Local and Global Recursive Ridge Regression for Online Residual Correction of Frozen Multivariate Forecasters under Label Delay

Takumi Fujimoto; Hiroaki Nishi

cs.LGOct 21, 2026

Local and Global Recursive Ridge Regression for Online Residual Correction of Frozen Multivariate Forecasters under Label Delay

Authors: Takumi Fujimoto, Hiroaki Nishi

Organizations: School of Science for Open and Environmental Systems, Graduate School of Science and Technology, Keio University, Japan · Department of System Design, Faculty of Science and Technology, Keio University, Japan

Abstract

A frozen multi-horizon forecaster can be corrected online from its own residuals. However, under label delay the target of a block arrives only after the whole horizon has passed, and the corrector's retained state grows with the number of channels. Aiming to combine a small retained state with a large error reduction, we propose DRiC (DCT Ridge Corrector), which fits a ridge regression of the leading discrete cosine transform (DCT) coefficients of the next residual block on those of the frozen forecast and of the latest input block, with sufficient statistics updated recursively. Its local variant, DRiC-L, keeps one such regression per channel; its global variant, DRiC-G, pools all channels into one regression, whose statistics do not depend on the channel count. A guard tracks a running scale of the frozen forecast's residual for each channel; it skips updates on blocks far outside that scale and clips the issued correction. We evaluate on seven datasets (four ETT series, Weather with its missing-value codes filled, Electricity, and Traffic) with DLinear, PatchTST, and iTransformer backbones, against 10 naive, gradient-trained, and closed-form methods. DRiC-L with eight DCT coefficients (DRiC-L8) reduces the mean squared error (MSE) by 14.55% on average over 126 conditions, against 12.98% for ELF, the most accurate comparator, while retaining 15-17 times less state, although ELF is more accurate on Traffic. DRiC-G retains 14.6 kB on the 321-channel Electricity data and 18.9 kB on the 862-channel Traffic data and is the most accurate method at or below its state on both. However, 5 to 7 of the 10 comparators, all with at least 10 times its state, are more accurate there. On the unfilled Weather file, DRiC-L8 without the guard has on average 8.0 times the frozen forecast's MSE, whereas with the guard it reduces the MSE by 7.94%. Code: https://github.com/keiotakmin/dric.

Figures & tables

Figure 1: One block of DRiC-L8 on ETTh1 (PatchTST, refit, seed 0): the block and channel whose MSE reduction is the median over all blocks and channels of this condition. Top: at the origin of block $t$ , the corrector maps the leading DCT coefficients of the newest input block and of the frozen forecast through the ridge map, clips the predicted residual coefficients to the guard scale, and issues the corrected forecast. Bottom: once the target is complete, the residual’s coefficients update the statistics if the guard admits the block. Curves, coefficient bars, the clipping bound, and the guard ratio are those of this block; in this typical block neither the clipping nor the guard is active. The clipping bound and the guard use the same running scale $s$ of the channel. Dashed arrows: the update, which is used from the next issue on. The heatmaps are schematic.

Figure 2: Statistics of the two variants. DRiC-L keeps one $(2K+2)\times(2K+2)$ Gram matrix and right-hand side per channel $c$ , fitted on that channel’s blocks only, with the $K$ leading coefficients of the forecast and of the newest input block and the last input value in $\phi_{t,c}$ . DRiC-G pools the blocks of all $C$ channels into one $q_{\mathrm{g}}\times q_{\mathrm{g}}$ Gram matrix and right-hand side ( $q_{\mathrm{g}}=42$ ), with $J=8$ coefficients of the forecast and of all four input blocks and the last input value in $\phi^{\mathrm{g}}_{t,c}$ . Sizes are in double precision; both variants also keep one guard scale per channel (8 bytes). The heatmaps are schematic.

Figure 3: Issue and update events for the OT column of the standard Weather file around its missing-value codes, with the frozen DLinear forecast (refit, seed 0) and the guard decisions of DRiC-L8 for this channel. Labels name where the codes are; the admitted block with codes in its input window lies just below the threshold. The forecast of a block is issued at its origin; its update happens one horizon later. Arrows mark forecasts that leave the plotted range.

Dataset	Channels	Length	Sampling	Evaluation blocks
ETTh1	7	17,420	1 h	362
ETTh2	7	17,420	1 h	362
ETTm1	7	69,680	15 min	1,451
ETTm2	7	69,680	15 min	1,451
Weather	21	52,696	10 min	1,097
Electricity	321	26,304	1 h	548

Table 1: Evaluated datasets. Evaluation blocks are the complete 24-sample targets scored per condition.

