On the Encoding of Calendar Features in Covariate-Informed Time Series Foundation Models

Alexander Kreusel; Maximilian Beichter; Alexandra Nikoltchovska; Benjamin Schäfer; Ralf Mikut; Frederik Zahn; Veit Hagenmeyer

cs.LGOct 9, 2026

On the Encoding of Calendar Features in Covariate-Informed Time Series Foundation Models

Authors: Alexander Kreusel, Maximilian Beichter, Alexandra Nikoltchovska, Benjamin Schäfer, Ralf Mikut, Frederik Zahn, Veit Hagenmeyer

Organizations: Institute for Automation and Applied Informatics Karlsruhe Institute of Technology Karlsruhe, Germany

Abstract

Time series foundation models (TSFMs) increasingly offer covariate support, with calendar information being the most common type of categorical covariate in practice, as it is available for every point in time. However, how these features are encoded is largely treated as an implementation detail rather than as an object of study, despite encoding being one of the few design choices left for pretrained models at inference time. With the present study, we demonstrate that the way information is propagated into the model substantially affects forecast accuracy of TSFMs. Therefore, we systematically compare seven calendar encodings, including three conventional target-agnostic representations (ordinal, one-hot, and cyclic) and four target encodings, across four state-of-the-art TSFMs on nine datasets from multiple domains in a zero-shot rolling evaluation. The nine datasets are influenced by social activities, such as weekends and holidays, making calendar information particularly relevant for forecasting. We find that encoding choice affects forecast accuracy across all models, with MASE differences of up to 9.9 % on non-holidays and 83 % on holidays within a single model. Effectiveness and utilization vary substantially across models. Improvements are concentrated around holidays, where calendar information provides signals unavailable from target history, while some model encoding combinations also improve non-holiday forecasts. A SHAP-based attribution analysis locates the contribution of the covariates relative to the autoregressive component and explains why some model-encoding combinations degrade performance. Our findings suggest that current benchmarks for covariate-informed TSFMs can distort model comparisons by imposing fixed encoding strategies.

Figures & tables

Encoding	Holiday	Representation	Dim.	Assumption
ordinal h,W + bin H	binary flag	ordinal (0–23; 0–6) $+$ flag	3	–
one-hot h,W + bin H	binary flag	one-hot $(24{+}7)$ $+$ flag	32	–
cyclic h,W + bin H	binary flag	sin/cos $+$ flag	5	Hour and week follow fixed periodicity.
$T_{h,W}(H=\mathrm{Sun})$	$\rightarrow$ Sunday	joint target encoding	1	Public holiday $\approx$ Sunday.
$T_{h,W}(H=7)$	8th weekday	joint target encoding	1	Holidays are distinct from Sundays.
$T_{h,W,H}$	binary flag	joint target encoding	1	Captures weekday-specific holiday effects.

Table 1: Calendar encodings. All variants use hour (h), weekday (W), and holiday (H) information, differing in their representation and combination.

		All days		Holidays only		Non-holidays only
Model	Encoding	MASE	SQL	MASE	SQL	MASE	SQL
Baselines	seasonal-naive (168h)	1.0362	0.9984	2.9358	2.6207	0.9682	0.9404
Baselines	$T_{h,W}(H=Sun)_{naive}$	1.1050	0.9468	1.5369	1.3182	1.0882	0.9322
Chronos-2	w/o covariates	0.6903	0.5534	2.2299	1.9242	0.6353	0.5044
	ordinal h,W + bin H	0.6760	0.5381	1.5630	1.2854	0.6440	0.5111
	one-hot h,W + bin H	0.7099	0.5690	2.0244	1.7294	0.6626	0.5272

Table 2: Overall results across all datasets, split by day type (mean over 9 datasets).

Figure 1: Pairwise win rate in MASE, as the share of the nine datasets won by the row encoding.

Figure 2: MASE change relative to the TSFM w/o covariates, averaged over nine datasets. Vertical lines indicate the mean, and horizontal lines the interquartile range (25th–75th percentile). Most models improve on holidays independently of the encoding, while results on non-holidays are mixed.

Figure 3: Share of the mean absolute SHAP value for the

T_{h,W}(H=7)

and

T_{h,W}+T_{h,H}

encodings, averaged across datasets. Covariates contribute 20–34 % on holidays but at most 14 % on non-holidays. With

T_{h,W}+T_{h,H}

, most holiday contribution moves to

T_{h,H}

, except for TiRex-2.

