*3.3. Least Squares Based Beale's MoN*

Consider the scalar function *h* for polynomial nonlinearity, as described in (1). As described in Beale's MoN, we choose *m* points *xi*, *i* = 1, 2, . . . , *m* in th neighborhood of *x*. Let

$$y\_i = ax\_i^n, \quad i = 1, 2, \dots, m. \tag{42}$$

An affine mapping as approximation to *yi* is given by

$$L(\mathbf{x}\_i) = A + B\mathbf{x}\_i, \quad i = 1, 2, \dots, m. \tag{43}$$

We compute *A* and *B* by minimizing the cost function

$$J(A,B) := \sum\_{i=1}^{m} (y\_i - A - Bx\_i)^2 \tag{44}$$

by the method of least squares (LS) [2,39]. The LS minimization of the cost function yields [52]

$$
\vec{B} = (\mathbb{C}\_{xy} - \mathfrak{x}\ \vec{y}) / (\mathbb{C}\_{xx} - \mathfrak{x}^2),
\tag{45}
$$

$$
\hat{A} = \mathcal{Y} - \hat{B}\mathfrak{x},
\tag{46}
$$

where

$$\mathfrak{x} = \frac{1}{m} \sum\_{i=1}^{m} x\_{i\prime} \quad \mathfrak{y} = \frac{1}{m} \sum\_{i=1}^{m} y\_{i\prime} \tag{47}$$

$$\mathbb{C}\_{xx} = \frac{1}{m} \sum\_{i=1}^{m} x\_i^2, \quad \mathbb{C}\_{xy} = \frac{1}{m} \sum\_{i=1}^{m} x\_i y\_i. \tag{48}$$

Then we can use the affine mapping with *A*ˆ and *B*ˆ in Beale's MoN.
