## 1 Introduction
However, when the nonlinear profile contains significantly unsmooth features (e.g., jumps or non-differentiable points), it might produce undesirable smoothness properties [1]. To deal with the unsmooth features found in complicated profiles, wavelet transform has been advocated by various authors. There are many successful applications [1; 11; 12; 13] of using wavelets for nonlinear profile monitoring. Wavelets are highly competent at function approximation and avoid the problems associated with nonparametric estimators. A typical wavelet-based approach involves selecting wavelet coefficients and constructing a test statistic for profile monitoring. In SPC, process monitoring is often classified into two phases: phase I and phase II. As noted in [14], the main objective of phase I is to analyze collected data to comprehend the  
Figure 1: Over temperature profile studied by [PERSON] et al. [2]. The blue line represents the in-control (IC) profile and the red line represents the out-of-control (OOC) profile. distributional characteristics and to determine whether a process is stable. After eliminating samples associated with any assignable causes in phase I, one can correctly estimate the process parameters associated with the IC state and use them in designing control charts for phase II application. The major objective of phase II is to detect changes in the process rapidly on the basis of the parameters estimated in phase I. [PERSON] et al. [15] identified two significant challenges that need to be addressed in the phase I analysis of nonlinear profiles. The first challenge is the high data dimensionality of nonlinear profiles. First of all, direct analysis of high-dimensional data is infeasible. The second challenge that comes up is the possible presence of outliers in a phase I dataset. The presence of outliers in a historical dataset can have a deleterious effect on phase I parameter estimation. Therefore, recognizing data from the OOC process (called the outlying profiles) and extracting data from the IC process (called the normal profiles) are crucial tasks. The inclusion of OOC phase I data can lead to a biased estimation of the process parameters, which affects the decision function and ultimately leads to increases in the numbers of false alarms and missed detections in phase II monitoring. In this study, an anomaly-detection-based method for nonlinear profile monitoring is proposed to deal with the above-mentioned challenges. The concept of anomaly detection is quite similar to that of phase I and phase II applications in SPC. The relationship between the entire profile monitoring procedure and anomaly detection is illustrated in Figure 2. The main aim of phase I is to estimate the control chart parameters using a set of historical data, which usually comprise data from the IC and OOC conditions. In phase II application, we assume control chart parameters are known. In practical applications, the phase II control chart parameters can be determined using the analysis results of phase I. The more accurate the estimation of the control chart parameters in phase I, the better the performance of the phase II control chart. Anomaly detection involves finding outlier values in a series of data. Novelty detection and outlier-detection algorithms can be applied to identify anomalies [16; 17]. In machine learning, the term anomaly detection is often used synonymously with outlier  
Figure 2: Comparison between statistical process control (SPC) and anomaly detection. UCL, CL, and LCL represent the upper control limit, center line, and lower control limit, respectively. UCL\({}^{*}\), CL\({}^{*}\), and LCL\({}^{*}\) denote the modified upper control limit, center line, and lower control limit, respectively. detection [18; 19; 20; 21]. However, some authors distinguish these two terms [16; 17]. In this study, we make a distinction between the aforementioned two terms. In outlier detection, the objective is to detect the preexisting outliers in a dataset.