Projection-Based Alignment with a Fractional Trajectory Metric for Time Series Forecasting

Time-series forecasting is predominantly trained with point-wise losses such as mean squared error (MSE), although multi-step forecasting requires predicting an entire future trajectory with joint temporal structure. Such objectives emphasize sample-wise accuracy but do not explicitly account for distributional structure across forecast horizons. We propose ProDA, a projection-based and model-agnostic objective that aligns the joint distributions of history-future and history-forecast trajectories through the Cramér-Wold discrepancy. While this provides a distribution-sensitive criterion, its pairwise geometry is governed by Euclidean trajectory distances, which can be dominated by slowly varying components under non-stationarity. To address this, ProDA incorporates fractional differencing directly into the alignment objective, inducing a structured trajectory metric that attenuates low-frequency discrepancies and relatively emphasizes localized temporal variation. Theoretically, we show that the induced Cramér-Wold kernel remains characteristic under the fractional transformation and derive its spectral reweighting, establishing that the geometry of distributional alignment can be modified without sacrificing distributional identifiability. Extensive experiments across multiple forecasting backbones, benchmark datasets, and zero-shot transfer settings demonstrate consistent improvements over standard training objectives and recent objective-centric baselines.