Fitting Imbalanced Uncertainties in Multi-output Time Series Forecasting

We focus on multi-step ahead time series forecasting with the multi-output strategy. From the perspective of multi-task learning (MTL), we recognize imbalanced uncertainties between prediction tasks of different future time steps. Unexpectedly, trained by the standard summed Mean Squared Error (MSE) loss, existing multi-output forecasting models may suffer from performance drops due to the inconsistency between the loss function and the imbalance structure. To address this problem, we reformulate each prediction task as a distinct Gaussian Mixture Model (GMM) and derive a multi-level Gaussian mixture loss function to better fit imbalanced uncertainties in multi-output time series forecasting. Instead of using the two-step Expectation-Maximization (EM) algorithm, we apply the self-attention mechanism on the task-specific parameters to learn the correlations between different prediction tasks and generate the weight distribution for each GMM component. In this way, our method jointly optimizes the parameters of the forecasting model and the mixture model simultaneously in an end-to-end fashion, avoiding the need of two-step optimization. Experiments on three real-world datasets demonstrate the effectiveness of our multi-level Gaussian mixture loss compared to models trained with the standard summed MSE loss function. All the experimental data and source code are available at https://github.com/smallGum/GMM-FNN.