# Model-Based and Model-Free Point Prediction Algorithms for Locally Stationary Random Fields
## 6 Model-Free vs. Model-Based Inference: Empirical Comparisons
### Simulation: Additive Model with Stationary 2D AR Errors  
Let a random field be generated using the 2D AR process as below:  
\[\begin{split} W_{\underline{t}}=W_{t_{1},t_{2}}& =0.25W_{t_{1}-1,t_{2}-1}+0.2W_{t_{1}-1,t_{2}+1}\\ &-0.05W_{t_{1}-2,t_{2}}+v_{t_{1},t_{2}}\end{split} \tag{49}\]  
Let this field be generated over the region defined by \(0\leq t_{1}\leq n_{1}\) & \(0\leq t_{2}\leq n_{2}\) where \(n_{1}=101,n_{2}=101\). The NSHP limits are set from \((101,101)\) to \((50,50)\), this defines the region \(E_{L\underline{t}}\) as shown in Figure 2. The data \(Y_{\underline{t}}\) is generated using the additive model in Equation (1) with trend specified as \(\mu(\underline{t})=\mu(t_{1},t_{2})=\sin(4\pi\frac{t_{2}-1}{t_{2}-1})\) where \(0\leq t_{1}\leq n_{1}\) & \(0\leq t_{2}\leq n_{2}\). Here \(v_{t_{1},t_{2}}\) are i.i.d. \(N(0,\tau^{2})\) where \(\tau=0.1\). Let \(t_{1}=50,t_{2}=50\) where point prediction is performed. Bandwidths for all Model-Based, Model-Free, and baseline predictors are calculated using cross-validation as described in Section 5.  
Results for point prediction using mean square error (MSE) over all MB and MF methods are shown in Table 1. A total of 100 realizations of the dataset were used for measuring point prediction performance. From this table it can be seen that MB-LL is the best point predictor. This is expected since the data was generated by a 2D AR model which is the same used in MB-LL prediction. In addition, the estimation is performed at the boundary of the random field with a strong linear trend as shown in Figure 3 where LL regression is expected to perform the best. In addition, it can be observed that MF-LLM performs the best among all MF point predictors and approaches the performance of MB-LL. This shows that monotonicity correction in the LLM distribution estimator has minimal effect on the center of the distribution that is used for point prediction.  
In addition, comparing the performance of MB-LL versus its corresponding baseline LL and that of MF-LLH and MF-LLM versus their corresponding baselines LLH and LLM, respectively, show that the baseline estimators underperform as they do not take into  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Prediction Method** & **Residual Type** & **MSE** \\ \hline MB-LC & P & \(1.488\times 10^{-2}\) \\ \hline  & F & \(1.520\times 10^{-2}\) \\ \hline MB-LL & P & \(1.393\times 10^{-2}\) \\ \hline  & F & \(1.400\times 10^{-2}\) \\ \hline MF-LC & P & \(1.530\times 10^{-2}\) \\ \hline  & F & \(1.549\times 10^{-2}\) \\ \hline MF-LLH & P & \(1.471\times 10^{-2}\) \\ \hline  & F & \(1.515\times 10^{-2}\) \\ \hline MF-LLM & P & \(1.414\times 10^{-2}\) \\ \hline  & F & \(1.456\times 10^{-2}\) \\ \hline LL & Not Applicable & \(1.488\times 10^{-2}\) \\ \hline LLH & Not Applicable & \(1.651\times 10^{-2}\) \\ \hline LLM & Not Applicable & \(1.455\times 10^{-2}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Point Prediction performance for 2D AR dataset.  
account the spatial dependence present in the data either by estimating the coefficients of the 2D AR model (Equation (6) as in the Model-Based case) or by estimating the autocovariance matrix \(\hat{\Gamma}_{n}^{AR}\) (Model-Free case).