Figure 4: Hourly forecasts and SHAP values for a holiday and a non-holiday from m_dense with

T_{h,W}+T_{h,H}

. Upper panels: median forecast, target, and SHAP contributions relative to the base prediction

f(\emptyset)

. Lower panels:

T_{h,W}

and

T_{h,H}

contributions on a separate scale. Units: vehicles per hour. On the holiday,

T_{h,H}

lowers the forecast for all models except TiRex-2, which overpredicts.

Appendix figures & tables37 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 5: The ordinal h,W + bin H encoding with its three resulting covariates.

Figure 6: The one-hot h,W + bin H encoding with its 32 resulting covariates.

Figure 7: The cyclic h,W + bin H encoding with its five resulting covariates.

Figure 8: The

T_{h,W}(H=\mathrm{Sun})

target encoding using the categories hour and weekday, with holidays mapped to Sunday.

Figure 9: The

T_{h,W}(H=7)

target encoding using the categories hour and weekday, with holidays treated as a separate weekday.

Figure 10: The

T_{h,W,H}

target encoding using the categories hour, weekday and holiday.

Figure 11: The

T_{h,W}+T_{h,H}

encoding consisting of two separate target encodings.

Name	Category	Description	Test Period	Reference
m_dense	Traffic	Madrid traffic intensity aggregated over 30 sensors.	2019-01 – 2019-12	Jiang et al. (2024)
mta_subway	Transport	New York City subway ridership aggregated over all stations.	2025-01 – 2025-12	Metropolitan Transportation Authority (2024b) ; Metropolitan Transportation Authority (2025b)
mta_bus	Transport	New York City bus ridership aggregated over all stations.	2025-01 – 2025-12	Metropolitan Transportation Authority (2024a) ; Metropolitan Transportation Authority (2025a)
TransnetBW	Energy	Total load of the TransnetBW control area in Germany.	2025-01 – 2025-12	ENTSO-E (2026)
FeederBW	Energy	Active power of three individual low-voltage feeders (#033, #035 and #091) from German distribution system operator NetzeBW.	2024-04 – 2025-03	Treutlein et al. (2026)
Alicante	Water	Average water consumption of 1,007 consumers in Alicante.	2016-01 – 2016-12	DAIAD Project (2017)

Appendix

Table 3: Overview of the benchmark datasets used in this work.

		All days		Holidays only		Non-holidays only
Model	Encoding	MASE	SQL	MASE	SQL	MASE	SQL
Baselines	seasonal-naive (168h)	1.0106	0.9876	3.8760	3.4173	0.9132	0.9050
Baselines	$T_{h,W}(H=Sun)_{naive}$	0.9404	0.8174	1.4057	1.1195	0.9246	0.8072
Chronos-2	w/o covariates	0.5667	0.4661	3.2231	2.8733	0.4764	0.3843
	ordinal h,W + bin H	0.5385	0.4364	2.0874	1.7720	0.4858	0.3910
	one-hot h,W + bin H	0.5715	0.4689	2.7856	2.5112	0.4962	0.3995

Appendix

Table 4: Results for the m_dense dataset, split by date-type.

Figure 12: MASE change against the model w/o covariates on the m_dense dataset.

Figure 13: SHAP contributions for all models on the m_dense dataset, exemplarily depicted for the

T_{h,W}(H=7)

and

T_{h,W}+T_{h,H}

encodings.

		All days		Holidays only		Non-holidays only
Model	Encoding	MASE	SQL	MASE	SQL	MASE	SQL
Baselines	seasonal-naive (168h)	1.0399	1.1594	4.3610	3.9833	0.9075	1.0468
Baselines	$T_{h,W}(H=Sun)_{naive}$	1.0595	0.9450	2.8182	2.5015	0.9893	0.8829
Chronos-2	w/o covariates	0.7549	0.6086	3.2626	2.8593	0.6549	0.5188
	ordinal h,W + bin H	0.7616	0.6089	2.5726	2.2009	0.6894	0.5454
	one-hot h,W + bin H	0.8271	0.6689	3.2127	2.8555	0.7320	0.5817

Appendix

Table 5: Results for the mta_subway dataset, split by date-type.

Figure 14: MASE change against the model w/o covariates on the mta_subway dataset.

