Abstract
We consider measures of nonlinearity (MoNs) of a polynomial curve in two-dimensions (2D), as previously studied in our Fusion 2010 and 2019 ICCAIS papers. Our previous work calculated curvature measures of nonlinearity (MoNs) using (i) extrinsic curvature, (ii) Bates and Watts parameter-effects curvature, and (iii) direct parameter-effects curvature. In this paper, we have introduced the computation and analysis of a number of new MoNs, including Beale’s MoN, Linssen’s MoN, Li’s MoN, and the MoN of Straka, Duník, and S̆imandl. Our results show that all of the MoNs studied follow the same type of variation as a function of the independent variable and the power of the polynomial. Secondly, theoretical analysis and numerical results show that the logarithm of the mean square error (MSE) is an affine function of the logarithm of the MoN for each type of MoN. This implies that, when the MoN increases, the MSE increases. We have presented an up-to-date review of various MoNs in the context of non-linear parameter estimation and non-linear filtering. The MoNs studied here can be used to compute MoN in non-linear filtering problems.
Highlights
The Kalman filter (KF) [1,2,3,4] is an optimal estimator (in the minimum mean square error (MMSE)sense) for a filtering problem with linear dynamic and measurement models with additive Gaussian noise
Can we show that the UKF, cubature KF (CKF), or particle filter (PF) gives better results than the extended Kalman filter (EKF), when the degree of nonlinearity (DoN) is high?
In [35], we showed analytically and through Monte Carlo simulations that affine mappings with positive slopes exist among the logarithm of the extrinsic curvature, Bates and Watts parameter-effects curvature, direct parameter-effects curvature, MSE, and Cramér-Rao lower bound (CRLB)
Summary
The Kalman filter (KF) [1,2,3,4] is an optimal estimator (in the minimum mean square error (MMSE). The normalized MoN that was proposed in [43] was calculated for non-linear filtering problems, including one with the nearly constant turn motion and a non-linear measurement model [45], a video tracking problem using PF [46], and a hypersonic entry vehicle state estimation problem [47]. In these cases, the normalized MoN were rather low. The n−dimensional identity matrix, m−dimensional null matrix, and m × n null matrix are denoted by In , 0m , and 0m×n , respectively
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