Validation of low-dimensional models using diffusion maps and harmonic averaging

This paper is concerned with efficient methods of dynamic model validation under the conditions when the underlying dynamics is high-dimensional, the data is high- dimensional, or when significant noise is present. In such cases there is a need to separate the geometric properties of the data from the noise. The approach we propose is based on a mapping of data into a low-dimensional subspace which requires no a priori knowledge of their qualitative behavior or dimension. Model comparison is then based on the intrinsic geometry of data sets and their temporal characteristics. We construct a diffusion map on the graph of the experimental data set as established in the work of Coifman, Lafon, et al. We then use the Nystrom extension to map the model data set into the frame of the experimental data, thereby allowing the two to be easily compared. We apply this to both face image data and to experiments and candidate models of a combustion experiment exhibiting limit-cycling oscillations.