# Model-Based and Model-Free Point Prediction Algorithms for Locally Stationary Random Fields
###### Abstract  
The Model-Free Prediction Principle has been successfully applied to general regression problems, as well as problems involving stationary and locally stationary time series. In this paper, we demonstrate how Model-Free Prediction can be applied to handle random fields that are only locally stationary such as pixel values over an image or satellite data observed on an ocean surface, i.e., they can be assumed to be stationary only across a limited part over their entire region of definition. We construct novel one-step-ahead Model-Based and Model-Free point predictors and compare their performance using synthetic data as well as images from the CIFAR-10 dataset. In the latter case, we demonstrate that our best Model-Free point prediction results outperform those obtained using Model-Based prediction. kernel smoothing; linear predictor; random fields; nonstationary series; point prediction 10.3390/00000-0000-0000-0000-0000-00000-  
images are shown in [17]. Local Linear based nonparametric estimators of such random fields and their theoretical properties have been discussed in [18; 19]. In this paper, we assume a locally stationary model for random fields \(Y_{\underline{t}}\in R\) defined over \(\underline{t}\in S\) where \(S\subset Z^{d}\), \(d=2\). Given data \(Y_{\underline{t}_{k}},Y_{\underline{t}_{2}},\ldots,Y_{\underline{t}_{n}}\), our objective is to perform point prediction for a _future_ unobserved data point \(Y_{\underline{t}_{n+1}}\). Here, \(\underline{t}_{1},\underline{t}_{2},\ldots,\underline{t}_{nr},\underline{t}_ {n+1}\in Z^{2}\) denote the coordinates of the random field over the 2D index set \(D\) and the notion of a _future_ datapoint over a coordinate of a random field for purposes of predictive inference over \(\underline{t}\in Z^{2}\) is defined in Section 2. The usual approach for dealing with locally stationary series is to assume that the data can be decomposed as the sum of three components:  
\[\mu(\underline{t})+S_{\underline{t}}+W_{\underline{t}}\]  
where \(\mu(\underline{t})\) is a deterministic trend function, \(S_{\underline{t}}\) is a seasonal (periodic) series, and \(\{W_{\underline{t}}\}\) is (strictly) stationary with mean zero. This type of decomposition has been proposed for time series [20] and can also be used for the decomposition of locally stationary random field data. The seasonal (periodic) component, be it random or deterministic, can be easily estimated and removed and having completed that, the 'classical' decomposition simplifies to the following model with the additive trend, i.e.,  
\[Y_{\underline{t}}=\mu(\underline{t})+W_{\underline{t}} \tag{1}\]  
which can be generalized to accommodate a coordinate-changing variance as well, i.e.,  
\[Y_{\underline{t}}=\mu(\underline{t})+\sigma(\underline{t})W_{\underline{t}}. \tag{2}\]  
In both above models, the series \(\{W_{\underline{t}}\}\) is assumed to be (strictly) stationary, weakly dependent, e.g., strong mixing, and satisfying \(E(W_{\underline{t}})=0\); in model (2), it is also assumed that \(var(W_{\underline{t}})=1\). The deterministic functions \(\mu(\cdot)\) and \(\sigma(\cdot)\) are unknown and can be assumed to belong to a class of functions that is either finite-dimensional (parametric) or not (nonparametric). In this paper, we focus on the nonparametric case and assume that \(\mu(\cdot)\) and \(\sigma(\cdot)\) have some degree of smoothness i.e., change smoothly (and slowly) with \(\underline{t}\). Models (1) and (2) can be used to capture the first two moments of the locally stationary random field; however it may be the case that the skewness and/or kurtosis of \(Y_{\underline{t}}\) changes with \(\underline{t}\). In addition it may also be the case that the correlation \(\operatorname{Corr}(Y_{\underline{t}_{j}},Y_{\underline{t}_{j+1}})\) changes smoothly (and slowly) with \(\underline{t}_{j}\in Z^{2}\). To address this more general case we propose a methodology for point prediction of locally stationary random fields that does not rely on simple additive models such as (1) and (2).