# Model-Based and Model-Free Point Prediction Algorithms for Locally Stationary Random Fields
## 4 Model-Free Point Prediction  
For the Model-Based case, Equation (2) accounts for spatially-changing mean and variance of \(Y_{\underline{t}}\). More generally, however, it may happen that the random field \(\{Y_{\underline{t}}\) for \(\underline{t}\in Z^{2}\}\) has a nonstationarity in its third (or higher moment), and/or in some other feature of its \(m\)th marginal distribution. This is addressed by the Model-Free Prediction Principle of [PERSON] (2013, 2015). The key towards Model-Free inference is to be able to construct an invertible transformation \(H_{\underline{n}}:\underline{Y}_{\underline{t}_{n}}\mapsto\underline{c}_{n}\) where \(\underline{Y}_{\underline{t}_{n}}=(Y_{\underline{t}_{1}},Y_{\underline{t}_{2} },\ldots,Y_{\underline{t}_{n}})\) denotes the random field data under consideration and \(\underline{c}_{n}=(\epsilon_{1},\ldots,\epsilon_{n})^{\prime}\) is a random vector with i.i.d. components. In order to do this in our context, let some \(m\geq 1\), and denote by \(\mathcal{L}(Y_{\underline{t}_{k}},Y_{\underline{t}_{k-1}},\ldots,Y_{\underline{ t}_{k-m+1}})\) the \(m\)th marginal of \(Y_{\underline{t}_{k}}\), i.e., the joint probability law of the vector \((Y_{\underline{t}_{k}},Y_{\underline{t}_{k-1}},\ldots,Y_{\underline{t}_{k-m+1}})^ {\prime}\). We assume that \(\mathcal{L}(Y_{\underline{t}_{k}},Y_{\underline{t}_{k-1}},\ldots,Y_{\underline{ t}_{k-m+1}})\) changes smoothly (and slowly) with \(\underline{t}_{k}\) in order to use nonparametric smoothing for estimation. In this case, \(\{Y_{\underline{t}_{k}},\ \underline{t}_{k}\in Z^{2}\}\) can be defined over a 2D index-set \(D\) and the set \((Y_{\underline{t}_{k}},Y_{\underline{t}_{k-1}},\ldots,Y_{\underline{t}_{k-m+1}})\) can be considered to be lexicographically ordered as discussed previously in Section 2. Similar to the framework proposed in [9] in order to ensure both the smoothness and data-based consistent estimation of \(\mathcal{L}(Y_{\underline{t}_{k}},Y_{\underline{t}_{k-1}},\ldots,Y_{\underline {t}_{k-m+1}})\) we assume that, for all \(\underline{t}_{k}\),  
\[Y_{\underline{t}_{k}}=\mathbf{f}_{\underline{t}_{k}}(W_{\underline{t}_{k}},W_{ \underline{t}_{k-1}},\ldots,W_{\underline{t}_{k-m+1}}) \tag{31}\]  
for some function \(\mathbf{f}_{\underline{t}_{k}}(w)\) that is smooth in both arguments \(\underline{t}_{k}\) and \(w\), and some strictly stationary and weakly dependent, univariate series \(W_{\underline{t}_{k}}\) where without loss of generality, it is assumed that \(W_{l_{k}}\) is a Gaussian series. Here model (2) is a special case of Equation (31) with \(m=1\), and the function \(\mathbf{f}_{l_{k}}(w)\) being affine/linear in \(w\). Therefore, for comparison with the Model-Based case of Equation (2), in this section, we focus on the case \(m=1\). For reference Model-Free estimators for point prediction and prediction intervals in the case of locally stationary time series for \(m=1\) have been discussed in [9]. Below, we describe the steps necessary to construct the invertible transformation \(H_{n}\) required to perform Model-Free point prediction for locally stationary random fields for the case \(m=1\). **Step 1: Transformation to uniform samples**  
With \(m=1\) let \(D_{l}(y)=P\{Y_{l}\leq y\}\) denote the first marginal distribution of the random field \(\{Y_{l}\}\). Applying the probability integral transform we have  
\[U_{l}=D_{l}(Y_{l})\;\;\text{for}\;t=\underline{t}_{1},\ldots,\underline{t}_{n} \tag{32}\]  
Here, \(U_{l_{1}},\ldots,U_{l_{n}}\) are random variables having distribution Uniform \((0,1)\). In this case, it is assumed that \(D_{l}(y)\) is (absolutely) continuous in \(y\) for all \(\underline{t}\). Therefore, we can use either local constant or local linear fitting to estimate it.