# Model-Based and Model-Free Point Prediction Algorithms for Locally Stationary Random Fields
## 6 Model-Free vs. Model-Based Inference: Empirical Comparisons  
The point prediction performance of the Model-Free and Model-Based predictors described above are empirically compared using simulated as well as real-life data. The Model-Based local constant and local linear methods are denoted as MB-LC and MB-LL, respectively. Model-Based predictors MB-LC and MB-LL are described in Section 3. The Model-Free methods using local constant, local linear (Hansen) and local linear (Monotone) are denoted as MF-LC, MF-LLH, and MF-LLM. Model-Free predictors are described in Section 4. Both fitted and predictive residuals as described in Sections 3 and 4 are used for point prediction and their performance as indicated by Mean Squared Error (MSE) is used to compare the various estimators.  
**Baseline comparisons:** Besides the above MB and MF estimators we also provide three baselines estimators for comparison as described below:  
* Model-Based estimation of \(Y_{\mathbb{I}_{k+1}}\) involves nonparametric mean and variance estimation followed by estimating the conditional expectation \(E(W_{\mathbb{I}_{k+1}}|\underline{W}_{\mathbb{I}_{k}})\) which involves calculating the coefficients of the 2D AR model as given by Equation (6). In this case as a baseline we have included results for both the synthetic and real-life datasets using only local linear estimation of the mean i.e., in this case the \(L_{2}\)-optimal predictor of \(Y_{\mathbb{I}_{k+1}}\) is given by: \[\hat{Y}_{\mathbb{I}_{k+1}}=\mu(\mathbb{I}_{k+1})\] (47) Here, \(\mu(\mathbb{I}_{k+1})\) is calculated using local linear fitting based on Equations (23)-(30) (in this case regular and predictive fitting are the same as stated in Remark 2.2 in [9]). In Tables 1 and 2 this estimator is shown as **LL**.
* Model-Free estimation of \(Y_{\mathbb{I}_{k+1}}\) involves nonparametric estimation of the first marginal distribution followed by estimating the autocovariance matrix \(\hat{\Gamma}_{n}^{AR}\). In this case, as a baseline we have included results using only local linear estimation of the uniformizing transformation as given by Equations (34) and (35) (Local Linear Hansen) and Algorithm 1 (Monotone Local Linear Distribution Estimation). For the same reasons as stated above, regular and predictive fitting are also the same in this case. The \(L_{2}\)-optimal predictor of \(Y_{\mathbb{I}_{k+1}}\) in this case is given by: \[\hat{Y}_{\mathbb{I}_{k+1}}=\frac{1}{M}\,\sum_{i=1}^{M}D_{\mathbb{I}_{k+1}}^{-1 }(u_{i})\] (48)Here, \(u_{1},\ldots,u_{M}\sim U[0,1]\) where \(M\) is some large integer, \(U\) is the uniform distribution and \(D_{l_{n+1}}\) is estimated by using \(\tilde{D}^{LLH}_{l_{n+1}}(\cdot)\) or \(\tilde{D}^{LLM}_{l_{n+1}}(\cdot)\). In Tables 1 and 2 these estimators are shown as **LLH, LLM**.  
The code for all algorithms used for the synthetic and real-life datasets as discussed in this paper can be found under [[https://github.com/srinjoyd/randomfields_pp](https://github.com/srinjoyd/randomfields_pp)]([https://github.com/srinjoyd/randomfields_pp](https://github.com/srinjoyd/randomfields_pp)) (accessed on 24 July 2023).