What are the two main ideas behind the FEDformer model for long-term time series forecasting, and how do they address the issue of distribution mismatch between predictions and ground truth?

The two main ideas behind FEDformer are incorporating a seasonal-trend decomposition approach into the transformer-based model and combining Fourier analysis with the transformer by applying it in the frequency domain instead of the time domain. These ideas address the distribution mismatch by helping the model capture the global properties and statistics of the time series as a whole, so its predictions follow a distribution closer to the ground truth than point-wise attention methods.

FEDformer's design is motivated by the observation that point-wise attention and prediction in transformers can make each timestep prediction independently, causing the model to fail to maintain the global properties and statistics of the whole time series. The first idea is to embed a widely used seasonal-trend decomposition approach directly into the transformer architecture, using Mixture Of Experts Decomposition blocks to separate seasonal and trend components. This decomposition mechanism helps bring the distribution of the prediction close to that of the ground truth, as verified by the Kolmogorov-Smirnov distribution test. The second idea is to combine Fourier analysis with the transformer by performing representation learning in the frequency domain rather than the time domain, using Frequency Enhanced Blocks and Frequency Enhanced Attention. This frequency-domain processing helps the transformer better capture the global properties of time series. FEDformer also randomly selects a subset of frequency components, including both low and high frequencies, rather than discarding all high-frequency information, because time series have sparse representations on a Fourier basis and a random subset can preserve most of the information in the transformed matrix. In distribution tests on ETTm1 and ETTm2, FEDformer is the only model whose forecasting output is not rejected as coming from a different distribution than the input sequence, with P-values larger than 0.01 in all cases.