Figure 15: SHAP contributions for all models on the mta_subway dataset, exemplarily depicted for the

T_{h,W}(H=7)

and

T_{h,W}+T_{h,H}

encodings.

		All days		Holidays only		Non-holidays only
Model	Encoding	MASE	SQL	MASE	SQL	MASE	SQL
Baselines	seasonal-naive (168h)	1.0352	1.1499	4.4528	4.0390	0.8989	1.0346
Baselines	$T_{h,W}(H=Sun)_{naive}$	1.0809	0.9837	2.7476	2.4613	1.0144	0.9248
Chronos-2	w/o covariates	0.7804	0.6293	3.5664	3.1018	0.6693	0.5307
	ordinal h,W + bin H	0.7569	0.6019	2.3343	1.9043	0.6940	0.5499
	one-hot h,W + bin H	0.8092	0.6540	3.4036	2.9105	0.7057	0.5640

Appendix

Table 6: Results for the mta_bus dataset, split by date-type.

Figure 16: MASE change against the model w/o covariates on the mta_bus dataset.

Figure 17: SHAP contributions for all models on the mta_bus dataset, exemplarily depicted for the

T_{h,W}(H=7)

and

T_{h,W}+T_{h,H}

encodings.

		All days		Holidays only		Non-holidays only
Model	Encoding	MASE	SQL	MASE	SQL	MASE	SQL
Baselines	seasonal-naive (168h)	1.0373	0.8887	3.5117	2.9987	0.9532	0.8170
Baselines	$T_{h,W}(H=Sun)_{naive}$	1.4602	1.1108	1.1270	0.8393	1.4715	1.1200
Chronos-2	w/o covariates	0.4850	0.3885	2.3853	2.0609	0.4205	0.3316
	ordinal h,W + bin H	0.4284	0.3353	0.7864	0.6258	0.4162	0.3254
	one-hot h,W + bin H	0.4765	0.3781	1.6213	1.3350	0.4376	0.3456

Appendix

Table 7: Results for the TransnetBW dataset, split by date-type.

Figure 18: MASE change against the model w/o covariates on the TransnetBW dataset.

Figure 19: SHAP contributions for all models on the TransnetBW dataset, exemplarily depicted for the

T_{h,W}(H=7)

and

T_{h,W}+T_{h,H}

encodings.

		All days		Holidays only		Non-holidays only
Model	Encoding	MASE	SQL	MASE	SQL	MASE	SQL
Baselines	seasonal-naive (168h)	1.0841	1.0291	2.2640	2.0940	1.0474	0.9960
Baselines	$T_{h,W}(H=Sun)_{naive}$	1.1726	1.0772	1.4250	1.2242	1.1648	1.0727
Chronos-2	w/o covariates	0.8007	0.6230	2.0622	1.7281	0.7615	0.5886
	ordinal h,W + bin H	0.7736	0.6029	1.3490	1.0991	0.7557	0.5875
	one-hot h,W + bin H	0.8060	0.6299	1.7825	1.4414	0.7756	0.6047

Appendix

Table 8: Results for the FeederBW_033 dataset, split by date-type.

Figure 20: MASE change against the model w/o covariates on the FeederBW_33 dataset.

Figure 21: SHAP contributions for all models on the FeederBW_33 dataset, exemplarily depicted for the

T_{h,W}(H=7)

and

T_{h,W}+T_{h,H}

encodings.

		All days		Holidays only		Non-holidays only
Model	Encoding	MASE	SQL	MASE	SQL	MASE	SQL
Baselines	seasonal-naive (168h)	1.0562	0.9302	2.1083	1.8575	1.0235	0.9014
Baselines	$T_{h,W}(H=Sun)_{naive}$	0.9036	0.7487	0.9751	0.7881	0.9013	0.7475
Chronos-2	w/o covariates	0.7308	0.5812	1.5818	1.3653	0.7043	0.5569
	ordinal h,W + bin H	0.7293	0.5775	1.4074	1.1544	0.7082	0.5596
	one-hot h,W + bin H	0.7479	0.5948	1.5148	1.2989	0.7241	0.5729

Appendix

Table 9: Results for the FeederBW_035 dataset, split by date-type.

Figure 22: MASE change against the model w/o covariates on the FeederBW_35 dataset.

Figure 23: SHAP contributions for all models on the FeederBW_35 dataset, exemplarily depicted for the

T_{h,W}(H=7)

and

T_{h,W}+T_{h,H}

encodings.

