## 2 Materials and Methods
### Methodology  
#### 2.3.1 The PLUS Model and Accuracy Verification  
The PLUS model contains a land expansion analysis strategy (LEAS) and a CA model that includes a patch-generation mechanism based on multi-type random seeds (CARS). The LEAS transforms the mining of transition rules of each land use type into a binary  
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classification problem (obtain the change and inertia probabilities of each land use type), which can be solved with many data mining methods [31]. The LEAS transform needs land use data of two periods to overlay and extract changing grids from later data, then selected random sampling points. Then the random forest classification (RFC) algorithm analyzes the relationship between the growth of each kind of land use types and various driving factors to obtain the transition rules by calculating the growth probability. In this research, 16 driving factors (Table 1) were chosen to calculate the probability of occurrence for each landscape type among these factors. The CARS module is a Cellular Automata (CA) model base on multi-type random patch seeds. The important property of CA is that they demonstrate the spatial and dynamic process [24]. The CA model in PLUS model has been improve with multi-type random patch seeds and threshold descent rules. Neighborhood weight represents the expansion intensity of the land type, that is, the expansion ability of each land type driven by external factors. [PERSON], [PERSON] [43] conducted dimensionless processing on the variation of Total Area (one of the landscape metrics, TA) of each land type, making its threshold between 0 and 1. Combined with the requirements of neighborhood weight on data structure, it is concluded that the dimensionless value of TA variation conforms to the requirements of neighborhood weight both in terms of parameter meaning and data structure. Formula is as follows:  
\[X^{*}=\frac{X_{i}-\min(X)}{\max(X)-\min(X)} \tag{1}\]  
where \(X=(X_{1},\cdots,X_{i})\); \(X^{*}\) is the \(i^{th}\) normalized value. The Markov chain model uses the Markov process method to project future land which has been widely used in the prediction of land demands [30; 31; 43]. This method is characterized by both high operational precision and high prediction accuracy. It can be presented using the following equation as:  
\[S_{t+1}=P_{ij}\times S_{t},\;\;P_{ij}\in[0,1) \tag{2}\]  
where \(S_{t+1}\) and \(S_{t}\) are the landscape composition at times \(t\) and \(t+1\), respectively; \(P_{ij}\) is the transition probability that land use type \(i\) is converted to land use type \(j\) at time \(t\). Firstly, the PLUS model was used to simulate the land use in 2020 with the land use in 2000 as the starting year and 2010 as the end. The simulation results were compared with the real 2020 land use map via a Kappa test, as follows:  
\[k=\frac{p_{o}-p_{e}}{p_{p}-p_{e}} \tag{3}\]  
where \(p_{o}\) represents overall accuracy which is calculated by dividing the sum of the correctly classified samples in each category by the total number of samples; \(p_{e}\) represents the correct proportion of the model in the random case; \(p_{p}\) represents the proportion of the correct simulation in the case of ideal classification which is 1 normally. The verification results revealed that the overall accuracy and kappa coefficient reached 0.8268 and 0.747233, respectively. This demonstrates that the accuracy of the model is substantial, and the PLUS model can be used for future simulation reliably in Beijing. Secondly, data from 2010 to 2020 were used to simulate future land use under different scenarios. In this study, different scenarios included the baseline scenario, cropland protection scenario and ecological protection scenario [35]. For the Chinese government's policy, these scenarios were all simulated over the planning constraints zones and development zones. The baseline scenario is the basis of considering other constraint conditions for simulation of land use change in urban agglomerations.