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EPOC: Endpoint-Preserving Online Correction With Compressed Residual State for Multi-Horizon Time Series Forecasting

Takumi Fujimoto, Hiroaki Nishi

arXiv:2609.30929v2cs.LG

TL;DR

Completed forecast errors can inform later predictions, but storing whole residual blocks costs auxiliary state and compressed coefficients can omit individual errors. EPOC retains low-order DCT coefficients and the preceding block’s endpoint for online correction. Across matched fixed-base conditions, it reduces error while retaining a compact auxiliary state.

  • Problem

    A fixed multi-horizon forecaster must choose which completed-block errors to retain between forecasts, since full residual blocks require length-proportional state and compressed coefficients lose individual values.

    Source: §1 · pp. 1–2

  • Method

    EPOC retains low-order DCT coefficients and the preceding residual block’s endpoint, sharing that endpoint across component-wise online ridge regressions that also use current-forecast coefficients.

    Source: §1 · p. 2

  • Results

    Across 96 matched fixed-base conditions, EPOC reduces MSE by 15.40% and MAE by 9.35% versus the uncorrected base, retaining median auxiliary arrays of 6,352 B.

    Source: §7 · p. 15

  • Takeaways & Limitations

    The comparison identifies an accuracy–storage operating point, with the choice depending on the series and available state budget.

    Source: §6 · p. 14

    Evidence covers eight series, two base forecasters, input length 96, horizon 24, and updates after complete target blocks; peak memory, latency, and energy remain unmeasured.

    Source: §6 · p. 15

Abstract

from arXiv · show

Completed forecasts provide residual feedback, but retaining full residual blocks increases auxiliary state. We propose Endpoint-Preserving Online Correction (EPOC) with a compressed residual state. It stores low-order discrete cosine transform (DCT) coefficients and the final value of the preceding residual block. Each channel shares the endpoint across component-wise online ridge regressions using current-forecast coefficients. We evaluate eight multivariate series, DLinear and PatchTST, three seeds, and two training variants: 96 matched fixed-base conditions at a 24-step horizon. EPOC achieves mean condition-wise reductions in mean squared error (MSE) and mean absolute error (MAE) of 15.40% and 9.35% from the uncorrected base, respectively, with a median of 6,352 B in retained auxiliary arrays. EPOC also outperforms the $\delta$-Adapter, COSA, FAC, and OMPB in paired MSE on most conditions while retaining less state. Full ELF achieves the largest mean MSE reduction, 19.29%, but its median retained state is 474,048 B ($\times$75 relative to EPOC). Equal-size summary controls favor the endpoint by 1.65--2.20% in paired MSE; a coefficient-reconstructed endpoint yields similar accuracy to the observed endpoint, highlighting its role as a shared input. Increasing the retained DCT component count from 4 to 8 adds 1.00 percentage point of MSE reduction for 5,728 B. On jointly trained bases, EPOC lowers MSE by 16.69--20.15% relative to globally blended TEFL-style adapters applied to the same base. Code and numerical records are available at [https://github.com/keiotakmin/endpoint-preserving-residual-correction](https://github.com/keiotakmin/endpoint-preserving-residual-correction).

1 Introduction

Completed residuals can guide a fixed forecaster’s next prediction, but retaining full blocks costs state while compressed summaries omit individual errors. EPOC pairs low-order residual DCT coefficients with the preceding block’s endpoint and uses them for online correction.

1 Introduction

  • A complete residual block preserves horizon-specific errors but requires state proportional to its length, while a few transform coefficients reduce state but lose individual values.Blocks can have identical retained DCT coefficients yet different final errors; the endpoint is the latest observed error for the next nonoverlapping forecast.Source: §1 · pp. 1–2
  • EPOC retains low-order DCT coefficients and the uncompressed final residual value, sharing that endpoint across component-wise online ridge regressions that also use current-forecast coefficients.The correction is reconstructed in a fixed DCT subspace, and updates occur after the full target block arrives.Source: §1 · p. 2
  • The evaluation covers eight multivariate series, DLinear and PatchTST, three seeds, and two base-training variants across 96 matched fixed-base conditions.A separate experiment compares EPOC with TEFL-style residual adapters on the same jointly trained bases.Source: §1 · p. 2
  • Across 96 matched fixed-base conditions, EPOC reduces MSE by 15.40% and MAE by 9.35% versus the uncorrected base, retaining median auxiliary arrays of 6,352 B.Full ELF achieves a larger mean MSE reduction of 19.29% but retains 474,048 median bytes; EPOC retains about one seventy-fifth as much state.Source: §1 · p. 2
  • Equal-size residual-summary controls, a coefficient-reconstructed endpoint, and input ablations test the endpoint’s contribution under matched base forecasts and correction dimensions.The experiments also examine how retained component count affects the accuracy–storage choice.Source: §1 · p. 2

2 Related Work

Related work spans residual feedback, calibration of fixed forecasters, and sequential adaptation. EPOC focuses on which information from a completed residual block to retain alongside compressed coefficients.