		All days		Holidays only		Non-holidays only
Model	Encoding	MASE	SQL	MASE	SQL	MASE	SQL
Baselines	seasonal-naive (168h)	0.9918	0.8589	2.4438	2.1058	0.9467	0.8202
Baselines	$T_{h,W}(H=Sun)_{naive}$	1.1284	0.8961	1.0713	0.8874	1.1302	0.8964
Chronos-2	w/o covariates	0.6666	0.5269	1.5522	1.3245	0.6391	0.5021
	ordinal h,W + bin H	0.6586	0.5193	1.1473	0.8890	0.6434	0.5079
	one-hot h,W + bin H	0.6811	0.5404	1.4729	1.2183	0.6565	0.5193

Appendix

Table 10: Results for the FeederBW_091 dataset, split by date-type.

Figure 24: MASE change against the model w/o covariates on the FeederBW_91 dataset.

Figure 25: SHAP contributions for all models on the FeederBW_91 dataset, exemplarily depicted for the

T_{h,W}(H=7)

and

T_{h,W}+T_{h,H}

encodings.

		All days		Holidays only		Non-holidays only
Model	Encoding	MASE	SQL	MASE	SQL	MASE	SQL
Baselines	seasonal-naive (168h)	0.9210	0.9390	1.4840	1.4474	0.9019	0.9217
Baselines	$T_{h,W}(H=Sun)_{naive}$	0.7462	0.6987	0.6702	0.6958	0.7488	0.6988
Chronos-2	w/o covariates	0.6474	0.5266	0.9571	0.7988	0.6369	0.5174
	ordinal h,W + bin H	0.6624	0.5368	0.9088	0.7497	0.6540	0.5296
	one-hot h,W + bin H	0.6710	0.5450	0.9388	0.7900	0.6619	0.5367

Appendix

Table 11: Results for the Alicante dataset, split by date-type.

Figure 26: MASE change against the model w/o covariates on the Alicante dataset.

Figure 27: SHAP contributions for all models on the Alicante dataset, exemplarily depicted for the

T_{h,W}(H=7)

and

T_{h,W}+T_{h,H}

encodings.

		All days		Holidays only		Non-holidays only
Model	Encoding	MASE	SQL	MASE	SQL	MASE	SQL
Baselines	seasonal-naive (168h)	1.1499	1.0432	1.9203	1.6437	1.1215	1.0211
Baselines	$T_{h,W}(H=Sun)_{naive}$	1.4535	1.2436	1.5922	1.3471	1.4484	1.2398
Chronos-2	w/o covariates	0.7803	0.6308	1.4783	1.2061	0.7546	0.6096
	ordinal h,W + bin H	0.7748	0.6241	1.4739	1.1732	0.7490	0.6039
	one-hot h,W + bin H	0.7989	0.6413	1.4869	1.2040	0.7736	0.6206

Appendix

Table 12: Results for the uci_bike dataset, split by date-type.

Figure 28: MASE change against the model w/o covariates on the uci_bike dataset.

Figure 29: SHAP contributions for all models on the uci_bike dataset, exemplarily depicted for the

T_{h,W}(H=7)

and

T_{h,W}+T_{h,H}

encodings.

	All days		Holidays only		Non-holidays only
Model	MASE	SQL	MASE	SQL	MASE	SQL
Chronos-2	0.2994	0.2348	0.5473	0.4363	0.2902	0.2274
TabPFN-TS-3	0.0277	0.0208	0.0267	0.0194	0.0277	0.0208
TiRex-2	0.3033	0.2480	0.2631	0.2121	0.3048	0.2493
TimesFM-3	0.0824	0.0645	0.0774	0.0612	0.0826	0.0646

Appendix

Table 13: Oracle encoding (target fed in as a known covariate): overall results across all datasets, split by day type (mean over 9 datasets).

		All days		Holidays only		Non-holidays only
Model	Encoding	MASE	SQL	MASE	SQL	MASE	SQL
Baselines	seasonal-naive (168h)	1.0437	0.9788	3.0978	2.7752	0.9703	0.9147
Baselines	$T_{h,W}(H=Sun)_{naive}$	0.9804	0.8290	1.4629	1.2491	0.9617	0.8127
Chronos-2	w/o covariates	0.6970	0.5607	2.3688	2.0581	0.6373	0.5072
	ordinal h,W + bin H	0.6883	0.5507	1.8060	1.5326	0.6483	0.5156
	one-hot h,W + bin H	0.7198	0.5788	2.2567	1.9568	0.6646	0.5293

Appendix

Table 14: Overall results across all datasets at context 2048, split by day type (mean over 9 datasets).

