*Article* **Analysis of Polynomial Nonlinearity Based on Measures of Nonlinearity Algorithms**

**Mahendra Mallick 1,\*,†,‡ and Xiaoqing Tian <sup>2</sup>**


Received: 31 March 2020; Accepted: 10 June 2020; Published: 17 June 2020

**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 Simandl. 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.

**Keywords:** polynomial curve in 2D; measures of nonlinearity (MoNs); extrinsic curvature; Beale's MoN; Linssen's MoN; Bates and Watts parameter-effects curvature; direct parameter-effects curvature; Li's MoN; MoN of Straka, Duník, and Simandl; maximum likelihood estimator (MLE); ˘ Cramér-Rao lower bound (CRLB)
