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.

[1]  Steven W. Zucker,et al.  Diffusion Maps and Geometric Harmonics for Automatic Target Recognition (ATR). Volume 2. Appendices , 2007 .

[2]  Richard A. Davis,et al.  Introduction to time series and forecasting , 1998 .

[3]  S T Roweis,et al.  Nonlinear dimensionality reduction by locally linear embedding. , 2000, Science.

[4]  J. Peraire,et al.  OPTIMAL CONTROL OF VORTEX SHEDDING USING LOW-ORDER MODELS. PART II-MODEL-BASED CONTROL , 1999 .

[5]  Alberto Tesi,et al.  A Frequency Method for Predicting Limit Cycle Bifurcations , 1997 .

[6]  I. Mezic,et al.  Comparison of Systems using Diffusion Maps , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[7]  Prashant G. Mehta,et al.  Linear and nonlinear analysis of controlled combustion processes. I. Linear analysis , 1999, Proceedings of the 1999 IEEE International Conference on Control Applications (Cat. No.99CH36328).

[8]  P. Holmes,et al.  Turbulence, Coherent Structures, Dynamical Systems and Symmetry , 1996 .

[9]  Ann B. Lee,et al.  Geometric diffusions as a tool for harmonic analysis and structure definition of data: diffusion maps. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[10]  I. Jolliffe.  Principal Component Analysis , 2002 .

[11]  Jeffrey M. Cohen,et al.  Active Control of Combustion Instabilities in Gas Turbine Engines for Low Emissions. Part I: Physics-Based and Experimentally Identified Models of Combustion Instability , 2000 .

[12]  Mikhail Belkin,et al.  Laplacian Eigenmaps for Dimensionality Reduction and Data Representation , 2003, Neural Computation.

[13]  J. Tenenbaum,et al.  A global geometric framework for nonlinear dimensionality reduction. , 2000, Science.

[14]  Ronald R. Coifman,et al.  Data Fusion and Multicue Data Matching by Diffusion Maps , 2006, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[15]  Bernhard Schölkopf,et al.  Nonlinear Component Analysis as a Kernel Eigenvalue Problem , 1998, Neural Computation.

[16]  Andrzej Banaszuk,et al.  Comparison of systems with complex behavior , 2004 .

[17]  Wr Graham,et al.  OPTIMAL CONTROL OF VORTEX SHEDDING USING LOW-ORDER MODELS. PART I-OPEN-LOOP MODEL DEVELOPMENT , 1999 .

[18]  Tim Lieuwen,et al.  Experimental Investigation of Limit Cycle Oscillations in an Unstable Gas Turbine Combustor , 2000 .