1 Introduction

As part of an experimental process, it is common to face up to situations where only a limited amount of data is available and the estimation of values between consecutive data points is needed, either for predicting results or inferring conclusions from data. Although this problem is traditionally approached by function approximation methods based on polynomial interpolation, a preferred alternative can be found in the approximation techniques based on piecewise-linear models ^{7, 16, 17, 5, 4} which consists in connecting consecutive data points by straight line segments through a continuous function. However, piecewise-linear models have the shortcomings of having zero curvature between data points, exhibiting abrupt changes at the breakpoints, and having undefined derivatives. It is precisely the lack of differentiability what limits their application in such cases where, as a result of derivative computations, the function dnf(x)dxn (where n indicates a derivative order and f(x) is an approximate function) becomes undefined or zero. With the purpose of incorporating the derivation capability into piecewise-linear functions, the strategy of smoothing the basis-function in a piecewise-linear model is adopted. In accordance with numerical simulations performed in a later section, it can be demonstrated the efficacy of this proposal. Although there are many relevant piecewise-linear models, in this paper the High Level Canonical model developed by Pedro-Julian et al. ¹² is chosen as a study case due to the fact that it can be constructed by a well-known algorithmic methodology, and by such methodology, its basis-function can easily be identified.

The paper is organized as follows. In Section 2, the absolute-value function that serves as kernel for the High Level Canonical piecewise-linear model is presented. Section 3 describes the methodology for constructing smooth-piecewise functions. Section 4 shows the application of such methodology by illustrative examples (for one- and two-dimensional domains). In Section 5 a comparative analysis and discussion about the curve fitting accuracy that can be achieved through the proposed strategy is exposed. The comparison is done among polynomial, splines, smooth-piecewise, and standard piecewise-linear approximation techniques. Finally, Section 6 presents the concluding remarks of this work.

2 Absolute-value Basis-function

A relationship between piecewise-linear models and the absolute-value function is established and reported in many references. Mathematical expressions such as the canonical form of Chua-Kang ^{3, 15, 1, 2}, the High Level Canonical representation of Pedro-Julian et al. ^{11, 8}, the Guzelis ⁶ model, and the Kahlert ^{13, 14} description are emblematic examples where this function appears implicitly. Specifically, in the High Level Canonical model the basis-function (also expressed in terms of absolute-value functions) is outlined and easily identified in a detailed methodology by which the approximation function can be constructed. Moreover, this construction methodology is widely reported in literature ^{10, 9}. For these reasons, the smoothing technique which is proposed in this paper takes as reference the basis-function γ(f_i , f_j) used in the Pedro-Julian et al. ⁸ model.

where f_i and f_j are linear equations used to sketch the linear partitions of the function domain.

3 Methodology for the Construction of Smooth-Piecewise Functions

Based on the methodology of Pedro-Julian et al. ⁸, our proposal for the construction of smooth-piecewise functions can be summarized as follows.

where the plus function (x)₊ can be approximated by

After combining equation (3) with equation (2), the following smooth approximation is obtained:

After numerical simulations on equation (4), a slight deviation from the absolute-value function can be observed. In order to obtain more accurate fitting, a constant β is included as

However, taking into account the logarithmic change-of-base formula log⁡(x)=ln⁡(x)ln⁡(10), equation (5) can be recast as follows:

being α = 10⁶ and β = 2.3 appropriate values to achieve small tolerance results. An absolute-value function approximation for α = 4, 6, 8, ..., 20 and β = 2.3 can be observed in Figure 1. In this figure the thin traces (dashed lines) correspond to these approximations while the thickest one corresponds to the exact absolute-value function y(x) = |x|. The thin uppermost curve represents the plot for (α = 4, β = 2.3) and the lowest curve for (α = 20, β = 2.3).

Fig. 1 Approximations for β = 2.3 and α-values, α = 4, 6, 8, ...,20 

For a one-dimensional domain only vertical line partition equations will be presented and k = 0, 1, 2, ... , (s₁ - 1).

where n indicates the number of vertexes. For the one-dimensional case, the above procedure is practically the same with the only difference of evaluating two classes of vertexes (zero- and one-, type).

This step is indistinctly implemented in the construction of one-dimensional and two-dimensional functions.

4 Simulation Results

In this section, three examples to illustrate the effectiveness of the proposed smoothing strategy are presented. We use the software Maple Release 15 to build the mathematical models and plot the approximate functions.

Example 1: Let the sequence of data points D = {(0,0), (1,2), (2,1), (3,3), (4, -1), (5,5)} be used to determine a smooth-piecewise function y_s(x) in the range [0,5] of x. In order to have a comparative reference, the methodology of Pedro-Julian et al.[8] is used to obtain a piecewise-linear function that satisfies the condition of constructing new data points within the range of [0, 5]. As a result, the function y(x) is obtained.

