## 3 Linear Regression Prediction-Based DMOEA  
In this section, to capture various environment change patterns and exploit the inter-correlations of \(PF\) points (optimal solutions) of the DMOP, we design an LR-based method to predict positions of some points of \(PS\) at the new environment based on their historical positions. In this paper, we consider that the position of each \(PF\) point at the next time is decided by that of all \(PF\) points in history, as exhibited in Figure 1. Due to the high complexity of predicting all \(PF\) points (the entire \(PS\)), we are concerned about a few key points on \(PF\). To be specific, we set key points as \(11\) statistical points (\(10*k\)-th percentile, \(k=0,1,\ldots,10\)) and one center point of the \(PF\), in this paper. Our proposed method is also compatible with other kinds of key points, e.g., knee points, that we will be concerned about for a better prediction. Below we detail our proposed LR-based method. At the beginning of time \(t+1\), we have historical \(PS\) and \(PF\) at all times before \(t\), and need to predict the new \(PS\) and \(PF\) at \(t+1\). Because the manifold of \(PF\) is ever-changing for DMOPs in the real world, we focus on the prediction of key points on the \(PF\) in this paper, which is a powerful guide in updating the population for the DMOEA to adapt to the environmental changes and is the basic idea employed by almost all existing prediction approaches of DMOEAs. We use \(X_{k}^{(t)}\) (\(k=0,1,\ldots,10\)) to represent the \(10*k\)-th percentile of the \(PS\) at \(t\), and \(X_{k}^{(t)}\) (\(k=11\)) to represent the center of the \(PS\) at \(t\), i.e., \(X_{11}^{(t)}=\sum_{X\in PS^{(t)}}X/|PS^{(t)}|\). By LR technology, we can establish the relational model of key points between two continuous environments, as Equation (9). \(A_{k}\) (\(k\in[0,11]\)) and \(B\) are the parameters that will be learned by LR, where \(A_{k}^{(t+1)}\) is an \(n\times n\) matrix and \(B\) is a \(n\) dimensional vector. \(n\) represents the dimension of the search space of the concerned DMOP. \(\widetilde{X_{k}^{(t+1)}}\) represents the estimated/predicted position of \(X_{k}^{(t+1)}\). \[\widetilde{X_{k}^{(t+1)}}=\sum_{k\in[0,11]}A_{k}\cdot X_{k}^{(t)}+B,k\in[0,11]. \tag{9}\]  
The samples used for learning the LR model at time \(t+1\) include \(\langle X^{(t^{\prime})},X^{(t^{\prime}+1)}\rangle\), \(t^{\prime}=0,\ldots,t-1\), which can be easily achieved from the historical \(PS\)s and \(P\)Fs, and where \(X^{(t^{\prime})}\) and \(X^{(t^{\prime}+1)}\) correspond to \(X_{k}^{(t)}\) and \(\widetilde{X_{k}^{(t+1)}}\) (\(k\in[0,11]\)) in Equation (9), respectively. Based on the least-squares method, LR is to learn parameters \(A_{k}^{(t)}\) and \(B^{(t)}\) to minimize the square error (SE) between predicted values and real values for samples. Thus, learning the LR model is solving the minimization problem (10) for the \(k\)-th key point. This optimization problem can be solved by setting the partial derivatives for the parameters to \(0\). But in such a way, the derived prediction model is easily overfitting. Therefore, L2 regularization is applied to raise the ability to extend the model, and the optimization  
Figure 1: The diagram of exploiting the inter-correlations of key points by the LR-based prediction method for the DMOP. The solid curve is the \(PS\) in the time labelled below. The circle represents the key point. The dotted line presents the motion trajectory of the key point. The solid line infers the inter-correlation. objective function (loss function) is transformed into Equation (12). \(\lambda\) is the regularization coefficient that is generally set to a number less than 0.01 [6; 7].