*3.8. MoN of Straka, Duník, and Simandl ˘*

Straka, Duník, and Simandl presented two local MoNs in [ ˘ 48,49]. Given the estimate *x*ˆ and variance *σ*<sup>2</sup> *<sup>x</sup>* , these MoNs use a number of points *χi*, *i* = 1, 2, ... , *m* in the neighborhood of *x*ˆ. We analyze the first MoN proposed by the authors. The transformed points by the non-linear function *h* are given by

$$z\_i = h(\chi\_i), \; i = 1, 2, \dots, m. \tag{68}$$

Define

$$\mathbf{Z} := \begin{bmatrix} z\_1 & z\_2 & \dots & z\_m \end{bmatrix}',\tag{69}$$

$$\mathbf{X} := \begin{bmatrix} \ \chi\_1 & \ \chi\_2 & \ \cdots & \ \chi\_m \end{bmatrix}'. \tag{70}$$

A linear approximation to **Z** is **X***θ*, where *θ* is a scalar parameter to be estimated. The cost function that is proposed in [48,49] to determine *θ* is given by

$$J\_1(\theta) := (\mathbb{Z} - \mathbf{X}\theta)' \mathbf{W}(\mathbb{Z} - \mathbf{X}\theta),\tag{71}$$

where the weight-matrix **W** is given by

$$\mathbf{W} = \text{diag}(d\_1, d\_2, \dots, d\_m), \tag{72}$$

$$d\_i = \left(\chi\_i - \hat{\mathfrak{x}}\right)^2, \; i = 1, 2, \dots, m. \tag{73}$$

The LS estimate [39] that minimizes the cost function is given by

$$\boldsymbol{\theta}\_{\rm LS} = (\mathbf{X}^\prime \mathbf{W} \mathbf{X})^{-1} \mathbf{X}^\prime \mathbf{W} \mathbf{Z}. \tag{74}$$

For this problem, the LS estimate in (74) reduces to

$$\theta\_{\rm LS} = \left(\sum\_{i=1}^{m} d\_i \chi\_i^2\right)^{-1} \sum\_{i=1}^{m} d\_i \chi\_i z\_i. \tag{75}$$

The cost function *J*<sup>1</sup> evaluated at ˆ *θ*LS is treated as a local MoN *η*, given by

$$
\eta = f\_1(\hat{\theta}\_{\rm LS}).\tag{76}
$$

**Remark 6.** *We have calculated the average MoN for the bearing-only filtering [27], GMTI [32], and video filtering [34] problems. The MoN is presented in the table below (Table 2). From this table we find that the degree of nonlinearity of the bearing-only filtering problem is about two orders of magnitude higher than that of the GMTI or video filtering problem. This implies that a simple filter, such as the EKF or UKF, is sufficient for the GMTI or video filtering problem, but an advanced filter, such as the PF, is needed for the BOF [17] problem.*

**Table 2.** MoNs for the bearing-only, GMTI, and video filtering problems.

