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

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.

Climate change, primarily driven by carbon dioxide (CO2) emissions, requires accurate forecasting tools to support effective mitigation policies and sustainable development strategies. Existing forecasting approaches typically rely on centralized data collection, which is often restricted by privacy regulations and the distributed nature of emission data across countries and industrial sectors. This paper proposes a novel federated hybrid forecasting framework that integrates ARIMA-based trend modeling, GARCH-based volatility modeling, LSTM-Attention temporal representation learning, and XGBoost prediction within a privacy-preserving federated learning environment. The proposed framework enables collaborative learning among distributed clients without requiring the exchange of raw data. Experimental evaluation across 14 clients demonstrates strong forecasting performance, achieving client R2 values between 0.50 and 0.97 with an average of 0.73, RMSE values ranging from 0.06 to 2.35 with an average of 1.21, and MAPE values between 1.5% and 11.3% with an average of 6.5%. The results indicate that the proposed framework provides an accurate, scalable, and regulation-compliant solution for collaborative carbon-emission forecasting.

While Transformer-based architectures have established themselves as a dominant paradigm in Multivariate Time Series Forecasting (MTSF), their core self-attention mechanism inherently functions as a low-pass filter, systematically smoothing out high-frequency signals vital for sharp local changes. Recent advancements have increasingly incorporated frequency-domain operations to address this bias, however, most existing designs rely on fixed spectral bases and apply sequence-wise (uniform) modulation, implicitly assuming a time-invariant frequency response. This overlooks a key property of real-world series that their spectral characteristics often evolve over time, making uniform modulation insufficient for capturing fine-grained temporal dynamics. To tackle these limitations, we propose FAiT, a Frequency-Aware inverted Transformer. Specifically, FAiT rectifies the spectral bias internally through Inverted Attention, which interprets the attention map as a learnable low-pass operator and constructs a dedicated complementary high-pass branch by inverting the attention matrix to recover attenuated transient signals. Furthermore, FAiT introduces Dynamic Temporal-Frequency Modulation (DTFM), which synthesizes instance-conditioned weights to adaptively re-calibrate the energy of spectral sub-bands, enabling fine-grained control over evolving multi-scale patterns. Extensive experiments on widely used benchmarks demonstrate that FAiT consistently outperforms state-of-the-art Transformer-based and frequency-enhanced baselines, while maintaining computational efficiency.