2.1 Residual feedback for multi-horizon forecasts

  • ResCAL estimates future traffic-forecasting errors from observed residuals and graph signals [1], while TEFL uses the most recent fully observed residual vector in a jointly trained adapter [2].EPOC examines explicitly retaining the final value when that residual block is compressed.Source: §2.1 · pp. 2–3

2.2 Calibration of fixed forecasters

  • Fixed-forecaster methods include online forecasts from Fourier features [3], target-statistic correction, spectral calibration [5], and input/output adapters [8].COSA, FAC [5], and OMPB [9] use distinct forms of gated forecast correction or calibration; EPOC instead uses prior residual coefficients and endpoint with current-forecast coefficients.Source: §2.2 · p. 3

2.3 Sequential adaptation and retained information

  • Sequential-adaptation methods may update or combine forecasting models, whereas EPOC keeps the backbone fixed and uses completed residuals as auxiliary-correction features.The design draws on DCT compression, ridge regression, and sequential least-squares forgetting; the retained input combines endpoint, residual coefficients, and current-forecast coefficients.Source: §2.3 · p. 3

3 EPOC: Endpoint-Preserving Online Correction

EPOC issues corrections from information available at forecast time, then updates its regression and blending states only after the full target block is observed. It combines a compressed residual summary with the preceding residual’s final value, and retains a compact set of auxiliary arrays.

3.1 Forecasting and feedback order

  • The fixed base issues each block’s forecast before its target arrives; only after the complete target is observed does EPOC update its regression, blending state, and next-block summary.The experiments use L = 96 and H = 24, with forecast origins advancing by H samples.Source: §3.1 · p. 4, Figure 1 · p. 5

3.2 Residual representation and online correction

  • EPOC retains low-order DCT coefficients and the uncompressed final value of each completed base-residual block, sharing that endpoint across a channel’s component regressions.The correction lies in the span of the first K fixed DCT basis vectors; K sets its output dimension.Source: §3.2 · pp. 4–6
  • The separately retained endpoint can distinguish residual blocks with identical retained DCT coefficients, because changing the endpoint need not change those coefficients.The forecasting effect of using an observed rather than coefficient-reconstructed endpoint is evaluated in Fig. 3.Source: §3.2 · p. 5
  • The online ridge regressions use exponentially weighted observations, with λ = 1 and ρ = 2^-1/128, corresponding to a 128-completed-block weight half-life.Their sufficient statistics are updated after feedback without retaining previous blocks.Source: §3.2 · pp. 5–6

3.3 Retained-state accounting

  • At H = 24, K = 4, and W = 32, the median retained auxiliary-array state across 96 conditions is 6,352 B.The count excludes the base forecaster, buffers, temporary features, solver workspace, runtime overhead, peak RAM, and execution time.Source: §3.3 · p. 7

4 Experimental Setup

The evaluation uses eight multivariate series with fixed DLinear and PatchTST bases, comparing correction methods under matched forecasts and targets. A separate jointly trained-base experiment compares EPOC with TEFL-style residual adapters, and the paper reports normalized-scale errors using specified aggregation procedures.

4.1 Data and common evaluation protocol

  • The eight series comprise four ETT datasets, Appliances Energy Prediction, and three BDG2 sites; forecasts and losses include every retained channel.The evaluation uses L = 96 samples, H = 24 samples, and origins spaced 24 samples apart; these sample counts correspond to different durations across datasets.Source: §4.1 · p. 7
  • The fixed-base evaluation comprises eight series, two architectures, three seeds, and two training variants, yielding 96 conditions with shared forecasts, targets, and origins within each condition.Each method forecasts before the target arrives and updates only after the complete target block is observed.Source: §4.1 · pp. 7–8

