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Nonparametric Reconstruction of Vector Fields From Noisy Observations of Their Flow Curves

Any state-space equation is identified by a vector field that maps the values of the state vector into the values of its derivative. Any solution to a state-space equation is called a flow curve of its associated vector field. This paper focuses on reconstruction of a vector field from noisy measurements of its flow curves generated by a physical process. A nonparametric least squares scheme is developed to estimate a vector field along each of its empirical flow curves. This scheme is formulated as a linear quadratic tracking problem with a well known solution from the theory of optimal control. By solving this problem for several empirical flow curves, the vector field is constructed along them, and using a nearest neighbor interpolation method presented here, the vector field is estimated at those points not lying on the flow curves. The overall procedure is demonstrated by reconstruction of the magnetic force around a magnet from the motion trajectories of a magnetic particle moving inside the magnetic field.