranks from 0.814 to 1.058. This diversity suggests that in- terval quality is not determined solely by how accurately re- trieved futures estimate the conditional mean. Weather is the clearest exception. It has the smallest measured shift, 0.060, and Online Split Conformal achieves the best IS90 for both embeddings. Under such stable con- ditions, an expanding residual calibration pool transfers ef- fectively to the test period and can produce particularly sharp intervals. Conversely, Exchange has the largest shift, 0.288. Residual-based and delayed adaptive calibration methods either under-cover at long horizons or substantially widen their intervals, whereas KReF retains comparatively strong coverage with sharper intervals. This contrast sug- gests that KReF is especially useful when globally accumu- lated residuals transfer poorly across time. Given the limited number of datasets, however, this should be interpreted as an empirical pattern rather than a causal conclusion. Dataset Sea- sonal Noise/sig- nal Shift (Wβ) Sim-rank HC RFF ETTh2 0.362 0.247 0.114 0.965 0.945 ETTm2 0.376 0.243 0.111 0.916 0.843 ETTh1 0.518 0.378 0.074 1.011 1.058 ETTm1 0.519 0.377 0.072 0.963 1.005 Weather 0.402 0.475 0.060 0.901 0.939 Exchange 0.118 0.045 0.288 0.822 0.814 Table 10: Characteristics of the six benchmark datasets. Appendix-E: Oracle Analysis of Retrieval Headroom. To distinguish the limitations of whole-future matching from the information available in the historical archive, we con- duct a single-neighbor oracle analysis at H=96. At forecast origin i, the admissible archive ν΄ ν contains the training and validation samples together with earlier test futures that have already been fully observed. Let ν ν βν ν»ΓνΆ denote the benchmark-standardized query future, and let B be a block of horizon-channel cells. The oracle selects ν ν,ν΅ β =ννν min νβν΄ ν 1 |ν΅| β(ν ν,β,ν βν ν,β,ν ) 2 (β,ν)βν΅ and directly copies the corresponding block from that sin- gle historical sample: ν Μ ν,β,ν =ν ν ν,ν΅ β ,β,ν , ( β,ν ) βν΅ Trajectory routing uses one block containing all ν»Γ νΆ cells and therefore selects one historical future for the complete prediction. Horizon-wise routing uses H blocks, selecting one sample independently for each horizon while jointly matching all channels. Channel-wise routing uses C blocks, selecting one complete H-step trajectory inde- pendently for each channel. No averaging, weighting, or fit- ting is performed. The assembled predictions are then evaluated using ex- actly the same MSE and MAE as in the main experiments, averaged over all test samples, horizons, and channels. Both matching and evaluation are performed in the globally standardized benchmark space. The reported errors are therefore directly comparable to the point-forecasting re- sults in the main experiments, although the oracle selection itself uses target-future information. Table 11 reveals substantial information beyond whole- trajectory matching. Across the six datasets, horizon-wise routing reduces mean MSE from 0.1954 to 0.0302 and mean MAE from 0.2940 to 0.1096, corresponding to reductions of 84.5% and 62.7%. Channel-wise routing reduces them to 0.0866 and 0.1786. The four ETT datasets exhibit particu- larly consistent horizon-wise gains, with MSE reductions between 93.2% and 94.2%, indicating that their archives contain close analogues for individual future states even when no single historical continuation matches the complete horizon. The preferred routing granularity is dataset dependent. Exchange benefits considerably more from channel-wise routing, which reduces MSE by 82.5%, whereas horizon- wise routing reduces it by only 30.4%. Weather benefits from both decompositions, attaining its lowest MSE under horizon-wise routing and its lowest MAE under channel- wise routing. These contrasting patterns suggest that reusa- ble historical information may be organized along different temporal and variable-specific structures across datasets. Because the true query future determines the selected samples, these oracle results are difficult to realize directly without an effective routing mechanism. However, they are not strict upper bounds on retrieval-based forecasting: the analysis is restricted to one historical sample per block and does not consider multi-neighbor aggregation, adaptive weighting, interpolation, local fitting, or learned retrieval. The results instead provide an