# A Spatial Fuzzy Co-Location Pattern Mining Method Based on Interval Type-2 Fuzzy Sets

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Innovation

- (1)
- In the fuzzy processing of corresponding attribute information based on the original type-1 fuzzy membership function, a granular type-2 fuzzy membership function based on an elliptic curve is generated from the evaluation data. We adopt a gradual method to adjust the parameters of the function so that the type-2 membership function not only meets the connectivity [11] but also makes its confidence reach the given threshold (set to 85%).
- (2)
- Based on the abovementioned granular type-2 fuzzy membership function, we define the concepts of fuzzy membership interval, upper bound participation ratio, lower bound participation ratio of fuzzy features, upper bound participation index, and lower bound participation index for FCP mining. Further, to smooth the “sharp boundary” of pattern prevalence and reduce the deviation caused by the subjective understanding of the prevalence degree of FCPs, we propose the concepts of absolutely prevalent FCPs, FCPs with a prevalence tendency degree, and absolutely non-prevalent FCPs.
- (3)
- Based on spatial cliques, a method of mining FCPs is proposed. First, an algorithm for generating spatial cliques from spatial datasets is presented, and then an Apriori-like algorithm and two pruning strategies are proposed to discover the absolutely prevalent FCPs and FCPs with a prevalence tendency degree. In addition, we apply interval type-2 fuzzy sets to traditional co-location pattern mining algorithms and form an FCPs mining method based on interval type-2 fuzzy sets and the traditional Join-based algorithm, as well as another FCPs mining algorithm based on interval type-2 fuzzy sets and the traditional Joinless algorithm.

## 3. Related Definitions and Theorems

#### 3.1. Definitions

**Definition**

**1**

**(Data**

**fuzzification).**

**(x).**

_{F}**Definition**

**2**

**(Granular**

**evaluation**

**data).**

**Definition**

**3**

**(Fuzzy**

**feature).**

**Fuzz_Fea**= {Fuzz_${f}_{1}$, Fuzz_${f}_{2}$, …, Fuzz_${f}_{n}$}. A fuzzy feature is represented as Fuzz_${f}_{i}$(1 $\le $i $\le $n).

**Definition**

**4**

**(Neighborhood**

**relationship).**

**Fuzz_Fea**is a set of features and

**S**is the set of their instances, for any two instances, s

_{1}and s

_{2}, if the Euclidean distance between the two instances is less than the threshold min_dist given by the user—that is, distance(s

_{1}, s

_{2}) ≤ min_dist—then the two instances satisfy the neighborhood relationship, which is represented as R(s

_{1}, s

_{2}).

**Definition**

**5**

**(Fuzzy**

**co-location**

**pattern).**

**Fuzz_c**is a subset of the set of fuzzy features, that is,

**Fuzz_c**$\subseteq $

**Fuzz_Fea**. The size of an FCP is the number of fuzzy features of the pattern, which is expressed as |

**Fuzz_c**|.

**Definition**

**6**

**(Row-instance**

**and**

**table-instance).**

**S**={s

_{1}, s

_{2}, …, s

_{k}} where any two instances are neighbors.If

**S**contains all the fuzzy features of an FCP

**Fuzz_c**and there is no subset of

**S**that can contain all the fuzzy features of

**Fuzz_c**, then

**S**is called a row-instance of

**Fuzz_c**. All row-instances of

**Fuzz_c**constitute the table-instance of

**Fuzz_c**.

**Definition**

**7**

**(Absolute**

**row-instance**

**and**

**absolute**

**table-instance).**

**Definition 8 (Upper bound participation ratio and lower bound participation ratio).**

**Fuzz_c**= {Fuzz_${f}_{1}$, Fuzz_${f}_{2}$, …, Fuzz_${f}_{k}$}, the lower bound participation ratio of the fuzzy feature Fuzz_${f}_{i}$(i∈{1, 2, … k}) is expressed as PR(

**Fuzz_c**, Fuzz_${f}_{i}$), which is defined as the ratio of the sum of the lower bound membership degrees of the instances that appear non-repeatedly in the table-instance of the FCP

**Fuzz_c**to the sum of the lower bound membership degrees of all the instances of Fuzz_${f}_{i}$. The upper participation ratio is expressed as$\overline{PR}$(

**Fuzz_c**, Fuzz_${f}_{i}$), which is defined as the ratio of the sum of the upper bound membership degrees of the instances that appear non-repeatedly in the table-instance of the FCP

