## 3 Materials and Methods
### Bidirectional Long Short-Term Memory Model for LUI/LC Prediction  
The LSTM model is considered the advanced model of RNN, where the long-term dependencies can be learned for the sequence prediction problems. The long-term vanishing-gradient problems are prevented by using the LSTM models. The key elements of the LSTM model are input, forget, and output gate [55; 56; 57]. Figure 11 displays the working principle of the LSTM model. In Figure 11, the vector operations represent the element-wise multiplication \((*)\), and element-wise summation \((+)\) respectively. The time step (t) indicates the length of the input sequence in all the Equations (21)-(26). Equation (21) shows the mathematical expression of the forget gate, where \(f_{t}\) represents the memory gate's output at time t, \(\sigma\) represents the sigmoid function (\(0<\sigma<1\)), \(W_{f}\) represents the weight value of ANN, \(h_{t-1}\) is the output value of the previous cell, \(x_{t}\) represents the input values, and \(b_{f}\) denotes the bias weight values of the ANN. At the output of the equation, the value 1 will keep the information and the value 0 will forget the information  
\[f_{t}=\sigma\left(W_{f}\,*\,[h_{t-1},x_{t}]+b_{f}\right) \tag{21}\]  
\[I_{t}=\sigma\,\left(W_{i}\,*\,[h_{t-1},x_{t}]+b_{i}\right) \tag{22}\]  
\[\widetilde{c_{t}}=tanh(W_{c}\,*\,[h_{t-1},x_{t}]+b_{c}) \tag{23}\]  
Figure 10: LST map for the area of Javadi Hills for the years (**a**) 2012 and (**b**) 2015. In Equation (22), \(I_{t}\) represents the output of the input gate, \(\sigma\) represents the sigmoid function, \(W_{i}\) represents the weight values stored in the memory of ANN, \(h_{t-1}\) is the output value of the previous cell, \(x_{t}\) represents the input values, and \(b_{i}\) denotes the bias weight values of the ANN. In Equation (23), \(\widetilde{c_{t}}\) represents the output of ANN with the normalized \(tanh\) function that outputs the value between \(-1\) and \(+1\), \(W_{c}\) represents the weight values stored in the memory of ANN, \(h_{t-1}\) is the output value of the previous cell, \(x_{t}\) represents the input values, and \(b_{c}\) denotes the bias weight values of the ANN. \[C_{t}=C_{t-1}\;*\;f_{t}+i_{t}\;*\;\widetilde{c_{t}} \tag{24}\]  
\[O_{t}=\sigma\;(W_{O}\;*\;[h_{t-1},x_{t}]+b_{O}) \tag{25}\]  
\[h_{t}=O_{t}\;*\;tanh\;(C_{t}) \tag{26}\]  
Equation (24) shows the mathematical expression of the updated gate, where the memory is updated. The ANN learns the stored or forgotten information from the memory and then updates the newly added information from Equations (21)-(23). Equation (25) shows the mathematical expression of the output gate, where \(W_{O}\) represents the weight values stored in the memory of ANN, \(h_{t-1}\) is the output value of the previous cell, \(x_{t}\) represents the input values, and \(b_{O}\) denotes the bias weight values of the ANN. The output value, \(h_{t}\), was calculated in Equation (26). The uniform LU/LC classes were generated through the Vision Transformer classification model, and the features of the LST map were extracted for the years 2012 and 2015. In this research work, we used the spatial features of the LST map and the LU/LC change classification map for evaluating the LU/LC prediction map, using the Bi-LSTM model. The idea of Bi-LSTM is to process the sequence data in both forward and backward directions. The Bi-LSTM algorithm was used in our research for extracting the spatial and temporal features of the fifteen-year time-series data from 2012 to 2027 for the area of Javadi Hills. Figure 12 displays the working principle of the Bi-LSTM prediction model. Figure 11: LSTM model. The inputs of the Bi-LSTM are given as the 3D vectors (samples, time steps, and features) for producing both spatial and temporal information. The samples define the number of the input LU/LC map (\(L\) (\(j_{m,n}\))) of size (\(m\ *\ n\)) with defined labels (\(j\)) for training and validation.