Correct Degree Selection for Koopman Mode Decomposition

Fourier Decomposition (FD) and Koopman Mode Decomposition (KMD) are important tools for time series data analysis, applied across a broad spectrum of applications. Both aim to decompose time series functions into superpositions of countably many wave functions, with strikingly similar mathematical foundations. These methodologies derive from the linear decomposition of functions within specific function spaces: FD uses a fixed basis of sine and cosine functions, while KMD employs eigenfunctions of the Koopman linear operator. A notable distinction lies in their scope: FD is confined to periodic functions, while KMD can decompose functions into exponentially amplifying or damping waveforms, making it potentially better suited for describing phenomena beyond FD’s capabilities. However, practical applications of KMD often show that despite accurate approximation of training data, its prediction accuracy is limited. This paper clarifies that this issue is closely related to the number of wave components used in decomposition, referred to as the degree of a KMD. Existing methods use predetermined, arbitrary, or ad hoc values for this degree. We demonstrate that using a degree different from a uniquely determined value for the data allows infinite KMDs to accurately approximate training data, explaining why current methods, which select a single KMD from these candidates, struggle with prediction accuracy. Furthermore, we introduce mathematically supported algorithms to determine the correct degree. Simulations verify that our algorithms can identify the right degrees and generate KMDs that can make accurate predictions, even with noisy data.