**Fuzz_c**to the sum of the upper bound membership degrees of all the instances of Fuzz_${f}_{i}$, namely:

**Definition**

**9**

**(Upper**

**bound**

**participation**

**index**

**and**

**lower**

**bound**

**participation**

**index).**

**Fuzz_c**is expressed as $\overline{PI}$(

**Fuzz_c**), which is defined as the minimum of the upper bound participation ratio of all fuzzy features Fuzz_${f}_{i}$ in

**Fuzz_c**, which is:

**Fuzz_c**is expressed as PI(

**Fuzz_c**), which is defined as the minimum of the lower bound participation ratio of all fuzzy features Fuzz_${f}_{i}$ in

**Fuzz_c**, which is:

**Definition**

**10**

**(Prevalent**

**fuzzy**

**co-location**

**pattern).**

**Fuzz_c**has an upper bound participation index and a lower bound participation index. For an FCP

**Fuzz_c**when given a prevalence threshold min_prev, if $\overline{PI}$(

**Fuzz_c**) < min_prev,

**Fuzz_c**is called an absolutely non-prevalent pattern. If PI(

**Fuzz_c**) ≥ min_prev,

**Fuzz_c**is called an absolutely prevalent pattern. If PI(

**Fuzz_c**) <min_prev $\le $$\overline{PI}$(

**Fuzz_c**),

**Fuzz_c**is called an FCP with prevalence tendency degree, and the prevalence tendency degree of the fuzzy pattern is:

**Example**

**1.**

**Fuzz_c**= {A.Cu(M), B.Cu(H), C.Cu(M)}. Here, the fuzzy feature B.Cu(H) represents the B functional area with a high copper content. The size of the FCP is 3. According to the granular type-2 fuzzy membership function constructed in Section 4.1, we obtain the interval membership of the attributes of the instances in Table 1, as shown in Table 2. As can be seen from Table 2, the instances of fuzzy feature A.Cu(M) are A.1, A.2, and A.3; the instance of fuzzy feature B.Cu(H) is B.1; and the instance of fuzzy feature C.Cu(M) is C.1. If A.2, B.1, and C.1 are neighbors, {A.2, B.1, C.1} is a row-instance of the FCP

**Fuzz_c**. According to Definition 8, the lower boundary participation ratio of these three fuzzy features is PR(

**Fuzz_c**, A.Cu(M)) = 0.2256, PR(

**Fuzz_c**, B.Cu(H)) = 1, PR(

**Fuzz_c**, C.Cu(M)) = 1 and the upper bound participation ratio of these three fuzzy features are $\overline{\mathrm{PR}}$(

**Fuzz_c**, A.Cu(M)) = 0.2866,$\overline{\mathrm{PR}}$(

**Fuzz_c**, B.Cu(H)) = 1,$\overline{\mathrm{PR}}$(

**Fuzz_c**, C.Cu(M)) = 1, so the lower participation index is PI(

**Fuzz_c**) = 0.2256 and the upper bound participation index is $\overline{\mathrm{PI}}$(

**Fuzz_c**) = 0.2866. If the prevalence threshold min_prev = 0.2, then PI(

**Fuzz_c**) > min_prev, that is,

**Fuzz_c**is absolutely prevalent. If the prevalence threshold min_prev = 0.3, then $\overline{\mathrm{PI}}$(

**Fuzz_c**) < minprev and

**Fuzz_c**is absolutely non-prevalent. If the prevalence threshold min_prev = 0.25, then $\overline{\mathrm{PI}}$(

**Fuzz_c**) > minprev > PI(

**Fuzz_c**) and the prevalence tendency degree

**θ**of

**Fuzz_c**is

**θ**= 0.6.

#### 3.2. Lemma and Theorem

**Lemma**

**1.**

**Proof.**

**Fuzz_c**, then when fuzzy pattern

**Fuzz_c’**$\subseteq $

**Fuzz_c**, instance ${s}_{1}$

**Fuzz_c’**must also be included in the row-instances of

**Fuzz_c’**. According to Definitions 8 and 9, the denominator of the upper bound participation ratio (or lower bound participation ratio) of the same fuzzy feature in the pattern and pattern

**Fuzz_c**is the same, and the numerator decreases with the increase of the size of the pattern. Therefore, the upper bound participation ratio (or lower bound participation ratio) of the fuzzy features decreases monotonically. According to the definition of the upper bound participation index and lower bound participation index, the upper bound participation index (or lower bound participation index) is the minimum of the upper bound participation ratio (or lower bound participation ratio) of all fuzzy features in an FCP. Therefore, the upper bound participation index (or lower bound participation index) decreases monotonically with the increase of the size of the pattern. □

