# A Spatial Fuzzy Co-Location Pattern Mining Method Based on Interval Type-2 Fuzzy Sets
## 4 Methods  
### A Mining Method Based on Spatial Cliques  
We propose a method based on spatial cliques to mine all prevalent FCPs. Our method first uses a method based on a prefix tree to generate spatial cliques from spatial datasets; then an Apriori-like algorithm is used to generate candidate FCPs, find row-instances of candidates from the generated cliques, and then filter prevalent FCPs. In this section, we discuss this approach in six subsections: generating granular type-2 fuzzy membership functions from type-1 fuzzy membership functions based on granular data (Section 4.1.1), materializing neighborhood relationships (Section 4.1.2), enumerating cliques (Section 4.1.3), generating candidate FCPs (Section 4.1.4), finding table-instances of candidates from cliques (Section 4.1.5), and filtering prevalent FCPs (Section 4.1.6). 1.1 Generating Granular Type-2 Fuzzy Membership Functions from Type-1 Fuzzy Membership Functions Based on Granular Data  
When the specific membership degree of a specific value under a fuzzy set is evaluated, the deviation is often large. However, when the interval membership degree of an interval value in a fuzzy set is evaluated, the deviation can be greatly reduced. Therefore, we use the expert evaluation method to evaluate the interval membership degree of interval values under a fuzzy set to form granular data. Furthermore, according to the original type-1 fuzzy membership function, we generate a granular type-2 fuzzy membership function based on elliptic curves and gradually adjust the parameter of the granular type-2 fuzzy membership function by means of equal expansion. 1. Count the granular evaluation data. In this process, we need to choose the appropriate length of the interval membership degree and the appropriate length of the evaluation interval data so that experts can make accurate judgments. We take the interval membership degree belonging to the fuzzy set Cu(M) as an example when the copper content is between 10 and 50. The interval length of the content we chose to evaluate is 5 and the length of interval membership is 0.1. If the length of the evaluated data is small, we set the interval length to 2--for example, when the copper content is between 6-20. We have counted the data of 1000 geologists who evaluated the interval value of copper content under the fuzzy set Cu.(M). The number in the particle indicates the number of evaluators who consider that an interval value corresponds to the interval membership. The darker the color is, the more people are evaluated in this interval. Figure 2a shows the number of evaluators of interval memberships. The original type-1 fuzzy membership function is drawn based on the granular evaluation data. In this example, according to the construction method in [4], we obtain the type-1 fuzzy membership function of heavy metal copper. We draw the type-1 fuzzy membership function, which is expressed as \(y=\frac{x-a}{b-a}=\frac{x-10}{40}\), as shown in Figure 2b. Here, \(a\) is the lowest value that experts believe the heavy metal content tends to belong to the middle content, and \(b\) is the value that experts believe the copper content must belong to the middle content. 1. Constructing a type-2 membership function based on the original type-1 membership function. The granular type-2 fuzzy membership function we proposed is represented by the upper bound fuzzy membership function and the lower bound fuzzy membership function.