**3. Measures of Nonlinearity**

To explain the key concepts of nonlinearity, consider the scalar function *h*(*x*) = 5 sin (4*x*)/*x* shown in Figure 1. We observe in Figure 1 that the function is nearly linear at A and E. If we draw a tangent to the curve at A and E, then the curve is close to the tangent in the neighborhood of A and E. However, tangents to the curve at points B, C, and D differ by large amounts from the curve in the neighborhood of these points. The tangent represents an affine approximation to the curve at a point. We observe that, among points B, C, and D, the curve bends the most at B and the least at point D. If we draw a circle (called the osculating circle) at these points, then the radius of the circle can be used to judge nonlinearity. The rate of bending is high when the radius of the circle is small. In differential geometry [37,38], the curvature *κ* is inverse of the radius of the osculating circle and, hence, curvature can be viewed as a measure of nonlinearity. The radii of the osculating circles at A and E are nearly infinity and, hence, the curvatures are nearly zero. From Figure 1, we observe that, in general, the nonlinearity of a function can vary with *x*. Hence, the nonlinearity is a local measure. If the second derivative of a function is non-zero, then the function is non-linear.

In [35,40], we analyzed the CMoN of a polynomial scalar function *h* of a non-random variable *x*, as described in Section 2.1. The CMoN were based on the extrinsic curvature using differential geometry, Bates and Watts parameter-effects curvature, and direct parameter-effects curvature. In this paper, we study the following MoNs:


*Sensors* **2020**, *20*, 3426


**Figure 1.** *y* = 5 sin(4*x*)/*x* versus *x*.

If a MoN has a high value, then the nonlinearity is high and if it has a low value, then the Therefore, it is impossible to compare them based on numerical values. We can only study their variations.

Consider the *m*-dimensional vector non-linear function **h** of the non-random *n*−dimensional parameter **x**. Let **x**ˆ be a known estimate of **x**. Using the Taylor series expansion of **h**(**x**) about **x**ˆ and keeping the first order term gives

$$\mathbf{h(x)} \approx \mathbf{T(x)} = \mathbf{h(\hat{x})} + \mathbf{\dot{H}(x-\hat{x})},\tag{34}$$

where **T**(**x**) represents the tangent plane approximation (an affine mapping) to **h**(**x**) and

$$\mathbf{H} = \frac{d\mathbf{h}(\mathbf{x})}{d\mathbf{x}}\Big|\_{\mathbf{x}=\mathbf{k}}.\tag{35}$$

If *m* > *n*, then **h** is an *n*−dimensional manifold embedded in an *m*−dimensional space [37,38]. The tangent plane is tangent to the surface **h** at **x**ˆ. The concept of tangent plane is used in Beale's MoN, Linssen's MoN, Bates and Watts parameter-effects curvatures [21,25], and direct parameter-effects curvature [44].

For polynomial nonlinearity, the CMoN using differential geometry is calculated at the true value *x* and, hence, it is non-random. The Bates and Watts parameter-effects curvature, direct parameter-effects curvature, Beale's MoN, Li's MoN, and the MoN of Straka et al. are calculated while using an estimate *x*ˆ of *x*. The estimate *x*ˆ is obtained from a measurement model involving the measurement function *h*. Since *x* is a scalar, we need one or more scalar measurements to estimate *x*. Table 1 summarizes features of various MoNs.


**Table 1.** Features of Various MoNs

The CMoN using differential geometry [36–38] is calculated at the true value *x*, whereas the Bates and Watts parameter-effects curvature [21,25,26], direct parameter-effects curvature [29], Beale's MoN, Li's MoN, and the MoN of Straka et al. are calculated while using an estimate *x*ˆ of *x*. The estimate *x*ˆ is obtained from a measurement model involving the measurement function *h*. Since *x* is a scalar, we need one or more scalar measurements to estimate *x*. Next, we describe various MoN.
