## 2 Methodology
### Phase 3. Land Cover/Use Change Model
#### 2.4.2 Steps for Generating a Land-Use Model  
This section presents the steps for generating a land-use model. The steps are shown in Figure 4.  
Step 1. Calculating the transition matrices. A transition matrix describes the changes in a system over discrete periods, where the value of any variable in a given period is the sum of fixed percentages of its value in the previous period. The change matrices can be single or multiple. Simple change matrices refer to transition rates for a given period (for example, 27 years if it were between 1992 and 2019; see Table 2), while multiple change matrices refer to annual rates of change. Calculating these matrices is essential because they are needed for validation and to generate future scenarios. A model is used in several time units, calculating a fixed gross rate per time unit (year), and dividing the accumulated change in the period by the number of units (each unit is one year) the period comprises. Step 2. The weights of evidence (WoE) model is a Bayesian geostatistical method that represents the empirical association of spatial factors related to land use and change. It is used to calculate the impact of binary factors on changes in land use. This method can be extended to multiple-category maps by treating one category versus all other categories combined in each case. In our case, the categorical variables (HM, CES and HCA) are already binary, while the continuous variables (distance to the beach and distance to the historical vehicle traffic route) should be categorized into ranges. For example, if we want to study the effect of distance to the beach on the change from \"Historical Use\" to \"New Housing\" and from \"Historical Use\" to \"Commercial Use,\" suppose the variable distance to the beach ranges from 0 to 800 m. We can categorize this distance into three ranges: close (0-20 m), medium (21-60 m) and far (61-800 m). This allows us to study the WoE of all combinations between ranges and land-use changes, such as the following:  
* The association of close distance to the beach with the change from \"Historical Use\" to \"New Housing\". * The association of medium distance to the beach with the change from \"Historical Use\" to \"New Housing\". * The association of far distance to the beach with the change from \"Historical Use\" to \"New Housing\". * The association of close distance to the beach with the change from \"Historical Use\" to \"Commercial Use\". * The association of medium distance to the beach with the change from \"Historical Use\" to \"Commercial Use\". * The association of far distance to the beach with the change from \"Historical Use\" to \"Commercial Use\". \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{4}{c}{**1992**} \\ \cline{2-6}
**1970** & **NUV** & **SE** & **NUE** & **NUEC** & **Total** \\ \hline HV & 0.03\% & 0.02\% & 0.02\% & 0.00\% & 0.07\% \\ HEAI & 1.36\% & 1.12\% & 0.00\% & 0.22\% & 2.70\% \\ HE & 0.00\% & 0.00\% & 1.40\% & 0.00\% & 1.40\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Transition matrix from 1970 to 1992. Figure 4: Creation phases of a spatio-temporal simulation model. The limits of the ranges are defined by a line-generalizing algorithm [39] that aims to preserve the structure of the data. Then, the WoE (W) are calculated in Equation (1). \[W^{+}=\ln\left(\frac{\frac{A1}{A1+A2}}{\frac{A3}{A3+A4}}\right),\quad W^{-}=\ln \left(\frac{\frac{A2}{A1+A2}}{\frac{A3}{A3+A4}}\right) \tag{1}\]  
\(A1\) is the number of cells with changes where the factor is present; \(A2\) is the number of cells with changes where the factor is not present; \(A3\) is the number of cells where the factor is given and there are no changes; \(A4\) is the number of cells where there are no changes, and the factor is not present. By calculating the contrast weight given by \((C)=(W^{+})-(W^{-})\), which measures the association/repulsion effect, results close to zero indicate that the analyzed variable has no effect.