Comparison of Systems using Diffusion Maps

In this paper we propose an efficient method of comparing data sets obtained from either an experiment or simulation of dynamical system model for the purpose of model validation. The proposed approach is based on comparing the intrinsic geometry and the associated dynamics linked with the data sets, and requires no a priori knowledge of the qualitative behavior or the dimension of the phase space. The approach to data analysis is based on constructing a diffusion map defined on the graph of the data set as established in the work of Coifman, Lafon, et al.[5], [10]. Low dimensional embedding is done via a singular value decomposition of the approximate diffusion map. We propose some simple metrics constructed from the eigenvectors of the diffusion map that describe the geometric and spectral properties of the data. The approach is illustrated by comparison of candidate models to data of a combustion experiment that shows limit-cycling acoustic oscillations.