  Delay embedding---a method for reconstructing dynamical systems by delay
coordinates---is widely used to forecast nonlinear time series as a model-free
approach. When multivariate time series are observed, several existing
frameworks can be applied to yield a single forecast combining multiple
forecasts derived from various embeddings. However, the performance of these
frameworks is not always satisfactory because they randomly select embeddings
or use brute force and do not consider the diversity of the embeddings to
combine. Herein, we develop a forecasting framework that overcomes these
existing problems. The framework exploits various "suboptimal embeddings"
obtained by minimizing the in-sample error via combinatorial optimization. The
framework achieves the best results among existing frameworks for sample toy
datasets and a real-world flood dataset. We show that the framework is
applicable to a wide range of data lengths and dimensions. Therefore, the
framework can be applied to various fields such as neuroscience, ecology,
finance, fluid dynamics, weather, and disaster prevention.
