Improvements to Measures of Nonlinearity Computation for Polynomial Nonlinearity

Abstract

In our Fusion 2010 paper, we calculated curvature measures of nonlinearity (CMoN) of a polynomial curve in 2D using differential geometry, Bates and Watts and direct parameter-effects curvatures. The parameter-effects curvatures require the Jacobian and Hessian of the measurement function evaluated at the estimated parameter. Previously we obtained the maximum likelihood (ML) estimate of the parameter $x$ by numerical optimization. In this paper, we present analytic expressions for the ML estimate and associated variance. We show through Monte Carlo simulations that the variance of the estimator and the Cramer-Rao lower bound (CRLB) are nearly the same for different powers of x. We also find that the bias error is small and the mean square error (MSE) is close to the CRLB and variance of the ML estimate. Our numerical results show that the average normalized estimation error squared (ANEES) lies within the 99% confidence interval most of the time, indicating that the variance is consistent with the estimation error.