optimistic reference for the information exposed by finer-grained routing and motivate blockwise, horizon-conditioned, channel-adaptive, and au- toregressive retrieval beyond the current KReF design. Dataset Metric Trajec- tory Horizon- wise Channel- wise ETTh1 MSE 0.3722 0.0217 0.1998 MAE 0.4268 0.1095 0.3119 ETTh2 MSE 0.2013 0.0137 0.1131 MAE 0.3089 0.0816 0.2224 ETTm1 MSE 0.1962 0.0128 0.0883 MAE 0.3108 0.0840 0.2046 ETTm2 MSE 0.1135 0.0075 0.0537 MAE 0.2316 0.0600 0.1480 Exchange MSE 0.1546 0.1076 0.0271 MAE 0.2910 0.2427 0.1137 Weather MSE 0.1347 0.0180 0.0374 MAE 0.1946 0.0800 0.0708 Table 11: Archive-oracle errors under different routing granularities at H=96. Appendix-F: Boundary Analysis of Lookback-PIT and Directional Tempering. This section gives a boundary- space interpretation of Lookback-PIT and explains why sep- arate expansion and shrinkage rates can improve empirical tail calibration. We use one-dimensional notation for clarity; in the full multivariate setting, a single query-specific mid- CDF is constructed by pooling all valid lookbackβchannel coordinates and is then applied coordinatewise to every fu- ture cell. F.1. Equivalence between Lookback-PIT and empirical quantile selection. For query i, let the retrieved future val- ues at one coordinate be sorted as ν ν,(1) β€...β€ν ν,(νΎ) , with weights ν ν,(ν) . Their cumulative base masses are νΆ ν,ν =βν€ ν,(ν) ν ν=1 ,νΆ ν,0 =0. To match the finite-sample implementation at the end- points, we use the endpoint-completed empirical mid-CDF. Applying the midrank definition for 0 < u < 1 and setting νΊ ν ( 0 ) =0 and νΊ ν ( 1 ) =1. The endpoint completion ensures that the transformed masses form a proper probability dis- tribution. Lookback-PIT transforms the cumulative future masses into νΆ Μ ν,ν =νΊ ν (νΆ ν,ν ); consequently, the adjusted mass assigned to the k-th sorted future equals the increment between two adjacent transformed cumulative masses. For target probability p, the selected support index is ν ν β ( ν ) =νννν:νΊ ν (νΆ ν,ν )β₯ν. Equivalently, define the smallest admissible base proba- bility level as ν’ ν β ( ν ) =ννννΆ ν,ν :νΊ ν (νΆ ν,ν )β₯ν. The resulting Lookback-PIT boundary therefore satisfies ν ν νΏν΅ ( ν ) =ν ν ν΅ (ν’ ν β ( ν ) ). In the continuous or interpolated representation, the same identity can be written using the left generalized inverse, ν ν νΏν΅ ( ν ) =ν ν B (νΊ ν β ( ν ) ),νΊ ν β ( ν ) =infν’:νΊ ν ( ν’ ) β₯ν. Thus, Lookback-PIT is equivalent to directly selecting the smallest nominal probability level whose empirical look- back mass reaches p. For an equal-tailed interval with nom- inal coverage alpha, the rule is applied at ν β = 1βνΌ 2 ,ν + =1β 1βνΌ 2 . Both tails therefore use the smallest admissible nominal level under the standard left generalized inverse. This is not an additional heuristic, but follows from the conventional empirical-quantile definition. The equivalence is exact at the transformed cumulative-mass level. The equivalence is exact at the transformed cumulative- mass level. Numerically, we linearly interpolate the two re- trieved futures bracketing the target probability rather than always returning the upper support value. This interpretation also explains the geometry of the adjustment: central look- back ranks tend to contract the interval, two-tailed ranks tend to expand it, and one-sided ranks can translate it. F.2. Boundary-space analysis of expansion and shrink- age. Lookback ranks need not transfer perfectly to future ranks. We therefore temper outward and inward PIT move- ments separately. Let r index a validation prediction coor- dinate, ν ν be an orientation sign, and ν ββ,+ denote the lower and upper tails, with ν ν =β1 for the lower tail and ν ν =1 for the upper tail. Let ν΅ ν,ν B ,ν΅ ν,ν PIT ,ννν ν΅ ν,ν β denote the base, Lookback-PIT, and oracle target boundaries. The ora- cle boundary is used only for analysis and attains the desired conditional tail probability. Define outward-oriented PIT and oracle displacements by ν₯ ν,ν =ν ν (ν΅ ν,ν PIT βν΅ ν,ν B ), ν¦ ν,ν =ν ν (ν΅ ν,ν β βν΅ ν,ν B ). Under this convention, a positive displacement is outward and a negative displacement is inward for either tail. For nonnegative expansion and shrinkage rates b and c, define ν ν,ν ( ν₯ ) =νν₯ ( ν₯β₯0 ) ,ν ν,ν ( ν₯ ) =νν₯ ( ν₯<0 ) . The tempered boundary is ν΅ ν,ν ν,ν =ν΅ ν,ν ν΅ +ν ν ν ν,ν (ν₯ ν,ν ), and its absolute oracle boundary residual is