Time Series Forecasting with Covariates

See all 19 papers

Oct 1, 2026

Foundations without Fundamentals: Zero-Shot Blind Spots in Time Series FMs

Despite the success of Time Series Foundation Models (TSFMs) on broad benchmarks, their ability to internalize basic temporal logic, especially in settings supported by exogenous covariates, remains under-examined. We introduce SimpleTimeBench, a diagnostic univariate and multivariate "unit test" suite for primitives such as monotonic trends, periodic signals and leading indicator covariates, scenarios where near-perfect forecasts should be trivial. Surprisingly, prominent multivariate TSFMs (Chronos-2, Moirai and Toto) frequently produce suboptimal zero-shot forecasts for these inputs. While fine-tuning Chronos-2 improves its behaviour on specific tasks, we show that this adaptation degrades performance on other fundamental patterns rather than enhancing its generalizable foundational capabilities. This reveals a gap between pre-training scale and basic temporal reasoning, suggesting that current TSFMs could potentially lack the inductive biases needed to capture simple predictable functions. We further demonstrate that these failures are not merely synthetic curiosities: they persist in real-world sensor forecasting, where TSFMs consistently underutilize leading indicators available in observed covariates. This inability to capture simple relationships limits the practical utility and reliability of current multivariate models.

Nafiseh Ghoroghchian, Haipeng Zhang, Shuyi Han +2

Layer6 AI

Multivariate Time Series Forecasting Zero-Shot Time Series Forecasting Time Series Reasoning Exogenous Time Series Forecasting Time Series Forecasting Forecasting Benchmarks+1

Ask PDF

May 12, 2026

Investigating simple target-covariate relationships for Chronos-2 and TabPFN-TS

Time Series Foundation Models (TSFMs) have recently achieved state-of-the-art performance, often outperforming supervised models in zero-shot settings. Recent TSFM architectures, such as Chronos-2 and TabPFN-TS, aim to integrate covariates. In this paper, we design controlled experiments based on simple target-covariate relationships to assess this integration capability. Our results show that TabPFN-TS captures these relationships more effectively than Chronos-2, especially for short horizons, suggesting that the strong benchmark performance of Chronos-2 does not automatically translate into optimal modeling of simple covariate-target dependencies.

Gaspard Berthelier, Mariia Baranova, Andrei-Tiberiu Pantea +4

EDF R&D, Palaiseau · Inria Sophia-Antipolis, Université Côte d’Azur · Université Paris Cit'e

Multivariate Time Series Forecasting Time Series Forecasting Time Series Forecasting with Covariates Time-Series Foundation Models

Ask PDF

Sep 29, 2026

Latent Inference-Time Guidance of Time Series Foundation Models

Time Series Foundation Models (TSFMs) currently provide state-of-the-art results in forecasting tasks. They are available out-of-the-box and rely on in-context learning to make their predictions, which makes the quality of their performance highly sensitive to the user-selected lookback, covariates, horizon and training data distributions. In practise, the quality of the forecasts are variable but complementary, which highlights the need for a principled ensembling approach, rather than selecting the best context. This paper introduces Latent Inference-Time Guidance for TSFMs, which adaptively combines a pool of TSFM forecasts through a time-dependent latent space with independent components. The framework comes equipped with identifiability and reconstruction guarantees, whilst maintaining the off-the-shelf aspect of foundation models. We provide experiments on datasets at various frequencies and from multiple domains: these show that the approach is competitive with traditional ensembling approaches.

Chloé Hashimoto-Cullen, Amaury Durand, Laurent Bozzi +3

LPSM, Sorbonne Université · EDF R&D · UCL

Time Series Forecasting Inference-Time Guidance Time-Series Foundation Models Ensemble Learning

Ask PDF

Abstract

Time Series Forecasting with Covariates

Explore similar work

Foundations without Fundamentals: Zero-Shot Blind Spots in Time Series FMs

Investigating simple target-covariate relationships for Chronos-2 and TabPFN-TS

Latent Inference-Time Guidance of Time Series Foundation Models