**Theorem**

**1.**

**Proof.**

**Fuzz_c**, if a k size pattern

**Fuzz_c’**$\supseteq $

**Fuzz_c**, according to

**Lemma 1**,$\overline{\mathrm{PI}}$(

**Fuzz_c’**) ≤ $\overline{\mathrm{PI}}$(

**Fuzz_c**). If this k-1 size pattern

**Fuzz_c**is absolutely non-prevalent, then $\overline{\mathrm{PI}}$(

**Fuz_c**) < min_prev. Therefore, if $\overline{\mathrm{PI}}$(

**Fuz_c’**) ≤ $\overline{\mathrm{PI}}$(

**Fuz_c**) < min_prev, then k size pattern

**Fuzz_c’**is also absolutely non-prevalent. □

## 4. Methods

#### 4.1. A Mining Method Based on Spatial Cliques

#### 4.1.1. Generating Granular Type-2 Fuzzy Membership Functions from Type-1 Fuzzy Membership Functions Based on Granular Data

- (1)
- Count the granular evaluation data.

- (2)
- The original type-1 fuzzy membership function is drawn based on the granular evaluation data.

- (3)
- Constructing a type-2 membership function based on the original type-1 membership function.

- (4)
- Determine the parameters of the granular type-2 fuzzy membership function.

- (5)
- Removing unconnected areas in the FOU.

#### 4.1.2. Generating FCP Mining Cliques (FCPM-Cliques)

**Definition**

**11**

**(Small**

**neighbor**

**instance**

**set).**

^{’}$\in S$|F

_{j}< F

_{i}$\cap $R(s, s

^{’})} is defined as a small neighbor instance set of s, where F

_{j}is the feature of s

^{’}. SNs(s) is a set of all the instances that are smaller than s and have spatial neighbor relationships with s.

**Definition**

**12**

**(Big**

**neighbor**

**instance**

**set).**

^{’}$\in S$| F

_{j}> F

_{i}$\cap $R(s, s

^{’})} is defined as a big neighbor instance set of s, where F

_{j}is the feature of s

^{’}. BNs(s) is a set of all the instances that are bigger than s and have spatial neighbor relationships with s.

**Definition**

**13**

**(NDM-Based**

**clique**

**tree**

**(N-tree**

**for**

**short)).**

**Strategy 1:**If the size of the clique |cli| = 1, SNs(s)$\cap $BNs(${s}_{1}$) = $\varnothing $, s and cli can form a new clique because s $\in $ BNs(${s}_{1}$) and ${s}_{1}\in $ SNs(s).

**Strategy 2:**If the size of the clique |cli| > 1, SNs(s)$\cap $BNs(${s}_{1}$) $\supseteq $ {${s}_{2}$,${s}_{3}$,…,${s}_{n}$}, s and cli can form a new clique because $\forall {s}_{i}\in $cli and there is R(s,${s}_{i}$).

**Strategy 3**

**:**If the size of the clique |cli| > 1, cli’ = SNs(s)$\cap $BNs(${s}_{1}$) $\subseteq $ {${s}_{2}$,${s}_{3}$,…,${s}_{n}$}, s and ${s}_{1}\cup $cli’ can form a new clique.

**Strategy 4:**If the size of the clique |cli| > 1, SNs(s)$\cap $BNs(${s}_{1}$)$\not\subset ${${s}_{2}$,${s}_{3}$,…,${s}_{n}$} and {${s}_{2}$,${s}_{3}$,…,${s}_{n}$}$\not\subset $ SNs(s)$\cap $BNs(${s}_{1}$), but cli’ = SNs(s)$\cap $BNs(${s}_{1}$)$\cap $ {${s}_{2}$,${s}_{3}$,…,${s}_{n}$}$\ne \varnothing $, s and cli’ $\cup {s}_{1}$ can form a new clique.

Algorithm 1: Neighborhood Driven Method. |

Input:nbsl: neighborhood relationship list (including SNs and BNs of each instance) S: a set of instances Output:list of N-cliques Steps:1. nTree = Initialize N-tree; 2. For each instance s In nbsl.Instances Do 3. queue = SNs(s); 4. While NotEmpty(queue) Do 5. head = nTree.AddHeadNode(queue.Out); 6. bodies = GetCurrentBodies(head); 7. relation = queue∩BNs(head); 8. If bodies == null || relation == null Then 9. head.AddNode(s) 10. Else 11. For Each list l in bodies do 12. If l == relation Then 13. s and cli can form a new clique; 14. Else If relation ⊇ l Then 15. s and cli can form a new clique; 16. Else If relation ⊆ l Then 17. s and relation ∪s _{1} can form a new clique;18. Else If NotEmpty(l ∩ relation) Then 19. s and relation ∩ l ∪s _{1} can form a new clique;20. End If 21. End For 22. End If 23. End While 24. End For |