exactly |ν΅ ν,ν ν,ν βν΅ ν,ν β |=|ν ν,ν (ν₯ ν,ν )βν¦ ν,ν |. Pooling both tails and all validation coordinates gives ν 2 (ν,ν)=β|ν ν,ν (ν₯ ν,ν )βν¦ ν,ν | ν ββ,+ν A single-rate transformation instead has ν 1 (ν)=β|νν₯ ν,ν βν¦ ν,ν | ν ββ,+ν Because the two-rate family contains every single-rate transformation as b=c=a, ννν ν,νβ₯0 ν 2 ( ν,ν ) β€ννν νβ₯0 ν 1 ( ν ) β€ν 1 ( 1 ) , where a=1 is untempered Lookback-PIT. Separate rates can therefore attain a boundary residual no larger than the best shared rate or pure PIT. To characterize strict improve- ment, split the residual into outward and inward components, ν νν’ν‘ ( ν ) =β|νν₯ ν,ν βν¦ ν,ν | ν₯ ν,ν β₯0 ν νν ( ν ) =β|νν₯ ν,ν βν¦ ν,ν | ν₯ ν,ν <0 By construction, the two-rate objective separates into an outward component and an inward component: ν 2 ( ν,ν ) = ν νν’ν‘ ( ν ) +ν νν ( ν ) . In contrast, the shared-rate objective re- quires both components to use the same rate: ν 1 ( ν ) = ν νν’ν‘ ( ν ) +ν νν ( ν ) . If the two directional objectives have no common minimizer and donβt equal 1, i.e., νννmin νβ₯0 ν νν’ν‘ ( ν ) β©νννmin νβ₯0 ν νν ( ν ) =β , νννmin νβ₯0 ν 1 ( ν ) β 1 then ννν ν,νβ₯0 ν 2 ( ν,ν ) <ννν νβ₯0 ν 1 ( ν ) <νννν 1 ( 1 ) . Hence, directional tempering can strictly reduce the ora- cle boundary residual whenever outward and inward look- back signals have different transfer reliability. Sharing b and c across the upper and lower tails is a structural regularizer: after orientation, both tails contribute evidence about expan- sion and contraction, which may reduce estimation variance relative to four tail-specific parameters. Connection to quantile and coverage residuals. Boundary and nominal-level residuals are not numerically identical, but they are locally coupled. Suppose the PIT and oracle boundaries can be represented ν΅ ν,ν B =ν ν B ( ν ν ) ,ν΅ ν,ν PIT = ν ν B (ν’ ν,ν ν ),ν΅ ν,ν β =ν ν B (ν’ ν,ν ν ), where ν’ ν,ν ν is the lookback-im- plied probability level and ν’ ν,ν ν is its oracle future counter- part. The oracle level ν’ ν,ν ν is introduced only for analysis and is never accessed by the forecasting algorithm. If ν ν B is lo- cally differentiable, the mean-value theorem gives ν₯ ν,ν =ν ν (ν΅ ν,ν PIT βν΅ ν,ν B )=ν ν ν ν Bβ² (ΞΎ ν,ν ν )(ν’ ν,ν ν βν ν ), y ν,ν =ν ν (ν΅ ν,ν * βν΅ ν,ν B )=ν ν ν ν Bβ² (ΞΎ ν,ν ν )(ν’ ν,ν ν βν ν ), for intermediate levels ΞΎ ν,ν ν and ΞΎ ν,ν ν . Boundary displacement is therefore a locally inverse-density-scaled quantile-level displacement: the two have the same direction, while value- space analysis also accounts for the local scale of the pre- dictive distribution. Let νΉ ν ν be the conditional future CDF with νΉ ν ν (ν΅ ν,ν β )=ν ν . Suppose its density is locally bounded by ν ν,ν β€ν ν ν (ν§)β€ ν ν,ν betweenν΅ ν,ν ν,ν and ν΅ ν,ν β .Since|ν΅ ν,ν ν,ν βν΅ ν,ν β |= |ν ν,ν (ν₯ ν,ν )βν¦ ν,ν | , the mean-value theorem gives 0< ν ν,ν |ν ν,ν (ν₯ ν,ν )βν¦ ν,ν |β€|νΉ ν ν (ν΅ ν,ν ν,ν )βν ν | β€ν ν,ν |ν ν,ν (ν₯ ν,ν )βν¦ ν,ν |<+β. Thus, under local regularity, boundary residual and tail- probability residual are equivalent up to density-dependent constants. In particular, reducing the boundary residual con- trols the corresponding tail-probability error; summing the two tail errors yields a bound-on interval-coverage error. This links the boundary-space result to the empirical cali- bration objective used in our method. Because oracle boundaries and conditional densities are unavailable in practice, we select b and c directly on valida- tion data by minimizing ν½ ν£νν ( ν,ν ) = | eΜ val β (ν,ν)βνΎ | + | eΜ val + (ν,ν)βνΎ | . where νΎ= 1βνΌ 2 and the two empirical terms are the lower- and upper-tail miss rates under (b, c). Mean interval width breaks ties. The boundary-residual analysis is therefore an oracle explanation for why directional rates can help, whereas the implementation directly optimizes their observ- able coverage consequence. The unrestricted analysis requires only nonnegative rates. In the reported implementation, νβ1,1.25,1.5,2,3 and νβ0,0.25,0.5,0.75,1 are reliability-oriented regularizers rather than theoretical requirements: outward PIT correc- tions are not attenuated, inward corrections are not ampli- fied, val-test drift is relieved by grid search hyperparameters and the validation search space is reduced. The optional ex- pansion cap is likewise a robustness safeguard against iso- lated extreme boundary movements and is not required by the inequalities above.