  This paper addresses the problem of multi-step time series forecasting for
non-stationary signals that can present sudden changes. Current
state-of-the-art deep learning forecasting methods, often trained with variants
of the MSE, lack the ability to provide sharp predictions in deterministic and
probabilistic contexts. To handle these challenges, we propose to incorporate
shape and temporal criteria in the training objective of deep models. We define
shape and temporal similarities and dissimilarities, based on a smooth
relaxation 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 including
shApe and TimE), a new objective for deterministic forecasting, that explicitly
incorporates two terms supporting precise shape and temporal change detection.
For probabilistic forecasting, we introduce STRIPE++ (Shape and Time diverRsIty
in Probabilistic forEcasting), a framework for providing a set of sharp and
diverse forecasts, where the structured shape and time diversity is enforced
with a determinantal point process (DPP) diversity loss. Extensive experiments
and ablations studies on synthetic and real-world datasets confirm the benefits
of leveraging shape and time features in time series forecasting.