**Definition**

**14**

**(Complete**

**clique**

**set).**

**Lemma**

**3.**

**Proof.**

**Lemma 3**, we know that N-cliques are complete. Therefore, the cliques generated by the NDM are correct and complete.

#### 4.1.3. Generating Candidate FCPs

Algorithm 2: The algorithm for generating candidate FCPs based on buckets. |

Input:$\left(1\right)\mathrm{Fuzzy}\mathrm{feature}\mathrm{set}\mathit{Fuzz}\_\mathit{Fea}=\{\mathrm{Fuzz}{f}_{1}$$,\mathrm{Fuzz}{f}_{2}$$,\dots ,\mathrm{Fuzz}{f}_{n}$}$\left(2\right)\mathrm{Feature}\mathrm{set}\mathit{Fea}\_\mathit{List}=\{Fe{a}_{1}$$,Fe{a}_{2}$$,\dots ,Fe{a}_{n}$} (3) Instance set of fuzzy features S = {s_{1}, s_{2},…, s_{n}} (4) Spatial proximity relation R (5) Minimum distance threshold min_dist (6) Minimum prevalence threshold min_prev Output:${C}_{k+1}$: k + 1 size candidate FCPsStep:1.if (k == 1) 2. for each Fuzz_f in Fuzz_Fea do 3. SetId( Fuzz_f) 4. Initialize_bucket( Fuzz_f) 5.else if (k == 2) 6. for each Fuzz_f_1 in one feature7. for each Fuzz_f_2 in other feature8. Generate a candidate pattern from Fuzz_f_1 and Fuzz_f_29. for each Fea in Fea_List 10. for each Fuzz_f_1 in the first attribute11. for each Fuzz_f_2 in the second attribute12. Generate a candidate pattern from Fuzz_f_1 and Fuzz_f_213. Put fuzzy patterns with the same features and the same attributes into the same bucket 14.else 15. for(i = 1; i < bucket_count; i++) 16. for each fuzzy pattern in bucket[i] 17. for(j = i + 1; j < bucket_count; j++) 18. for each fuzzy pattern in bucket[j] 19. if the first k – 1 fuzzy features of the two k-size fuzzy patterns are the same 20. Generate a fuzzy pattern c_{k+1}21. if (check(k − 1, c_{k+1}, bucket))22. c_{k+1} is a new candidate fuzzy pattern23. Put fuzzy patterns with the same features and the same attributes into the same bucket |

#### 4.1.4. Find the Row-Instances of the Candidate FCPs by the Column-Filter Method

Algorithm 3: The column-filter method. |

Input:$\mathrm{A}\mathrm{candidate}\mathrm{FCP}\mathit{Fuzz}\_\mathit{c}=\{\mathrm{Fuzz}{f}_{1}$$,\mathrm{Fuzz}{f}_{2}$$,\dots ,\mathrm{Fuzz}{f}_{n}$} Table-instance Table_c of co-location pattern that contains the candidate FCPOutput:Table-instances of candidate FCP Table[n]Step:1. Table = Table_c;$2.\mathrm{for}\mathrm{each}\mathrm{fuzzy}\mathrm{feature}\mathrm{Fuzz}{f}_{i}$ in the candidate FCP Fuzz_c3. for each row-instance in Table4. if (the row-instance doesn’t contain Fuzz_f_i) 5. Table.Delete(the row-instance);6. end for 7. Table[i] = Table;8. end for 9. if i == n 10. Table[i] is the table-instance of the candidate FCP |

**Table_c**, a table-instance of the co-location pattern that consists of the features of the candidate FCP. Steps 2–8: For all row-instances of the table-instance in

**Table_c**, we judge whether they have the first fuzzy feature of the candidate FCP. For any one of the row-instances, if it does not have the first fuzzy feature, we delete it; in this way, we filter out all the row-instances with the first fuzzy feature. Then, we judge whether these row-instances have the second fuzzy feature and filter out the row-instances that have both the first fuzzy feature and the second fuzzy feature, and so on. Steps 9–10: If the selected row-instance has all the fuzzy features of the candidate FCP, then the row-instance is a row-instance of the candidate FCP and all row-instances of the candidate FCP make up its table-instance.

#### 4.1.5. Filtering Prevalent FCPs

**Definitions 8**and

**9**, we can calculate the upper bound participation index and lower bound participation index of the FCP. Then, according to the upper bound participation index, the lower bound participation index, and the prevalence threshold, we can filter prevalent FCPs. Therefore, we propose two pruning strategies to filter complete and correct prevalent FCPs from candidate FCPs.

