# A Spatial Fuzzy Co-Location Pattern Mining Method Based on Interval Type-2 Fuzzy Sets
## 4 Methods
### A Mining Method Based on Spatial Cliques
When the original type-1 membership function is a decreasing function, the formula is expressed as:  
\[\text{Upper bound function }\overline{u}(x)=\sqrt[q]{1-\left(\frac{x-a}{b-a} \right)^{g}}.\] \[\text{Lower bound function }\underline{u}(x)=1-\sqrt[q]{1-\left(\frac{b-x}{b-a} \right)^{g}}\left(a\leq x\leq b\right)\]  
When the original type-1 membership function is an increasing function, the formula is expressed as:  
\[\text{Upper bound function }\overline{u}(x)=\sqrt[q]{1-\left(\frac{b-x}{b-a} \right)^{g}}\] \[\text{Lower bound function }\underline{u}(x)=1-\sqrt[q]{1-\left(\frac{x-a}{b-a} \right)^{g}}\left(a\leq x\leq b\right)\text{ where }a\text{ is the lowest value that experts start to believe has a tendency to belong to the fuzzy set and }b\text{ is the value that experts believe must belong to the fuzzy set. The area covered by the upper bound function and the lower bound function is expressed as the FOU. Here, \(g\) determines the degree of expansion and contraction of the elliptic curve. The larger the value of \(g\), the larger the FOU; the smaller the value of \(g\), the smaller the FOU. In this example, when \(g=1.1\), our granular type-2 fuzzy membership function is shown in Figure 3a. Figure 2: (a) Number of evaluators in interval data; (b) Type-1 membership function. A type-2 fuzzy membership function needs to satisfy two conditions: 1 The FOU between the upper bound fuzzy membership function and the lower bound fuzzy membership function must be connected; and 2 the proportion of data contained in the FOU to all data must reach a given threshold, that is, the confidence must reach a given threshold, which we set at 85% in this article. The granular data represents that, for any specific value in interval data, the possibility of taking any specific membership value in the interval membership is the same. Therefore, for interval data, the confidence of the type-2 membership function is \(\sum_{i=1}^{n}S_{i}\ *\ x\). Here, \(n\) is the number of granules in the interval data, \(S_{i}\) is the ratio of the area of the \(i\)th granule in the FOU to the area of the \(i\)th granule, and \(x\) are the number of evaluators in this granule. Within a range, the average of the confidence of all interval data is the confidence of the granular type-2 fuzzy membership function in this range. (4) Determine the parameters of the granular type-2 fuzzy membership function. We use a gradual approach to determine the unique parameter \(g\) in the granular type-2 fuzzy membership function. First, we set an increment \(\alpha=0.1\), when \(g=1+k_{1}\alpha\) (\(k_{1}=1,\) 2,..., \(n\)). Then, we observe whether the FOU is continuous. If there is \(\psi\) (\(\psi\in\{1,\) 2,..., \(n\}\)) when \(g=1+\psi\alpha\), the FOU is continuous; when \(g=1+(\psi+1)\alpha\), the FOU is not continuous. Then, set the gradual value to one-tenth of the original (i.e., \(\beta=0.1\alpha\)), \(g=1+\psi\alpha+k_{2}\beta\) (\(k_{2}\in\{1,\) 2,..., \(10\}\)), and so on, until the accuracy reaches the user's requirement and the FOU is connected. This process is shown in Table 3.  
When \(g=1.283\), the FOU of the granular type-2 fuzzy membership function is connected. At this time, the data in the FOU accounts for 85.13% of the total evaluation data, that is, the confidence of FOU is 0.8513, which reaches the default threshold. This is shown in Figure 3b. The type-2 membership function we finally obtain from the granular data is shown in Figures 4 and 5. Figure 3: (**a**) Type-2 membership function when \(g=1.1\); (**b**) Type-2 membership function when \(g=1.283\).