4.2 Comparators and learned comparison design

  • The main fixed-base comparison includes Static and six correction arms, while all correction arms use the same bounded global blending rule over 32 completed blocks.The base remains fixed, including when FAC calibrates its input.Source: §4.2 · p. 8
  • The δ-Adapter, COSA, FAC, and OMPB use distinct bounded-adapter, target-context, frequency-calibration, and Bayesian residual-head designs, while full ELF uses cumulative Fourier-coordinate ridge regression.The learned TEFL-style adapter comparison is reported separately from the 96 fixed-base conditions.Source: §4.2 · pp. 8–9
  • A separate TEFL-style adapter experiment covers six series, both architectures, and three seeds, comparing corrections on the same jointly trained base under a shared blending rule.It also includes a separately trained base-only run to test the benefit of the joint adapter procedure.Source: §4.2 · pp. 8–9

4.3 Metrics and aggregation

  • For each metric and condition, reduction from Static and paired reduction from a comparator use different denominators; MSE and MAE are calculated separately.Overview values average 96 condition-wise reductions and use the median retained numerical-array bytes; aggregation order differs for the jointly trained experiment.Source: §4.3 · p. 9

5 Results

Across fixed-base conditions, EPOC improves on Static and most correction comparators while retaining less state than the full-residual ELF. Equal-size controls favor the endpoint, and increasing the retained DCT component count improves the accuracy–storage trade-off.

5.1 Accuracy and retained state

  • 15.40% mean MSE and 9.35% mean MAE reductions from Static; EPOC retains 6,352 B median, while full ELF reaches 19.29% MSE reduction at 474,048 B.EPOC has lower paired MSE than the δ-Adapter, COSA, FAC, and OMPB in 84, 77, 58, and 91 of 96 conditions, respectively; it retains about one seventy-fifth of ELF’s state.Source: §5.1 · pp. 9–10, Table 3 · p. 10
  • EPOC has the smallest retained state among the six plotted corrections in all three channel groups, while group means can hide series-specific rankings.On Fox, the δ-Adapter, COSA, and FAC outperform EPOC with both backbones; ELF has the lowest MSE on Fox and Panther with substantially more state.Source: §5.1 · p. 11, Figure 2 · p. 11
  • On jointly trained bases, EPOC lowers MSE by 16.69–20.15% and MAE by 8.82–11.29% relative to globally blended adapters on the same base.The comparison is against adapters that also improve over their paired base-only runs.Source: §5.1 · p. 11, Table 4 · p. 12

5.2 Endpoint selection and input construction

  • At equal regression dimension and retained size, the endpoint yields 2.20%, 1.65%, and 1.96% lower paired MSE than the first residual, middle residual, and block mean.Across joint-base groups, endpoint MSE gains range from 4.70% to 6.00% with MAE warmup and 5.43% to 6.94% with MAE+SF warmup.Source: Table 5 · p. 12, §5.2 · p. 12
  • 15.40% mean MSE reduction comes from the complete input, versus 13.37% without the endpoint and 11.34% with endpoint-only regression; endpoint-only is beaten in all 96 paired conditions.The projected endpoint yields 14.98% mean MSE reduction, close to 15.40% with the observed endpoint.Source: §5.2 · p. 12, Figure 3 · p. 13

5.3 Accuracy–storage and setting sensitivity

  • Increasing K from 4 to 8 raises mean MSE reduction from 15.40% to 16.40% for 5,728 additional retained bytes.With K = 4 fixed, changes to ridge strength, forgetting half-life, or blending window produce mean MSE reductions between 14.47% and 15.43%.Source: §5.3 · p. 13, Figure 4 · p. 14

6 Discussion

EPOC offers a compact accuracy–storage trade-off, with the endpoint and retained-component count shaping the correction. The reported evaluation is limited to specified series, forecasters, horizon, and complete-block updates.

