Connection between Differential Geometry and Estimation Theory for Polynomial Nonlinearity in 2D Mahendra Mallick Georgia Tech Research Institute (GTRI) Georgia Institute of Technology Atlanta, GA 30332, U.S.A. mahendra.mallick@gtri.gatech.edu Yanjun Yan Department of Electrical Engineering and Computer Science, Syracuse University Syracuse, NY 13244, U.S.A. yayan@syr.edu Sanjeev Arulampalam Defence Science and Technology Organisation PO Box 1500, Edinburgh SA 5111, Australia Sanjeev.Arulampalam@dsto.defence.gov.au Aditya Mallick Department of Mathematics University of California at Los Angeles Los Angeles, CA 90095, U.S.A. amallick@ucla.edu Abstract – A relationship between differential geometry and estimation theory was lacking until the work of Bates and Watts in the context of nonlinear parameter estimation. They used differential geometry based curvature measures of nonlinearity (CMoN), namely, the parameter-effects and in- trinsic curvatures to quantify the degree of nonlinearity of a general multi-dimensional nonlinear parameter estimation problem. However, they didn’t establish a relationship be- tween CMoN and the curvature in differential geometry. We consider a polynomial curve in two dimensions and for the first time show analytically and through Monte Carlo simu- lations that affine mappings with positive slopes exist among the logarithm of the curvature in differential geometry, Bates and Watts CMoN, and mean square error. Keywords: Differential Geometry, Extrinsic Curvature, Parameter-effects Curvature, Degree of Nonlinearity, Poly- nomial Nonlinearity, Curvature Measures of Nonlinearity, Mean Square Error, Cram´ er-Rao Lower Bound. 1 Introduction Suppose we have a linear parameter estimation problem for a multi-dimensional non-random parameter with additive Gaussian measurement noise [1],[2]. Then the maximum likelihood (ML) estimate of the parameter can be calculated exactly and an exact analytic expression for the covariance of estimation error is possible [1],[2]. Exact confidence in- tervals for the components of the true parameter can be also calculated [2]. However, if the parameter estimation prob- lem is nonlinear, then the parameter and the covariance of estimation error cannot be calculated exactly in all cases. A nonlinear parameter estimation algorithm commonly uses a linearization approximation or tangent plane approximation [3],[4] in the Taylor series expansion of the nonlinear mea- surement function about an estimate. If the tangent plane approximation is valid, then the ML estimate is asymptot- ically unbiased, has asymptotic minimum mean square er- ror (MMSE) (hence equal to the Cram´ er-Rao lower bound (CRLB) [5]), and is asymptotically normal [4]. However, if the tangent plane approximation is not valid, then these three asymptotic properties don’t hold. Secondly, the confidence intervals for the true parameter are not accurate. For exam- ple, a 95% confidence intervals for a component of the true parameter does not contain the true value 95% of the time. The validity of the tangent plane approximation depends on the degree of nonlinearity [6, 7], [4]. Bates and Watts [6, 7] developed quantitative measures for the degree of nonlinearity of a nonlinear parame- ter estimation problem using differential geometry based CMoN, the parameter-effects and intrinsic curvatures. The parameter-effects curvature depends on the type of parame- terization used for the nonlinear measurement function [3], [4]. Any nonlinear one-to-one mapping of the original pa- rameter can change the parameter-effects curvatures. The intrinsic curvature is an intrinsic property of the nonlinear measurement function and does not depend on the type of parameterization used. Bates and Watts CMoN depend on the Jacobian and Hessian of the nonlinear function at an esti- mate and hence are random variables. Therefore, Bates and Watts curvatures show a great deal of variation with mea- surements, and the mean or median of the Bates and Watts curvature can be regarded as a curvature measure for the nonlinear estimation problem. The curvature or more precisely the extrinsic curvature of curve in 2D at a point in the XY plane is a well known quantity in differential geometry [8], [9], [10]. The intrin- sic curvature of such a curve is exactly zero. Similarly, the Gaussian and mean curvatures of a two-dimensional surface in 3D are also well known quantities [8]. The Gaussian and mean curvatures are intrinsic and extrinsic in nature, respec- tively. Extending these ideas to higher dimensions, we can