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This paper addresses the problem of multi-step time series forecasting fornon-stationary signals that can present sudden changes. Currentstate-of-the-art deep learning forecasting methods, often trained with variantsof the MSE, lack the ability to provide sharp predictions in deterministic andprobabilistic contexts. To handle these challenges, we propose to incorporateshape and temporal criteria in the training objective of deep models. We defineshape and temporal similarities and dissimilarities, based on a smoothrelaxation of Dynamic Time Warping (DTW) and Temporal Distortion Index (TDI),that enable to build differentiable loss functions and positive semi-definite(PSD) kernels. With these tools, we introduce DILATE (DIstortion Loss includingshApe and TimE), a new objective for deterministic forecasting, that explicitlyincorporates two terms supporting precise shape and temporal change detection.For probabilistic forecasting, we introduce STRIPE++ (Shape and Time diverRsItyin Probabilistic forEcasting), a framework for providing a set of sharp anddiverse forecasts, where the structured shape and time diversity is enforcedwith a determinantal point process (DPP) diversity loss. Extensive experimentsand ablations studies on synthetic and real-world datasets confirm the benefitsof leveraging shape and time features in time series forecasting.

Deep Time Series Forecasting with Shape and Temporal Criteria