6 Discussion

  • EPOC reduces mean MSE by 15.40% relative to Static with 6,352 median retained bytes; full ELF reaches 19.29% with about 75 times as many bytes.EPOC uses less state than the δ-Adapter, COSA, FAC, and OMPB and has lower paired MSE than each in a majority of conditions.Source: §6 · p. 14
  • The endpoint gives 1.65–2.20% lower paired MSE than equal-size first-, middle-, or mean-value summaries, while a projected endpoint reaches 14.98% mean MSE reduction versus 15.40% observed.Without an endpoint input, mean MSE reduction falls to 13.37%; the complete input lowers paired MSE by 4.55% versus endpoint-only regression.Source: §6 · p. 14
  • The complete correction combines residual coefficients, a shared endpoint feature, and current-forecast coefficients in its component regressions.The endpoint is shared across components; each regression also receives the corresponding current-forecast coefficient.Source: §6 · pp. 14–15
  • Increasing K from 1 to 4 raises mean MSE reduction from 11.13% to 15.40%, and increasing K from 4 to 8 adds 1.00 percentage point for 5,728 B.K = 4 is a compact reference setting, not the most accurate tested setting; the preferred K depends on storage available per channel.Source: §6 · p. 15
  • The evaluation covers eight series, two base forecasters, input length 96, horizon 24, and updates after complete target blocks.Retained-byte counts cover persistent auxiliary numerical arrays; peak memory, latency, and energy on target devices remain unmeasured.Source: §6 · p. 15

7 Conclusion

The conclusion presents EPOC as a compressed residual correction for fixed multi-horizon forecasters. Across the reported fixed-base conditions, it combines lower errors with a compact retained state, while the choice of storage and accuracy depends on the comparison and setting.

7 Conclusion

  • Across 96 matched fixed-base conditions, EPOC reduces mean condition-wise MSE and MAE from Static by 15.40% and 9.35%, respectively, with 6,352 B median retained state.It uses less auxiliary state than the δ-Adapter, COSA, FAC, and OMPB and yields lower paired MSE than each in most conditions.Source: §7 · p. 15
  • Full ELF achieves a 19.29% mean MSE reduction from Static with about 75 times EPOC’s state, while EPOC lowers MSE by 16.69–20.15% versus globally blended TEFL-style adapters on jointly trained bases.Equal-size residual-summary controls favor the endpoint by 1.65–2.20% in paired MSE.Source: §7 · p. 15
  • The projected-endpoint and input-ablation controls indicate that sharing an endpoint-like feature across component regressions accounts for much of the measured gain.Increasing K from 4 to 8 adds 1.00 percentage point of mean MSE reduction for 5,728 B of additional retained state.Source: §7 · p. 15

A Supplementary Numerical Records

The supplementary materials provide numerical records and instructions for reconstructing the main-table aggregates. They also identify the learned-adapter records and the common evaluation settings.

A Supplementary Numerical Records

  • Table S1 collects common evaluation and joint-training settings, and code and numerical records are available in the project repository.For Table 4 same-base percentages, average seed losses within groups, calculate 100(1 − mboundary/madapter_global), then average across 12 series–backbone pairs.Source: Table S1 · p. 17, A Supplementary Numerical Records · p. 17
  • tableS01_all_conditions.csv contains 96 fixed-base-condition rows; its mse and mae fields describe EPOC, while static and elf prefixes identify the uncorrected base and globally blended full ELF.The independent_mse field is the coefficient-only correction, percentage suffixes name reference methods, and trace_file points to archived block losses.Source: A Supplementary Numerical Records · p. 17
  • The seven-arm table03_all_conditions.csv contains 672 method-by-condition rows behind Table 3, and its archived Proposed label denotes EPOC.The Table 3 overview is reconstructed by averaging reduction percentages across 96 rows per method and taking median state_bytes.Source: A Supplementary Numerical Records · p. 17
  • The learned-adapter records contain 864 rows across six series, two backbones, two warmups, three seeds, three widths, and four forecast variants; a companion file contains 288 width-64 rows.The jointly trained-base files distinguish uncorrected, raw-adapter, globally blended adapter, and globally blended EPOC arms; bdg2 denotes BDG2 (Rat).Source: A Supplementary Numerical Records · p. 17

B Setting Sensitivity

The sensitivity analysis varies one setting at a time on 96 fixed-base conditions, with other settings held to the reference configuration. Increasing retained components improves mean MSE reduction while increasing retained state.

B Setting Sensitivity

  • Raising K from 1 to 8 increases mean MSE reduction from 11.13% to 16.40% and median retained state from 2,056 to 12,080 B.Table S2 lists 13 one-factor settings; percentages average condition-wise reductions from Static with global blending, and bytes are median retained arrays.Source: B Setting Sensitivity · p. 18, Table S2 · p. 18
  • With K = 4, tested settings yield mean MSE reductions of 14.47–15.43%; the shortest tested forgetting half-life, 32 blocks, gives 15.43%, while no forgetting gives 14.47%.Changing ridge strength, forgetting half-life, or blending window is evaluated one factor at a time.Source: B Setting Sensitivity · p. 18

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