**Pruning**

**strategy**

**1**

**(Fuzzy**

**feature**

**pruning):**

**Proof.**

**c**, its upper bound participation index $\overline{\mathrm{PI}}$(

**c**) = $mi{n}_{i=1}^{k}${$\overline{\mathrm{PR}}$(

**c**, Fuzz_${f}_{i}$)} and $\overline{\mathrm{PR}}$(

**c**, Fuzz_${f}_{i}$) is the upper bound participation ratio of the ith fuzzy feature in

**c**. If the upper bound participation ratio of the fuzzy feature Fuzz_${f}_{i}$ is less than the given prevalence threshold, that is, $\overline{\mathrm{PI}}$(

**c**) = $mi{n}_{i=1}^{k}${$\overline{\mathrm{PR}}$(

**c**, Fuzz_${f}_{i}$

**c**)} < min_prev, the upper bound participation index of is less than the given prevalence threshold min_prev. In this case, according to Definition 10, is absolutely non-prevalent. □

**Pruning**

**strategy**

**2**

**(Absolute**

**table-instance**

**pruning):**

**c**, if the lower bound participation index of its absolute table-instance API(

**c**) is greater than or equal to the given prevalent threshold min_prev, then the FCP is an absolutely prevalent FCP.

**Proof.**

**c**, its absolute table-instance is included in its table instance, so the lower bound participation index of pattern

**c**is greater than that of its absolute table-instance API(

**c**), that is, PI(

**c**) $\ge $(

**c**) API(

**c**). If API(

**c**)$\ge $ min_prev, then PI(

**c**) $\ge $ API(

**c**) $\ge $ min_prev, according to Definition 10, c is absolutely prevalent. □

#### 4.1.6. Time Performance Analysis

#### 4.2. Extending Traditional Algorithms to Discover Prevalent FCPs

#### 4.2.1. An Extended Join-Based Algorithm for Mining Prevalent FCPs

Algorithm 4: The row-filter method. |

Input:cps: the set of candidate FCPs${\mathit{T}}_{\mathit{i}\mathit{n}\mathit{s}}$: the table-instance of the co-location pattern consisting of non-recurring features in the candidate FCP Output:all row-instances of candidate FCPs Steps:$1.\mathrm{For}\mathrm{each}\mathrm{candidate}\mathrm{FCP}c{p}_{i}$ In cps Do$2.\mathrm{suppose}c{p}_{i}$$=\{\mathrm{Fuzz}{f}_{1}$$,\mathrm{Fuzz}{f}_{2}$$,\dots ,\mathrm{Fuzz}{f}_{j}$$\};\mathrm{here},\mathrm{Fuzz}{f}_{m}$ $(1\le m\le j)\mathrm{is}\mathrm{a}\mathrm{fuzzy}\mathrm{feature},\mathrm{the}\mathrm{set}\mathrm{of}\mathrm{the}\mathrm{non}-\mathrm{recurring}\mathrm{features}\mathrm{of}\mathrm{all}\mathrm{fuzzy}\mathrm{features}\mathrm{in}c{p}_{i}$$\mathrm{is}\mathrm{sf}=\{{f}_{1}$$,{f}_{2}$$,\dots ,{f}_{k}$} (1 ≤ k ≤ j); $3.\mathrm{For}\mathrm{each}\mathrm{row}-\mathrm{instance}{R}_{ins}$$\mathrm{In}{T}_{ins}$ Do 4. if |sf | == 1 $5.{f}_{1}\_satisfy$$.\mathrm{instance}=\mathrm{filter}(\mathrm{instances}\mathrm{of}{f}_{1}$ that have all the fuzzy features in the candidate fuzzy pattern simultaneously) $6.c{p}_{i}$$\_\mathrm{table}\_\mathrm{instance}.\mathrm{Add}({f}_{1}\_satisfy$.instance) 7. else $8.{R}_{ins}\_satisfy$$=\mathrm{filter}({R}_{ins}$ that have all the fuzzy features in the candidate fuzzy pattern simultaneously) $9.c{p}_{i}$$\_\mathrm{table}\_\mathrm{instance}.\mathrm{Add}({R}_{ins}\_satisfy$) 10. End For 11. End For |

