# Coal Seam Thickness Prediction Based on Transition Probability of Structural Elements
## 3 TTP-GPR for Coal Seam Thickness Prediction
### Structural Element Transition Probability  
For a \(3\times 3\) structural element \(st=\{z_{1},z_{2},\ldots,z_{9}|z_{i}\in R^{+}\}\) that contains nine coal seam points \(z_{i}\), and the spatial distribution is mainly composed of one unit distance point and \(\sqrt{2}\) unit distance point in eight directions. Where, one unit distance and \(\sqrt{2}\) unit distance points contain four points, respectively. In the process of iteration prediction in the marching direction, we use the structural elements \(\text{st}_{i}\) to predict the coal seam thickness in the point i, and predict the next coal seam thickness in turn. The structural element moves in the marching direction with the iteration prediction. Due to the structural elements movement in the prediction process, there is data overlap between two structural elements that move one step and two steps. Figure 4 shows the data overlap diagram of one step moving. In Figure 4, the point marked \"\(1\)\" indicates the data distribution point in the initial state, and the point labeled \"\(2\)\" indicates the data distribution point under the state of moving one step. The point labeled \"\(1/2\)\" indicates that the data distribution point overlapped by both states. Figure 3: Radial Basis Function (RBF-Kriging) algorithm interpolated diagram (blue points). Figure 2: Change of prediction mode diagram, (**a**) is the prediction mode in the marching direction, (**b**) is our prediction mode through structural elements. It used the red color data to predict the green unknown data. Due to the interdependencies among structural elements, the structural element of the next state is determined by the structural elements of the current state and previous states. For the Markov chain, the state of the next moment of the system is uniquely determined by the current state and it does not depend on its past [17]. Thus, the transition probabilities that were calculated by the Markov chain are used to measure the influence of these points, which have the equal distance to a prediction point in a structural element, on this prediction point. For computing the transition probability, we adopt the method of the Markov process [18] that needs to divide the data state that depended on some criterion for computing the transition probability. Assuming that the structural elements \(St=\{st_{i-1},st_{i},st_{i+1}|1\leq i\leq n\}\) has interdependencies, the state \(S=\{s_{1},s_{2},\ldots,s_{i},\ldots,s_{m}|1\leq i\leq m\}\) of each data point in the structural elements is expressed as the state division:  
\[\min\left|h_{st_{i+1,i}}-h_{st_{i,j}}\right|\leq\Delta h\leq\max\left|h_{st_{i +1,i}}-h_{st_{i-1,i}}\right|,1\leq i\leq n,1\leq j\leq 9, \tag{12}\]  
where m is the number of state, \(h_{st_{i,j}}\) is the coal seam thickness value at each point in the structural elements, and \(\Delta h\) is the maximum difference of coal seam thickness within a state. Notice that the rationality of the state division must be maintained for each coal seam thickness data. If the state division is too fine, then each point data is in a separated state. As a result, the calculation amount increases and the minimal changes in the data will cause the chain reaction in the prediction process to not be conducive in maintaining the stability of the model. If the state division is too rough, all of the data in the structure elements is in a single state, and there is no spatial distribution information used in the prediction of the coal seam thickness. Thus, the maximum and minimum values between the two structural elements need to be dynamically recalculated after movement. It contains nine points, that is eight controlling points and a prediction point in the center in a structural element, and each point would be divided to a state s. Assuming the state of controlling point is \(s_{i}\) and the prediction point is \(s_{j}\), and the each controlling point and prediction point form a state pair, like the two yellow state in the Figure 5, is a state pair.