(9)

In accordance with the construction methodology presented in the previous section, a smooth function y_s(x) can be determined by replacing the γ basis-function by its logarithmic approximation reported in equation (6). With aid of Maple software, the resulting smooth expression can be simplified in a more compact algebraic form as

With P=ln⁡(1+eGix+Hi)+ln⁡(1+eLix+Mi), where the function parameters for a constant K = -8.54611 are summarized in Table 1.

In Figure 2 the dashed line shows the curve of the function y(x) while the solid line depicts the curve for the smooth-piecewise function y_s(x).

Fig. 2 Function approximation: piecewise-linear (equation (9), dashed line) and smooth-piecewise (equation (10), solid line) 

An error between the piecewise-linear function and its smooth version can be estimated by the percent error formulation ϵ=100×1-ys(x)y(x). e = 100 x Figure 3 shows a curve of the percent error ϵ(x).

Fig. 3 Percent error between y_s(x) and y(x) 

From this figure two important results must be underlined: the error is distributed along each linear segment and a precise fitting is achieved at the breakpoints locations. Moreover, an interesting property can be observed if the differentiability of functions y(x) and y_s(x) is put under testing. As can be noted in Figure 4, for the function y(x) only the first order derivative is defined (except in the breakpoints) while the n-order derivative (for all n ≥ 1) is zero or not defined.

Fig. 4 First and second derivatives of y(s) 

In contrast, it must be noted that the existence of the higher order derivatives is guaranteed in the smooth function due to the fact that its second derivative is continuous everywhere.

As an example, Figure 5 shows curves for the first and second order derivatives of y_s(x).

Fig. 5 First and second derivatives of y_s(x) 

An important observation regarding to equation (10) is that it can be decomposed into two component functions which can be derived from the parameters reported in Table 1: one for i = 1,···,20, denoted as y_l(x), and another for i = 21,···,30, expressed as y_nl(x). This means that the smooth piecewise function can be expressed as y_s(x) = y_l(x) + y_nl(x), with a nonlinear behavior of the characteristic curve of y_s(x) mainly contained in y_nl(x), and a quasilinear behavior strongly related to y_l(x). This allows to simplify the smooth-piecewise function in a more compact expression by approximating y_l(x) with a line equation y_le(x) (for this example, y_le(x) = 2.0666667x + 14.84611591). As expected, a good approximation of y_s(x) is obtained by y_nl(x) + y_le(x). Considering this result, equation (10) can be expressed (also in reference to Table 1) in a more compact form:

In Figure 6 graphs of y_s(x), y_l(x), y_nl(x), and y_le(x) functions are shown.

Fig. 6 Components of the smooth-piecewise function y_s(x) 

Example 2: In this example it is assumed that the only input data that is available to construct an interpolation function z (x, y) is the set of points D.

D={(0,0,1),(1,0,0),(2,0,2),(3,0,0),(0,1,0),(0,2,1), (3,3,0),(1,1,0),(1,2,0),(1,3,2),(2,1,1),(2,2,1),(2,3,0), (3,1,0),(3,2,0),(3,1,3)}.

By following the traditional construction methodology of Pedro-Julian et al. ⁸, a two dimensional piecewise-linear function z=fHS[S](x,y) can be obtained. Such function is defined over a compact domain S equally partitioned by a simplicial boundary configuration H_S. In accordance with the methodology presented in the previous section, a smooth version of fHS[S](x,y) can be obtained if the same construction methodology is applied and the absolute-value basis-function is replaced by the exponential approximation of equation (6). The resulting smooth function has the property of removing the abrupt variation at the breakpoints while preserving the waveform pattern. This fact ensures the differentiability of the function. In Figure 7 and Figure 8, surfaces for the piecewise-linear function z(x, y)=fHS[S](x,y) and its smooth version are shown.

Fig. 7 Surface for the two-dimensional simplicial piecewise-linear function obtained from the input data D of Example 2 

Fig. 8 Surface for the two-dimensional smooth-piecewise function obtained from the input data D of Example 2 

Similarly to what was done in the one-dimensional example, a precise curve fitting was achieved at the breakpoint coordinates while the maximum deviation between the curves which are depicted in Figure 7 and Figure 8 is observed inside of each simplex in the XY plane. In Figure 9 this effect can be graphically appreciated. Additionally, it must be observed that in contrast with the original piecewise-linear function where the second and higher order derivatives are null, in the smooth function this problem is overcome. In order to illustrate this numerical advantage, surfaces for the first and second derivative of z(x, y)=fHS[S](x,y) are shown in Figure 10 and Figure 11, respectively.