	MSE reduction vs. frozen forecast (%)								Retained state
	All 126			Mean per dataset
Method	Mean	Panels	Median	ETT	Weather*	Elec.	Traffic	Weather	ETT	Traffic
Persistence	1.31	0.84	0.16	1.93	1.21	0.00	0.24	$-$ 0.61	528 B	528 B
Seasonal naive	3.73	2.34	2.13	5.58	0.85	0.24	2.71	$-$ 2.33	528 B	528 B
TAFAS	7.35	7.91	5.52	6.60	7.10	7.95	9.99	2.27	833 kB	103 MB
PETSA	7.64	7.67	6.50	7.60	6.41	7.62	9.04	5.65	195 kB	21.1 MB

Table 2: MSE reduction relative to the frozen forecast (%) and retained state. All 126: conditions (7 datasets

\times

3 backbones

\times

3 seeds

\times

legacy/refit) with the filled Weather file (Weather*); Mean: over the conditions; Panels: over the 12 dataset–backbone panels, with the four ETT series as one dataset; Median: over the conditions. Per-dataset columns: means over 72 ETT or 18 conditions. Weather is the standard file with its missing-value codes and is not part of All 126. State: largest over the three backbones. Bold marks the best value in each accuracy column.

Figure 4: Mean MSE reduction against retained state for each dataset (rows) and backbone (columns). Each point averages 24 ETT conditions or 6 conditions of the other datasets. Stars: the correctors of this paper. The dashed line is the best reduction achievable at or below each state among the methods shown. Weather* is the filled Weather file. The horizontal axis is logarithmic.

Configuration	All (%)	$\Delta$	Weather (%)	State, ETT	State, Traffic
DRiC-L8	14.55		7.94	19.8 kB	2.18 MB
+ global regression	13.73	$-$ 0.82	7.03	29.7 kB	2.19 MB
$-$ guard	12.80	$-$ 1.75	$-$ 700.38	19.8 kB	2.17 MB
$-$ input block and last value	11.19	$-$ 3.36	6.02	8.74 kB	816 kB
all four input blocks	13.53	$-$ 1.02	7.44	71.6 kB	8.55 MB
$K=4$	13.17	$-$ 1.37	8.57	7.50 kB	664 kB

Table 3: Components of the correctors. All: mean MSE reduction over the 126 conditions (%);

\Delta

: mean paired difference from the first row of each group (percentage points); Weather: standard file. Without the input block and the last value, the local regression reads only the forecast’s coefficients.

		Skipped updates		Effect of the guard (points)
Dataset	Channel updates	Mean	Range	DRiC-L8	DRiC-L4	DRiC-G
ETTh1	2,534	0.0	0–0	0.00	0.00	0.00
ETTh2	2,534	0.0	0–0	0.00	0.00	0.00
ETTm1	10,157	0.3	0–2	0.00	0.00	0.00
ETTm2	10,157	0.8	0–3	0.00	0.00	0.00
Weather*	23,037	161.7	36–330	+9.61	+3.08	+4.13

Table 4: The guard by dataset. Channel updates: channels

\times

evaluation blocks per condition. Skipped updates: mean and range over the 18 conditions. Effect: mean paired difference in MSE reduction between each corrector and its version without the guard (percentage points). Weather: standard file.

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Setting	DRiC-L4	DRiC-L8	DRiC-G
default ( $\tau=10$ , $n_{1/2}=128$ , $\lambda=1$ )	13.17 (8.57)	14.55 (7.94)	9.17 (3.29)
$\tau=5$	12.94 (8.58)	14.29 (8.25)	8.95 (3.84)
$\tau=20$	13.35 (8.82)	14.79 (7.40)	9.10 (2.51)
$\tau=40$	13.45 (9.66)	13.97 (3.04)	8.78 (1.29)
$\tau=80$	12.52 (5.87)	–	–
$\tau=160$	11.57 (0.79)	–	–

Appendix

Table 5: One-at-a-time sensitivity: mean MSE reduction over the 126 conditions (%), with the standard Weather file in parentheses.

n_{1/2}

: half-life of the regressions in blocks.

Dataset	DLinear	PatchTST	iTransformer
ETTh1	0.396	0.438	0.404
ETTh2	0.145	0.181	0.157
ETTm1	0.263	0.317	0.288
ETTm2	0.100	0.117	0.096
Weather*	0.289	0.251	0.240
Electricity	0.193	0.152	0.131

Appendix

Table 6: MSE of the frozen forecasts (standardized scale), mean over the three seeds and the legacy and refit bases. Weather: standard file.

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

Takumi Fujimoto, Hiroaki Nishi

School of Science for Open and Environmental Systems, Graduate School of Science and Technology, Keio University, Yokohama 223-8522, Japan · Department of System Design, Faculty of Science and Technology, Keio University, Yokohama 223-8522, Japan

Online Time Series Forecasting Multivariate Time Series Forecasting Residual Learning Time Series Forecasting

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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

δ

-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.

Abstract

Multivariate Time Series Forecasting

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