## 1 Introduction  
In statistical process control (SPC), control charts are applied to determine whether a manufacturing or business process is in statistical control. They have been widely used to monitor changes in quality characteristics over time. Control charts can be used to monitor various types of data such as mean, variance, reject rate, and defect rate of a process. In certain circumstances, we need to simultaneously monitor two or more related quality characteristics. In this case, multivariate control charts are often used to detect changes in the mean vector and covariance matrix of various quality characteristics. In some applications of SPC, the quality of a process or product (called response \(y\)) is functionally dependent on one or more explanatory variables (also called independent variables, \(x\)). A single observation of an in-control (IC) process is composed of \(n\) pairs of \((x,y)\) data that can be expressed as \(y=f^{0}(x)+\varepsilon\), where \(f^{0}\) is a known function and \(\varepsilon\) is random noise with a mean of 0 and a standard deviation of \(\sigma\)[1]. The plotted data will take the shape of a curve, which is known as a profile in the literature. Figure 1 depicts the oven temperature profile studied by [PERSON] et al. [2]. The temperatures plotted in this profile were measured over approximately equal time intervals by sensors installed in an oven. Inthe aforementioned profile, the dependent variable is temperature, and the independent variable is sampling time. The aim of profile monitoring is to determine when a profile has changed, as this will indicate an out-of-control (OOC) condition, with assignable causes of variation that must be identified and eliminated. Many control chart methods have been developed to address the profile monitoring task. In [3], the authors provided an in-depth review of examples of profiles and discussed various profile monitoring methods. Recently, [PERSON] et al. [4] conducted a survey of the literature on profile monitoring. They analyzed related papers by using different metrics and recommended directions for future research. Many studies have been conducted on linear profiles, whose functions are assumed to be simple straight lines. Most existing methods for linear profile monitoring involve monitoring changes in the intercept and slope and variations in the standard deviation of error by using control chart methods [5; 6; 7; 8]. [PERSON] et al. [1] pointed out that monitoring the parameters of a linear curve is not efficient because the process may have changed so that the resulting curve is no longer linear, but using a straight line fit may obtain the same parameter estimates. Thus, one might be unable to discern the lack of linearity [1]. As an alternative to a linear profile, a nonlinear profile may be more appropriate to describe the quality of a process or product in some situations. A successful application of a multivariate nonlinear profile using the parametric form can be found in [9]. However, there are some drawbacks to using the parametric profile modeling approach. Similar to linear modeling, the nonlinear profile may have changed, but the estimating method may obtain the same parameter estimates. In addition, it is not easy to determine a suitable parametric form to fit a nonlinear profile [1]. Recently, nonparametric approaches have received considerable attention in monitoring nonlinear profiles. They have great benefits in modeling complicated profiles. For instance, [PERSON] et al. [10] used a smoothing spline to monitor the thickness variation of a semiconductor wafer. [PERSON] et al. [9] used nonparametric forms for monitoring nonlinear profiles. They also applied the spline smoothing method to nonlinear profile data. The spline smoothing method can be thought of as a nonparametric estimator for smoothing data. It has greater flexibility than parametric methods.