# A Spatial Fuzzy Co-Location Pattern Mining Method Based on Interval Type-2 Fuzzy Sets
## 1 Introduction  
Spatial co-location pattern mining is an important branch of spatial data mining. A spatial co-location pattern is a subset of spatial features whose instances are often neighbors in space. For example, [hotel, restaurant] may be a prevalent co-location pattern because hotels and restaurants often neighbor each other in a city center. Spatial co-location pattern mining is widely used in the fields of botany [1], geographic information science [2, 3, 4], geology [5], urban facilities distribution, etc. For example, geologists may be interested in neighboring minerals in the same area [4]. In recent years, many algorithms for mining spatial co-location patterns have been developed. These algorithms can be divided into two types. Apriori-like algorithms [1, 5, 6, 7, 8] are one type. This algorithm first generates the candidate co-location patterns, then spends a lot of time generating row-instances of the candidate co-location patterns and checking their prevalence. Another algorithm is the algorithm based on prefix trees [2, 3, 9, 10, 11, 12, 13, 14]. This algorithm generates maximal cliques from spatial datasets to discover prevalent spatial co-location patterns, which is generally more efficient than an Apriori-like algorithm. We have noticed that spatial data are often accompanied by attribute information expressed by specific values, such as the content of heavy metals in a certain geographical location [4]. However, existing spatial co-location pattern mining algorithms mainly focus on the location information of spatial data but ignore their attribute information. In recent years, some researchers have used type-1 fuzzy sets to process the location information of spatial data and mine fuzzy co-location patterns (FCPs). However, type-1 fuzzy sets have a large amount of uncertainty. Type-2 fuzzy sets can model both intra-individual uncertainty and inter-individual uncertainty, while type-1 fuzzy sets can only model intra-individual uncertainty. Every expert can model a type-1 fuzzy set (intra-individual uncertainty), and the type-1 fuzzy sets of different experts are often different. Type-2 fuzzy sets can be used to fuse these different type-1 fuzzy sets, so as to consider the uncertainty between individuals. In recent years, people have made great progress in the study of type-2 fuzzy sets [15; 16; 17; 18; 19; 20]. Many researchers have applied fuzzy set theory to association rule mining. [PERSON] et al. [21] used type-2 fuzzy sets to mine multi-level association rules. [PERSON] et al. [22] used type-2 fuzzy sets to discover fuzzy frequent item sets efficiently. [PERSON] et al. [23] used type-2 fuzzy sets and genetic algorithms to mine fuzzy rules in high-dimensional classification tasks. [PERSON] et al. [24] and [PERSON] et al. [25] introduced fuzzy set theory into spatial co-location pattern mining and proposed a new metric method. [PERSON] et al. [26] used a hierarchy to represent fuzzy association rules. [PERSON] et al. [27] quickly discovered fuzzy frequent item sets from quantitative database based on type-2 fuzzy sets. [PERSON] et al. [28] mined the relationship between different types of crime rate based on fuzzy association rules. [PERSON] et al. [29] and [PERSON] et al. [30] combined a fuzzy set and rough set and proposed a new method of mining fuzzy association rules. In addition, fuzzy set theory is also widely used in attribute data mining. [PERSON] et al. [31] and [PERSON] et al. [32] introduced fuzzy weight into fuzzy multi-attribute mining. [PERSON] et al. [33] used interval type-2 fuzzy sets to solve multi-attribute decision-making in sponge city construction. [PERSON] et al. [34] and [PERSON] et al. [35] applied ordered fuzzy sets to multi-attribute decision-making systems. In recent years, many type-2 fuzzy membership functions have been proposed, such as ladder type, triangular type, Gaussian type and S type [36; 37], \(\pi\) type [38], etc.