# Model-Based and Model-Free Point Prediction Algorithms for Locally Stationary Random Fields
## 2 Causality of Random Fields  
Given the random field observations \(Y_{\underline{t}_{1}},\ldots,Y_{\underline{t}_{n}}\), our goal is to perform predictive inference for the \"next\" unknown datapoint \(Y_{\underline{t}_{n+1}}\). In this context, a definition of causality is necessary to specify the random field coordinate \(\underline{t}_{n+1}\) where predictive inference will be performed. For this purpose, we adopt the framework proposed in [15] and consider random fields discussed in this paper to be defined over a subset of the nonsymmetric half-plane (NSHP) denoted as \(H_{\infty}\). Figure 1 shows an NSHP centered at (0, 0). The NSHP can also be centered at any other point \(\underline{t}\) as follows:  
\[NSHP(\underline{t})=\underline{t}+\underline{s}\quad\forall\underline{s}\in NSHP (0,0) \tag{3}\]  
Such nonsymmetric half-planes have been used previously for specifying causal 2D AR models [15]. In such cases, a causal 2D AR model with \(H_{p}\subset H_{\infty}\) can be defined as below in Equation (4) where the set \(H_{p}\) is termed as the region of support (ROS) of the 2D AR model. Here \(H_{p}=\{(j,k)\mid j=1,2,\ldots,p\) and \(k=0,\pm 1,\ldots,\pm p\}\cup\{(0,k)\mid k=1,2,\ldots,p\}\) and \(v_{t_{1}t_{2}}\) is a 2D white noise process with mean 0 and variance \(\sigma^{2}>0\).  
\[Y_{t_{1}t_{2}}=\sum_{(j,k)\in H_{p}}\beta_{j,k}Y_{t_{1}-j,t_{2}-k}+v_{t_{1}t_ {2}} \tag{4}\]  
Based on [23], a 2D AR process with ROS \(S\) is causal if there exists a subset \(C\) of \(Z^{2}\) satisfying the following conditions:  
* The set \(C\) consists of 2 rays emanating from the origin and the points lie between the rays;
* The angle between the 2 rays is strictly less than 180 degrees;
* \(S\subset C\) In this case, since \(H_{p}\subset H_{\infty}\) satisfies these conditions the 2D AR process denoted by (4) is causal. We use this framework to describe a causal random field defined over the NSHP and perform predictive inference on the same. Given this, our setup for point prediction of random fields is described as below.  
Consider random field data \(\{Y_{\underline{t}},\,\underline{t}\in E\}\) where \(E\) can be any finite subset of \(Z^{2}\) for, e.g., \(E_{\underline{t}}=\{\underline{t}\in Z^{2}\text{ with }\,\underline{u}=(n_{1},n_{2})\}\). Our goal is predictive inference at \(\underline{t}=(t_{1},t_{2})\) where \(0<t_{1}<n_{1}\) & \(0<t_{2}<n_{2}\). This \"future\" value \(Y_{t_{1}t_{2}}\) is determined using data defined over the region as shown in Figure 2:  
\[E_{\underline{t}\underline{t}}=NSHP(\underline{t})\cap E_{\underline{u}}\]  
Both Model-Based and Model-Free causal inference for \(Y_{t_{1}t_{2}}\) are performed using the data specified over this region \(E_{\underline{t}\underline{t}}\). We consider predictive inference at \(Y_{\underline{t}}=Y_{t_{1}t_{2}}\) given the data (\(Y_{\underline{s}}\mid\underline{s}\prec\underline{t}\) & \(\underline{s}\in E_{\underline{t}\underline{u}}\)) where the symbol \(\prec\) denotes lexicographical ordering on the region of support of the random field, i.e., \((a_{k},b_{k})\prec(a_{k+1},b_{k+1})\) if and only if either \(a_{k}<a_{k+1}\) or \((a_{k}=a_{k+1}\) and \(b_{k}<b_{k+1})\)[15]. In the subsequent discussion  
Figure 1: Nonsymmetric Half-Plane.  
the lexicographically ordered \"past\" data \(Y_{\underline{s}}\) will be denoted as \(Y_{\underline{t}_{1}},Y_{\underline{t}_{2}},\ldots,Y_{\underline{t}_{n}}\) and point \(p\) prediction will be performed at \(Y_{\underline{t}}=Y_{\underline{t}_{n+1}}\).