## 2 Methods
### PIHCP Evaluation (LASSO)  
To determine the optimal number of potential predictive risk factors for the PIHCP model, the LASSO process was performed with cross-validation. PIHCP predictive values for each patient were calculated using the following formula [13], as outlined in our previous reports [10,14,15]:  
\[PIHCP=\frac{1}{1+e^{-S}}\text{ where }S=\beta_{0}+\sum\
olimits_{i=1}^{n}\beta_{i} \cdot x_{i} \tag{1}\]  
where \(\beta_{i}\) is the regression coefficient, xi represents the different predictive factors, n is the number of predictive factors in the model, and \(\beta_{0}\) is a constant coefficient.  
The details of the LASSO-based model were presented in previous reports [7,8,14,16]. The following equation was used to identify the predictive factors and shrink the coefficients:  
\[\argmin_{\beta}||Y-X\beta||^{2}\text{ subject to }||\beta||=\sum\
olimits_{j=0}^{d} \abs{\beta_{j}}\ \leq\ t \tag{2}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline
**No.** & **Factors** & **Range or Classification** & **Median or Frequency** & **Correlation** \\ \hline
1 & Age & 21–77 & 46 & \(-\)0.001 \\
2 & IPL & 0–10 & 0 & 0.197 \\
3 & AHA Peels & 0–11 & 0 & 0.458 \\
4 & Sonophoresis & 0–41 & 0 & 0.175 \\
5 & Skin Care & 0–96 & 0 & \(-\)0.129 \\
6 & TF & 2–28 & 6 & \(-\)0.018 \\
7 & LD & 600–1000 & 750 & \(-\)0.002 \\
8 & Melasma & 0, 1 * & 81, 44 & 0.969 \\
9 & FSPT & 0, 1, 2 & 85, 27, 13 & \#1 \\
10 & ZN & 0, 1 * & 115, 10 & \(-\)1.271 \\
11 & Lentigines & 0, 1 * & 45, 80 & \(-\)0.378 \\
12 & Acnes & 0, 1 * & 109, 16 & 1.052 \\
13 & Pores & 0, 1 * & 85, 40 & 0.773 \\
14 & LM & 0, 1 * & 85, 40 & 1.431 \\
15 & IEL & 0, 1, 2 & 95, 21, 9 & \#2 \\ \hline \end{tabular} Abbreviation: IPL: intense pulse light; AHA Peels: \(\alpha\)-hydroxyl acid peels; TF: HELIOS II Treatment Frequency; LD: Laser Dosage; FSPT: Fitzpatrick skin phototype; ZN: Zygomatic Nevus; LM: Laser Mode; IEL: Immediate Endpoints of Laser, * 0 = No, 1 = Yes; \#1: F(0) = 0, F(1) = \(-\)1.533, F(2) = \(-\)1.175, \#2: I(0) = 0, I(1) = \(-\)2.379, I(2) = \(-\)0.615.  
\end{table}
Table 2: Predictive factors in the post inflammatory hyperpigmentation dataset.  
where \(t\) represents the tuning parameters that can be determined by cross-validation and d indicates the number of variables selected. The details were reported previously [7; 14; 17].  
An online MATLAB (MathWorks, Natick, MA, USA) package (LASSOGLM function) was downloaded and applied to fit the LASSO model on a HP Workstation Z640 (Hewlett-Packard, US). The LASSO method was performed with cross-validation (10 times) and the parameters (\(\lambda\) and \(\alpha\) values) were set based on the previous studies [3; 4; 18]. The default settings used for \(\lambda\)and \(\alpha\) (the elasticnet mixing parameter) were set to 1. In this study, the PIHCP probability model was derived from multiple logistic regression. In medical research, dependent variables are often referred to as outcome variables; independent variables are risk factors. The relevant details can be found in previous studies [10; 14].  
The system's performance measures were tested using the Nagelkerke R\({}^{2}\) and Omnibus tests, scaled Brier score, area under the receiver operating characteristic curve (AUC), Hosmer-Lemeshow (HL) test, and negative predictive value (NPV) [15; 19; 20]. The data were processed using SPSS Statistics 19 (SPSS Inc., Chicago, IL, USA) and MATLAB R2013a (MathWorks, Natick, MA, USA). In all analyses, \(p<0.05\) was taken to indicate statistical significance.