FTimeXer: Frequency-aware Time-series Transformer with Exogenous variables for Robust Carbon Footprint Forecasting

Qingzhong Li; Yue Hu; Zhou Long; Qingchang Ma; Hui Ma; Jinhai Sa

cs.LGFeb 16, 2026

FTimeXer: Frequency-aware Time-series Transformer with Exogenous variables for Robust Carbon Footprint Forecasting

Authors: Qingzhong Li, Yue Hu, Zhou Long, Qingchang Ma, Hui Ma, Jinhai Sa

Organizations: School of Software, Xinjiang University, Urumqi, China · Xinjiang Key Laboratory of Intelligent Computing and Smart Applications, School of Software, Xinjiang University, Urumqi 830046, China

Abstract

Accurate and up-to-date forecasting of the power grid's carbon footprint is crucial for effective product carbon footprint (PCF) accounting and informed decarbonization decisions. However, the carbon intensity of the grid exhibits high non-stationarity, and existing methods often struggle to effectively leverage periodic and oscillatory patterns. Furthermore, these methods tend to perform poorly when confronted with irregular exogenous inputs, such as missing data or misalignment. To tackle these challenges, we propose FTimeXer, a frequency-aware time-series Transformer designed with a robust training scheme that accommodates exogenous factors. FTimeXer features an Fast Fourier Transform (FFT)-driven frequency branch combined with gated time-frequency fusion, allowing it to capture multi-scale periodicity effectively. It also employs stochastic exogenous masking in conjunction with consistency regularization, which helps reduce spurious correlations and enhance stability. Experiments conducted on three real-world datasets show consistent improvements over strong baselines. As a result, these enhancements lead to more reliable forecasts of grid carbon factors, which are essential for effective PCF accounting and informed decision-making regarding decarbonization.

Figures & tables

Model	Magnolia				California_CT2				NowYork_Greenidge
Model	$R^{2}\uparrow$	MSE $\downarrow$	RMSE $\downarrow$	MAE $\downarrow$	$R^{2}\uparrow$	MSE $\downarrow$	RMSE $\downarrow$	MAE $\downarrow$	$R^{2}\uparrow$	MSE $\downarrow$	RMSE $\downarrow$	MAE $\downarrow$
GRU	0.819	47.552	6.896	4.681	0.744	671.380	25.911	17.911	0.678	22.816	4.777	3.320
LSTM	0.830	44.796	6.693	4.627	0.809	501.571	22.396	13.993	0.667	23.630	4.861	2.913
Transformer	0.837	42.880	6.548	4.423	0.787	557.731	23.616	15.068	0.707	20.817	4.563	2.575
Informer [ 18 ]	0.858	37.398	6.115	5.076	0.781	574.756	23.974	19.119	0.724	19.580	4.425	2.442
TimeXer [ 9 ]	0.873	33.377	5.777	3.512	0.822	468.237	21.639	12.215	0.726	19.430	4.408	2.338

TABLE I: Performance comparison on three datasets.

Setting	Magnolia				California_CT2				NowYork_Greenidge
Setting	$R^{2}\uparrow$	MSE $\downarrow$	RMSE $\downarrow$	MAE $\downarrow$	$R^{2}\uparrow$	MSE $\downarrow$	RMSE $\downarrow$	MAE $\downarrow$	$R^{2}\uparrow$	MSE $\downarrow$	RMSE $\downarrow$	MAE $\downarrow$
Baseline	0.879	33.759	5.782	3.594	0.816	498.366	22.216	12.444	0.747	18.265	4.247	2.356
Masking 10%	0.881	33.165	5.732	3.544	0.823	470.250	21.572	12.325	0.751	17.919	4.207	2.317
Masking 20%	0.883	32.472	5.673	3.505	0.828	455.400	21.235	12.177	0.754	17.672	4.188	2.297
Masking 30%	0.887	31.581	5.594	3.465	0.836	435.600	20.770	11.930	0.759	17.424	4.158	2.277
Masking 40%	0.884	32.274	5.653	3.495	0.831	450.450	21.117	12.078	0.756	17.573	4.168	2.287

TABLE II: Ablation study on Magnolia, California_CT2, and NewYork_Greenidge datasets.

Panel of Figure 4 — Fig. 2: Visualization results on Magnolia, California_CT2, and NewYork_Greenidge datasets.

Time Series Forecasting

Time series forecasting predicts future values from ordered observations using statistical, neural, and probabilistic models to support decisions under uncertainty.

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In many time series forecasting settings, the target time series is accompanied by exogenous covariates, such as promotions and prices in retail demand; temperature in energy load; calendar and holiday indicators for traffic or sales; and grid load or fuel costs in electricity pricing. Ignoring these exogenous signals can substantially degrade forecasting accuracy, particularly when they drive spikes, discontinuities, or regime and phase changes in the target series. Most current time series foundation models (e.g., Chronos, Sundial, TimesFM, TimeMoE, TimeLLM, and LagLlama) ignore exogenous covariates and make forecasts solely from the numerical time series history, thereby limiting their performance. In this paper, we develop ApolloPFN, a prior-data fitted network (PFN) that is time-aware (unlike prior PFNs) and that natively incorporates exogenous covariates (unlike prior univariate forecasters). Our design introduces two major advances: (i) a synthetic data generation framework that injects realistic temporal patterns, structural changes, and exogenous dependencies into the PFN prior; and (ii) time-aware architectural modifications that embed inductive biases needed to exploit temporal context. We demonstrate that ApolloPFN outperforms existing baselines across several forecasting benchmarks with exogenous information, including M5, electric price forecasting, UCI Air Quality, and Solar Energy datasets.

Abstract

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