# Comparing a Fuzzy Hybrid Approach with Invariant MGCFA to Study National Identity
## 4 Methods
### Fuzzy Hybrid Analysis
#### 4.2.1 The First Three Steps  
The vagueness of the information is introduced with the use of fuzzy sets, which are based on the membership function \(\mu_{A}(x)\), which are used to proxy the relative truth that exist in the statements that \(x\in A\)[47; 32; 48]. [PERSON] [47] contended that fuzzy sets provide the perfect framework to deal with problems in which the source of imprecision is related to the absence of accurate and well-defined indicators rather than the presence of random variables. [PERSON] [49] added that \"FST provides a strict mathematical framework (there is nothing fuzzy about fuzzy set theory!) in which vague conceptual phenomena can be precisely and rigorously studied (p. 5)\". The fuzzy sets that are used in our study are Triangular Fuzzy Numbers (TFNs) that will contain the information matrix of the national identity responses. [PERSON] et al. [56] review the studies that use the keywords 'TOPSIS' or 'technique for order preference by similarity ideal solution' and 'development' and 'fuzzy', and the authors concluded that TFNs are still the most common fuzzy sets used by researchers when they deal with uncertainty and vague information. The TFNs are also known as type-1 fuzzy sets. The TFNs' \(\widetilde{\mathbf{A}}\) are usually parametrized with a 3-uple (\(\mathbf{a_{1}},\ \mathbf{a_{2}},\ \mathbf{a_{3}}\)) in which the extremes of the interval represent the thresholds that determine the possible values for the information, and the mid-point expresses the most likely value. There are different representations that have been used in different applications. Our study will be based on the following representation (1 = (0,0,50); 2 = (30,50,70); 3 = (50,70,90); 4 = (70,100,100)) that has already been used by [57]. The selected TFNs imply that there is less fuzziness when citizens answer that a particular indicator is considered very important to be a [country] national. In the rest of the cases, it is assumed that the vagueness degree is higher. It can also be seen that TFNs are characterized because their possible values belong to the interval [0, 100]. In each of the categories, the information provided is vague as all the consecutive ordinal semantic points are represented by 3-uples that intersect in some interval. For example, the interval (\(30,50\)) is in the intersection of the first two points (not at all important and not very important). As said, the most likely values are 0 and 50, respectively. Fuzzy Set Logic Algebra facilitates the aggregation of TFNs. Therefore, it is straightforward to calculate the average TFN for a particular population segment that can be determined by some variable of interest [58]. The properties of algebra guarantee that the average of TFNs is also a TFN. In this study, we will examine 143 different socio-demographic segments that have their origin in the categories obtained for thirteen covariates used in the study, and the average TFN can be obtained for each of these segments of interest. Therefore, a matrix (8, 143) of TFN will be the information matrix that will be used in subsequent steps of the method. This matrix is known as the TFN information matrix, and it contains a lot of information which is difficult to analyse at first glance. Therefore, a defuzzification of the matrix is carried out to synthesize the information before applying other methods such as, for example, the technique for order of preference by similarity to ideal solutions (TOPSIS) [8]. For this reason, the method is known as FHA. Therefore, we transform the fuzzy information matrix into a plausible or more credible real number or crisp value information matrix as uncertainty and information vagueness has been adequately handled. [PERSON] [59] provided a defuzzification method by calculating the weighted average of the 3-uple that represent the respective TFN of the fuzzy information matrix, giving more importance to the value that, according to fuzzy ideas, contains more truth.