*3.5. Frequency Ratio*

One of the statistical bivariate models is FR, which is widely used in modeling environmental hazards as a geospatial assessment tool for quantifying the potential relationship

between dependent and independent variables [63]. The FR value for a certain class from a given factor can be calculated using:

$$FR = \frac{\frac{N\_{pix}(X\_i)}{\frac{\sum\_{i=1}^{m} X\_i}{\sum\_{i=1}^{m} X\_i}}}{\frac{N\_{pix}\left(X\_j\right)}{\sum\_{j=1}^{m} N\_{pix}\left(X\_j\right)}}\tag{21}$$

where *Npix*(*Xi*) is the number of pixels in each class of each factor with land subsidence locations. *X*.*Npix Xj* is the number of pixels of *Xj* factor, *m* is the number of classes in the *Xi* factor, and *n* is the number of factors in the study area [64].

#### *3.6. ANFIS with Meta-Heuristic Algorithms*

In ANFIS, parameter adjustment and the creation of a basic fuzzy system are done by combining traditional methods and then back error propagation. In this research, ICA and GWO were used as meta-heuristic algorithms to enhance the results of the ANFIS system and also to tweak the parameters of membership functions [52,65]. First, using input and target data, the FIS is created by the ANFIS model. Next, the membership functions are optimized and adjusted by meta-heuristic algorithms, and the output for the ANFIS (y) model is computed by [66]:

$$
\varepsilon = t - y \tag{22}
$$

$$RMSE = \sqrt{mean(e)^2} \tag{23}$$

where *t* is the target data, *y* is a function of input data and optimized FIS, and *e* is the error function that should be minimized. When the final conditions are met with the best output, the optimization process stops; otherwise, the membership function optimization is repeated.
