## 2 Method  
Consider a categorical random variable (RV) \(C\left(x_{0}\right)\) which can take one out of \(K\) mutually exclusive and collectively exhaustive states \(c\left(x_{0}\right)\in\left\{1,\cdots,K\right\}\) at any arbitrary location with coordinate \(x_{0}\); when there is no other information, we assume that the probability mass function (PMF) of RV \(C\left(x_{0}\right)\) is stationary and can be approximated by the \(K\) global class proportions \(\pi_{1},\cdots,\pi_{K}\). A central task in prediction and simulation of categorical random fields is the estimation of the conditional PMF of \(C\left(x_{0}\right)\) in the presence of observed class labels \(c\left(x_{1}\right),\cdots,c\left(x_{N}\right)\) available at \(N\) locations \(x_{1},\cdots,x_{N}\). Our task can now be re-stated as that of estimating the conditional PMF \(P\left\{C\left(x_{0}\right)\middle|c\left(x_{1}\right),\cdots,c\left(x_{N} \right)\right\}\). To simplify notation, we use \(A\) and \(D_{1},\cdots,D_{N}\) to represent the events in sample spaces of \(C\left(x_{0}\right)\) and \(C\left(x_{1}\right),\cdots,C\left(x_{N}\right)\), respectively, and \(\overline{A}\) to denote the complementary event of \(A\). ANNs offer an approach to geostatistical simulation with the possibility of automatic recognition of the correlation structure. We consider \(f_{i}=P\left(D_{j}\middle|A\right),\)\(j\in\left\{1,\)\(2,\cdots,i\right\}\) as the input node, and thus the pre-posterior (multi-point) probabilities \(y_{i}=P\left(D_{i}\middle|AD_{1}\cdots D_{i-1}\right),\)\(i=2,\cdots,N\) can be regarded as the desired output node. To obtain the target (multi-point) posterior probability \(P\left(A\middle|D_{1}D_{2}\cdots D_{N}\right)\), we need to train \(N-1\) ANNs, which is called the MANN. The continuously differentiable sigmoid functions are the most common form of activation function in ANNs [17], and the learning process is achieved by using feed forward and back propagation algorithms. Let \(\hat{y}_{i}=\hat{P}\left(D_{i}\middle|AD_{1}\cdots D_{i-1}\right)\) be the actual output obtained on one particular iteration after \(f_{i}\) has been fed forward through the network. The current total error, which is defined as a function of the connection weights, is:  
\[E(W)=\sum\
olimits_{i=2}^{N}(y_{i}-\hat{y}_{i})^{2} \tag{1}\]  
where \(W\) is the matrix of connection weights. In most traditional neural networks, the desired output nodes are considered as known response variables. Thus, the central work of an ANN is to train the appropriate weights in hidden layers. However, the desired outputs \(y_{i}=P\left(D_{i}\middle|AD_{1}\cdots D_{i-1}\right),\)\(i=2,\cdots,N\) in our model cannot be obtained from the available training set directly. We use MANN with interaction noise to solve this problem. Just as MCRF, Tau \(\left(\tau\right)\) model [18] and Nu \(\left(v\right)\) expression [19] suggested, \(P\left(D_{i}\middle|AD_{1}\cdots D_{i-1}\right)\) is primarily determined by \(P\left(D_{i}\middle|A\right)\). We assume that  
\[\ln\frac{P\left(D_{i}\middle|AD_{1}\cdots D_{i-1}\right)}{P\left(D_{i}\middle| A\right)}=\varepsilon_{i} \tag{2}\]  
where \(\ln\) denotes the natural logarithm, \(\varepsilon_{i}\) is a random noise which capsules the interaction information from \(D_{1}\) to \(D_{i}\). Spearman correlation coefficients are used to measure the interdependence from \(P\left(D_{1}\middle|A\right)\) to \(P\left(D_{i}\middle|A\right)\). In this method, the one that has the most significant correlation with \(P\left(D_{i}\middle|A\right)\) is chosen to be the input node.