Piecewise Linear Regression

a piecewise linear model would require three different functions to explain this curve functions A, B, and C. 

The goal in piecewise regression is to model a nonlinear model by linear pieces, for example, by conducting a microbial inactivation study that is nonlinear. 

This function is easy to model using an indicator variable, because the function is merely two piecewise equations. Only one additional x, is required. 

FIGURE 9.11 (a) Thermal death curve, B. stearothermopMlus spores, (b) Piecewise model points, thermal death curve, (c) Piecewise fit, thermal death curve. 

Example 9.2 In a steam sterilization experiment, three replicate biological indicators B. stearothermophilus spore vials) are put in the sterilizer s cold spot over the course of 17 times of exposure. Each biological indicator has an inoculated population of 1 x 10 CPU spores per vial. The resulting data are displayed in Table 9.22. 

---

When the averages are plotted separately (see Figure 8.7), raie can see that they provide a much different picture than that of the averages pooled. Sex of the subject was confounding in this evaluation. Also, note the interaction. The slopes of A and B are not the same at any point. We will return to this example when we discuss piecewise linear regression using dummy variables. 

Figure 2.33 Albite dissolution rate vs. pH at temperature between 25 and 300 °C. The data at 100, 200 and 300 °C (rhombs, triangles and squares) are after HeUman, 1994 at 25 °C (circles) are after Chou and WoUast, 1985. Grey curves are results of nonlinear regression. Dashed straight lines are results of piecewise-linear regression at 100 °C. Black curves are sums of piecewise-linear regression (Palandri and Kharaka, 2004)...

To obtain the isosteres, one must interpolate the pressure at a given quantity adsorbed at each temperature. To avoid large interpolation errors, particularly at low pressures, the data were fitted piecewise to well behaved polynomial functions by a least squares method. Typical isosteres are shown in Figure 5. The slope of an isostere was determined by linear regression on the interpolated points. The resulting isosteric heat curves are shown in Figure 6. 

In Chapter 9, we discuss piecewise multiple regressions with dummy variables, but the use of linear splines can accomplish the same thing. Knots, again, are the points of the regression that link two separate linear splines (see Figure 7.17). 

In general, a linear model is naturally assumed to be justified. If tests [for coefficients of determination, goodness of fit (GOF) and lack of fit (LOF) see below] show this to be improbable, a nonlinear calibration must be selected, in the simplest ca.se by using quasi-linear models (consideration of higher powers of X polynomials), or piecewise linear calibration can be performed [22]. (A survey of various basic regression models can be found in [23]). 

Qiu, P. The local piecewisely linear kernel smoothing procedure forfitting regression surfaces. 

Given a set of experimental data, we look for the time profile of A (t) and b(t) parameters in (C.l). To perform this key operation in the procedure, it is necessary to estimate the model on-line at the same time as the input-output data are received [600]. Identification techniques that comply with this context are called recursive identification methods, since the measured input-output data are processed recursively (sequentially) as they become available. Other commonly used terms for such techniques are on-line or real-time identification, or sequential parameter estimation [352]. Using these techniques, it may be possible to investigate time variations in the process in a real-time context. However, tools for recursive estimation are available for discrete-time models. If the input r (t) is piecewise constant over time intervals (this condition is fulfilled in our context), then the conversion of (C.l) to a discrete-time model is possible without any approximation or additional hypothesis. Most common discrete-time models are difference equation descriptions, such as the Auto-.Regression with eXtra inputs (ARX) model. The basic relationship is the linear difference equation ... 

However, such a highly derived process would likely make the data abstract. A preferred way, and a much simpler method, is to perform a piecewise regression, using indicator or dummy variables. We employ that method in a later chapter, where we make separate functions for each linear portion of the data set. 

Polynomial regression models are useful in situations in which the curvilinear response function is too complex to linearize by means of a transformation, and an estimated response function fits the data adequately. Generally, if the modeled polynomial is not too complex to be generalized to a wide variety of similar studies, it is useful. On the other hand, if a modeled polynomial overfits the data of one experiment, then, for each experiment, a new polynomial must be built. This is generally ineffective, as the same type of experiment must use the same model if any iterative comparisons are required. Figure 7.1 presents a dataset that can be modeled by a polynomial function, or that can be set up as a piecewise regression. It is impossible to linearize this function by a simple scale transformation. 

The extension of the piecewise regression to more complex designs is straightforward. For example, in bioequivalence studies, absorption and elimination rates are often evaluated over time, and the collected data are not linear. Figure 9.17 shows one possibility. 

---