# Fuzzy Multivariate Regression Models for Seismic Assessment of Rocking Structures
## 2 Methods
### Formal Analysis
#### 2.3.1 Fuzzy Linear Regression  
Classical regression analysis is a widely used statistical method for modeling the variation of the dependent output \(Y\) regarding the variation of one or more input variables \(X\)[59; 60]. Its utilization is vital for minimizing the difference between the predicted and observed values of the dependent variable. The general form of classical regression with many explanatory variables is [61]:  
\[Y=\alpha_{0}+\alpha_{1}X_{1}+\ldots+\alpha_{n}X_{n}+u \tag{4}\]  
where \(u\) is the disturbance term that represents the error between the variables. The utilization of classical regression can be applied in many cases where a predictive model of the output is required. Regression analysis was particularly used for the hotel industry's quality analysis [62]. The results were compared with other methods, such as rough set and cluster analysis, and it was concluded that the fitted values closely matched the actual values of the regression model. Also, [PERSON] et al. [63] used multiple linear regression to evaluate the soil strength in Bangladesh. According to the results, the prediction model proficiently captured the inherent patterns within the training data and generated precise forecasts. However, classical regression has some limitations if the variables are highly correlated or the relationship between the parameters is nonlinear. These assumptions can lead to inaccurate results and, thus, poor predictions.  
The fuzzy linear regression (FLR) method extends classical logic in which the input and output parameters are expressed in fuzzy sets. More specifically, it manages the perceptual vagueness associated with the prediction process by applying fuzzy numbers. In contrast with the classical regression method, which assigns crisp values to variables, FLR introduces the concept of membership functions to evaluate the degree of association of an element to a set. The equation of the fuzzy linear regression method has the following form [64; 65; 66]:  
\[Y=A_{0}+A_{1}X_{1}+\ldots+A_{n}X_{n} \tag{5}\]  
where \(A_{n}=(r_{n},c_{n})\) express the fuzzy numbers in which \(r_{n}\) denotes the central point that the membership function attains a value of 1 and \(c_{n}\) represents the extent of the values. The function that determines the degree of membership is described as [67]:\[\mu_{A}(x)=L\left(\frac{x-r}{c}\right) \tag{6}\]  
where L(x) represents the reference function of a fuzzy number that adheres to the following conditions [68]:  
\[L(x)=L(-x) \tag{7}\]  
\[L(0)=1 \tag{8}\]  
\[L(1)=0 \tag{9}\]  
\[L(x)\text{ is decreasing in }[0,\text{ }\infty) \tag{10}\]  
\[L(x)\text{ possesses invertibility across the interval }[0,\text{ }1] \tag{11}\]  
Hence, the construction of the membership function for the linear possibility Equation (6) is achieved as follows [69]:  
\[\mu_{Y_{j}}\left(Y_{j}\right)=L\left[\frac{y_{j}-\left(r_{0}+\sum_{i=1}^{n}r_{ i}x_{ij}\right)}{c_{0}+\sum_{i=1}^{n}c_{i}|x_{ij}|}\right] \tag{12}\]  
in which i represents the count of inputs, and j represents the count of sets. Thus, the degree h is computed to encompass the data within the estimated output Y, denoted as [70]:  
\[\mu_{Y_{j}}\left(Y_{j}\right)\geq h \tag{13}\]  
In our study, given that the fuzzy numbers were triangular, the reference function L(x) exhibited the following structure [71]:  
\[L(x)=max(0,1-|x|) \tag{14}\]  
and consequently, it led to the formulation of the following linear programming problem [42]:  
\[J=min\{mc_{0}+\sum_{j=1}^{m}\sum_{i=1}^{n}c_{i}|x_{ij}|\} \tag{15}\]  
\[y_{j}\geq r_{0}+\sum_{i=1}^{n}r_{i}x_{ij}-(1-h)\left(c_{0}+\sum_{i=1}^{n}c_{i} |x_{ij}|\right) \tag{16}\]  
\[y_{j}\leq r_{0}+\sum_{i=1}^{n}r_{i}x_{ij}+(1-h)\left(c_{0}+\sum_{i=1}^{n}c_{i} |x_{ij}|\right) \tag{17}\]  
\[c_{i}\geq 0,\text{ i}=1,2,\ldots,n \tag{18}\]  
the Equations (16)-(18) were employed to facilitate the minimization process of the objective function (15).