#### 4.2.2. An Extended Joinless Algorithm for Mining Prevalent FCPs

Algorithm 5: The extended Joinless algorithm. |

Input:$\left(1\right)\mathrm{Fuzzy}\mathrm{feature}\mathrm{set}\mathit{Fuzz}\_\mathit{Fea}=\{\mathrm{Fuzz}\_{f}_{1}$$,\mathrm{Fuzz}{f}_{2}$$,\dots ,\mathrm{Fuzz}{f}_{n}$}(2) Instance set of fuzzy features S = {s_{1},s_{2},…,s_{n}}$\left(3\right)\mathrm{A}\mathrm{set}\mathrm{of}\mathrm{star}\mathrm{neighbors}\mathrm{of}\mathrm{the}\mathrm{feature}\mathrm{Fuzz}\_{f}_{i}$$:\{s{n}_{{f}_{1}}$$,s{n}_{{f}_{2}}$$,\dots ,s{n}_{{f}_{n}}$} (4) Spatial proximity relation R (5) Minimum distance threshold min_dist (6) Minimum prevalence threshold min_prev Output: Prevalent FCP set Fre_PVariable:k: The size of FCPs ${C}_{k}$: candidate k size FCPs $S{I}_{k}$: Star instance set of k size candidate patterns $C{I}_{k}$: Group instance set of k size candidate patterns $P{I}_{k}$: The set of star instances have all fuzzy features in candidate patterns simultaneously ${P}_{k}$: prevalent k-size FCPs MemSum_up[k]: The sum of upper bound membership degrees of all instances of all fuzzy features in k size FCP.MemSum_down[k]: The sum of lower bound membership degrees of all instances of all fuzzy features in k size FCP.Step:1. SN = gen_star_neighbor( Fea,S,R);2. ${P}_{1}$ = Fuzz_Fea; k = 1;3. while(${P}_{k}$ $\ne $ $\varnothing $){ 4. ${C}_{k+1}$ = gen_candidate(${P}_{k}$); 5. for i in 1 to n 6. for x where ${f}_{i}$=$c{f}_{1}$,$c{f}_{1}$ is the first feature of ${C}_{k+1}$ 7. $S{I}_{k+1}$= gen_star_instance(${C}_{k+1}$, x); 8. end for 9. end for 10. if t = 1 (t is the number of non recurring features in a candidate fuzzy pattern) $11.P{I}_{k+1}$= The instances of the only feature that have all fuzzy feature of the candidate fuzzy co-location pattern; $12.C{I}_{k+1}$$=P{I}_{k+1}$; 13. if t = 2 $14.P{I}_{k+1}$$=\mathrm{The}\mathrm{row}-\mathrm{instances}\mathrm{in}S{I}_{k+1}$ which have all fuzzy features of the candidate fuzzy co-location pattern; $15.C{I}_{k+1}$$=P{I}_{k+1}$; 16. else do $17.P{I}_{k+1}$$=\mathrm{The}\mathrm{star}\mathrm{instances}S{I}_{t}$ that have all fuzzy features of the candidate fuzzy co-location pattern; $18.{C}_{k+1}$$=\mathrm{filter}\_\mathrm{coarse}\_\mathrm{prevalent}\_\mathrm{colocations}({C}_{k+1}$$,P{I}_{k+1}$, minprev); $19.C{I}_{k+1}$ $=\mathrm{gen}\_\mathrm{clique}\_\mathrm{instances}({C}_{k+1}$$,P{I}_{k+1}$); 20. end do $21.{P}_{k+1}$$=\mathrm{filter}\_\mathrm{prevalent}\_\mathrm{colocations}({C}_{k+1}$$,C{I}_{k+1}$, min_prev); 22. k = k + 1; $23.\mathit{Fre}\_\mathit{P}=\mathit{Fre}\_\mathit{P}\cup $${P}_{k+1}$; 24. } |

