*3.2. Beale's MoN*

Consider the nonlinear measurement model for the non-random *n*-dimensional parameter **x**

$$\mathbf{z} = \mathbf{h}(\mathbf{x}) + \mathbf{v},\tag{39}$$

where **z**, **h**, and **v** are the measurement, non-linear measurement function, and measurement noise, respectively. Let **x**ˆ be an estimate of **x**. Subsequently, a Taylor series expansion of **h**(**x**) about

**x**ˆ and keeping the first order term is as (34). Suppose we choose *m* vectors **x***i*, *i* = 1, 2, ... , *m* in the neighborhood of **x**. Then Beale's first empirical MoN [18] is given by

$$\hat{N}\_{\mathbf{x}} = \rho^2 \frac{\sum\_{i=1}^{m} ||\mathbf{h}(\mathbf{x}\_i) - \mathbf{T}(\mathbf{x}\_i)||^2}{\sum\_{i=1}^{m} ||\mathbf{h}(\mathbf{x}\_i) - \mathbf{h}(\hat{\mathbf{x}})||^4},\tag{40}$$

where *ρ* is the standard radius and it is defined by

$$\rho^2 := \|\mathbf{z} - \mathbf{h}(\mathbf{\hat{x}})\|^2 / (n(N - n)). \tag{41}$$

Guttman and Meeter [19] observed that the empirical MoN underestimates severe nonlinearity. When *m* approaches infinity, the empirical MoN *N*ˆ **<sup>x</sup>** approaches the theoretical MoN *N***x**.
