# Model-Based and Model-Free Point Prediction Algorithms for Locally Stationary Random Fields
## 4 Model-Free Point Prediction
Using **local constant fitting** a smooth estimator can be defined as:  
\[\bar{D}_{\underline{k}}(y)=\sum_{i=1}^{T}\Lambda(\frac{y-Y_{\underline{t}_{i }}}{h_{0}})\tilde{K}(\frac{\underline{t}_{k}-\underline{t}_{i}}{b}) \tag{33}\]  
where \(\tilde{K}(\frac{\underline{t}_{k}-\underline{t}_{i}}{b})=K(\frac{\underline{ t}_{k}-\underline{t}_{i}}{b})/\sum_{j=1}^{T}K(\frac{\underline{t}_{k}- \underline{t}_{j}}{b})\), \(\Lambda(y)\) is a smooth distribution function, which is strictly increasing with density \(\lambda(y)>0\) i.e., \(\Lambda(y)=\int_{-\infty}^{y}\lambda(s)ds\) and \(h_{0}\) is a secondary bandwidth. Furthermore, as in Section 3, we can let \(T=k\) or \(T=k-1\) leading to a **fitted vs. predictive** way to estimate \(D_{\underline{k}}(y)\) using \(\bar{D}_{\underline{k}}(y)\). Similar to the Model-Based case, we will assume throughout that \(K(\cdot)\) is a nonnegative, symmetric 2D Gaussian kernel function for which the diagonal values are set to the bandwidth \(b\) and the off-diagonal terms are set to \(0\). Note that the kernel estimator (33) is one-sided for the same reasons discussed before in Section 3. Cross-validation is used to determine the bandwidths \(h_{0}\) and \(b\); details are described in Section 5. Since point prediction is performed on the boundary of the random field one can also consider **local linear estimation** as an alternative to the local constant-based smoothing approach. \(\bar{D}_{\underline{k}}(y)\) as defined in Equation (33) is the Nadaraya-Watson smoother of the variables \(v_{1},\ldots,v_{n}\) where \(v_{i}=\Lambda(\frac{y-Y_{\underline{t}_{i}}}{h_{0}})\). It is possible to define \(\bar{D}_{\underline{k}}^{LL}(y)\) which is the local linear estimator of \(D_{\underline{k}}(y)\) based on the smoothed variables \(\Lambda(\frac{y-Y_{\underline{t}_{i}}}{h_{0}})\). This estimator is expected to have a smaller bias than \(\bar{D}_{\underline{t}_{i}}(y)\). However, there is no guarantee that this will be a proper distribution function as a function of \(y\), i.e., being nondecreasing in \(y\) with a left limit of \(0\) and a right limit of \(1\) as discussed in [26]. A proposed solution put forward by [PERSON] [30] involves a straightforward adjustment to the local linear estimator of a conditional distribution function that maintains its favorable asymptotic properties. The local linear version of \(\bar{D}_{\underline{k}}(y)\) adjusted via [PERSON]'s (2004) proposal is given as follows:  
\[\bar{D}_{\underline{k}}^{LLH}(y)=\frac{\sum_{i=1}^{T}w_{i}^{\circ}\Lambda( \frac{y-Y_{\underline{t}_{i}}}{h_{0}})}{\sum_{i=1}^{T}w_{i}^{\circ}}. \tag{34}\]  
The weights \(w_{i}^{\circ}\) are derived from weights \(w_{i}\) described in Equations (22) and (30) for the fitted and predictive cases where  
\[w_{i}^{\circ}=\left\{ \begin{array}{ll}0&\text{when}\;\;w_{i}<0\\ w_{i}&\text{when}\;\;w_{i}\geq 0\end{array} \right. \tag{35}\]As with Equation (33), we can let \(T=k\) or \(T=k-1\) in the above, leading to a fitted vs. predictive local linear estimators of \(D_{lk}(y)\) using \(\bar{D}_{lk}^{LLH}(y)\). One problem with the local linear estimator described above is that it replaces negative weights with zeros, and then renormalizes the nonzero weights. However, if estimation is performed on the boundary (as in the case with one-step ahead prediction of random fields), negative weights are crucially needed in order to ensure the extrapolation takes place with minimal bias. To address this problem, we modify the original, possibly non-monotonic local linear distribution estimator \(\bar{D}_{lk}^{LL}(y)\) to construct a monotonic version denoted by \(\bar{D}_{k}^{LLM}(y)\). The **Monotone Local Linear Distribution Estimator**\(\bar{D}_{lk}^{LLM}(y)\) can be constructed by Algorithm 1 as given below [31]. 1.