# A Spatial Fuzzy Co-Location Pattern Mining Method Based on Interval Type-2 Fuzzy Sets
## 4 Methods
### Extending Traditional Algorithms to Discover Prevalent FCPs
#### 4.2.2 An Extended Joinless Algorithm for Mining Prevalent FCPs  
For a circle with a distance threshold of the neighborhood relationship as its radius, instances in the circle have a star neighborhood relationship. The traditional Joinless algorithm uses the star partition model to materialize the neighborhood relationship of spatial datasets; so, in the process of generating row-instances of co-location patterns, it uses the instance lookup operations instead of a large number of Join operations. ```
**Algorithm 4** The row-filter method. We improve the traditional Joinless algorithm and propose an extended Joinless algorithm (Algorithm 5) to mine prevalent FCPs:

Step 1: spatial data sets are materialized into disjoint star neighbor sets. Step 4: \(k+1\) size candidate FCPs are generated from \(k\) size prevalent FCPs. Steps 5-9: star instances of candidate FCPs are generated from star neighbor sets, which requires that the first feature of candidate FCPs is the same as the feature type of the central object of the star neighbor sets. For example, the star instances of FCP {A.Cu(M), B.Zn(M), C.Cu(H)} are found from the neighbor set of feature A. Steps 10-12: show that if the number of features that do not repeatedly appear in the candidate FCP is 1, the instances of the unique feature that have all fuzzy features in the candidate FCP simultaneously are selected as the row-instances of the candidate FCP. Steps 13-15: show that if the number of features that do not repeatedly appear in the candidate FCP is 2, the star instances with size-2 that have all the fuzzy features of the candidate FCP simultaneously are selected as the row-instances of the candidate FCP. Since the spatial proximity relation is symmetrical, a star instance with size-2 is a clique instance. Steps 16-20: show that if the number of features that do not repeatedly appear when a candidate FCP is \(n\) (\(n\geq 3\)), the star instances with size-\(n\) that have all fuzzy features of the candidate FCP are temporarily selected as the row-instances of the candidate fuzzy co-location patterns. Then, we check whether the upper bound participation index of the candidate FCP is lower than the given prevalence threshold _min_prev_; if so, the candidate FCP is non-prevalent. If the upper bound participation index of the candidate FCP is greater than _min_prev_, we check whether the star instances of the FCP are clique instances, that is, whether each instance of a star instance are neighbors to each other; if so, the star instance is a clique instance, and if not, the star instance is discarded. Step 21: we calculate the upper bound participation index and lower bound participation index of candidate FCPs and compare this with the given prevalence threshold _min_prev_. We then obtain the absolutely prevalent FCPs and the FCPs with prevalence tendency degrees. ```
Input:\((1)\) Fuzzy feature set \(\textbf{Fuzz\_Fea}=\{\textbf{Fuzz\_}f_{1},\textbf{Fuzz\_}f_{2},\ldots,\textbf{Fuzz \_}f_{n}\}\) (2) Instance set of fuzzy features \(\textbf{S}=\{\textbf{s}_{1}\textbf{s}_{2},\ldots,\textbf{s}_{n}\}\) (3) A set of star neighbors of the feature \(\textbf{Fuzz\_}f_{i}\):\(\left\{\,sn_{f_{1}},\,sn_{f_{2}},\ldots,\,sn_{f_{n}}\right\}\) (4) Spatial proximity relation R (5) Minimum distance threshold \(min\_dist\) (6) Minimum prevalence threshold \(min\_prev\) Output: Prevalent FCP set \(\textbf{Fre\_P}\) Variable: \(k\): The size of FCPs \(C_{k}\): candidate \(k\) size FCPs \(SI_{k}\): Star instance set of \(k\) size candidate patterns \(CI_{k}\): Group instance set of \(k\) size candidate patterns \(PI_{k}\): The set of star instances have all fuzzy features in candidate patterns simultaneously \(P_{k}\): prevalent \(k\)-size FCPs \(\textbf{MemSum\_up}[k]\): The sum of upper bound membership degrees of all instances of all fuzzy features in \(k\) size FCP. \(\textbf{MemSum\_down}[k]\): The sum of lower bound membership degrees of all instances of all fuzzy features in \(k\) size FCP. Step: 1. SN = gen_star_neighbor(_Fea_,S,R); 2.