# Model-Based and Model-Free Point Prediction Algorithms for Locally Stationary Random Fields
## 3 Model-Based Point Prediction  
We adopt the time-changing mean and variance model as given by Equation (2). The \(L_{2}\)-optimal predictor of \(Y_{\underline{t}_{n+1}}\) given the data \(Y_{\underline{s}}=\underline{Y}_{\underline{t}_{n}}=(Y_{\underline{t}_{1}}, \ldots,Y_{\underline{t}_{n}})^{\prime}\) is the conditional expectation \(E(Y_{\underline{t}_{n+1}}|\underline{Y}_{\underline{t}_{n}})\). Using model (2) and assuming that \(\underline{W}_{\underline{t}_{n}}\) is weakly dependent it can be shown that [9]:  
\[E(Y_{\underline{t}_{n+1}}|\underline{Y}_{\underline{t}_{n}})=\mu(\underline{t} _{n+1})+\sigma(\underline{t}_{n+1})E(W_{\underline{t}_{n+1}}|\underline{W}_{ \underline{t}_{n}}). \tag{5}\]  
From the above equation, we can see that for Model-Based point prediction we need to estimate the conditional expectation \(E(W_{\underline{t}_{n+1}}|\underline{W}_{\underline{t}_{n}})\) as well as the coordinate changing trend and variance, i.e., \(\mu(\underline{t}_{n+1})\) and \(\sigma(\underline{t}_{n+1})\). **Estimating the conditional expectation**: This is performed by fitting a (causal) AR\((p,q)\) model to the data \(\underline{W}_{\underline{t}_{n}}=(W_{\underline{t}_{1}},\ldots,W_{\underline{t }_{n}})\) with \(p,q\) chosen by minimizing AIC, BIC or a related criterion as described in [15]. Using this framework involves estimating the coefficients of the following 2D AR model defined over a ROS \(H_{p}\) as described in Section 2:  
\[W_{r,s}=\sum_{(j,k)\in H_{p}}\beta_{j,k}W_{r-j,s-k}+v_{r,s} \tag{6}\]  
Here, \(v_{r,s}\) is a 2D white noise process, i.e., an uncorrelated sequence, with mean 0 and variance \(\sigma^{2}>0\). For our point prediction problem setting this implies that  
\[\bar{E}(W_{\underline{t}_{n+1}}|\underline{W}_{\underline{t}_{n}})=\sum_{(j, k)\in H_{p}}\beta_{j,k}W_{\underline{t}_{n_{1}}-j,\underline{t}_{n_{2}}-k} \tag{7}\]  
Figure 2: Prediction point for NSHP. In this drawing, NSHP(\(\underline{t}\)) denotes the nonsymmetric half-plane centered at \(\underline{t}=(t_{1},t_{2})\) covering the hashed area. \(E_{\underline{t}}\) denotes the finite subset of \(Z^{2}\) marked by the red boundary. The intersection of the two gives \(E_{\underline{t},\underline{t}}\). Point prediction is performed at \(\underline{t}=(t_{1},t_{2})\). **Estimating the trend and variance**: This can be performed by using kernel smoothing [24; 25; 26] using 2D kernels, i.e., Nadaraya-Watson (NW) estimation. In addition, since predicting \(Y_{\underline{t}_{k+1}}\) is essentially a boundary problem it is also possible to use local linear fitting, which has been reported to have smaller bias than kernel smoothing for such estimation problems [26; 27; 28]. For time series problems, \(\{Y_{t},\ t\in Z\}\) local linear nonparametric estimation can approximate the trend locally by a straight line whereas for the case of random fields \(\{Y_{t},\ t\in Z^{2}\}\) discussed in this paper local linear estimation can be used to approximate the trend locally with a plane. In order to estimate \(E(W_{\underline{t}_{k+1}}|\underline{W}_{\underline{t}_{k}})\), the stationary data \(W_{\underline{t}_{1}},\ldots,W_{\underline{t}_{n}}\) needs to be estimated. In this case, \(W_{\underline{t}}\) has to be calculated in a one-sided manner for all points including those at the center of the dataset else the obtained values \(\hat{W}_{\underline{t}_{1}},\ldots,\hat{W}_{\underline{t}_{n}}\) will not be stationary, which leads to incorrect estimation of the conditional expectation of \(W_{\underline{t}_{k+1}}\). The one-sided kernel smoothed and local linear estimators used for calculating \(\hat{W}_{\underline{t}_{1}},\ldots,\hat{W}_{\underline{t}_{n}}\) can be defined in two ways as shown in the equations below on NW-Regular, NW-Predictive, LL-Regular and LL-Predictive fitting.