## 5. Results

#### 5.1. Results for Synthetic Datasets

- (1)
- The effect of distance thresholds on the results

- (2)
- The effect of prevalence threshold on the results

- (3)
- The effect of the number of fuzzy features on the results

- (4)
- The effect of the number of instances on the results

- (5)
- The effect of the number of attributes on the results

#### 5.2. Results for Real Datasets

## 6. Discussion

- (1)
- By constructing granular type-2 fuzzy membership functions, we reduce the deviation caused by the uncertainty of type-1 fuzzy membership functions. In this paper, the membership degrees of the attributes of spatial instances are expressed by interval values, and then the mined FCPs have an upper bound participation index and a lower bound participation index. We propose a method to measure the prevalence degree of FCPs when there is uncertainty in the fuzzy membership function, which provides a basis for judging the influence of the uncertainty of the fuzzy membership function on the results of mining FCPs.
- (2)
- Due to the uncertainty of fuzzy membership functions, we find FCPs whose lower bound participation index is less than the prevalence threshold but the upper bound participation index is higher than the threshold. This shows that in all the possible values of the participation index of these FCPs, some values are higher than the prevalence threshold and some values are lower. That is to say, such an FCP has a certain tendency to be prevalent, but also has a certain tendency to be non-prevalent; we regard this as a kind of potential prevalent FCP. For example, in the real dataset 1, the FCP {A.Cu(M), C.Cu(M)} has a lower bound membership of 0.1567, an upper bound membership of 0.3513, and a prevalence tendency degree of 0.2636. We can find that when there is uncertainty in the fuzzy membership function, {A.Cu(M), C.Cu(M)} has a 26.36% tendency degree to be prevalent. However, in the traditional method based on type-1 fuzzy sets, the participation index of the FCP is 0.2384 when the prevalence threshold is 0.3; we can only find that it is non-prevalent and cannot find the influence of the uncertainty of the fuzzy membership function on its participation index and prevalence degree. When using type-1 fuzzy sets, we cannot find the potential association between the middle concentration of copper in the industrial area and the middle concentration of copper in the living area, even if the association is very significant.
- (3)
- We use the prevalence tendency degree to measure the prevalence degree of FCPs. This allows us to find the FCPs whose lowest possible value of participation index is higher than the prevalence threshold. These FCPs are still prevalent when there is uncertainty in the fuzzy membership function, so we define them as absolutely prevalent FCPs and we regard them as a kind of stable and prevalent FCP. However, in the traditional method based on type-1 fuzzy sets, it is difficult to find this kind of FCP according to the participation index and the prevalence threshold. For example, in the real dataset 1, an absolutely prevalent FCP {B.Cu(M), B.Zn(L), C.Cu(M), C.Zn(L)} has a lower bound membership of 0.3933 and an upper bound membership of 0.738. We can easily find that the pattern is stably prevalent and we can find the influence of the uncertainty of the fuzzy membership function on the prevalence degree of FCPs. However, in the traditional method based on type-1 fuzzy sets, the participation index of the FCP is 0.5726 and it is difficult to judge whether this pattern is prevalent when there is uncertainty in the fuzzy membership function.

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**(

**a**) Type-2 membership function when $g$ = 1.1; (

**b**) Type-2 membership function when $g$ = 1.283.

**Figure 10.**(

**a**) Effect of the neighbor distance thresholds; (

**b**) Pruning degree of the two strategies.

Functional Area | The Number of Sampling Points | The Content of Copper | The Content of Zinc |
---|---|---|---|

A | 1 | 62 | 226 |

A | 2 | 31 | 105 |

A | 3 | 46 | 136 |

B | 1 | 86 | 183 |

B | 2 | 29 | 112 |

B | 3 | 45 | 150 |

B | 4 | 57 | 124 |

C | 1 | 63 | 210 |

D | 1 | 51 | 155 |

Functional Area | Number of Sampling Points | Cu(L) | Cu(M) | Cu(H) | Zn(L) | Zn(M) | Zn(H) |
---|---|---|---|---|---|---|---|

A | 1 | 0 | [0.4403, 0.7456] | [0.0104, 0.0831] | 0 | 0 | 1 |

A | 2 | 0 | [0.3614, 0.6848] | 0 | 0 | [0.5414, 0.8298] | 0 |

A | 3 | 0 | [0.8, 0.9591] | 0 | 0 | [0.5191, 0.8146] | 0 |

B | 1 | 0 | 0 | [0.283, 0.5931] | 0 | 0 | [0.3905, 0.7023] |

B | 2 | 0 | [0.3152, 0.6386] | 0 | 0 | [0.7138, 0.9252] | 0 |

B | 3 | 0 | [0.7633, 0.9455] | 0 | 0 | [0.2495, 0.5656] | 0 |

B | 4 | 0 | [0.5786, 0.8991] | 0 | 0 | [0.8315, 0.9695] | 0 |

C | 1 | 0 | [0.4069, 0.717] | [0.0174, 0.1141] | 0 | 0 | 1 |

D | 1 | 0 | [0.9169, 0.9896] | 0 | 0 | [0.1702, 0.4586] | [0.0335, 0.1698] |

$g$ | 1.1 | 1.2 | 1.3 | 1.4 | 1.5 | 1.6 | 1.7 | 1.8 | 1.9 | 2.0 |

Is it connected? | Yes | Yes | No | No | No | No | No | No | No | No |

$g$ | 1.21 | 1.22 | 1.23 | 1.24 | 1.25 | 1.26 | 1.27 | 1.28 | 1.29 | 1.3 |

Is it connected? | Yes | Yes | Yes | Yes | Yes | Yes | Yes | Yes | No | No |

$g$ | 1.281 | 1.282 | 1.283 | 1.284 | 1.285 | 1.286 | 1.287 | 1.288 | 1.289 | 1.29 |