Fig. 9 Deviation between the piecewise-linear and the smooth curve obtained from the input data D of Example 2 

Fig. 10 Derivatives of function z(x, y)=fHS[S](x,y). (a) First order derivative. (b) Second order derivative 

Fig. 11 Derivatives of function z(x, y)=fHS[S](x,y): (a) First order derivative, (b) Second order derivative 

Example 3: This example illustrates a potential application of the smoothing proposal in a typical problem of control systems engineering. The black curve of Figure 12 represents the transient response of a second-order system for a unit-impulse excitation. Consider that we are interested in determining the time t_p when the maximum peak of this response occurs. Let us consider that the analytical transfer function is not accessible and only a set of equally spaced breakpoints (discrete measurements) is the input data available to construct an approximate function. By following the construction strategy exposed in section 3, the two approximations depicted in Figure 12 can be obtained: piecewise-linear c_pwl(t) (red) and smooth-piecewise c_s(t) (blue).

Fig. 12 Approximate functions for the transient response c(t): smooth-piecewise c_s(t), piecewise-linear c_pwl(t) 

In Figure 12, the corresponding smooth and piecewise-linear functions are expressed in the form

and

with the parameters summarized in Table 2 and Table 3, respectively.

From this set of data, we may determine the peak time (t_p) by differentiating the approximate function (c_pwl(t) or c_s(t)) with respect to time (t) and setting this derivative equal to zero

or

However, for the piecewise-linear function a numerical solution cannot be reached, hence the smooth function c_s(t) appears as the better approximation alternative. After solving equation (15) with Maple software we obtain t_p = 1.193s, which clearly corresponds to the peak time. The piecewise-linear c_pwl(t) and the smooth-piecewise c_s(t) approximations of the impulse transient response, as well as their derivatives with respect to time, are shown in Figure 13.

Fig. 13 Approximate functions and derivatives for the transient response: piecewise-linear and smooth-piecewise 

In this figure, it is important to observe the notorious difference that exists, from the point of view of function continuity, between the derivatives of these two approximate functions: while dcs(t)dt is completely connected at all points, dcpwl(t)dt has discontinuities at the breakpoints (plotted as the red points).

5 Comparative Discussion

In this section, a comparative discussion about the curve fitting performance of the proposed method against other strategies such as polynomial and cubic spline techniques is presented. To start this analysis, it is important to mention that a better curve fitting can be achieved by the smooth-piecewise method, this is because it always imposes on the function the condition of passing through the input data coordinates. Similarly to the piecewise-linear functions, the continuity is also preserved in all the function domain but unlike the piecewise-linear reference, the smoothing proposal adds the function differentiation capability. In contrast, although in polynomial interpolation the function continuity is guaranteed, curve fitting is not very precise what results in a remarkable curve deviation that increases the derivative growing very quickly. An alternative approach to minimizing the curve fitting error is the spline interpolation which consists in restricting the approximation to low degree polynomials (typically, like in our comparative analysis, polynomials of third order degree are preferred) over partitioned sections along the function domain. However, in spite of showing a better curve fitting in comparison to the polynomial counterpart, the best performance is clearly obtained by the smooth-piecewise proposal. To emphasize this characteristic, the smooth-piecewise, polynomial, and spline approximate functions for D_C = {(0, 5), (1, 3), (2,10), (3,0), (4, 3), (5,1), (6,10)} are shown in Figure 14.

Fig. 14 Approximate functions for the input data Dc: piecewise-linear, smooth-piecewise, polynomial, and splines 

The error in these approximations can be estimated by the deviation e among the smooth-piecewise, polynomial, and splines curves with respect to the piecewise-linear reference. Such deviations are depicted in Figure 15.

Fig. 15 Curve deviation with respect to the piecewise-linear reference: smooth-piecewise, polynomial, and splines 

6 Conclusion

In this paper, the proof-of-concept related to obtaining smooth-piecewise functions by replacing the basis-function used in the construction methodology of the High Level Canonical piecewise-linear model was demonstrated by numerical simulations. In accordance with such simulations, smooth-piecewise functions not only preserve curve fitting accuracy but also incorporate derivation capability. By illustrative examples, it was observed that the error between the original piecewise-linear curve and its smoothing version is uniformly distributed along each linear partition, being more pronounced approximately before and after its middle location and extremely reduced around the breakpoints. Moreover, two very important observations must be highlighted: the smooth function can be decomposed into two representations (one nonlinear and other quasilinear), and the number of terms of the resulting smooth function can significantly be reduced due to the fact that a great number of them can be approximated by a line equation. Although its potential application to practical engineering problems was illustrated in Example 3, now our ongoing work centers on exploring alternative applications for this smoothing strategy.