TimeSQL: Improving Multivariate Time Series Forecasting with Multi-Scale
  Patching and Smooth Quadratic Loss
Site Mo Haoxin Wang Bixiong Li Songhai Fan Yuankai Wu Xianggen Liu
Time series is a special type of sequence data, a sequence of real-valued
random variables collected at even intervals of time. The real-world
multivariate time series comes with noises and contains complicated local and
global temporal dynamics, making it difficult to forecast the future time
series given the historical observations. This work proposes a simple and
effective framework, coined as TimeSQL, which leverages multi-scale patching
and smooth quadratic loss (SQL) to tackle the above challenges. The multi-scale
patching transforms the time series into two-dimensional patches with different
length scales, facilitating the perception of both locality and long-term
correlations in time series. SQL is derived from the rational quadratic kernel
and can dynamically adjust the gradients to avoid overfitting to the noises and
outliers. Theoretical analysis demonstrates that, under mild conditions, the
effect of the noises on the model with SQL is always smaller than that with
MSE. Based on the two modules, TimeSQL achieves new state-of-the-art
performance on the eight real-world benchmark datasets. Further ablation
studies indicate that the key modules in TimeSQL could also enhance the results
of other models for multivariate time series forecasting, standing as
plug-and-play techniques.