Is it connected? | Yes | Yes | Yes | No | No | No | No | No | No | No |

SNs | Instance | BNs |
---|---|---|

- | A.1 | {B.1} |

- | A.2 | {B.2} |

- | A.3 | {B.3, B.4, C.1, D.1} |

{A.1} | B.1 | - |

{A.2} | B.2 | - |

{A.3} | B.3 | {C.1} |

{A.3} | B.4 | {C.1, D.1} |

{A.3, B.3, B.4} | C.1 | {D.1} |

{A.3, B.3, B.4, C.1} | D.1 | - |

Phase 1 | Phase 2 | Phase 3 | ||
---|---|---|---|---|

Star Instances of Feature A | A | {A, B} | {A.Cu(M), B.Cu(M)} | {A.Cu(M), A.Zn(M)} |

A.1, B.2, C.2 A.2, B.3, C.1 A.3, B.5, C.2 A.4, C.3 | A.1 A.2 A.3 A.4 | A.1, B.2 A.2, B.3 A.3, B.5 | A.1, B.2 A.2, B.3 | A.1 |

Prevalent Fuzzy Co-Location Pattern | Prevalence Tendency Degree | Lower Participation Index and Upper Participation Index | Participation Index for the Traditional Method |
---|---|---|---|

{A.Cu(L), C.Cu(L)} | 0.6125 | 0.0709, 0.6621 | 0.2312 |

{A.Cu(L), D.Cu(L)} | 0.2328 | 0.036, 0.3801 | 0.1169 |

{A.Cu(M), C.Cu(M)} | 0.2636 | 0.1567, 0.3513 | 0.2384 |

{D.Cu(M), D.Zn(H)} | 1 | 0.4781, 0.9135 | 0.6599 |

{D.Cu(L), D.Zn(M)} | 0.9096 | 0.2414, 0.8896 | 0.6681 |

{A.Zn(L), A.Cu(M), C.Zn(L)} | 0.2636 | 0.1567, 0.3513 | 0.2384 |

{A.Zn(L), C.Zn(L), D.Cu(M)} | 0.2167 | 0.1749, 0.3346 | 0.2492 |

{B.Cu(L), B.Zn(L), C.Zn(L)} | 0.8586 | 0.2597, 0.5448 | 0.4342 |

{B.Cu(M), B.Zn(L), C.Cu(M), C.Zn(M)} | 0.7039 | 0.1804, 0.5844 | 0.3358 |

{B.Cu(M), B.Zn(L), C.Cu(M), C.Zn(L)} | 1 | 0.3933, 0.738 | 0.5726 |

Prevalent Fuzzy Co-Location Pattern | Prevalence Tendency Degree | Lower Participation Index and Upper Participation Index | Participation Index for the Traditional Method |
---|---|---|---|

{A.Cu(M), A.Zn(M)} | 0.7965 | 0.2438, 0.5199 | 0.3661 |

{A.Zn(H), D.Zn(H)} | 1 | 0.4114, 0.966 | 0.8528 |

{A.Cu(M), D.Zn(H)} | 0.8208 | 0.2438, 0.5575 | 0.3661 |

{E.Cu(L), E.Zn(L)} | 1 | 0.3657, 0.4952 | 0.4425 |

{A.Cu(M), A.Zn(M), D.Cu(M)} | 0.7965 | 0.2438, 0.5199 | 0.3661 |

{A.Cu(M), A.Zn(H), D.Zn(M)} | 0.9047 | 0.2828, 0.4633 | 0.3853 |

{A.Cu(M), A.Zn(M), D.Cu(M), D.Zn(H)} | 0.7965 | 0.2438, 0.5199 | 0.3661 |

{A.Cu(M), A.Zn(H), D.Cu(M), D.Zn(H)} | 0.9047 | 0.2828, 0.4633 | 0.3853 |

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Guo, J.; Wang, L.
A Spatial Fuzzy Co-Location Pattern Mining Method Based on Interval Type-2 Fuzzy Sets. *Appl. Sci.* **2022**, *12*, 6259.
https://doi.org/10.3390/app12126259

**AMA Style**

Guo J, Wang L.
A Spatial Fuzzy Co-Location Pattern Mining Method Based on Interval Type-2 Fuzzy Sets. *Applied Sciences*. 2022; 12(12):6259.
https://doi.org/10.3390/app12126259

**Chicago/Turabian Style**

Guo, Jinyu, and Lizhen Wang.
2022. "A Spatial Fuzzy Co-Location Pattern Mining Method Based on Interval Type-2 Fuzzy Sets" *Applied Sciences* 12, no. 12: 6259.
https://doi.org/10.3390/app12126259