# 2010 Research Summary

## Statistical System Identification

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Anil Jayanti Aswani, Claire Tomlin, Peter Bickel^{1} and Mark Biggin^{2}

National Science Foundation CCR-0225610

Local linearization techniques are an important class of nonparametric system identification. Identifying local linearizations in practice involves solving a linear regression problem that is ill-posed. The problem can be ill-posed either if the dynamics of the system lie on a manifold of lower dimension than the ambient space or if there are not enough measurements of all the modes of the dynamics of the system. We describe a set of linear regression estimators that can handle data lying on a lower-dimension manifold. These estimators differ from previous estimators, because these estimators are able to improve estimator performance by exploiting the sparsity of the system – the existence of direct interconnections between only some of the states – and can work in the “large p, small n” setting in which the number of states is comparable to the number of data points. Our system identification procedure, which consists of a presmoothing step and a regression step, can be applied to data taken from a quadrotor helicopter or other data sets. These data sets show that our procedure can outperform existing procedures, when compared using various technical metrics. We also apply this to study biological networks taken from the embryogenesis of Drosophila melangoster.

Figure 1: Many important problems in system identification and statistics can be abstracted into a local regression problem on an embedded submanifold with dimension less than that of the ambient space.

- [1]
- Aswani, A., Bickel, P., and Tomlin, C. (2009), Statistics for sparse, high-dimensional, and nonparametric system identification, In Proceedings of the IEEE ICRA2009: 2133-2138.
- [2]
- Aswani, A., Bickel, P., and Tomlin, C. (2009), Regression on Manifolds: Estimation of the exterior derivative. Submitted to Annals of Statistics.

^{1}UC Berkeley - Statistics

^{2}LBNL - BDTNP